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Figueroa Jr. +robertojr.figueroa@up.edu.ph +University of the Philippines Open University, Philippines + +Florinda Amparo Palma Gil +floripg@tufs.ac.jp +Tokyo University of Foreign Studies, Japan + +Hiroshi Taniguchi +htaniguchi@up.edu.ph +University of the Philippines Open University, Philippines + +Joshze Rica Esguerra +jlesguerra2@up.edu.ph +University of the Philippines Open University, Philippines + +Abstract: When educational institutions worldwide scrambled for ways to continue their classes +during lockdowns caused by the COVID-19 pandemic, the use of information and communication +technology (ICT) for remote teaching has become widely considered to be a potential solution. As +universities raced to implement emergency remote teaching (ERT) strategies in Japan, some have +explored innovative interventions other than webinar platforms and learning management systems +to bridge the gap caused by restricted mobility among teachers and learners. One such innovation is +virtual reality (VR). VR has been changing the landscape of higher education because of its ability +to "teleport" learners to various places by simulating real-world environments in the virtual world. +Some teachers, including the authors of this paper, explored integrating VR into their activities to +address issues caused by geographical limitations brought about by the heightened restrictions in +2020. Results were largely encouraging. However, rules started relaxing in the succeeding years as +more people got vaccinated. Thus, some fully online classes in Japan shifted to blended learning as +they moved toward fully returning to in-person classes prompting educators to modify how they +implemented their VR-based interventions. This paper describes how a class of university students +in Japan who were taking a Filipino language course experienced a VR-based intervention in +blended mode, which was originally prototyped during the peak of the ERT era. Moreover, +adjustments and comparisons regarding methodological idiosyncrasies and findings between the +fully online iteration and the recently implemented blended one are reported in detail. + +Keywords: virtual reality, immersive open pedagogies, immersive learning + +INTRODUCTION + +Background of the Study + +During lockdowns caused by the COVID-19 pandemic, universities raced to implement emergency remote +teaching (ERT) strategies in Japan. Some have explored innovative interventions other than webinar platforms and +learning management systems to bridge the gap caused by restricted mobility among teachers and learners. One such +innovation is virtual reality (VR). VR has been changing the landscape of higher education because of its ability to +"teleport" learners to various places by simulating real-world environments in the virtual world. To fill in the gap +brought by geographical limitations due to heightened restrictions in 2020, educators at Tokyo University of Foreign +Studies (TUFS) explored integrating VR in teaching the Filipino Language to first year Japanese students (Figueroa +et al., 2022). + +INTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN +GOVERNANCE, EDUCATION AND BUSINESS +Vol. 4, No. 1, 2022 +ISSN 2686-0694 (Print) +e-ISSN 2721-0030 (Online) + +IJITGEB, Vol. 4 No.1, 2022, pp. 29-41, ISSN 2686-0694, e-ISSN 2721-0030 30 + +The Filipino language was first taught in Japan at the Osaka University of Foreign Studies, now Osaka +University in 1983 followed by TUFS in 1992 (Laranjo, 2020). These universities offer an entire major course in the +Filipino language and Philippine-related courses. Before the pandemic, classes were held using traditional in-person +classroom-based or blended pedagogy using a learning management system (LMS). Students were encouraged to join +short-term language classes abroad during the long spring and summer vacations or to join one-year student exchange +programs with affiliated universities abroad. These programs not only provided a more immersive experience for the +learners as they used the language and interacted with the native speakers of the language they were studying, but they +also increased their motivation to apply and experience first-hand what they learned inside the classroom. + +Therefore, when the short-term visits and student exchange programs were canceled due to the stricter rules +at the height of the pandemic in 2020, a photo-based VR tour lessons on Filipino vocabulary at TUFS was created to +provide students with an immersive way of learning Filipino language and experience the Filipino culture at the +comfort of their homes while being unable to physically visit the Philippines (Figueroa et al., 2022). However, rules +started relaxing in 2021 and 2022 when vaccines were introduced. Thus, some fully online classes in Japan shifted to +blended learning. The same happened at TUFS. With favorable feedback from students in 2020, the photo-based VR +tour lessons on Filipino vocabulary were consequently integrated even in the blended offering of the course in 2021 +and 2022. + +Research Questions + +With the new changes, the procedure on how the photo-based VR tour lessons were incorporated into the +Filipino Language course at TUFS was revised to fit the course’s evolving context. This paper aims to compare +experience and related outcomes between the fully online classes in 2020 and the blended-learning implementation in +2022 by answering the following research questions. + +1. How different were the satisfaction, presence, and interest felt and experienced by learners between +groups who used VR tours and those who did not in each tour in 2022? +2. How different were the satisfaction and presence felt by learners who used the VR tour-based lessons +between 2020 and 2022? + +RESEARCH DESIGN & METHODS + +Duration and Nature of the Study + +This longitudinal study compared 2020 and 2022 implementations of VR tour lessons. The lessons in both 2020 +and 2022 spanned two months. Described as a cognitive innovation, the 2020 pilot of the VR tour lessons followed +the design-based research approach where iterative design and implementation cycles were adjusted and modified +based on the data collected and analyzed from each cycle. + +Context Comparison + +The two implementations had slightly different contexts. Table 1 shows slight nuances and similarities between +the 2020 and 2022 implementations including the profile of students and how the classes were conducted. The +participants were Japanese university students who were enrolled in the Philippine Studies Program. +There were 15 student participants in the 2020 implementation and 12 participants in the 2022 +implementation. In 2020, all the photo-based VR tour lessons were held online, while the 2022 classes were both held +online and during face-to-face classes. In the same year, students were divided into three groups - high immersion +group (used VR goggles), moderate immersion group (did not use VR goggles but used the VR tours) and low +immersion group (did not use VR goggles and VR tours; only used photo-based PowerPoint tours). In the 2022 +implementation, the students were only divided into two groups, but both groups were able to experience the photo- +based VR tours while using VR Goggles and the photo-based tours presented in PowerPoint presentations. + + + + + + +INTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN +GOVERNANCE, EDUCATION AND BUSINESS +Vol. 4, No. 1, 2022 +ISSN 2686-0694 (Print) +e-ISSN 2721-0030 (Online) + +IJITGEB, Vol. 4 No.1, 2022, pp. 29-41, ISSN 2686-0694, e-ISSN 2721-0030 31 +Table 1 + +Contextual Data of the 2020 and 2022 Implementations + +Variable + 2020 Implementation +2021 Implementation +Number of Students +15 +12 +Year Level +First Year +First Year +Mode + Fully Online (Synchronous) +Blended (Alternating Online and In-Person) + Activity Groupings + 3 (Immersive Tour, Non Immersive +Tour, PowerPoint) +2 (Immersive Tour, PowerPoint) + Group Composition +Group 1: 5 students +Group 2: 5 students +Group 3: 5 students +Group 1: 6 students +Group 2: 6 students + +Sequence of Activities + + + +Figure 1. Procedural Diagram of the 2020 and 2022 Implementations as Illustrated in Figueroa et al. (2022) + +The sequence of activities were the same in both the 2020 and 2021 implementations as shown in Fig. 1. +The procedural diagram was directly lifted from Figueroa et al. (2022). As illustrated, a survey was given to students +at the beginning of the semester before they could experience the VR or PowerPoint presentation tours. The steps in +the darker square represent activities that are conducted in class. There were six classes conducted in both +Only in 2020 + +Preparation +Pre-VR Tour Survey +Pre-Test +LESSON +Implementation +x 6 times +(*1) +Post-Test +After-classSurvey +(*1) After each lesson, the teacher +and +two +other +researchers +discussed +and +wrotedowntheir observations +Post-semesterSurvey +Reflection +Focus Group +DiscussionsINTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN +GOVERNANCE, EDUCATION AND BUSINESS +Vol. 4, No. 1, 2022 +ISSN 2686-0694 (Print) +e-ISSN 2721-0030 (Online) + +IJITGEB, Vol. 4 No.1, 2022, pp. 29-41, ISSN 2686-0694, e-ISSN 2721-0030 32 +implementations, which included a pre-test, the lesson proper that involved the tours, a post-test, and an after class +survey. At the end of the semester, students were asked to reflect on the whole experience through a post-semester +survey and focus group discussions. The only difference during the 2022 implementation was that there was no more +focus group discussion conducted. + +Group Configuration +Another major difference between the two implementations is the grouping configuration. Three groups +were formed in 2020 (high, medium, and low). The high immersion group consisted of students who experienced VR +tours using their smart phones with VR goggles delivered to their homes. The medium immersion group consisted of +students who experienced VR tours without the VR goggles. The low immersion group consisted of students who +experienced PowerPoint-based tours with the same content as the VR tours. The grouping was only changed once, +after the first lesson where some students reported their smartphones not working with the goggles. However, in the +five succeeding lessons, the groupings and their assigned activities did not change (Figueroa et al., 2022). In contrast, +the implementation in 2022 only involved two groups. As illustrated in Fig. 2, in the first three lessons, Group 1 +experienced VR tours with goggles (VR Group) while Group 2 experienced PowerPoint-based tours (Non-VR Group). +In the second three lessons, Group 2 became the VR group and Group 1 became the Non-VR Group. This was done +so that all the students may be able to experience both types of activities. + + + + + + + + + + + + + + + +Figure 2. Group Configuration in 2022 Implementation + +Platform Selection for Immersive Open Pedagogical Activities + +In this section, we shall describe the platforms used in the two iterations of the study. Kuula is a web-based +software that makes it easy to create 360° virtual tours. The free basic plan allows level correction and retouching of +images, while paid plans ranging from 16 to 48 US Dollars per month include audio support, unlimited uploads, +unlisted and password-protected tours, custom icons and fonts, and analytics (Kuula, n.d.). + +A free alternative to Kuula with audio support is StorySpheres, a website created by Grumpy Sailor with the +help of Google’s Creative Lab in 2014 (Story Spheres, n.d). A user must upload 1 JPG/JPEG image and at least 1 +MP3 audio file, with the total size of all files below 15 MB. In addition to having a background sound, audio hotspots +can easily be added and positioned using the slider, as shown in Fig. 3. + + +Group 1 +Group 2 +VRTour1withVRGoggles +PPTTour1 +VR Tour2withVRGoggles +PPTTour2 +VRTour3withVRGoggles +PPTTour3 +PPTTour4 +VRTour4withVRGoggles +PPTTour5 +VR Tour5withVRGoggles +PPTTour6 +VRTour6withVRGogglesINTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN +GOVERNANCE, EDUCATION AND BUSINESS +Vol. 4, No. 1, 2022 +ISSN 2686-0694 (Print) +e-ISSN 2721-0030 (Online) + +IJITGEB, Vol. 4 No.1, 2022, pp. 29-41, ISSN 2686-0694, e-ISSN 2721-0030 33 + +Figure 3. Using and Positioning Hotspots to Play Audio Narrations in Story Spheres + + +For those with HTML and JavaScript knowledge, A-Frame (https://aframe.io/) is a notable option for more +freedom in developing 360° tours. It is a web framework based on top of HTML for building VR experiences with +only text-editing software and a web browser needed. When developing a virtual tour, JavaScript can be used to +change the image, music, and hotspot locations upon the click of a user. Since it requires coding, it will allow for more +freedom and customization in the tours. For example, all paid features in Kuula can be done in A-Frame, with the only +limitation being the learning curve. A finished A-Frame project can be deployed to a user’s server for personal or +company branding, or online Integrated Development Environments (IDEs) with hosting such as Glitch +(https://glitch.com/). Table 2 shows a comparative summary of the main features of the three platforms presented in +this section. + +Converting from Kuula to A-Frame + + Kuula was a viable option in the 2020 implementation because of its capability to facilitate rapid prototyping. +However, because of the recurring costs of maintaining a paid account, A-Frame was chosen to migrate the developed +VR tours for sustainability and was eventually used in the 2022 implementation. +The first step was to retrieve the 360° images from Kuula by clicking the Download link at the bottom of the +Edit pane and then saving the image. Recognizable faces on all photos were blurred using Adobe Fresco. The +narrations had to be recorded using Audacity since the Kuula platform did not allow audio files to be downloaded +from its tours. +The index page with portals used a 360° panoramic image as the initial source of the element. There +were multiple portals, each one an element with its source and the image representing the destination. +Behind it is a white to mimic an outline. Since A-Frame does not have support for non-alphanumeric text, +Japanese characters were added by importing a Multi-channel Signed Distance Font (MSDF) file that was generated +online. + + + +Upload audio +Uploadoneormoreaudiofiles. +Audiofilesmustbe: +..mp3 +Tip:Trylimitingtheaudiodata +ratetokeepyourfilessmall. +Onceuploaded,selectafilethenchoose +thetypeof audioto beeitherbackground +orhotspot. +Usethecontrolstoposition the audio +withinthe sphere. +Uploadaudio files* +PositionAudioSnippet +*requiredfield. +AudioFileName +1_2_3.mp3 +J1,2.3.mp3 +X +Horizontal Angle +0.117 +OBackground +OHot Spot +Vertical Angle +1.211 +Depth +74 +15%offileallowance +NEXTINTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN +GOVERNANCE, EDUCATION AND BUSINESS +Vol. 4, No. 1, 2022 +ISSN 2686-0694 (Print) +e-ISSN 2721-0030 (Online) + +IJITGEB, Vol. 4 No.1, 2022, pp. 29-41, ISSN 2686-0694, e-ISSN 2721-0030 34 +Table 2 + +Comparison of the VR Tour Platforms + +Platform +Price +Features +Limitations +Kuula +Free +● Retouch images +● Level correction +● Private tours +● Choose transition type +● Add images and hotspots +● Hotspots can open video/text +cards and URLs +● No audio support +● Max 100 uploads per month +● Max 25 images per batch upload +16-20 USD +per month +● Allows audio files +● Walkthrough mode +● Unlisted tours +● Custom icons and fonts +● 360° videos not supported +36-48 USD +per month +● Custom domain +● Password-protected tours +● Analytics +StorySpheres +Free +● Allows audio files in the +background or hotspot +● Stitching is not seamless +● Up to 15MB total file size +● Requires at least 1 audio file +● Video files are not supported +● Cannot add text +A-Frame +Free +● Allows for more freedom and +customization +● 360° videos supported +● Can host on own or cloud servers +● Requires coding +● Learning curve + +When a user hovers on a portal, there will be a preview (see Fig. 4) by displaying the name of the place and +temporarily changing the source with the use of JavaScript. The animation component was utilized to make +the transition smoother. Clicking a portal will redirect the browser to the tour of that location. Fig. 5 shows the interface +of the tour when entering VR mode on a mobile browser. + + +Figure 4. The User Interface Before and After Hovering on a Portal + +Aurora ForestINTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN +GOVERNANCE, EDUCATION AND BUSINESS +Vol. 4, No. 1, 2022 +ISSN 2686-0694 (Print) +e-ISSN 2721-0030 (Online) + +IJITGEB, Vol. 4 No.1, 2022, pp. 29-41, ISSN 2686-0694, e-ISSN 2721-0030 35 + +Figure 5. Viewing the Tour on a VR-Ready Mobile Phone + +Each tour includes multiple narrations that will play when its corresponding audio button is clicked. Audio +buttons are hotspots that are mapped with the help of the A-Frame Inspector (see Fig. 6), by dragging it to +the corresponding position and copying the coordinates to the position attribute in the code. The look-at component is +used to easily change the angle so that it will always face the user. When the button is clicked, the script will change +the sound attribute of the a-sky to the narration and toggle the player. + +Figure 6. Getting the Position Coordinates in A-Frame Inspector + +While the created tours are on separate web pages for easier sharing and access, another approach would be to +use a single webpage to host all tours. This can be done by using JavaScript to change the source of the tag +and the coordinates and identifiers of each audio hotspot with each click on the portal. However, since the tours were +non-contiguous and were presented separately, they were developed as separate pages. + +class +rayclick +position +63.49.000-2.000 +rotation +74.8783.107477 +1.Bayani +scale +6.0006.0001.000 +2 +Matapang +visible +3.Rebolusyunaryo +mixins +Addmixin. +COMPONENTS +Addcomponent. +GEOMETRY +LOOK-AT +MATERIAL +PLAY-MUSIC +心INTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN +GOVERNANCE, EDUCATION AND BUSINESS +Vol. 4, No. 1, 2022 +ISSN 2686-0694 (Print) +e-ISSN 2721-0030 (Online) + +IJITGEB, Vol. 4 No.1, 2022, pp. 29-41, ISSN 2686-0694, e-ISSN 2721-0030 36 +Data Collection + +The data used in this study include the results of six after-class surveys in the (1) 2020 implementation and +the (2) data collected during the photo-based VR tour lessons held in the first semester spanning from May to June in +2022. The results of the pre-test and post-test quizzes were not included as they were not included in the scope of the +study. All questionnaires contain both Likert-type items and open-ended questions. Data (1) was analyzed to answer +RQ1 while data (1) and (2) were compared to answer RQ2. Fig. 7 was a table lifted from an appendix of the previous +publication (Figueroa et al., 2022), which lists the after-class survey items used in both the 2020 and 2022 +implementations. Among these, only items two, four, and 12 were used for this study. Item two, which was boxed in +red in the figure, represented satisfaction. Item four, which was boxed in blue, represented interest and item 12, which +was boxed in green, represented presence. All the items were translated in the Japanese language. Face validity and +language expert consultation were conducted for the three items. While there was no other validity and reliability +tests conducted for the interest and satisfaction items, the presence item was a slightly modified version of the single- +item measure proposed and validated by Bouchard et al. (2004). + + +Figure 7. After-class Survey Questions in the 2020 and 2022 Implementations +Data Analysis + +To answer the first research question, summary statistics were generated for satisfaction, presence, and +interest among students of the two groups in each of the six lessons to see whether there are trends regarding +differences. Statistical significance was determined by performing the Mann-Whitney U test in each lesson using the +stats library in R (R Core Team, 2012). To answer the second question, summary statistics and boxplots were + +After-classSurveyQuestions +8.少了一体上、今俊今日語巢使确率法 +思？ +1.名前/二岁夕木一么(English:Name/Nickname +English: How much do you see yourself using the Filipino words you +learned today in the future after the tour? +低(Lowest)高(Highest) +1 ---2--- 3 --- 4 --- 5--- 6--- 7--- 8--- 9--- 10 +2.今回の体晚评俩？龙 +9VRの中良感 +English: How would you rate your experience? +English: What were the positive feelings you had during the VR tour? +良（Lowest））良（Highest） +1--- 2--- 3--- 4--- 5--- 6--- 7--- 8 ---9--- 10 +3.周の俩の理由述龙 +10.VRの感 +English:What'sthereasonforvourratinginnumber2? +English: What were the negativefeelings youhad during the VRtour? +4.一自体面百感？ +11.一避龙English:ChooseOne +English: How interested were you in the actual experience? +VR中、の真の感 +面百(Lowest)面白(Highest) +English: During the VR Tour, I felt like I was just looking at a photo. +1---2---3---4---5---6---7.-8---9---10 +本当体の感 +English: I felt like I was in an actual tour. +12.の享真、一本当体 +の感？ +来L龙加？ +English:Howmuch didyoufeel that youwereinthetourandnotjust +English: How much were you interested in the lesson's content (new +lookingataphoto? +words)? +感(Lowest)感(Highest +1---2--3---4---5---6--7.-8---9---10 +1--2---3---4---5--6-7..-8---9--- 10 +6.味持部分使？当法の心遵人下龙去 +12今俊の授の一体思 +来？世走思？ +English: What were the most interesting parts? +English: Would you like to do more of these tours in future online +classes? Why or why not? +7.の少了一体上、为老自身将来今日暂无 +13.老の他今回の体记阅寸多文下、提案、主老实尚等机 +语の语使の想像下享？の場面 +英语使思？ +English: Please share other comments, suggestions, or questions +English: Do you see yourself using the Filipino words you learned +regarding the whole experience. +today in the future after the tour? If yes, how?INTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN +GOVERNANCE, EDUCATION AND BUSINESS +Vol. 4, No. 1, 2022 +ISSN 2686-0694 (Print) +e-ISSN 2721-0030 (Online) + +IJITGEB, Vol. 4 No.1, 2022, pp. 29-41, ISSN 2686-0694, e-ISSN 2721-0030 37 +generated for satisfaction, presence, and interest among students of lessons two through five in 2020 and 2022. +Statistical significance per lesson was determined by performing the Mann-Whitney U. + +RESULTS + +RQ 1: How different were the satisfaction, presence, and interest felt and experienced by learners between +groups who used VR tours and those who did not in each tour in 2022? + +Lesson 1 + +Table 3 compares the medians of the satisfaction, presence, and interest ratings of students in the VR and +non-VR groups in lesson 1. + +Table 3 + +Comparison of Medians of Student Ratings between 2 Groups in Lesson 1 +Group +Satisfaction +Presence +Interest +VR (1) +10 +9.5 +10 +Non-VR (2) +8 +6 +7.5 + +The Mann Whitney U test indicated that satisfaction ratings were greater for students in the VR group (Mdn +=10) than those in the non-VR group (Mdn = 8) ,U = 35, p = .006. It also indicated that presence ratings were greater +for students in the VR group (Mdn = 9.5) than those in the non-VR group (Mdn = 6), U = 36, p =.004. However, +interest ratings were not statistically significantly different between the two groups. + +Lesson 2 + +Table 4 compares the medians of the satisfaction, presence, and interest ratings of students in the VR and +non-VR groups in lesson 2. + +Table 4 + +Comparison of Medians of Student Ratings between 2 Groups in Lesson 2 +Group +Satisfaction +Presence +Interest +VR (1) +10 +9.5 +10 +Non-VR (2) +8 +6 +7.5 + + +The Mann Whitney U test indicated that none of the three variables were statistically different between the +two groups. + +Lesson 3 + +Table 5 compares the medians of the satisfaction, presence, and interest ratings of students in the VR and +non-VR groups in lesson 3. + +Table 5 + +Comparison of Medians of Student Ratings between 2 Groups in Lesson 3 +Group +Satisfaction +Presence +Interest +VR (1) +10 +10 +10 +Non-VR (2) +8 +7 +8 + + +The Mann Whitney U test indicated that presence ratings were greater for students in the VR group (Mdn +=10) than those in the non-VR group (Mdn = 7) ,U = 31, p = .018. However, satisfaction and interest ratings were +not statistically significantly different between the two groups. + +INTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN +GOVERNANCE, EDUCATION AND BUSINESS +Vol. 4, No. 1, 2022 +ISSN 2686-0694 (Print) +e-ISSN 2721-0030 (Online) + +IJITGEB, Vol. 4 No.1, 2022, pp. 29-41, ISSN 2686-0694, e-ISSN 2721-0030 38 +Lesson 4 + +Table 6 compares the medians of the satisfaction, presence, and interest ratings of students in the VR and +non-VR groups in lesson 4. + +Table 6 + +Comparison of Medians of Student Ratings between 2 Groups in Lesson 4 +Group +Satisfaction +Presence +Interest +VR (2) +9.5 +9.5 +9 +Non-VR (1) +10 +10 +10 + + +The Mann Whitney U test indicated that none of the three variables were statistically different between the +two groups. + +Lesson 5 + +Table 7 compares the medians of the satisfaction, presence, and interest ratings of students in the VR and +non-VR groups in lesson 5. + +Table 7 + +Comparison of Medians of Student Ratings between 2 Groups in Lesson 5 +Group +Satisfaction +Presence +Interest +VR (2) +10 +10 +10 +Non-VR (1) +10 +10 +10 + + +The Mann Whitney U test indicated that none of the three variables were statistically different between the +two groups. + +Lesson 6 + +Table 8 compares the medians of the satisfaction, presence, and interest ratings of students in the VR and +non-VR groups in lesson 6. + +Table 8 + +Comparison of Medians of Student Ratings between 2 Groups in Lesson 6 +Group +Satisfaction +Presence +Interest +VR (2) +8.5 +8.5 +8 +Non-VR (1) +10 +10 +10 + + +`The Mann Whitney U test indicated that none of the three variables were statistically different between the +two groups. + +RQ 2: How different were the learning outcomes and attitudes of learners who used the VR tour-based lessons +between 2020 and 2022? + + +Fig. 8 compares 2020 and 2022 boxplots of satisfaction, presence, and interest ratings that were aggregated +across lessons two to six. It could be seen that the ratings of satisfaction, presence, and interest in 2022 were generally +higher than the ratings of the three variables in 2020. + +The Mann Whitney U test conducted per lesson confirmed this trend in the second and third lessons. In the +second lesson, there was a statistically significant difference in satisfaction ratings between 2020 (Mdn = 9) and 2022 +(Mdn = 10), U = 7.5, p = .02. In the same lesson, there is a statistically significant difference in presence ratings +between 2020 (Mdn = 8) and 2022 (Mdn = 10), U = 9.5, p = .05. + +INTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN +GOVERNANCE, EDUCATION AND BUSINESS +Vol. 4, No. 1, 2022 +ISSN 2686-0694 (Print) +e-ISSN 2721-0030 (Online) + +IJITGEB, Vol. 4 No.1, 2022, pp. 29-41, ISSN 2686-0694, e-ISSN 2721-0030 39 +In the third lesson, there was a statistically significant difference in satisfaction ratings between 2020 (Mdn += 8) and 2022 (Mdn = 10), U = 7.5, p = .02. In the same lesson, there is a statistically significant difference in presence +ratings between 2020 (Mdn = 8) and 2022 (Mdn = 10), U = 9.5, p = .003. None of the other lessons had statistically +significant differences in presence and satisfaction ratings. Furthermore, there were no statistically significant +differences in interest ratings between the two-year offerings. + + +Figure 8. Comparative Boxplots of Aggregated Ratings of Satisfaction, Presence, and Interest in 2020 and 2022 + + +DISCUSSION + +The findings revealed very enlightening trends in similarities and differences between the activities +implemented in 2020 and 2022. + +The Novelty of VR Tours +The statistically significant difference in presence, interest, and satisfaction between VR and Non-VR Groups +in the first lesson of the 2022 implementation showed that the VR tours piqued the students' interest, provided more +spatial presence, and gave them a better experience than in the PowerPoint-based tours. However, this was not evident +in the succeeding lessons. This may be explained by novelty, which was found to increase the interest among +participants and viewed by motivational researchers as one of its dimensions or components (Deci, 1992; Sun et al., +2008). However, novelty wanes through time (Spielberger & Starr, 1994). This may have happened in the succeeding +lessons. Unlike in the 2020 implementation where data supported interest in the succeeding lessons, data which could +support this trend in the 2022 implementation was yet to be analyzed, thereby posing a significant limitation of this +study. However, the findings of this study highlighted that a VR tour is a practical activity for gaining attention, which + +Satisfaction in 2020 and 2022 +Presence in 2020 and 2022 +Interest in 2020 and 2022 +0 +0 +16 +16 +1 +1 +9 +- +1 +T8 +8 +一 +1 + satisfaction +8 +1 +interest +- +1 +1 +1 +一 +7- +1 +1 +1 +1 +1 +1 +1 +1 +6i +1 +1 +1 +1 +7- +1 +1 +1 +1 +1 +6 +1 +/ +1 +1 +- +51 +1 +1 +一 +1 +1 +1 +1 +1 +1 +19 +L +一 +51 +4- +1 +1 +1 +1 +1 +1 +4 +。 +31 +51 +1 +1 +1 +2020 +2022 +2020 +2022 +2020 +2022 +year +year +yearINTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN +GOVERNANCE, EDUCATION AND BUSINESS +Vol. 4, No. 1, 2022 +ISSN 2686-0694 (Print) +e-ISSN 2721-0030 (Online) + +IJITGEB, Vol. 4 No.1, 2022, pp. 29-41, ISSN 2686-0694, e-ISSN 2721-0030 40 +was recommended as an initial step in effective teaching according to Gagne’s nine events of instruction (Schunk, +2012). + +In-Person Orientation Benefits +Another revelation was that 2022 implementation of the VR-based activities in blended mode yielded higher +presence and satisfaction ratings than that in the purely online mode in 2020. These findings showed the advantage +of conducting VR-based activities in blended settings compared to purely remote ones. The learning curve and +technical challenges in training students to use a VR device in a purely remote environment may have blunted the +motivational benefits that could have been obtained from using these novel technologies. The blended nature of the +classes in 2022 enabled the teacher to support students in using the VR devices in person. They could still access it +during the online sessions, but they were already well acquainted with the technology through the in-person +orientation. The importance of ensuring that students are comfortable in using an instructional technology has been +echoed by studies in technology readiness (Hubbard, 2013; Ngampornchai & Adams, 2016; Warden et al., 2022) . +Therefore, having an initial in-person session to help students get acquainted with VR devices and applications for +VR-based learning activities even in purely online learning settings would be extremely helpful as the technology is +still not that common. +The piloting and prototyping nature of the implementation in 2020 could also be attributed for this +observation. During that time, many of the problems encountered by students were still unknown and had to be +discovered. Those problems have already been addressed in the 2022 implementation. This confirms the practical +benefits of employing a design-based research approach in 2020, which was characterized by iterative cycles of design, +enactment, analysis, and redesign in a single setting over a period (Design-Based Research Collective [DBRC], 2003). + +CONCLUSION + +With many of the traditional universities embracing blended learning after implementing fully online classes +during the height of the COVID-19 pandemic, opportunities for improving students' experience in technology- +enhanced learning can be explored. In this paper, the findings from a study involving a method of learning a foreign +language in a remote teaching context through VR tours in 2020 and changes in the 2022 implementation were +presented. While limitations persist regarding generalizability and the need for qualitative data that could support +earlier findings, the study may provide practical insights regarding the advantage of in-person technical training and +the benefits of piloting a method using the design-based research approach. +REFERENCES +A-Frame. (n.d.). Retrieved September 2, 2022, from https://aframe.io/ +Bouchard, S., Robillard, G., St-Jacques, J., Dumoulin, S., Patry, M., & Renaud, P. (2004, November). Reliability and +validity of a single-item measure of presence in VR [Conference paper]. The 3rd IEEE International +Workshop on Haptic, Audio and Visual Environments and their Applications, Ottawa, ON, Canada. +Deci, E. L. (1992). The relation of interest to the motivation of behavior: A self-determination theory perspective. In +K.A. Renninger, S. Hidi, & A. Krapp (Eds.), The role of interest in learning and development (pp. 43-70). +Lawrence Erlbaum Associates. +Design-Based Research Collective. (2003). Designbased research: An emerging paradigm for educational inquiry. +Educational Researcher, 32(1), 5–8. +Figueroa, R. B., Palma Gil, F. A., & Taniguchi, H. (2022). Piloting virtual reality photo-based tours among students +of a Filipino language class: A case of emergency remote teaching in Japan. Avant: Trends in +Interdisciplinary Studies, 13(1). doi: 10.26913/avant.202208 +Glitch. (n.d.). Retrieved September 2, 2022, from https://glitch.com/ +Hubbard, P. (2013). Making a case for learner training in technology enhanced language learning +environments. Calico Journal, 30(2), 163-178. +Kuula. (n.d.). Pricing. Retrieved September 2, 2022, from https://kuula.co/ + +INTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN +GOVERNANCE, EDUCATION AND BUSINESS +Vol. 4, No. 1, 2022 +ISSN 2686-0694 (Print) +e-ISSN 2721-0030 (Online) + +IJITGEB, Vol. 4 No.1, 2022, pp. 29-41, ISSN 2686-0694, e-ISSN 2721-0030 41 +Laranjo, R. O. (2020). Mapping Philippine studies in Northeast Asia: A SWOT analysis of Southeast Asian Studies +programs from China, Japan and Korea. SUVANNABHUMI Multi-disciplinary Journal of Southeast Asian +Studies, 111-130. +Ngampornchai, A., & Adams, J. (2016). Students’ acceptance and readiness for E-learning in Northeastern +Thailand. International Journal of Educational Technology in Higher Education, 13(1), 1-13. +R Core Team. (2012). R: A language and environment for statistical computing. R Foundation for Statistical +Computing. Vienna, Austria. +Schunk, D. (2012). Learning theories an educational perspective (6th ed.). Pearson Education. +Spielberger, C. D., & Starr, L. M. (1994). Curiosity and exploratory behavior. n H. F. O'Neil Jr., & M. Drillings (Eds.), +Motivation: Theory and research (pp. 221-243). +Story Spheres. (n.d.). About. Retrieved September 2, 2022, from https://storyspheres.com/ +Sun, H., Chen, A., Ennis, C., Martin, R., & Shen, B. (2008). An examination of the multidimensionality of situational +interest in elementary school physical education. Research Quarterly for Exercise and Sport, 79(1), 62-70. +Warden, C. A., Yi-Shun, W., Stanworth, J. O., & Chen, J. F. (2022). Millennials’ technology readiness and self- +efficacy in online classes. Innovations in Education and Teaching International, 59(2), 226-236. + + diff --git a/09AzT4oBgHgl3EQf8v5L/content/tmp_files/load_file.txt b/09AzT4oBgHgl3EQf8v5L/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cdcf2e2451e5a11b1acb9e744e4ed39f7b1cb995 --- /dev/null +++ b/09AzT4oBgHgl3EQf8v5L/content/tmp_files/load_file.txt @@ -0,0 +1,625 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf,len=624 +page_content='INTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN GOVERNANCE, EDUCATION AND BUSINESS Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1, 2022 ISSN 2686-0694 (Print) e-ISSN 2721-0030 (Online) IJITGEB, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='1, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 29-41, ISSN 2686-0694, e-ISSN 2721-0030 29 Virtual Reality Photo-based Tours for Teaching Filipino Vocabulary in an Online Class in Japan: Transitioning into the New Normal Roberto B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Figueroa Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' robertojr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='figueroa@up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='ph University of the Philippines Open University, Philippines Florinda Amparo Palma Gil floripg@tufs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='jp Tokyo University of Foreign Studies, Japan Hiroshi Taniguchi htaniguchi@up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='ph University of the Philippines Open University, Philippines Joshze Rica Esguerra jlesguerra2@up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='ph University of the Philippines Open University, Philippines Abstract: When educational institutions worldwide scrambled for ways to continue their classes during lockdowns caused by the COVID-19 pandemic, the use of information and communication technology (ICT) for remote teaching has become widely considered to be a potential solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' As universities raced to implement emergency remote teaching (ERT) strategies in Japan, some have explored innovative interventions other than webinar platforms and learning management systems to bridge the gap caused by restricted mobility among teachers and learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' One such innovation is virtual reality (VR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' VR has been changing the landscape of higher education because of its ability to "teleport" learners to various places by simulating real-world environments in the virtual world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Some teachers, including the authors of this paper, explored integrating VR into their activities to address issues caused by geographical limitations brought about by the heightened restrictions in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Results were largely encouraging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' However, rules started relaxing in the succeeding years as more people got vaccinated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Thus, some fully online classes in Japan shifted to blended learning as they moved toward fully returning to in-person classes prompting educators to modify how they implemented their VR-based interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' This paper describes how a class of university students in Japan who were taking a Filipino language course experienced a VR-based intervention in blended mode, which was originally prototyped during the peak of the ERT era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Moreover, adjustments and comparisons regarding methodological idiosyncrasies and findings between the fully online iteration and the recently implemented blended one are reported in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Keywords: virtual reality, immersive open pedagogies, immersive learning INTRODUCTION Background of the Study During lockdowns caused by the COVID-19 pandemic, universities raced to implement emergency remote teaching (ERT) strategies in Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Some have explored innovative interventions other than webinar platforms and learning management systems to bridge the gap caused by restricted mobility among teachers and learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' One such innovation is virtual reality (VR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' VR has been changing the landscape of higher education because of its ability to "teleport" learners to various places by simulating real-world environments in the virtual world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' To fill in the gap brought by geographical limitations due to heightened restrictions in 2020, educators at Tokyo University of Foreign Studies (TUFS) explored integrating VR in teaching the Filipino Language to first year Japanese students (Figueroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' INTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN GOVERNANCE, EDUCATION AND BUSINESS Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1, 2022 ISSN 2686-0694 (Print) e-ISSN 2721-0030 (Online) IJITGEB, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='1, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 29-41, ISSN 2686-0694, e-ISSN 2721-0030 30 The Filipino language was first taught in Japan at the Osaka University of Foreign Studies, now Osaka University in 1983 followed by TUFS in 1992 (Laranjo, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' These universities offer an entire major course in the Filipino language and Philippine-related courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Before the pandemic, classes were held using traditional in-person classroom-based or blended pedagogy using a learning management system (LMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Students were encouraged to join short-term language classes abroad during the long spring and summer vacations or to join one-year student exchange programs with affiliated universities abroad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' These programs not only provided a more immersive experience for the learners as they used the language and interacted with the native speakers of the language they were studying, but they also increased their motivation to apply and experience first-hand what they learned inside the classroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Therefore, when the short-term visits and student exchange programs were canceled due to the stricter rules at the height of the pandemic in 2020, a photo-based VR tour lessons on Filipino vocabulary at TUFS was created to provide students with an immersive way of learning Filipino language and experience the Filipino culture at the comfort of their homes while being unable to physically visit the Philippines (Figueroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' However, rules started relaxing in 2021 and 2022 when vaccines were introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Thus, some fully online classes in Japan shifted to blended learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The same happened at TUFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' With favorable feedback from students in 2020, the photo-based VR tour lessons on Filipino vocabulary were consequently integrated even in the blended offering of the course in 2021 and 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Research Questions With the new changes, the procedure on how the photo-based VR tour lessons were incorporated into the Filipino Language course at TUFS was revised to fit the course’s evolving context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' This paper aims to compare experience and related outcomes between the fully online classes in 2020 and the blended-learning implementation in 2022 by answering the following research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' How different were the satisfaction, presence, and interest felt and experienced by learners between groups who used VR tours and those who did not in each tour in 2022?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' How different were the satisfaction and presence felt by learners who used the VR tour-based lessons between 2020 and 2022?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' RESEARCH DESIGN & METHODS Duration and Nature of the Study This longitudinal study compared 2020 and 2022 implementations of VR tour lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The lessons in both 2020 and 2022 spanned two months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Described as a cognitive innovation, the 2020 pilot of the VR tour lessons followed the design-based research approach where iterative design and implementation cycles were adjusted and modified based on the data collected and analyzed from each cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Context Comparison The two implementations had slightly different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Table 1 shows slight nuances and similarities between the 2020 and 2022 implementations including the profile of students and how the classes were conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The participants were Japanese university students who were enrolled in the Philippine Studies Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' There were 15 student participants in the 2020 implementation and 12 participants in the 2022 implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' In 2020, all the photo-based VR tour lessons were held online, while the 2022 classes were both held online and during face-to-face classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' In the same year, students were divided into three groups - high immersion group (used VR goggles), moderate immersion group (did not use VR goggles but used the VR tours) and low immersion group (did not use VR goggles and VR tours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' only used photo-based PowerPoint tours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' In the 2022 implementation, the students were only divided into two groups, but both groups were able to experience the photo- based VR tours while using VR Goggles and the photo-based tours presented in PowerPoint presentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' INTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN GOVERNANCE, EDUCATION AND BUSINESS Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1, 2022 ISSN 2686-0694 (Print) e-ISSN 2721-0030 (Online) IJITGEB, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='1, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 29-41,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' ISSN 2686-0694,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' e-ISSN 2721-0030 31 Table 1 Contextual Data of the 2020 and 2022 Implementations Variable 2020 Implementation 2021 Implementation Number of Students 15 12 Year Level First Year First Year Mode Fully Online (Synchronous) Blended (Alternating Online and In-Person) Activity Groupings 3 (Immersive Tour,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Non Immersive Tour,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' PowerPoint) 2 (Immersive Tour,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' PowerPoint) Group Composition Group 1: 5 students Group 2: 5 students Group 3: 5 students Group 1: 6 students Group 2: 6 students Sequence of Activities Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Procedural Diagram of the 2020 and 2022 Implementations as Illustrated in Figueroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' (2022) The sequence of activities were the same in both the 2020 and 2021 implementations as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The procedural diagram was directly lifted from Figueroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' As illustrated, a survey was given to students at the beginning of the semester before they could experience the VR or PowerPoint presentation tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The steps in the darker square represent activities that are conducted in class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' There were six classes conducted in both Only in 2020 Preparation Pre-VR Tour Survey Pre-Test LESSON Implementation x 6 times (*1) Post-Test After-classSurvey (*1) After each lesson, the teacher and two other researchers discussed and wrotedowntheir observations Post-semesterSurvey Reflection Focus Group DiscussionsINTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN GOVERNANCE, EDUCATION AND BUSINESS Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1, 2022 ISSN 2686-0694 (Print) e-ISSN 2721-0030 (Online) IJITGEB, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='1, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 29-41, ISSN 2686-0694, e-ISSN 2721-0030 32 implementations, which included a pre-test, the lesson proper that involved the tours, a post-test, and an after class survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' At the end of the semester, students were asked to reflect on the whole experience through a post-semester survey and focus group discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The only difference during the 2022 implementation was that there was no more focus group discussion conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Group Configuration Another major difference between the two implementations is the grouping configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Three groups were formed in 2020 (high, medium, and low).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The high immersion group consisted of students who experienced VR tours using their smart phones with VR goggles delivered to their homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The medium immersion group consisted of students who experienced VR tours without the VR goggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The low immersion group consisted of students who experienced PowerPoint-based tours with the same content as the VR tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The grouping was only changed once, after the first lesson where some students reported their smartphones not working with the goggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' However, in the five succeeding lessons, the groupings and their assigned activities did not change (Figueroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' In contrast, the implementation in 2022 only involved two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 2, in the first three lessons, Group 1 experienced VR tours with goggles (VR Group) while Group 2 experienced PowerPoint-based tours (Non-VR Group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' In the second three lessons, Group 2 became the VR group and Group 1 became the Non-VR Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' This was done so that all the students may be able to experience both types of activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Group Configuration in 2022 Implementation Platform Selection for Immersive Open Pedagogical Activities In this section, we shall describe the platforms used in the two iterations of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Kuula is a web-based software that makes it easy to create 360° virtual tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The free basic plan allows level correction and retouching of images, while paid plans ranging from 16 to 48 US Dollars per month include audio support, unlimited uploads, unlisted and password-protected tours, custom icons and fonts, and analytics (Kuula, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' A free alternative to Kuula with audio support is StorySpheres, a website created by Grumpy Sailor with the help of Google’s Creative Lab in 2014 (Story Spheres, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' A user must upload 1 JPG/JPEG image and at least 1 MP3 audio file, with the total size of all files below 15 MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' In addition to having a background sound, audio hotspots can easily be added and positioned using the slider, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Group 1 Group 2 VRTour1withVRGoggles PPTTour1 VR Tour2withVRGoggles PPTTour2 VRTour3withVRGoggles PPTTour3 PPTTour4 VRTour4withVRGoggles PPTTour5 VR Tour5withVRGoggles PPTTour6 VRTour6withVRGogglesINTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN GOVERNANCE, EDUCATION AND BUSINESS Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1, 2022 ISSN 2686-0694 (Print) e-ISSN 2721-0030 (Online) IJITGEB, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='1, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 29-41, ISSN 2686-0694, e-ISSN 2721-0030 33 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Using and Positioning Hotspots to Play Audio Narrations in Story Spheres For those with HTML and JavaScript knowledge, A-Frame (https://aframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='io/) is a notable option for more freedom in developing 360° tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' It is a web framework based on top of HTML for building VR experiences with only text-editing software and a web browser needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' When developing a virtual tour, JavaScript can be used to change the image, music, and hotspot locations upon the click of a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Since it requires coding, it will allow for more freedom and customization in the tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' For example, all paid features in Kuula can be done in A-Frame, with the only limitation being the learning curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' A finished A-Frame project can be deployed to a user’s server for personal or company branding, or online Integrated Development Environments (IDEs) with hosting such as Glitch (https://glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='com/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Table 2 shows a comparative summary of the main features of the three platforms presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Converting from Kuula to A-Frame Kuula was a viable option in the 2020 implementation because of its capability to facilitate rapid prototyping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' However, because of the recurring costs of maintaining a paid account, A-Frame was chosen to migrate the developed VR tours for sustainability and was eventually used in the 2022 implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The first step was to retrieve the 360° images from Kuula by clicking the Download link at the bottom of the Edit pane and then saving the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Recognizable faces on all photos were blurred using Adobe Fresco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The narrations had to be recorded using Audacity since the Kuula platform did not allow audio files to be downloaded from its tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The index page with portals used a 360° panoramic image as the initial source of the element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' There were multiple portals, each one an element with its source and the image representing the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Behind it is a white to mimic an outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Since A-Frame does not have support for non-alphanumeric text, Japanese characters were added by importing a Multi-channel Signed Distance Font (MSDF) file that was generated online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Upload audio Uploadoneormoreaudiofiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Audiofilesmustbe: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='.mp3 Tip:Trylimitingtheaudiodata ratetokeepyourfilessmall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Onceuploaded,selectafilethenchoose thetypeof audioto beeitherbackground orhotspot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Usethecontrolstoposition the audio withinthe sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Uploadaudio files* PositionAudioSnippet *requiredfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' AudioFileName 1_2_3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='mp3 J1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='mp3 X Horizontal Angle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='117 OBackground OHot Spot Vertical Angle 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='211 Depth 74 15%offileallowance NEXTINTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN GOVERNANCE, EDUCATION AND BUSINESS Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1, 2022 ISSN 2686-0694 (Print) e-ISSN 2721-0030 (Online) IJITGEB, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='1, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 29-41,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' ISSN 2686-0694,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' e-ISSN 2721-0030 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='34 Table 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='Comparison of the VR Tour Platforms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='Platform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='Price ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The animation component was utilized to make the transition smoother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Clicking a portal will redirect the browser to the tour of that location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 5 shows the interface of the tour when entering VR mode on a mobile browser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The User Interface Before and After Hovering on a Portal Aurora ForestINTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN GOVERNANCE, EDUCATION AND BUSINESS Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1, 2022 ISSN 2686-0694 (Print) e-ISSN 2721-0030 (Online) IJITGEB, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='1, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 29-41, ISSN 2686-0694, e-ISSN 2721-0030 35 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Viewing the Tour on a VR-Ready Mobile Phone Each tour includes multiple narrations that will play when its corresponding audio button is clicked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Audio buttons are hotspots that are mapped with the help of the A-Frame Inspector (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 6), by dragging it to the corresponding position and copying the coordinates to the position attribute in the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The look-at component is used to easily change the angle so that it will always face the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' When the button is clicked, the script will change the sound attribute of the a-sky to the narration and toggle the player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Getting the Position Coordinates in A-Frame Inspector While the created tours are on separate web pages for easier sharing and access, another approach would be to use a single webpage to host all tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' This can be done by using JavaScript to change the source of the tag and the coordinates and identifiers of each audio hotspot with each click on the portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' However, since the tours were non-contiguous and were presented separately, they were developed as separate pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' class rayclick position 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='000-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='000 rotation 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='8783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='107477 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='Bayani scale 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='0006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='000 2 Matapang visible 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='Rebolusyunaryo mixins Addmixin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' COMPONENTS Addcomponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' GEOMETRY LOOK-AT MATERIAL PLAY-MUSIC 心INTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN GOVERNANCE, EDUCATION AND BUSINESS Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1, 2022 ISSN 2686-0694 (Print) e-ISSN 2721-0030 (Online) IJITGEB, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='1, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 29-41, ISSN 2686-0694, e-ISSN 2721-0030 36 Data Collection The data used in this study include the results of six after-class surveys in the (1) 2020 implementation and the (2) data collected during the photo-based VR tour lessons held in the first semester spanning from May to June in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The results of the pre-test and post-test quizzes were not included as they were not included in the scope of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' All questionnaires contain both Likert-type items and open-ended questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Data (1) was analyzed to answer RQ1 while data (1) and (2) were compared to answer RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 7 was a table lifted from an appendix of the previous publication (Figueroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=', 2022), which lists the after-class survey items used in both the 2020 and 2022 implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Among these, only items two, four, and 12 were used for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Item two, which was boxed in red in the figure, represented satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Item four, which was boxed in blue, represented interest and item 12, which was boxed in green, represented presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' All the items were translated in the Japanese language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Face validity and language expert consultation were conducted for the three items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' While there was no other validity and reliability tests conducted for the interest and satisfaction items, the presence item was a slightly modified version of the single- item measure proposed and validated by Bouchard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' After-class Survey Questions in the 2020 and 2022 Implementations Data Analysis To answer the first research question, summary statistics were generated for satisfaction, presence, and interest among students of the two groups in each of the six lessons to see whether there are trends regarding differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Statistical significance was determined by performing the Mann-Whitney U test in each lesson using the stats library in R (R Core Team, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' To answer the second question, summary statistics and boxplots were After-classSurveyQuestions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='少了一体上、今俊今日語巢使确率法 思？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='名前/二岁夕木一么(English:Name/Nickname English: How much do you see yourself using the Filipino words you learned today in the future after the tour?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 低(Lowest)高(Highest) 1 ---2--- 3 --- 4 --- 5--- 6--- 7--- 8--- 9--- 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='今回の体晚评俩？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='龙 9VRの中良感 English: How would you rate your experience?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' English: What were the positive feelings you had during the VR tour?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 良（Lowest））良（Highest） 1--- 2--- 3--- 4--- 5--- 6--- 7--- 8 ---9--- 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='周の俩の理由述龙 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content="VRの感 English:What'sthereasonforvourratinginnumber2?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' English: What were the negativefeelings youhad during the VRtour?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='一自体面百感？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='一避龙English:ChooseOne English: How interested were you in the actual experience?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' VR中、の真の感 面百(Lowest)面白(Highest) English: During the VR Tour, I felt like I was just looking at a photo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1---2---3---4---5---6---7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='-8---9---10 本当体の感 English: I felt like I was in an actual tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='の享真、一本当体 の感？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 来L龙加？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=" English:Howmuch didyoufeel that youwereinthetourandnotjust English: How much were you interested in the lesson's content (new lookingataphoto?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' words)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 感(Lowest)感(Highest 1---2--3---4---5---6--7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='-8---9---10 1--2---3---4---5--6-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='.-8---9--- 10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='味持部分使？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='当法の心遵人下龙去 12今俊の授の一体思 来？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='世走思？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' English: What were the most interesting parts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' English: Would you like to do more of these tours in future online classes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='の少了一体上、为老自身将来今日暂无 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='老の他今回の体记阅寸多文下、提案、主老实尚等机 语の语使の想像下享？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='の場面 英语使思？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' English: Please share other comments, suggestions, or questions English: Do you see yourself using the Filipino words you learned regarding the whole experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' today in the future after the tour?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' If yes, how?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='INTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN GOVERNANCE, EDUCATION AND BUSINESS Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1, 2022 ISSN 2686-0694 (Print) e-ISSN 2721-0030 (Online) IJITGEB, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='1, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 29-41, ISSN 2686-0694, e-ISSN 2721-0030 37 generated for satisfaction, presence, and interest among students of lessons two through five in 2020 and 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Statistical significance per lesson was determined by performing the Mann-Whitney U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' RESULTS RQ 1: How different were the satisfaction, presence, and interest felt and experienced by learners between groups who used VR tours and those who did not in each tour in 2022?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Lesson 1 Table 3 compares the medians of the satisfaction, presence, and interest ratings of students in the VR and non-VR groups in lesson 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Table 3 Comparison of Medians of Student Ratings between 2 Groups in Lesson 1 Group Satisfaction Presence Interest VR (1) 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='5 10 Non-VR (2) 8 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='5 The Mann Whitney U test indicated that satisfaction ratings were greater for students in the VR group (Mdn =10) than those in the non-VR group (Mdn = 8) ,U = 35, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' It also indicated that presence ratings were greater for students in the VR group (Mdn = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='5) than those in the non-VR group (Mdn = 6), U = 36, p =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' However, interest ratings were not statistically significantly different between the two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Lesson 2 Table 4 compares the medians of the satisfaction, presence, and interest ratings of students in the VR and non-VR groups in lesson 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Table 4 Comparison of Medians of Student Ratings between 2 Groups in Lesson 2 Group Satisfaction Presence Interest VR (1) 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='5 10 Non-VR (2) 8 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='5 The Mann Whitney U test indicated that none of the three variables were statistically different between the two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Lesson 3 Table 5 compares the medians of the satisfaction, presence, and interest ratings of students in the VR and non-VR groups in lesson 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Table 5 Comparison of Medians of Student Ratings between 2 Groups in Lesson 3 Group Satisfaction Presence Interest VR (1) 10 10 10 Non-VR (2) 8 7 8 The Mann Whitney U test indicated that presence ratings were greater for students in the VR group (Mdn =10) than those in the non-VR group (Mdn = 7) ,U = 31, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' However, satisfaction and interest ratings were not statistically significantly different between the two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' INTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN GOVERNANCE, EDUCATION AND BUSINESS Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1, 2022 ISSN 2686-0694 (Print) e-ISSN 2721-0030 (Online) IJITGEB, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='1, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 29-41, ISSN 2686-0694, e-ISSN 2721-0030 38 Lesson 4 Table 6 compares the medians of the satisfaction, presence, and interest ratings of students in the VR and non-VR groups in lesson 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Table 6 Comparison of Medians of Student Ratings between 2 Groups in Lesson 4 Group Satisfaction Presence Interest VR (2) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='5 9 Non-VR (1) 10 10 10 The Mann Whitney U test indicated that none of the three variables were statistically different between the two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Lesson 5 Table 7 compares the medians of the satisfaction, presence, and interest ratings of students in the VR and non-VR groups in lesson 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Table 7 Comparison of Medians of Student Ratings between 2 Groups in Lesson 5 Group Satisfaction Presence Interest VR (2) 10 10 10 Non-VR (1) 10 10 10 The Mann Whitney U test indicated that none of the three variables were statistically different between the two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Lesson 6 Table 8 compares the medians of the satisfaction, presence, and interest ratings of students in the VR and non-VR groups in lesson 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Table 8 Comparison of Medians of Student Ratings between 2 Groups in Lesson 6 Group Satisfaction Presence Interest VR (2) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='5 8 Non-VR (1) 10 10 10 `The Mann Whitney U test indicated that none of the three variables were statistically different between the two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' RQ 2: How different were the learning outcomes and attitudes of learners who used the VR tour-based lessons between 2020 and 2022?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 8 compares 2020 and 2022 boxplots of satisfaction, presence, and interest ratings that were aggregated across lessons two to six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' It could be seen that the ratings of satisfaction, presence, and interest in 2022 were generally higher than the ratings of the three variables in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The Mann Whitney U test conducted per lesson confirmed this trend in the second and third lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' In the second lesson, there was a statistically significant difference in satisfaction ratings between 2020 (Mdn = 9) and 2022 (Mdn = 10), U = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='5, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' In the same lesson, there is a statistically significant difference in presence ratings between 2020 (Mdn = 8) and 2022 (Mdn = 10), U = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='5, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' INTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN GOVERNANCE, EDUCATION AND BUSINESS Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1, 2022 ISSN 2686-0694 (Print) e-ISSN 2721-0030 (Online) IJITGEB, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='1, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 29-41, ISSN 2686-0694, e-ISSN 2721-0030 39 In the third lesson, there was a statistically significant difference in satisfaction ratings between 2020 (Mdn = 8) and 2022 (Mdn = 10), U = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='5, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' In the same lesson, there is a statistically significant difference in presence ratings between 2020 (Mdn = 8) and 2022 (Mdn = 10), U = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='5, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' None of the other lessons had statistically significant differences in presence and satisfaction ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Furthermore, there were no statistically significant differences in interest ratings between the two-year offerings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Comparative Boxplots of Aggregated Ratings of Satisfaction, Presence, and Interest in 2020 and 2022 DISCUSSION The findings revealed very enlightening trends in similarities and differences between the activities implemented in 2020 and 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=" The Novelty of VR Tours The statistically significant difference in presence, interest, and satisfaction between VR and Non-VR Groups in the first lesson of the 2022 implementation showed that the VR tours piqued the students' interest, provided more spatial presence, and gave them a better experience than in the PowerPoint-based tours." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' However, this was not evident in the succeeding lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' This may be explained by novelty, which was found to increase the interest among participants and viewed by motivational researchers as one of its dimensions or components (Deci, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' However, novelty wanes through time (Spielberger & Starr, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' This may have happened in the succeeding lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Unlike in the 2020 implementation where data supported interest in the succeeding lessons, data which could support this trend in the 2022 implementation was yet to be analyzed, thereby posing a significant limitation of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' However, the findings of this study highlighted that a VR tour is a practical activity for gaining attention, which Satisfaction in 2020 and 2022 Presence in 2020 and 2022 Interest in 2020 and 2022 0 0 16 16 1 1 9 - 1 T8 8 一 1 satisfaction 8 1 interest - 1 1 1 一 7- 1 1 1 1 1 1 1 1 6i 1 1 1 1 7- 1 1 1 1 1 6 1 / 1 1 - 51 1 1 一 1 1 1 1 1 1 19 L 一 51 4- 1 1 1 1 1 1 4 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 31 51 1 1 1 2020 2022 2020 2022 2020 2022 year year yearINTERNATIONAL JOURNAL IN INFORMATION TECHNOLOGY IN GOVERNANCE, EDUCATION AND BUSINESS Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 1, 2022 ISSN 2686-0694 (Print) e-ISSN 2721-0030 (Online) IJITGEB, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 4 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='1, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' 29-41, ISSN 2686-0694, e-ISSN 2721-0030 40 was recommended as an initial step in effective teaching according to Gagne’s nine events of instruction (Schunk, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' In-Person Orientation Benefits Another revelation was that 2022 implementation of the VR-based activities in blended mode yielded higher presence and satisfaction ratings than that in the purely online mode in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' These findings showed the advantage of conducting VR-based activities in blended settings compared to purely remote ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The learning curve and technical challenges in training students to use a VR device in a purely remote environment may have blunted the motivational benefits that could have been obtained from using these novel technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The blended nature of the classes in 2022 enabled the teacher to support students in using the VR devices in person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' They could still access it during the online sessions, but they were already well acquainted with the technology through the in-person orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The importance of ensuring that students are comfortable in using an instructional technology has been echoed by studies in technology readiness (Hubbard, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Ngampornchai & Adams, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Warden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=', 2022) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Therefore, having an initial in-person session to help students get acquainted with VR devices and applications for VR-based learning activities even in purely online learning settings would be extremely helpful as the technology is still not that common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' The piloting and prototyping nature of the implementation in 2020 could also be attributed for this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' During that time, many of the problems encountered by students were still unknown and had to be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Those problems have already been addressed in the 2022 implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' This confirms the practical benefits of employing a design-based research approach in 2020, which was characterized by iterative cycles of design, enactment, analysis, and redesign in a single setting over a period (Design-Based Research Collective [DBRC], 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=" CONCLUSION With many of the traditional universities embracing blended learning after implementing fully online classes during the height of the COVID-19 pandemic, opportunities for improving students' experience in technology- enhanced learning can be explored." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' In this paper, the findings from a study involving a method of learning a foreign language in a remote teaching context through VR tours in 2020 and changes in the 2022 implementation were presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' While limitations persist regarding generalizability and the need for qualitative data that could support earlier findings, the study may provide practical insights regarding the advantage of in-person technical training and the benefits of piloting a method using the design-based research approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' REFERENCES A-Frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' (n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AzT4oBgHgl3EQf8v5L/content/2301.01908v1.pdf'} +page_content=' Retrieved September 2, 2022, from https://aframe.' 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Rastogi +Indian Institute of Management, Indore (India) + + + + + + + + + + +Correspondence address: + +Ajay Sharma, J-206, Academic Block, Indian Institute of Management Indore, Prabandh Shikhar, Rau- +Pithampur Road, Indore, M.P. (India) - 453556. Ph: +91-7312439622. E-mail: ajays@iimidr.ac.in; +ajaysharma87@gmail.com. + +Siddhartha K. Rastogi, B-101, Academic Block, Indian Institute of Management Indore, Rau-Pithampur +Road, Indore, M.P. (India) - 453556. Ph: +91-7312439534. E-mail: srastogi@iimidr.ac.in + + + + + + +Strategic Environmental Corporate Social Responsibility (ECSR) Certification and Endogenous +Market Structure + + + + +Abstract +This paper extends the findings of Liu et al. (2015, Strategic environmental corporate social +responsibility in a differentiated duopoly market, Economics Letters), along two dimensions. First, we +consider the case of endogenous market structure a la Vives and Singh (1984, Price and quantity +competition in a differentiated duopoly, The Rand Journal of Economics). Second, we refine the ECSR +certification standards in differentiated duopoly with rankings. We find that optimal ECSR certification +standards by NGO are the highest in Bertrand competition, followed by mixed markets and the lowest in +Cournot competition. Next, NGO certifier will set the ECSR standards below the optimal level. Also, we +show that given the ECSR certification standards, there is a possibility of both price and quantity +contracts choices by the firms in endogenous market structure. + + +JEL Classification: D43; L13; L22; M14 +Keywords: Corporate social responsibility certification; Differentiated duopoly; Environmental standards; +Price competition; Quantity competition + +Declaration of interest: The authors do not have any conflict of interests. + + + + + + + + + +1. Introduction + +Corporate Social Responsibility (CSR) has become a mainstream pursuit among the business activities of +firms in the past few years, wherein more than 30% (71% and 90%) of companies in the US (the UK and +Japan, respectively) adopted CSR reporting in 2013 (Kim et al., 2017). +Given the strategic importance of CSR activities as a non-core business pursuit and their significant +implication for costs, eco-labeling, certification, hallmarking etc. are the common ways of CSR signaling +especially for environmental outcomes. Though certification is not a perfect mechanism, it is sufficiently +trustworthy to convey useful information (Auriol and Schilizzi, 2015). +The certification can come from self or third-party and can be mandatory or optional. The existing +literature on the strategic aspects of third-party certification focuses on nature of competition and third- +party certifiers. Manasakis et al. (2013) suggest that the certification by alternative third parties differ +with respect to their objectives and has implications for certification standards. Liu et al. (2015) compares +the ECSR certification level in Cournot versus Bertrand competition and show that certification standards +are lower in Bertrand than Cournot competition. +Our contribution to this literature is two folds. First, we extend the analysis of Liu et al. (2015) by +endogenizing the market structure a la Singh and Vives (1984). If the firms have option of price or +quantity contracts, given the ECSR standards, then, what would be optimal choice for the firms? Second, +we refine the ECSR certification standards in this endogenous market structure by providing rankings and +then considering uniform standards. + +2. The Model +Based on Manasakis et al. (2013) and Liu et al. (2015), the utility function of a representative consumer is +𝑈 = (𝐴 + 𝑒1𝛼𝑠1)𝑞1 + (𝐴 + 𝑒2𝛼𝑠2)𝑞2 − (𝑞12 + 2𝛾𝑞1𝑞2 + 𝑞22) +2 + +where 𝑞𝑖 is output and 𝑠𝑖 is the level of ECSR, for firm 𝑖 (𝑖 = 1,2). The parameter 𝛾 ∈ (0,1) measures +the nature of products being substitutes (𝛾 > 0). The parameter 𝛼 ∈ (0,1) indicates the consumer’s +preference for firm’s ECSR activities. The firms choose ECSR as a strategic variable. Based on +Manasakis et al. (2013), we consider that ECSR activities can be informed to consumers through a + +credible signal. For the same, the firm seeks certification from a third-party NGO certifier who maximizes +Net Consumer Surplus (NCS). A firm can get the certification 𝑒𝑖 if it satisfies criteria of minimum level +of ECSR activities 𝑠 : +𝑒𝑖 = {0 𝑖𝑓 𝑠𝑖 < 𝑠 𝑎𝑛𝑑 𝑓𝑖𝑟𝑚 𝑖 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑟𝑒𝑐𝑒𝑖𝑣𝑒 𝑎 𝐸𝐶𝑆𝑅 𝑐𝑒𝑡𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 +1 𝑖𝑓 𝑠𝑖 ≥ 𝑠 𝑎𝑛𝑑 𝑓𝑖𝑟𝑚 𝑖 𝑟𝑒𝑐𝑒𝑖𝑣𝑒𝑠 𝑎 𝐸𝐶𝑆𝑅 𝑐𝑒𝑡𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 +} +It is important to note that a firm will consider doing ECSR activity only if it generates net positive +benefits. A firm would spend at 𝑠 (minimum ECSR for certification) and not beyond that. If both firms +choose to get the certification by spending 𝑠 on ECSR, the representative consumer’s utility would be: +𝑈 = (𝐴 + 𝑒1𝛼𝑠)𝑞1 + (𝐴 + 𝑒2𝛼𝑠)𝑞2 − (𝑞12 + 2𝛾𝑞1𝑞2 + 𝑞22) +2 + +The corresponding demand functions would be 𝑞𝑖 = +𝐴(1−𝛾)−𝑝𝑖+𝛾𝑝𝑗+𝛼𝑒𝑖𝑠−𝛼𝛾𝑒𝑗𝑠 +1−𝛾2 + and inverse demand +functions 𝑝𝑖 = 𝐴 − 𝑞𝑖 − 𝛾𝑞𝑗 + 𝛼𝑒𝑖𝑠, 𝑓𝑜𝑟 𝑖, 𝑗 = 1,2; 𝑖 ≠ 𝑗. +We assume that firms use same technology with cost of production as zero, without loss of generalization. +Also, one unit of output produces one unit of pollution emission. The NGO certifier will not charge any +fee for certification if firm complies with ECSR standards. The cost of ECSR for firms is 𝑠𝑖2. +The firm’s profit function is, 𝜋𝑖 = 𝑝𝑖𝑞𝑖 − 𝑒𝑖𝑠2, 𝑖 = 1,2. NGO certifier’s objective function is 𝑁𝐶𝑆 = + 𝐶𝑆 − +𝑑(𝑞1+𝑞2−𝑒1𝑠−𝑒2𝑠)2 +2 + where 𝐶𝑆 = +(𝑞12+2𝛾𝑞1𝑞2+𝑞22) +2 + ; and 𝑑 > 0 is the marginal environmental damage +due to emissions. + +3. The Game +The game is organized as follows. In the first stage, the firm decides to choose price or quantity contracts. +In the second stage, the certifier decides threshold level of ECSR for certification. Firms meeting the +threshold condition get the certification, otherwise not. In the third stage, firms choose the level of output +and prices to maximize their profits. +We solve the game using backward induction. +3.1. Product market competition + +In this stage, we analyze the four possible options: a) both firms choose prices (𝑝𝑝) i.e., Bertrand +competition; b) both firms choose quantities (𝑞𝑞) i.e., Cournot competition; c) one firm chooses price +(quantity) contract while the other firm chooses the quantity (price) contract i.e., 𝑝𝑞 (𝑞𝑝) outcomes. +We avoid providing the calculations for (a) and (b) option for the sake of brevity, as they are identical to +Liu et al. (2015). Please refer to the online appendix for the same. +Proposition 1: The NGO certifier will set the standards, 𝑠 = 𝑠𝑃𝑃𝑈 and 𝑠𝑄𝑄𝑈in the Bertrand (pp game) +and Cournot (qq game) respectively. +Proof: See online appendix. + +Next, both (c) and (d) will be identical in nature. Therefore, we only solve the pq game. + +𝑝𝑞 game (Price versus Quantity Contract) +We use the superscript PQ for price-quantity contract case i.e., firm 1 decides price while firm 2 decides +quantity. The outcomes in the product market with firms not adopting ECSR are +𝑞1 +𝑃𝑄𝑁 = +𝐴(2−𝛾−𝛾2) +4−3𝛾2 +; 𝑞2 +𝑃𝑄𝑁 = +𝐴(2−𝛾) +4−3𝛾2 ; 𝑝1 +𝑃𝑄𝑁 = +𝐴(2−𝛾−𝛾2) +4−3𝛾2 +; 𝑝2 +𝑃𝑄𝑁 = +𝐴(2−𝛾)(1−𝛾)(1+𝛾) +4−3𝛾2 +; 𝜋1 +𝑃𝑄𝑁 = +𝐴2(2−𝛾−𝛾2)2 +(4−3𝛾2)2 +; 𝜋2 +𝑃𝑄𝑁 = +𝐴2(2−𝛾)2(1−𝛾2) +(4−3𝛾2)2 + ; NCS𝑃𝑄𝑁 = +𝐴2(8−10𝛾2+3𝛾4−𝑑(4−𝛾(2+𝛾))2) +2(4−3𝛾2)2 + (5) +If the firms choose to opt for ECSR activities and get certification, the equilibrium outcomes would be, +𝑞1 +𝑃𝑄𝐶 = +(2−𝛾−𝛾2)(𝐴+𝛼𝑠) +4−3𝛾2 +; 𝑞2 +𝑃𝑄𝐶 = +(2−𝛾)(𝐴+𝛼𝑠) +4−3𝛾2 +; 𝑝1 +𝑃𝑄𝐶 = +(2−𝛾−𝛾2)(𝐴+𝛼𝑠) +4−3𝛾2 +; 𝑝2 +𝑃𝑄𝐶 = +(2−𝛾)(1−𝛾)(1+𝛾)(𝐴+𝛼𝑠) +4−3𝛾2 +; 𝜋1 +𝑃𝑄𝐶 = + +(𝐴2+2𝐴𝛼𝑠+𝛼2𝑠2)(2−𝛾−𝛾2)2−(4−3𝛾2) +2𝑠2 +(4−3𝛾2)2 + ; 𝜋2 +𝑃𝑄𝐶 +(𝐴2+2𝐴𝛼𝑠+𝛼2𝑠2)(2−𝛾)2(1−𝛾2)+((4−3𝛾2) +2 +(4−3𝛾2)2 + ; NCS𝑃𝑄𝐶 = +(2−𝛾2)(𝐴+𝛼𝑠)2 +8−6𝛾2 +− +𝐴2𝑑(−4+𝛾(2+𝛾))2 +2(4−3𝛾2)2 + + +For 𝑞1 +𝑃𝑄𝐶 > 𝑠 , 𝑠 < +2𝐴−𝐴𝛾−𝐴𝛾2 +4−2𝛼+𝛼𝛾−3𝛾2+𝛼𝛾2 and for 𝑞2 +𝑃𝑄𝐶 > 𝑠 , 𝑠 < +2𝐴−𝐴𝛾 +4−2𝛼+𝛼𝛾−3𝛾2 should be satisfied. For both +𝑞1 +𝑃𝑄𝐶 > 𝑠 𝑎𝑛𝑑 𝑞2 +𝑃𝑄𝐶 > 𝑠, 𝑠 < +2𝐴−𝐴𝛾−𝐴𝛾2 +4−2𝛼+𝛼𝛾−3𝛾2+𝛼𝛾2 must be satisfied. Further 𝑞2 +𝑃𝑄𝐶 > 𝑞1 +𝑃𝑄𝐶 holds for all +parametric values. +Firm 1 would be willing to adopt ECSR certification if 𝜋1 +𝑃𝑄𝐶 > 𝜋1 +𝑃𝑄𝑁 i.e., 𝑠 < 𝑠𝑃𝑄𝑈1 = +2𝐴−𝐴𝛾−𝐴𝛾2 +4−2𝛼+𝛼𝛾−3𝛾2+𝛼𝛾2 +holds. For firm 2, decision to adopt ECSR certification is chosen if 𝜋2 +𝑃𝑄𝐶 > 𝜋2 +𝑃𝑄𝑁 i.e. 𝑠 < 𝑠𝑃𝑄𝑈2 = +𝐴𝛼(2−𝛾−𝛾2)2 +(4−3𝛾2)2−𝛼2(2−𝛾−𝛾2)2 holds. +Also, comparing the upper threshold of spending on ECSR, we observe that firm 1 (choosing price) has +higher threshold than firm 2 (choosing quantity)’s ECSR spending, i.e., 𝑠𝑃𝑄𝑈1 > 𝑠𝑃𝑄𝑈2. +Lemma 1: In a price vs. quantity game, price setting firm has higher threshold for ECSR spending than +quantity setting firm. + +NGO certifier +Coming to second stage, we obtain the optimal choice of ECSR certification standard for NGO certifier +by evaluating +𝑑 𝑁𝐶𝑆𝑃𝑄𝐶 +𝑑𝑠 += 0. We get +𝑠𝑃𝑄∗ = 𝐴(𝛼(8 − 10𝛾2 + 3𝛾4) − 𝑑(−4 + 𝛾(2 + 𝛾))(8 − 6𝛾2 + 𝛼(−4 + 𝛾(2 + 𝛾)))) +𝛼2(−8 + 10𝛾2 − 3𝛾4) + 𝑑(8 − 6𝛾2 + 𝛼(−4 + 𝛾(2 + 𝛾)))2 + +𝑠𝑃𝑄∗ > 0 if 𝑑 > +𝛼2(8−10𝛾2+3𝛾4) +(8−6𝛾2+𝛼(−4+𝛾(2+𝛾)))2 holds. +Further, 𝑠𝑃𝑄∗ > 𝑠𝑃𝑄𝑈1 and 𝑠𝑃𝑄∗ > 𝑠𝑃𝑄𝑈2 when 𝑑 > +𝛼2(8−10𝛾2+3𝛾4) +(8−6𝛾2+𝛼(−4+𝛾(2+𝛾)))2. This means that certifier’s +optimal level of ECSR standard would be higher than the upper limit for the firms in price vs. quantity +competition and any firm will not spend on ECSR if a certifier sets the standard at 𝑠𝑃𝑄∗. NGO certifier +can set the ECSR standard for certification at either 𝑠 = 𝑠𝑃𝑄𝑈1or 𝑠 = 𝑠𝑃𝑄𝑈2 level for participation. If +𝑠 = 𝑠𝑃𝑄𝑈1 is chosen as ECSR standard, then only price-setting firm 1 will get the certification and +quantity-setting firm 2 will not get ECSR certification. Interestingly, profit of firm 2 will be higher than +firm 1. + +On the other hand, 𝑠 = 𝑠𝑃𝑄𝑈2 as ECSR standard leads to both firms getting the certification. In this case +also, firm 1’s profit would be lower than firm 2’s. +This indicates that quantity setting firm 2 has net advantage over price setting firm 1 irrespective of +whether firm 1 unilaterally get the ECSR certification or both firms get the certification. This is a new +result. +Therefore, to induce the firms in adopting the certification, the standard would be set at 𝑠 = 𝑠𝑃𝑄𝑈. +Further, we find that consumers and firms would benefit from such ECSR standard as compared to no +ECSR at all because NCSPQC > NCSPQN. + +Proposition 2: In a price vs. quantity competition, +a) NGO certifier will set the ECSR standard below the optimal level +b) if ECSR certification standard is set at 𝑠 = 𝑠𝑃𝑄𝑈1, then only price-setting firm will get the +certification, whereas quantity-setting firm 2 will not opt for certification. +c) if ECSR certification standard is set at 𝑠 = 𝑠𝑃𝑄𝑈2, then both firms will get the certification and it +is beneficial for both firms and consumers. + + +4. Comparison of ECSR Certification Standards + +Comparing the optimal ECSR standard, 𝑠 by the NGO certifier across endogenous market structure, we +observe that 𝑠𝑃𝑃∗ > 𝑠𝑃𝑄∗ > 𝑠𝑄𝑄∗ indicating Bertrand has the highest level followed by price-quantity and +lastly Cournot case. +Proposition 3: Across the spectrum of market structure, the NGO certifier’s optimal ECSR standard +rankings are 𝑠𝑃𝑃∗ > 𝑠𝑃𝑄∗ > 𝑠𝑄𝑄∗. +In all the cases, the NGO certifier is not able to implement the optimal level of ECSR standard because +the firms will not adopt such ECSR standards as that leads to lower profit for them. Therefore, the +certifier would choose a sub-optimal ECSR standard to incentivize the firms. Comparing these +equilibrium standards, we get the ranking in proposition 4. + +Proposition 4: The NGO certifier’s equilibrium ECSR standard rankings are 𝑠𝑃𝑄𝑈1 > 𝑠𝑄𝑄𝑈 > 𝑠𝑃𝑃𝑈 > +𝑠𝑃𝑄𝑈2. + +5. Endogenous Market Structure: Price or Quantity Contract +Now, we solve the first stage of the game where firms have options to choose price or quantity contracts +in the product market competition. For the sake of brevity, we do not consider the case where no firm +chooses ECSR certification. The outcome of that subgame will be identical to Singh and Vives (1984). + +Lemma 2 (Singh and Vives, 1984): In a product market competition for substitute goods1, with price and +quantity as strategic choices, firms choose quantity contracts as dominant strategies. + + +ECSR certification standards and market structure + +If firms opt for the ECSR certification, then the outcome of the subgame can differ from Singh and Vives +(1984). The certifier can choose a uniform standard irrespective of the nature of market competition, or +different standards based on nature of competition2. We only consider possibility of uniform ECSR +certification standards. +In case of uniform ECSR standard, there are four choices, 𝑠𝑃𝑄𝑈1 > 𝑠𝑄𝑄𝑈 > 𝑠𝑃𝑃𝑈 > 𝑠𝑃𝑄𝑈2 (see +Proposition 4). If the NGO certifier sets the lowest three ECSR certification standard, i.e., either 𝑠𝑃𝑄𝑈2 or +𝑠𝑃𝑃𝑈 or 𝑠𝑄𝑄𝑈, then the Nash equilibrium outcome of the game in Table 1, is {Quantity, Quantity}. On the +other hand, if the ECSR certifier sets the standard at the highest level possible i.e., 𝑠𝑃𝑄𝑈1, then there are +two Nash equilibria outcomes of the game {Price, Quantity} and {Quantity, Price}. + + +1 In this paper, we only consider substitute goods in the market offered by competing firms. +2 But from a real-world point of view, such standards may not be feasible due to monitoring issues, +discrimination, and mimicking behavior among firms. + + +Table 1: Price-Quantity Contract Game (with ECSR certification) + +Firm 2 +Firm 1 + +Price +Quantity +Price +𝜋1 +𝑃𝑃𝐶, 𝜋2 +𝑃𝑃𝐶 +𝜋1 +𝑃𝑄𝐶, 𝜋2 +𝑃𝑄𝐶 +Quantity +𝜋1 +𝑄𝑃𝐶, 𝜋2 +𝑄𝑃𝐶 +𝜋1 +𝑄𝑄𝐶, 𝜋2 +𝑄𝑄𝐶 + +Proposition 5: In a price-quantity contract game, +a) If a certifier decides, the uniform ECSR standard at either 𝑠𝑃𝑄𝑈2 or 𝑠𝑃𝑃𝑈 or 𝑠𝑄𝑄𝑈 level, the +subgame perfect Nash equilibrium is {Quantity, Quantity} +b) If the certifier decides, the uniform ECSR standard at 𝑠𝑃𝑄𝑈1, there are two subgame perfect Nash +equilibria {price, Quantity}, {Quantity, Price} +Proof: See online appendix + +6. Conclusion +In this paper, we analyze the relationship between endogenous market structure and strategic ECSR in a +differentiated duopoly. We show that NGO certifier will always set the ECSR standards below the +optimal level to ensure participation. In a price-quantity game, there is possibility of partial or full +compliance with ECSR standards. Lastly, while setting a uniform ECSR standards in endogenous market +structure, there is a possibility of Cournot outcome as well as mixed market outcome. + + + + + + + + + + +References + +Auriol, E. and Schilizzi, S.G.M. (2015) Quality signaling through certification in developing countries. +Journal of Development Economics. 116. 105-121. +Kim, S., Lee, S., and Matsumura, T. (2017) Corporate social responsibility and privatization policy in a +mixed oligopoly. MPRA Paper No. 79780. +Liu, C. C., Wang, L. F., and Lee, S. H. (2015) Strategic environmental corporate social responsibility in a +differentiated duopoly market. Economics Letters. 129. 108-111. +Manasakis, C., Mitrokostas, E., and Petrakis, E. (2013) Certification of corporate social responsibility +activities in oligopolistic markets. Canadian Journal of Economics. 46(1). 282-309. +Singh, N., and Vives, X. (1984) Price and quantity competition in a differentiated duopoly. The Rand +Journal of Economics. 546-554. + + + + + + + + + + + + + + +ONLINE APPENDIX + + +A1. A 𝑝𝑝 game (Bertrand Competition) +We use the superscript PPN to denote equilibrium outcome for firms not adopting ECSR in 𝑝𝑝 +game i.e., Bertrand competition, otherwise PPC. Solving the game, we get +𝑞1 +𝑃𝑃𝑁 = 𝑞2 +𝑃𝑃𝑁 = +𝐴 +2+𝛾−𝛾2 ; 𝑝1 +𝑃𝑃𝑁 = 𝑝2 +𝑃𝑃𝑁 = +𝐴(1−𝛾) +2−𝛾 ; 𝜋1 +𝑃𝑃𝑁 = 𝜋2 +𝑃𝑃𝑁 = +𝐴2(1−𝛾) +(2−𝛾)2(1+𝛾) ; NCSPPN = +𝐴2(1−2𝑑+𝛾) +(2+𝛾−𝛾2)2 (1) +If the firms decide to adopt for ECSR and get certification for the same, the outcomes are +𝑞1 +𝑃𝑃𝐶 = 𝑞1 +𝑃𝑃𝐶 = +𝐴 + 𝛼𝑠 +2 + 𝛾 − 𝛾2 ; 𝑝1 +𝑃𝑃𝐶 = 𝑝2 +𝑃𝑃𝐶 = +(1 − 𝛾)(𝐴 + 𝛼𝑠) +2 − 𝛾 +; 𝜋1 +𝑃𝑃𝐶 = 𝜋2 +𝑃𝑃𝐶 += +(1 − 𝛾)(𝐴 + 𝛼𝑠) +2 +(2 − 𝛾)(2 + 𝛾 − 𝛾2) − 𝑠2 ; +NCSPPC = +(𝐴+𝛼𝑠)2 +(2−𝛾)2(1+𝛾) − +2𝑑(𝐴+(𝛼+(−2+𝛾)(1+𝛾))𝑠)2 +(2+𝛾−𝛾2)2 + (2) +For a firm to be adopting ECSR, the certification threshold needs to be lower than the level of +pollution, otherwise the cost will be more than its benefits. For 𝑞𝑖 +𝑃𝑃𝐶 > 𝑠, 𝑠 < +𝐴 +2−𝛼+𝛾−𝛾2 should +be satisfied. +Comparing the profits of firms3 with or without ECSR, +𝜋1 +𝑃𝑃𝐶 − 𝜋1 +𝑃𝑃𝑁 = 𝐴2(1 − 𝛾) + 2𝐴𝛼(1 − 𝛾)𝑠 − (4 − 𝛼2(1 − 𝛾) − (3 − 𝛾)𝛾2)𝑠2 +(2 − 𝛾)2(1 + 𝛾) + + +3 Given the symmetry of the firms and their outcomes, we only compare the results of one firm, and it +holds for both of them. + +We observe that 𝜋1 +𝑃𝑃𝐶 − 𝜋1 +𝑃𝑃𝑁 > 0 if 𝑠 < 𝑠𝑃𝑃𝑈 = +2𝐴𝛼(1−𝛾) +4−𝛼2+𝛼2𝛾−3𝛾2+𝛾3 , which provides the upper +bound for the ECSR spending to adopt the ECSR certification. So, firms will spend strategically +on ECSR and get certification if 𝑠 < 𝑠𝑃𝑃𝑈.4 + +Optimal ECSR Certification Standard + +We obtain the optimal choice of ECSR certification standard in case of NGO certifier by +evaluating +𝑑 𝑁𝐶𝑆𝑃𝑃𝐶 +𝑑𝑠 += 0. We get, +𝑠𝑃𝑃∗ = 𝐴(𝛼 + 𝛼𝛾 − 2𝑑(𝛼 − (2 − 𝛾)(1 + 𝛾))) +2𝑑(𝛼 − (2 − 𝛾)(1 + 𝛾))2 − 𝛼2(1 + 𝛾) +𝑠𝑃𝑃∗ > 0 if 𝑑 > +𝛼2+𝛼2𝛾 +2(𝛼−(2−𝛾)(1+𝛾))2 holds. Further, 𝑠𝑃𝑃∗ > 𝑠𝑃𝑃𝑈 when 𝑑 > +𝛼2+𝛼2𝛾 +2(𝛼−(2−𝛾)(1+𝛾))2. This +means that an NGO certifier’s optimal level of ECSR standard would be higher that the upper +limit for the firms in Bertrand competition and a firm would not choose to spend on ECSR if a +certifier sets the standard at 𝑠𝑃𝑃∗. Therefore, to induce the firms, the standard would be set at 𝑠 = + 𝑠𝑃𝑃𝑈 by the NGO certifier. Further, we can also show that consumer and firms would benefit +from such ECSR standard as compared to no ECSR at all i.e., NCSPPC > NCSPPN. + +Proposition A1: An NGO certifier would set the ECSR standard below the optimal level if firms +engage in Bertrand competition and, it is beneficial for both firms and consumers in terms of +profit and net consumer surplus, respectively. + +A2. A 𝑞𝑞 game (Cournot Competition) + +4 Superscript U denotes upper bound. + +For Cournot game, we use the superscript QQ. The outcomes of the product market competition, +if firms do not adopt ECSR are, 𝑞1 +𝑄𝑄𝑁 = 𝑞2 +𝑄𝑄𝑁 = +𝐴 +2+𝛾 ; 𝑝1 +𝑄𝑄𝑁 = 𝑝2 +𝑄𝑄𝑁 = +𝐴 +2+𝛾 ; 𝜋1 +𝑄𝑄𝑁 = 𝜋2 +𝑄𝑄𝑁 = + +𝐴2 +(2+𝛾)2 ; NCS𝑄𝑄𝑁 = +𝐴2(1−2𝑑+𝛾) +(2+𝛾)2 + (3) +On the other hand, if the firms decide to adopt ECSR certification, the outcomes would be, +𝑞1 +𝑄𝑄𝐶 = 𝑞2 +𝑃𝑃𝐶 = +𝐴+𝛼𝑠 +2+𝛾 ; 𝑝1 +𝑄𝑄𝐶 = 𝑝2 +𝑄𝑄𝐶 = +𝐴+𝛼𝑠 +2+𝛾 ; 𝜋1 +𝑄𝑄𝐶 = 𝜋2 +𝑄𝑄𝐶 = +(𝐴+𝛼𝑠)2 +(2+𝛾)2 − 𝑠2; NCS𝑄𝑄𝐶 = +(1+𝛾)(𝐴+𝛼𝑠) +2−2𝑑(𝐴−(2−𝛼+𝛾)𝑠)2 +(2+𝛾)2 + (4) + +For 𝑞𝑖 +𝑄𝑄𝐶 > 𝑠 , 𝑠 < +𝐴 +2−𝛼+𝛾 should be satisfied. Further comparing the profits of firms with and +without adopting ECSR, +𝜋1 +𝑄𝑄𝐶 − 𝜋1 +𝑄𝑄𝑁 = 𝐴2 + 2𝐴𝛼𝑠 + (−2 + 𝛼 − 𝛾)(2 + 𝛼 + 𝛾)𝑠2 +(2 + 𝛾)2 + +A firm would profit from adopting ECSR if 𝜋1 +𝑄𝑄𝐶 > 𝜋1 +𝑄𝑄𝑁 i.e., when 𝑠 < 𝑠𝑄𝑄𝑈 = +2𝐴𝛼 +4−𝛼2+4𝛾+𝛾2. +This denotes the upper bound to spend on ECSR for certification. + +Optimal ECSR Certification Standard +We obtain the optimal choice of ECSR certification standard in case of NGO certifier by +evaluating +𝑑 𝑁𝐶𝑆𝑄𝑄𝐶 +𝑑𝑠 += 0. We get, +𝑠𝑄𝑄∗ = 𝐴(𝛼 + 𝛼𝛾 + 2𝑑(2 − 𝛼 + 𝛾)) +2𝑑(2 − 𝛼 + 𝛾)2 − 𝛼2(1 + 𝛾) +𝑠𝑄𝑄∗ > 0 if 𝑑 > +𝛼2+𝛼2𝛾 +2(2−𝛼+𝛾)2 holds. Further, 𝑠𝑄𝑄∗ > 𝑠𝑄𝑄𝑈 when 𝑑 > +𝛼2+𝛼2𝛾 +2(2−𝛼+𝛾)2. This means that a +certifier’s optimal level of ECSR standard would be higher than the upper limit for the firms in +Cournot competition and a firm would not choose to spend on ECSR if a certifier sets the +standard at 𝑠𝑄𝑄∗. Therefore, to induce the firms, the standard would be set at 𝑠 = 𝑠𝑄𝑄𝑈 by the + +NGO certifier. Further, we can also show that consumer and firms would benefit from such +ECSR standard as compared to no ECSR at all i.e., NCSQQC > NCSQQN. + +Proposition A2: An NGO certifier would set the ECSR standard below the optimal level if firms +engage in Cournot competition and, it is beneficial for both firms and consumers in terms of +profit and net consumer surplus, respectively. + +A3. Proof for Proposition 5 +Proof: +a) In all three cases, we observe that 𝜋1 +𝑄𝑃𝐶 > 𝜋1 +𝑃𝑃𝐶 and 𝜋1 +𝑄𝑄𝐶 > 𝜋1 +𝑃𝑄𝐶 for firm 1; and +𝜋2 +𝑃𝑄𝐶 > 𝜋2 +𝑃𝑃𝐶 and 𝜋2 +𝑄𝑄𝐶 > 𝜋2 +𝑄𝑃𝐶 for firm 2. This makes ‘Quantity contract’ as the +dominant strategy for both the firms and the Nash equilibrium is {Quantity, Quantity}. +b) In this case, we observe that 𝜋1 +𝑄𝑃𝐶 > 𝜋1 +𝑃𝑃𝐶 and 𝜋1 +𝑄𝑄𝐶 < 𝜋1 +𝑃𝑄𝐶 for firm 1; and 𝜋2 +𝑃𝑄𝐶 > +𝜋2 +𝑃𝑃𝐶 and 𝜋2 +𝑄𝑄𝐶 < 𝜋2 +𝑄𝑃𝐶 for firm 2. This means that there are no dominant strategies and +the best response of either firm is the choice of opposite strategy, and the Nash equilibria +are {Price, Quantity}, {Quantity Price}. + + + diff --git a/3tE1T4oBgHgl3EQfmAQH/content/tmp_files/load_file.txt b/3tE1T4oBgHgl3EQfmAQH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f710bb18b942622770009766b5e40b156bb817b4 --- /dev/null +++ b/3tE1T4oBgHgl3EQfmAQH/content/tmp_files/load_file.txt @@ -0,0 +1,274 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf,len=273 +page_content='Strategic Environmental Corporate Social Responsibility (ECSR) Certification and Endogenous Market Structure Ajay Sharma Indian Institute of Management, Indore (India) Siddhartha K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Rastogi Indian Institute of Management, Indore (India) Correspondence address: Ajay Sharma, J-206, Academic Block, Indian Institute of Management Indore, Prabandh Shikhar, Rau- Pithampur Road, Indore, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' (India) - 453556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Ph: +91-7312439622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' E-mail: ajays@iimidr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' ajaysharma87@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Siddhartha K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Rastogi, B-101, Academic Block, Indian Institute of Management Indore, Rau-Pithampur Road, Indore, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' (India) - 453556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Ph: +91-7312439534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' E-mail: srastogi@iimidr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='in Strategic Environmental Corporate Social Responsibility (ECSR) Certification and Endogenous Market Structure Abstract This paper extends the findings of Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' (2015, Strategic environmental corporate social responsibility in a differentiated duopoly market, Economics Letters), along two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' First, we consider the case of endogenous market structure a la Vives and Singh (1984, Price and quantity competition in a differentiated duopoly, The Rand Journal of Economics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Second, we refine the ECSR certification standards in differentiated duopoly with rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' We find that optimal ECSR certification standards by NGO are the highest in Bertrand competition, followed by mixed markets and the lowest in Cournot competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Next, NGO certifier will set the ECSR standards below the optimal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Also, we show that given the ECSR certification standards, there is a possibility of both price and quantity contracts choices by the firms in endogenous market structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' JEL Classification: D43;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' L13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' L22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' M14 Keywords: Corporate social responsibility certification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Differentiated duopoly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Environmental standards;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Price competition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Quantity competition Declaration of interest: The authors do not have any conflict of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Introduction Corporate Social Responsibility (CSR) has become a mainstream pursuit among the business activities of firms in the past few years, wherein more than 30% (71% and 90%) of companies in the US (the UK and Japan, respectively) adopted CSR reporting in 2013 (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Given the strategic importance of CSR activities as a non-core business pursuit and their significant implication for costs, eco-labeling, certification, hallmarking etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' are the common ways of CSR signaling especially for environmental outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Though certification is not a perfect mechanism, it is sufficiently trustworthy to convey useful information (Auriol and Schilizzi, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The certification can come from self or third-party and can be mandatory or optional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The existing literature on the strategic aspects of third-party certification focuses on nature of competition and third- party certifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Manasakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' (2013) suggest that the certification by alternative third parties differ with respect to their objectives and has implications for certification standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' (2015) compares the ECSR certification level in Cournot versus Bertrand competition and show that certification standards are lower in Bertrand than Cournot competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Our contribution to this literature is two folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' First, we extend the analysis of Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' (2015) by endogenizing the market structure a la Singh and Vives (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' If the firms have option of price or quantity contracts, given the ECSR standards, then, what would be optimal choice for the firms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Second, we refine the ECSR certification standards in this endogenous market structure by providing rankings and then considering uniform standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The Model Based on Manasakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' (2013) and Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' (2015), the utility function of a representative consumer is 𝑈 = (𝐴 + 𝑒1𝛼𝑠1)𝑞1 + (𝐴 + 𝑒2𝛼𝑠2)𝑞2 − (𝑞12 + 2𝛾𝑞1𝑞2 + 𝑞22) 2 where 𝑞𝑖 is output and 𝑠𝑖 is the level of ECSR, for firm 𝑖 (𝑖 = 1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The parameter 𝛾 ∈ (0,1) measures the nature of products being substitutes (𝛾 > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The parameter 𝛼 ∈ (0,1) indicates the consumer’s preference for firm’s ECSR activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The firms choose ECSR as a strategic variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Based on Manasakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' (2013), we consider that ECSR activities can be informed to consumers through a credible signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' For the same, the firm seeks certification from a third-party NGO certifier who maximizes Net Consumer Surplus (NCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' A firm can get the certification 𝑒𝑖 if it satisfies criteria of minimum level of ECSR activities 𝑠 : 𝑒𝑖 = {0 𝑖𝑓 𝑠𝑖 < 𝑠 𝑎𝑛𝑑 𝑓𝑖𝑟𝑚 𝑖 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑟𝑒𝑐𝑒𝑖𝑣𝑒 𝑎 𝐸𝐶𝑆𝑅 𝑐𝑒𝑡𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 1 𝑖𝑓 𝑠𝑖 ≥ 𝑠 𝑎𝑛𝑑 𝑓𝑖𝑟𝑚 𝑖 𝑟𝑒𝑐𝑒𝑖𝑣𝑒𝑠 𝑎 𝐸𝐶𝑆𝑅 𝑐𝑒𝑡𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 } It is important to note that a firm will consider doing ECSR activity only if it generates net positive benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' A firm would spend at 𝑠 (minimum ECSR for certification) and not beyond that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' If both firms choose to get the certification by spending 𝑠 on ECSR, the representative consumer’s utility would be: 𝑈 = (𝐴 + 𝑒1𝛼𝑠)𝑞1 + (𝐴 + 𝑒2𝛼𝑠)𝑞2 − (𝑞12 + 2𝛾𝑞1𝑞2 + 𝑞22) 2 The corresponding demand functions would be 𝑞𝑖 = 𝐴(1−𝛾)−𝑝𝑖+𝛾𝑝𝑗+𝛼𝑒𝑖𝑠−𝛼𝛾𝑒𝑗𝑠 1−𝛾2 and inverse demand functions 𝑝𝑖 = 𝐴 − 𝑞𝑖 − 𝛾𝑞𝑗 + 𝛼𝑒𝑖𝑠, 𝑓𝑜𝑟 𝑖, 𝑗 = 1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝑖 ≠ 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' We assume that firms use same technology with cost of production as zero, without loss of generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Also, one unit of output produces one unit of pollution emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The NGO certifier will not charge any fee for certification if firm complies with ECSR standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The cost of ECSR for firms is 𝑠𝑖2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The firm’s profit function is, 𝜋𝑖 = 𝑝𝑖𝑞𝑖 − 𝑒𝑖𝑠2, 𝑖 = 1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' NGO certifier’s objective function is 𝑁𝐶𝑆 = 𝐶𝑆 − 𝑑(𝑞1+𝑞2−𝑒1𝑠−𝑒2𝑠)2 2 where 𝐶𝑆 = (𝑞12+2𝛾𝑞1𝑞2+𝑞22) 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' and 𝑑 > 0 is the marginal environmental damage due to emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The Game The game is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' In the first stage, the firm decides to choose price or quantity contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' In the second stage, the certifier decides threshold level of ECSR for certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Firms meeting the threshold condition get the certification, otherwise not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' In the third stage, firms choose the level of output and prices to maximize their profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' We solve the game using backward induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Product market competition In this stage, we analyze the four possible options: a) both firms choose prices (𝑝𝑝) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', Bertrand competition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' b) both firms choose quantities (𝑞𝑞) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', Cournot competition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' c) one firm chooses price (quantity) contract while the other firm chooses the quantity (price) contract i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', 𝑝𝑞 (𝑞𝑝) outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' We avoid providing the calculations for (a) and (b) option for the sake of brevity, as they are identical to Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Please refer to the online appendix for the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Proposition 1: The NGO certifier will set the standards, 𝑠 = 𝑠𝑃𝑃𝑈 and 𝑠𝑄𝑄𝑈in the Bertrand (pp game) and Cournot (qq game) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Proof: See online appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Next, both (c) and (d) will be identical in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Therefore, we only solve the pq game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝑝𝑞 game (Price versus Quantity Contract) We use the superscript PQ for price-quantity contract case i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', firm 1 decides price while firm 2 decides quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The outcomes in the product market with firms not adopting ECSR are 𝑞1 𝑃𝑄𝑁 = 𝐴(2−𝛾−𝛾2) 4−3𝛾2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝑞2 𝑃𝑄𝑁 = 𝐴(2−𝛾) 4−3𝛾2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝑝1 𝑃𝑄𝑁 = 𝐴(2−𝛾−𝛾2) 4−3𝛾2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝑝2 𝑃𝑄𝑁 = 𝐴(2−𝛾)(1−𝛾)(1+𝛾) 4−3𝛾2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝜋1 𝑃𝑄𝑁 = 𝐴2(2−𝛾−𝛾2)2 (4−3𝛾2)2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝜋2 𝑃𝑄𝑁 = 𝐴2(2−𝛾)2(1−𝛾2) (4−3𝛾2)2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' NCS𝑃𝑄𝑁 = 𝐴2(8−10𝛾2+3𝛾4−𝑑(4−𝛾(2+𝛾))2) 2(4−3𝛾2)2 (5) If the firms choose to opt for ECSR activities and get certification, the equilibrium outcomes would be, 𝑞1 𝑃𝑄𝐶 = (2−𝛾−𝛾2)(𝐴+𝛼𝑠) 4−3𝛾2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝑞2 𝑃𝑄𝐶 = (2−𝛾)(𝐴+𝛼𝑠) 4−3𝛾2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝑝1 𝑃𝑄𝐶 = (2−𝛾−𝛾2)(𝐴+𝛼𝑠) 4−3𝛾2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝑝2 𝑃𝑄𝐶 = (2−𝛾)(1−𝛾)(1+𝛾)(𝐴+𝛼𝑠) 4−3𝛾2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝜋1 𝑃𝑄𝐶 = (𝐴2+2𝐴𝛼𝑠+𝛼2𝑠2)(2−𝛾−𝛾2)2−(4−3𝛾2) 2𝑠2 (4−3𝛾2)2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝜋2 𝑃𝑄𝐶 (𝐴2+2𝐴𝛼𝑠+𝛼2𝑠2)(2−𝛾)2(1−𝛾2)+((4−3𝛾2) 2 (4−3𝛾2)2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' NCS𝑃𝑄𝐶 = (2−𝛾2)(𝐴+𝛼𝑠)2 8−6𝛾2 − 𝐴2𝑑(−4+𝛾(2+𝛾))2 2(4−3𝛾2)2 For 𝑞1 𝑃𝑄𝐶 > 𝑠 , 𝑠 < 2𝐴−𝐴𝛾−𝐴𝛾2 4−2𝛼+𝛼𝛾−3𝛾2+𝛼𝛾2 and for 𝑞2 𝑃𝑄𝐶 > 𝑠 , 𝑠 < 2𝐴−𝐴𝛾 4−2𝛼+𝛼𝛾−3𝛾2 should be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' For both 𝑞1 𝑃𝑄𝐶 > 𝑠 𝑎𝑛𝑑 𝑞2 𝑃𝑄𝐶 > 𝑠, 𝑠 < 2𝐴−𝐴𝛾−𝐴𝛾2 4−2𝛼+𝛼𝛾−3𝛾2+𝛼𝛾2 must be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Further 𝑞2 𝑃𝑄𝐶 > 𝑞1 𝑃𝑄𝐶 holds for all parametric values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Firm 1 would be willing to adopt ECSR certification if 𝜋1 𝑃𝑄𝐶 > 𝜋1 𝑃𝑄𝑁 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', 𝑠 < 𝑠𝑃𝑄𝑈1 = 2𝐴−𝐴𝛾−𝐴𝛾2 4−2𝛼+𝛼𝛾−3𝛾2+𝛼𝛾2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' For firm 2, decision to adopt ECSR certification is chosen if 𝜋2 𝑃𝑄𝐶 > 𝜋2 𝑃𝑄𝑁 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝑠 < 𝑠𝑃𝑄𝑈2 = 𝐴𝛼(2−𝛾−𝛾2)2 (4−3𝛾2)2−𝛼2(2−𝛾−𝛾2)2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Also, comparing the upper threshold of spending on ECSR, we observe that firm 1 (choosing price) has higher threshold than firm 2 (choosing quantity)’s ECSR spending, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', 𝑠𝑃𝑄𝑈1 > 𝑠𝑃𝑄𝑈2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Lemma 1: In a price vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' quantity game, price setting firm has higher threshold for ECSR spending than quantity setting firm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' NGO certifier Coming to second stage, we obtain the optimal choice of ECSR certification standard for NGO certifier by evaluating 𝑑 𝑁𝐶𝑆𝑃𝑄𝐶 𝑑𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' We get 𝑠𝑃𝑄∗ = 𝐴(𝛼(8 − 10𝛾2 + 3𝛾4) − 𝑑(−4 + 𝛾(2 + 𝛾))(8 − 6𝛾2 + 𝛼(−4 + 𝛾(2 + 𝛾)))) 𝛼2(−8 + 10𝛾2 − 3𝛾4) + 𝑑(8 − 6𝛾2 + 𝛼(−4 + 𝛾(2 + 𝛾)))2 𝑠𝑃𝑄∗ > 0 if 𝑑 > 𝛼2(8−10𝛾2+3𝛾4) (8−6𝛾2+𝛼(−4+𝛾(2+𝛾)))2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Further, 𝑠𝑃𝑄∗ > 𝑠𝑃𝑄𝑈1 and 𝑠𝑃𝑄∗ > 𝑠𝑃𝑄𝑈2 when 𝑑 > 𝛼2(8−10𝛾2+3𝛾4) (8−6𝛾2+𝛼(−4+𝛾(2+𝛾)))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' This means that certifier’s optimal level of ECSR standard would be higher than the upper limit for the firms in price vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' quantity competition and any firm will not spend on ECSR if a certifier sets the standard at 𝑠𝑃𝑄∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' NGO certifier can set the ECSR standard for certification at either 𝑠 = 𝑠𝑃𝑄𝑈1or 𝑠 = 𝑠𝑃𝑄𝑈2 level for participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' If 𝑠 = 𝑠𝑃𝑄𝑈1 is chosen as ECSR standard, then only price-setting firm 1 will get the certification and quantity-setting firm 2 will not get ECSR certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Interestingly, profit of firm 2 will be higher than firm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' On the other hand, 𝑠 = 𝑠𝑃𝑄𝑈2 as ECSR standard leads to both firms getting the certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' In this case also, firm 1’s profit would be lower than firm 2’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' This indicates that quantity setting firm 2 has net advantage over price setting firm 1 irrespective of whether firm 1 unilaterally get the ECSR certification or both firms get the certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' This is a new result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Therefore, to induce the firms in adopting the certification, the standard would be set at 𝑠 = 𝑠𝑃𝑄𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Further, we find that consumers and firms would benefit from such ECSR standard as compared to no ECSR at all because NCSPQC > NCSPQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Proposition 2: In a price vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' quantity competition, a) NGO certifier will set the ECSR standard below the optimal level b) if ECSR certification standard is set at 𝑠 = 𝑠𝑃𝑄𝑈1, then only price-setting firm will get the certification, whereas quantity-setting firm 2 will not opt for certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' c) if ECSR certification standard is set at 𝑠 = 𝑠𝑃𝑄𝑈2, then both firms will get the certification and it is beneficial for both firms and consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Comparison of ECSR Certification Standards Comparing the optimal ECSR standard, 𝑠 by the NGO certifier across endogenous market structure, we observe that 𝑠𝑃𝑃∗ > 𝑠𝑃𝑄∗ > 𝑠𝑄𝑄∗ indicating Bertrand has the highest level followed by price-quantity and lastly Cournot case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Proposition 3: Across the spectrum of market structure, the NGO certifier’s optimal ECSR standard rankings are 𝑠𝑃𝑃∗ > 𝑠𝑃𝑄∗ > 𝑠𝑄𝑄∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' In all the cases, the NGO certifier is not able to implement the optimal level of ECSR standard because the firms will not adopt such ECSR standards as that leads to lower profit for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Therefore, the certifier would choose a sub-optimal ECSR standard to incentivize the firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Comparing these equilibrium standards, we get the ranking in proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Proposition 4: The NGO certifier’s equilibrium ECSR standard rankings are 𝑠𝑃𝑄𝑈1 > 𝑠𝑄𝑄𝑈 > 𝑠𝑃𝑃𝑈 > 𝑠𝑃𝑄𝑈2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Endogenous Market Structure: Price or Quantity Contract Now, we solve the first stage of the game where firms have options to choose price or quantity contracts in the product market competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' For the sake of brevity, we do not consider the case where no firm chooses ECSR certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The outcome of that subgame will be identical to Singh and Vives (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Lemma 2 (Singh and Vives, 1984): In a product market competition for substitute goods1, with price and quantity as strategic choices, firms choose quantity contracts as dominant strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' ECSR certification standards and market structure If firms opt for the ECSR certification, then the outcome of the subgame can differ from Singh and Vives (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The certifier can choose a uniform standard irrespective of the nature of market competition, or different standards based on nature of competition2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' We only consider possibility of uniform ECSR certification standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' In case of uniform ECSR standard, there are four choices, 𝑠𝑃𝑄𝑈1 > 𝑠𝑄𝑄𝑈 > 𝑠𝑃𝑃𝑈 > 𝑠𝑃𝑄𝑈2 (see Proposition 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' If the NGO certifier sets the lowest three ECSR certification standard, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', either 𝑠𝑃𝑄𝑈2 or 𝑠𝑃𝑃𝑈 or 𝑠𝑄𝑄𝑈, then the Nash equilibrium outcome of the game in Table 1, is {Quantity, Quantity}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' On the other hand, if the ECSR certifier sets the standard at the highest level possible i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', 𝑠𝑃𝑄𝑈1, then there are two Nash equilibria outcomes of the game {Price, Quantity} and {Quantity, Price}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 1 In this paper, we only consider substitute goods in the market offered by competing firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 2 But from a real-world point of view, such standards may not be feasible due to monitoring issues, discrimination, and mimicking behavior among firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Table 1: Price-Quantity Contract Game (with ECSR certification) Firm 2 Firm 1 Price Quantity Price 𝜋1 𝑃𝑃𝐶,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝜋2 𝑃𝑃𝐶 𝜋1 𝑃𝑄𝐶,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝜋2 𝑃𝑄𝐶 Quantity 𝜋1 𝑄𝑃𝐶,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝜋2 𝑄𝑃𝐶 𝜋1 𝑄𝑄𝐶,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝜋2 𝑄𝑄𝐶 Proposition 5: In a price-quantity contract game,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' a) If a certifier decides,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' the uniform ECSR standard at either 𝑠𝑃𝑄𝑈2 or 𝑠𝑃𝑃𝑈 or 𝑠𝑄𝑄𝑈 level,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' the subgame perfect Nash equilibrium is {Quantity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Quantity} b) If the certifier decides,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' the uniform ECSR standard at 𝑠𝑃𝑄𝑈1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' there are two subgame perfect Nash equilibria {price,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Quantity},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' {Quantity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Price} Proof: See online appendix 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Conclusion In this paper, we analyze the relationship between endogenous market structure and strategic ECSR in a differentiated duopoly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' We show that NGO certifier will always set the ECSR standards below the optimal level to ensure participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' In a price-quantity game, there is possibility of partial or full compliance with ECSR standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Lastly, while setting a uniform ECSR standards in endogenous market structure, there is a possibility of Cournot outcome as well as mixed market outcome.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 105-121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', and Matsumura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' (2017) Corporate social responsibility and privatization policy in a mixed oligopoly.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' (2015) Strategic environmental corporate social responsibility in a differentiated duopoly market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Economics Letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 108-111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Manasakis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', Mitrokostas, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', and Petrakis, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' (2013) Certification of corporate social responsibility activities in oligopolistic markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Canadian Journal of Economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 46(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 282-309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Singh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', and Vives, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' (1984) Price and quantity competition in a differentiated duopoly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The Rand Journal of Economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 546-554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' ONLINE APPENDIX A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' A 𝑝𝑝 game (Bertrand Competition) We use the superscript PPN to denote equilibrium outcome for firms not adopting ECSR in 𝑝𝑝 game i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', Bertrand competition, otherwise PPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Solving the game, we get 𝑞1 𝑃𝑃𝑁 = 𝑞2 𝑃𝑃𝑁 = 𝐴 2+𝛾−𝛾2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝑝1 𝑃𝑃𝑁 = 𝑝2 𝑃𝑃𝑁 = 𝐴(1−𝛾) 2−𝛾 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝜋1 𝑃𝑃𝑁 = 𝜋2 𝑃𝑃𝑁 = 𝐴2(1−𝛾) (2−𝛾)2(1+𝛾) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' NCSPPN = 𝐴2(1−2𝑑+𝛾) (2+𝛾−𝛾2)2 (1) If the firms decide to adopt for ECSR and get certification for the same, the outcomes are 𝑞1 𝑃𝑃𝐶 = 𝑞1 𝑃𝑃𝐶 = 𝐴 + 𝛼𝑠 2 + 𝛾 − 𝛾2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝑝1 𝑃𝑃𝐶 = 𝑝2 𝑃𝑃𝐶 = (1 − 𝛾)(𝐴 + 𝛼𝑠) 2 − 𝛾 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝜋1 𝑃𝑃𝐶 = 𝜋2 𝑃𝑃𝐶 = (1 − 𝛾)(𝐴 + 𝛼𝑠) 2 (2 − 𝛾)(2 + 𝛾 − 𝛾2) − 𝑠2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' NCSPPC = (𝐴+𝛼𝑠)2 (2−𝛾)2(1+𝛾) − 2𝑑(𝐴+(𝛼+(−2+𝛾)(1+𝛾))𝑠)2 (2+𝛾−𝛾2)2 (2) For a firm to be adopting ECSR, the certification threshold needs to be lower than the level of pollution, otherwise the cost will be more than its benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' For 𝑞𝑖 𝑃𝑃𝐶 > 𝑠, 𝑠 < 𝐴 2−𝛼+𝛾−𝛾2 should be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Comparing the profits of firms3 with or without ECSR, 𝜋1 𝑃𝑃𝐶 − 𝜋1 𝑃𝑃𝑁 = 𝐴2(1 − 𝛾) + 2𝐴𝛼(1 − 𝛾)𝑠 − (4 − 𝛼2(1 − 𝛾) − (3 − 𝛾)𝛾2)𝑠2 (2 − 𝛾)2(1 + 𝛾) 3 Given the symmetry of the firms and their outcomes, we only compare the results of one firm, and it holds for both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' We observe that 𝜋1 𝑃𝑃𝐶 − 𝜋1 𝑃𝑃𝑁 > 0 if 𝑠 < 𝑠𝑃𝑃𝑈 = 2𝐴𝛼(1−𝛾) 4−𝛼2+𝛼2𝛾−3𝛾2+𝛾3 , which provides the upper bound for the ECSR spending to adopt the ECSR certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' So, firms will spend strategically on ECSR and get certification if 𝑠 < 𝑠𝑃𝑃𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='4 Optimal ECSR Certification Standard We obtain the optimal choice of ECSR certification standard in case of NGO certifier by evaluating 𝑑 𝑁𝐶𝑆𝑃𝑃𝐶 𝑑𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' We get, 𝑠𝑃𝑃∗ = 𝐴(𝛼 + 𝛼𝛾 − 2𝑑(𝛼 − (2 − 𝛾)(1 + 𝛾))) 2𝑑(𝛼 − (2 − 𝛾)(1 + 𝛾))2 − 𝛼2(1 + 𝛾) 𝑠𝑃𝑃∗ > 0 if 𝑑 > 𝛼2+𝛼2𝛾 2(𝛼−(2−𝛾)(1+𝛾))2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Further, 𝑠𝑃𝑃∗ > 𝑠𝑃𝑃𝑈 when 𝑑 > 𝛼2+𝛼2𝛾 2(𝛼−(2−𝛾)(1+𝛾))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' This means that an NGO certifier’s optimal level of ECSR standard would be higher that the upper limit for the firms in Bertrand competition and a firm would not choose to spend on ECSR if a certifier sets the standard at 𝑠𝑃𝑃∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Therefore, to induce the firms, the standard would be set at 𝑠 = 𝑠𝑃𝑃𝑈 by the NGO certifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Further, we can also show that consumer and firms would benefit from such ECSR standard as compared to no ECSR at all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', NCSPPC > NCSPPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Proposition A1: An NGO certifier would set the ECSR standard below the optimal level if firms engage in Bertrand competition and, it is beneficial for both firms and consumers in terms of profit and net consumer surplus, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' A 𝑞𝑞 game (Cournot Competition) 4 Superscript U denotes upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' For Cournot game, we use the superscript QQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' The outcomes of the product market competition, if firms do not adopt ECSR are, 𝑞1 𝑄𝑄𝑁 = 𝑞2 𝑄𝑄𝑁 = 𝐴 2+𝛾 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝑝1 𝑄𝑄𝑁 = 𝑝2 𝑄𝑄𝑁 = 𝐴 2+𝛾 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝜋1 𝑄𝑄𝑁 = 𝜋2 𝑄𝑄𝑁 = 𝐴2 (2+𝛾)2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' NCS𝑄𝑄𝑁 = 𝐴2(1−2𝑑+𝛾) (2+𝛾)2 (3) On the other hand, if the firms decide to adopt ECSR certification, the outcomes would be, 𝑞1 𝑄𝑄𝐶 = 𝑞2 𝑃𝑃𝐶 = 𝐴+𝛼𝑠 2+𝛾 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝑝1 𝑄𝑄𝐶 = 𝑝2 𝑄𝑄𝐶 = 𝐴+𝛼𝑠 2+𝛾 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' 𝜋1 𝑄𝑄𝐶 = 𝜋2 𝑄𝑄𝐶 = (𝐴+𝛼𝑠)2 (2+𝛾)2 − 𝑠2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' NCS𝑄𝑄𝐶 = (1+𝛾)(𝐴+𝛼𝑠) 2−2𝑑(𝐴−(2−𝛼+𝛾)𝑠)2 (2+𝛾)2 (4) For 𝑞𝑖 𝑄𝑄𝐶 > 𝑠 , 𝑠 < 𝐴 2−𝛼+𝛾 should be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Further comparing the profits of firms with and without adopting ECSR, 𝜋1 𝑄𝑄𝐶 − 𝜋1 𝑄𝑄𝑁 = 𝐴2 + 2𝐴𝛼𝑠 + (−2 + 𝛼 − 𝛾)(2 + 𝛼 + 𝛾)𝑠2 (2 + 𝛾)2 A firm would profit from adopting ECSR if 𝜋1 𝑄𝑄𝐶 > 𝜋1 𝑄𝑄𝑁 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', when 𝑠 < 𝑠𝑄𝑄𝑈 = 2𝐴𝛼 4−𝛼2+4𝛾+𝛾2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' This denotes the upper bound to spend on ECSR for certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Optimal ECSR Certification Standard We obtain the optimal choice of ECSR certification standard in case of NGO certifier by evaluating 𝑑 𝑁𝐶𝑆𝑄𝑄𝐶 𝑑𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' We get, 𝑠𝑄𝑄∗ = 𝐴(𝛼 + 𝛼𝛾 + 2𝑑(2 − 𝛼 + 𝛾)) 2𝑑(2 − 𝛼 + 𝛾)2 − 𝛼2(1 + 𝛾) 𝑠𝑄𝑄∗ > 0 if 𝑑 > 𝛼2+𝛼2𝛾 2(2−𝛼+𝛾)2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Further, 𝑠𝑄𝑄∗ > 𝑠𝑄𝑄𝑈 when 𝑑 > 𝛼2+𝛼2𝛾 2(2−𝛼+𝛾)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' This means that a certifier’s optimal level of ECSR standard would be higher than the upper limit for the firms in Cournot competition and a firm would not choose to spend on ECSR if a certifier sets the standard at 𝑠𝑄𝑄∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Therefore, to induce the firms, the standard would be set at 𝑠 = 𝑠𝑄𝑄𝑈 by the NGO certifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Further, we can also show that consumer and firms would benefit from such ECSR standard as compared to no ECSR at all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=', NCSQQC > NCSQQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Proposition A2: An NGO certifier would set the ECSR standard below the optimal level if firms engage in Cournot competition and, it is beneficial for both firms and consumers in terms of profit and net consumer surplus, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' Proof for Proposition 5 Proof: a) In all three cases, we observe that 𝜋1 𝑄𝑃𝐶 > 𝜋1 𝑃𝑃𝐶 and 𝜋1 𝑄𝑄𝐶 > 𝜋1 𝑃𝑄𝐶 for firm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' and 𝜋2 𝑃𝑄𝐶 > 𝜋2 𝑃𝑃𝐶 and 𝜋2 𝑄𝑄𝐶 > 𝜋2 𝑄𝑃𝐶 for firm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' This makes ‘Quantity contract’ as the dominant strategy for both the firms and the Nash equilibrium is {Quantity, Quantity}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' b) In this case, we observe that 𝜋1 𝑄𝑃𝐶 > 𝜋1 𝑃𝑃𝐶 and 𝜋1 𝑄𝑄𝐶 < 𝜋1 𝑃𝑄𝐶 for firm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' and 𝜋2 𝑃𝑄𝐶 > 𝜋2 𝑃𝑃𝐶 and 𝜋2 𝑄𝑄𝐶 < 𝜋2 𝑄𝑃𝐶 for firm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} +page_content=' This means that there are no dominant strategies and the best response of either firm is the choice of opposite strategy, and the Nash equilibria are {Price, Quantity}, {Quantity Price}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tE1T4oBgHgl3EQfmAQH/content/2301.03291v1.pdf'} diff --git a/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf b/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7337ecb36946dded513e9e64257c5176f7279ca2 --- /dev/null +++ b/49FAT4oBgHgl3EQfFRw-/content/2301.08426v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6bed7268f0535987765d1b982e4347d66a93f74e4498258c83df5946c5921f0a +size 2026121 diff --git a/49FAT4oBgHgl3EQfFRw-/vector_store/index.pkl b/49FAT4oBgHgl3EQfFRw-/vector_store/index.pkl new file mode 100644 index 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Economics of Education, University of Bern +2 Swiss Institute for Empirical Economic Research, University of St.Gallen +3 Department of Economics, University of Potsdam +This version: January 30, 2023 +Abstract +We analyze the causal impact of positive and negative feedback on professional +performance. We exploit a unique data source in which quasi-random, naturally +occurring variations within subjective ratings serve as positive and negative feed- +back. The analysis shows that receiving positive feedback has a favorable impact +on subsequent performance, while negative feedback does not have an effect. These +main results are found in two different environments and for distinct cultural back- +grounds, experiences, and gender of the feedback recipients. The findings imply +that managers should focus on giving positive motivational feedback. +Keywords: Feedback, Performance, Causal Analysis, Cultural Background +∗Preliminary version. Do not quote or circulate without permission of one of the authors. Comments +are very welcome. We like to thank Swiss-ski (Swiss ski federation), Michel Roth, David Morris, Alexan- +der Mesch, and the DSV (German swimming federation) for valuable insights into the sports contests +from the perspective of (former) professional athletes. Also, we thank participants of the ESEA Confer- +ence, 2022, and the Berlin School of Economics Workshop, 2022, as well as Enzo Brox, Lisa Bruttel, and +Sandro Heiniger for their helpful comments and suggestions. +arXiv:2301.11776v1 [econ.GN] 27 Jan 2023 + +1 +Introduction +Providing performance feedback is one of the main tasks of managers and leaders (Morgeson, +DeRue, & Karam, 2010). One important aim of feedback is to create a favorable emotional +response. +At best, positive or negative feedback can motivate employees and increase their +productivity. In the worst case, it leaves the employees frustrated and unproductive. Therefore, +the question of how feedback impacts subsequent performance is of tremendous importance. +Consequently, numerous studies investigating the impact of feedback on creativity (Harrison +& Rouse, 2015; Itzchakov & Latham, 2020; Kim & Kim, 2020), the learning process of indi- +viduals and firms (Hattie & Timperley, 2007; Lee, Lee, & Kim, 2021) or motivation (Deci & +Casico, 1972; Fong, Patall, Vasquez, & Stautberg, 2019) emerged. In particular for positive and +negative feedback on performance or productivity, studies show the full range from favorable +to unfavorable effects (Eggers & Suh, 2019; Kluger & DeNisi, 1996; Podsakoff & Farh, 1989; +Sleiman, Sigurjonsdottir, Elnes, Gage, & Gravina, 2020; Waldersee & Luthans, 1994, etc). +The two major difficulties when investigating the impact of feedback on performance are +(1) observing truthful and trustworthy feedback in real-incentive situations and (2) quantifying +feedback and performance. While observational studies typically fail to satisfactorily tackle the +second difficulty, experimental studies cannot fulfill the first requirement. We are not aware of +any causal study in which both requirements are met together. +To address this common shortcoming, we exploit a unique setting to estimate the causal effect +of positive and negative feedback on subsequent performance. For this purpose, we use data +from professional sports: diving as the primary data source, and ski jumping for supplementary +analyses. In these sports, individuals’ performance is evaluated subjectively by a jury of seven +(or five) experienced judges according to precise rules. Each judge independently issues one +rating for the task performance (hereafter, "judges rating" or “rating”). Discarding the highest +and lowest rating(s), the common assessment of the jury is calculated from the average of the +three remaining ratings (hereafter, “jury performance assessment”).1 +Following the definition in Kluger and DeNisi (1996), stating that feedback is information +about one’s task performance provided by an external agent, we consider the deviation of the +discarded (highest and lowest) ratings from the jury’s performance assessment as feedback on +1Receiving the jury performance assessment can already be seen as a knowledge of results (Kluger +& DeNisi, 1996) intervention. The analysis of this knowledge of results, however, is beyond the scope of +this paper. +2 + +task performance. The discarded ratings are not relevant to the assessment of task performance, +but this additional information about judges’ general perceptions of performance provides feed- +back that can only work through the motivational channel on subsequent performance. Kluger +and DeNisi (1996) argue that the feedback sign depends on the relation between the performance +rating and a benchmark. In line with this, discarded ratings define quasi-randomly occurring +positive (negative) deviations from the jury performance evaluation that serve as positive (neg- +ative) feedback. No deviation from the benchmark implies neutral feedback. We describe the +evaluation and feedback process in more detail in Section 3.2, Figure 1. +We test several of the propositions from the model of the seminal work by Kluger and +DeNisi (1996) within a single framework. +In our setup, the feedback is truthful, accurately +observable, and from an external source. Feedback can impact subsequent performance only +through its motivational impact. +Performance is strongly incentivized and can be precisely +quantified. The performance is measured in non-artificial tasks that individuals are not only +familiar with but that are routine aspects of their work. What is particularly valuable from a +management perspective is that we can investigate the impact of feedback in an international +context. +Theoretically guided by the feedback intervention model (Kluger & DeNisi, 1996), we inves- +tigate the effect of positive and negative feedback on performance. Further, we investigate the +internal and external generalizability of the results. To assess internal generalizability, we can +use our extensive data to analyze whether situational (or personal) variables and task charac- +teristics moderate the effects of the feedback intervention on performance. The international +sample covering female and male individuals from more than 50 nations from 6 continents offer +the unique opportunity to analyze feedback effects for different cultural backgrounds and gender +within the same framework. To investigate external generalizability, we complement the main +findings with a second, independent setting. We investigate these aspects using both classical +statistical and causal machine learning methods. This is followed by analyses examining the +feedback interventions’ long-term, repetition, and spill-over effects. +Our analysis shows a performance-enhancing causal effect of positive feedback. The favorable +effect of positive feedback is found for recipients from different cultural backgrounds, experience +levels, and gender. We observe favorable effects even when individuals repeatedly receive positive +feedback. The impact of positive feedback is stronger when the relevance of the task is high. +3 + +In contrast to all this, negative feedback on average does not have an impact on performance. +Merely, the subgroup of the more experienced individuals benefits from negative feedback. +Our findings imply that managers can use positive feedback to enhance the performance of +their employees. Importantly, positive feedback can be given repeatedly on a regular basis. It +has a favorable impact irrespective of several relevant characteristics of the recipient and can be +universally applied in an international context. With our main finding we are in line with the +studies conducted by Azmat and Iriberri (2010), Bandiera, Larcinese, and Rasul (2015), Choi, +Johnson, Moon, and Oah (2018), and Itzchakov and Latham (2020) for positive feedback and +the meta-study by Fong et al. (2019) for negative feedback. We complement decades of research +that provides guidelines on how to optimally give feedback (Balcazar, Hopkins, & Suarez, 1985; +Alvero, Bucklin, & Austin, 2001; Sleiman et al., 2020). +2 +Theoretical framing +To provide a theoretical foundation for the later empirical analysis, we begin by describing the +concept of feedback. Then, we collect relevant empirical research and form predictions based on +propositions stated by Kluger and DeNisi (1996). +2.1 +The concept of feedback +Feedback exists in many forms. Kluger and DeNisi (1996) define feedback as "[...] actions taken +by (an) external agent (s) to provide information regarding some aspect (s) of one’s task per- +formance" (p. 255). Burgers, Eden, van Engelenburg, and Buningh (2015) distinguish between +elaborate and simple feedback. Elaborate feedback typically includes a lengthy explanation, +which provides a guide for learning. Simple feedback merely gives information, about whether +something was done right or wrong. Burgers et al. (2015) further distinguish between descrip- +tive, comparative, and evaluative feedback. Descriptive feedback – sometimes called objective +feedback (Johnson, 2013) – merely sums up behavior shown by the agent. Comparative feedback +uses the performance of other individuals as a reference. Evaluative feedback provides a judg- +ment of the performance. Villeval (2020) distinguishes between a cognitive and a motivational +perspective. The cognitive perspective rests on the assumption that individuals have imperfect +knowledge about their skills. Here, feedback serves as a signal used in an information-updating +4 + +process. The motivational perspective focuses on the impact of feedback on intrinsic motivation. +Individuals might receive feedback from one agent or several agents. Stone and Stone (1984) +find that receiving feedback from two sources instead of one source increases self-perceived task +competence. Related, there is a strand of literature analyzing multi-source feedback (Bailey +& Fletcher, 2002; Smither, London, & Reilley, 2005), also called 360 degree feedback (DeNisi +& Kluger, 2000). Finally, feedback can be with direct consequences or inconsequential. Often +feedback comes without direct (monetary) consequences. Still, research shows that agents also +react to irrelevant information (Abeler, Falk, Goette, & Huffman, 2011; Cason & Mui, 1998). +The focus of our paper lies on the impact of simple and evaluative feedback on subsequent +performance. The feedback is subjective in the sense that is created by subjective evaluation +based on objective guidelines. Our study focuses on the impact of single feedback embedded +in a multi-source evaluative process. The feedback has no further consequences besides that +it can motivate or demotivate the recipient. +One important distinction is between positive +and negative feedback. +We define positive feedback, sometimes called promotion-orientated +feedback (Carpentier & Mageau, 2013), as the expression that the evaluated performance is +above a certain reference point. We define negative feedback, sometimes called change-orientated +feedback (Carpentier & Mageau, 2013) or corrective feedback (Waldersee & Luthans, 1994), as +the expression that the rated performance is below the reference. +2.2 +Review and hypotheses +In their influential model, Kluger and DeNisi (1996) assume that there are no behavioral effects +when there is no discrepancy between the rating and the reference. Positive feedback increases +effort if the agent has the possibility to set new self-goals. Likewise, negative feedback leads +to an increase in effort. Similarly, Villeval (2020) argues that positive and negative feedback +fosters motivation. On the other hand, positive feedback can lead to a decrease in efforts, when +individuals have no possibility to set new goals (Kluger & DeNisi, 1996). Negative feedback can +discourage individuals when it threatens the self-perception of their competence (Fong et al., +2019). +Some empirical studies show a favorable impact of positive feedback. Choi et al. (2018) find +a better performance in a computerized task after purely positive feedback than in a baseline +treatment. Itzchakov and Latham (2020) report better performance in a brainstorming task +5 + +after positive than after neutral feedback. Bandiera et al. (2015) report that positive feedback +improves the performance of university students and Azmat and Iriberri (2010) that positive +relative rank feedback enhances the performance of high school students. Other studies, such +as Podsakoff and Farh (1989) reporting no impact of positive feedback on performance in an +object-listing task, find no influence of positive feedback. Waldersee and Luthans (1994) even +report an adverse impact of positive feedback on the performance of employees of fast food +restaurants. +Empirical work on the effect of negative feedback provides an ambiguous picture. Several +studies show a favorable impact of negative feedback. +As for positive feedback, Choi et al. +(2018) find an improved performance after purely negative feedback in comparison to a baseline +treatment. Azmat and Iriberri (2010) find a favorable effect of negative relative rank feedback. +Itzchakov and Latham (2020) report a positive impact of negative feedback on performance in a +brainstorming task. Podsakoff and Farh (1989) report a favorable impact of negative feedback +in an object-listing task. Waldersee and Luthans (1994) find a performance-enhancing effect of +negative feedback for employees of fast food restaurants. Some research, such as the meta-study +by Fong et al. (2019), shows no impact of negative feedback. Other studies show an unfavorable +impact. For example, Deci and Casico (1972) observe that a negative feedback group shows +lower motivation to conduct a puzzle task than a control group. +A reason for the ambiguity in reaction to negative feedback might be heterogeneity in the way +how individuals update their perception after receiving self-relevant information. Some research +finds that agents do not fully update their self-perception after negative information, while they +update their self-perception after observing a positive signal (Eil & Rao, 2011; Kuzmanovic, +Jefferson, & Vogeley, 2015; Möbius, Niederle, Niehaus, & Rosenblat, 2022; Sharot et al., 2012). +This would imply to find no reaction to negative feedback. Yet, other studies observe a rational +updating of beliefs for positive and negative information (Barron, 2021) or even an overweighting +of negative information (Coutts, 2019; Ertac, 2011), leaving this strand of empirical research +inconclusive. +We build our hypotheses on the theoretical model by Kluger and DeNisi (1996). We argue +that in the domain of professional performance, there is always the possibility to set more am- +bitious goals. This indicates that positive feedback might have a favorable impact. +6 + +Hypothesis 1 - Positive Feedback: +The performance is better after receiving positive feedback than after receiving +neutral feedback. +We follow Kluger and DeNisi (1996) and Villeval (2020) by assuming that also negative feed- +back has a performance-enhancing effect. We argue that in the field of professional performance, +individuals have a rather stable self-perception of confidence. +Hypothesis 2 - Negative Feedback: +The performance is better after receiving negative feedback than after receiving +neutral feedback. +A vital aspect that most empirical studies usually can barely answer is the question of the +generalizability of these hypotheses. Here, it is useful to distinguish between the two superor- +dinate layers of personal and task-specific characteristics by which effects could be moderated +(compare Fong et al. (2019), for example). +For task characteristics, our hypotheses more readily generalize when individuals’ responses +to feedback are inherently similar irrespective of the difficulty and importance of the task. +Difficult and easy tasks might be perceived differently (Moore & Healy, 2008), which can lead to +different perceptions of feedback (Pulford & Colman, 1997) and varying subsequent performance +(Vancouver & Tischner, 2004). Kluger and DeNisi (1996) argue that the reaction to feedback +is stronger the fewer cognitive resources are needed to perform the task. Likewise, performance +might differ depending on the importance of the task (Goller & Heiniger, 2022). Here, Kluger +and DeNisi (1996) argue that the effectiveness of feedback increases the more attention is on +the task. +Guided by the model predictions of Kluger and DeNisi (1996), we do not expect +generalizability across task characteristics. Accordingly, we expect stronger feedback effects on +performance for (relatively) easier tasks needing fewer cognitive resources and more important +tasks that require more attention. +Within the personal domain, three potential moderators seem highly relevant in modern +workplaces: cultural background, gender, and experience of the feedback recipients. The litera- +ture acknowledges that despite the high relevance of cultural differences in a globalized world, +non-WEIRD (not coming from Western, Educated, Industrialized, Rich, and Democratic coun- +tries) individuals are largely underrepresented in behavioral research (Henrich, Heine, & Noren- +zayan, 2010). For example, authors postulate differences in self-construals (Markus & Kitayama, +7 + +1991), in feedback seeking of individuals (Sully De Luque & Sommer, 2000) and in feedback re- +action of firms (Rhee, Alexandra, & Powell, 2020) between collectivistic and individualistic +cultures. +Bear, Cushenbery, London, and Sherman (2017) postulate and Berlin and Dargnies (2016), +respectively, Roberts and Nolen-Hoeksema (1994) observe different feedback reactions for women +than for men. Eggers and Suh (2019) find that the reaction of organizations to negative feedback +depends on the experience in the business area. Kluger and DeNisi (1996) propose differential +effects for individuals’ behavioral or psychological traits. +More relevant from a managerial +perspective is if those potentially moderating traits are associated with directly observable char- +acteristics of individuals in a company’s diverse context. We refrain from forming explicit ex- +pectations and leave the question of generalizability for different cultural backgrounds, genders, +and experience levels exploratory. +3 +Setting and data +We collect data on international competitions of two competitive sports. In the two sports, +namely, ski jumping and diving, athletes compete individually in multi-round competitions. In +each round, the athletes’ task execution is evaluated by multiple professional judges. +Besides the similarities, there are several specifics to each of the sports. In diving, athletes +acrobatically jump into the water. We use data on individual performances in three different +types of competitions: 1m springboard, 3m springboard, and 10m platform. The scoring consists +of two elements. First, each jump is rated by seven judges with respect to the proper execution. +Each judge can reward up to 10 style points (in increments of 0.5). The two highest and the two +lowest judges’ ratings are discarded for the jury performance assessment of the jump, for which +the remaining three judges’ ratings are summed up. Second, the jury performance assessment +is multiplied by the difficulty coefficient, which depends on the complexity of the jump and is +assigned to the jump according to the official rules.2 In competitions between women, points +are accumulated over five jumps, and in competitions between men, over six jumps. Depending +on the contest there are preliminary rounds and/or semi-finals and the final round. +2See +https://resources.fina.org/fina/document/2021/01/12/916f78f6-2a42-46d6-bea8 +-e49130211edf/2017-2021_diving_16032018.pdf for a current version of the rules (last accessed on +01/23/2023). +8 + +In the winter sport of ski jumping, athletes jump on skis after sliding down a ramp. Scoring +consists of four components. First, athletes receive points for the length of their jump. Second, +there are compensation points for the force and direction of the wind. Third, scoring depends on +the length of the ramp (gate points). Fourth, athletes receive up to 20 style points (in increments +of 0.5) for the flight and landing of the jump. The (style) ratings are independently rewarded by +five judges according to official rules.3 The worst and the best rating are discarded and the other +three are accounted for the athletes’ score of the round. In a typical competition, 50 athletes +start in the first round, of which the 30 best reach the final round. After the final round, both +jumps’ total scores are added to determine the winner and the succeeding rankings. +3.1 +Data sets +Table 1: Descriptive statistics +Diving +Ski jumping +Mean +Std. dev. +Mean +Std. dev. +Panel A: Treatments +Positive Feedback (deviation positive) +0.426 +(0.286) +0.316 +(0.262) +Negative Feedback (deviation negative) +0.477 +(0.320) +0.357 +(0.290) +Panel B: Outcomes +Score +68.737 +(14.557) +118.647 +(16.204) +Performance (rem. 3 judges’ ratings) +7.119 +(1.189) +17.771 +(0.744) +Performance (all 5 / 7 judges’ ratings) +7.110 +(1.182) +17.765 +(0.741) +Panel C: Covariates +Compatriot judge +0.248 +0.457 +Home event +0.099 +0.127 +Experience (Age in years) +22.429 +(3.789) +26.836 +(4.949) +Female +0.450 +Difficulty +3.211 +(0.331) +Distance +122.608 +(11.837) +Prev. Distance +123.940 +(11.143) +Prev. Difficulty +3.166 +(0.317) +Prev. Performance +7.270 +(0.958) +17.854 +(0.580) +N +13075 +4529 +Notes: Mean and standard deviation (in parentheses; for non-binary variables). rem. = +remaining. Some variables were only observed in one of the data sets. Full descriptive +statistics in Appendix Table 6. +3See +https://assets.fis-ski.com/image/upload/v1665482445/fis-prod/assets/ICR_Ski +_Jumping_2022_marked-up.pdf for a current version of the rules (last accessed on 01/23/2023). +9 + +The main analysis is conducted using data on official diving competitions from 2013 through +2017. This includes special events such as World Championships and the Summer Olympics. +Except for the first jump, each jump constitutes one observation. We exclude observations where +the rating points of the current or subsequent jump are at the lower or upper bound.4 Athletes +who stop competing during the contest are excluded, e.g., due to injury. +We conduct the analysis based on 13075 observations. +The data consists of the jumps +performed by 434 athletes from 54 countries in Africa, Asia, Europe, North America, Oceania, +and South America. +As visible in panel C of Table 1, roughly one-half of the athletes are +female and on average 22.4 years old. In 25 percent of the cases, at least one of the judges +has the same nationality as the task taker and about 10 percent of observations are at a home +event. Difficulty and previous difficulty of the jump are on average around 3.2, and (current and +previous) performance are on average around 7.1 to 7.3. +For our analysis on ski jumping, we have 4529 observations on events from the 2010/11 +through 2016/17 season (based on a collection conducted by Krumer, Otto, and Pawlowski +(2022)). Each observation refers to a second jump. Athletes who fail to qualify for the second +round are excluded. In 13 percent of the cases, athletes perform in their respective country of +birth. In 45 percent of the cases, one of the judges is of the same nationality as the performing +athlete. The average age is about 26.8 years. Jumps are on average about 123 meters and +(current and previous) performance are on average around 17.7 (see panels B and C of Table 1). +4To put it more concretely: We remove observations that have received an average score of 9.5 or +higher (19.5 in ski jumping), as well as those with an average score of less than 5 (14 in ski jumping). +Furthermore, we remove observations with individual scores of 3 or lower (14 in ski jumping), as these +are most likely to be crashes. All of these choices are robust to changes, and we show the robustness of +the results to data pre-processing in the results section. +10 + +3.2 +Variables +Figure 1: Illustration of the evaluation and feedback process +Notes: For a current task (on the right), feedback is given for the previous task (left). The broken +arrows represent our main hypotheses, i.e., the potential influence of feedback on performance +in the subsequent task. Task and individual characteristics (dotted square) potentially +moderate this effect. In the case of seven judges, the two highest and lowest ratings are +discarded, and only the most extreme ratings are used. See Section 5.5 for other specifications +used in the robustness checks. +Figure 1 describes the evaluation and feedback process in our setup. For the task execution +evaluation, each judge in the jury independently gives a numerical rating for the task execution +of the task taker. +The largest and smallest of those judges’ ratings are discarded and the +jury performance assessment is the mean of the remaining (three) judges’ ratings. The task +performance assessment quantifies the task performance result. +In our study, we focus on the discarded judges’ ratings that are not regarded for the jury’s +performance assessment and can affect subsequent performance only through their motivational +impact. Our treatment variables are constructed as deviations of the discarded judges’ ratings +from the jury performance assessment. More concrete, Deviation positive is constructed by sub- +tracting the jury performance assessment (the mean of the ratings in absence of the discarded +ratings) from the largest discarded judges’ rating. Deviation negative is constructed by sub- +tracting the smallest discarded judges’ rating from the jury performance assessment.5 We define +5Additionally, we construct and test two alternative specifications. All specifications can be found in +the full descriptive statistics in Appendix Table 6. Especially, for diving, there are two (highest/lowest) +judges’ ratings discarded. The base specification uses the most extreme judges’ ratings. Other specifi- +11 + +Highest +rating +Deviation +Taskandindividual +Positive +characteristics +Jury= +ratings +feedback +Panel +Jury +of +Y +Ratings +Judges' +performance +Judges +remaining +assessment +Deviation +(5 or 7) +Negative +feedback +Lowest +rating +Task +Task +taker +Taskexecution +performance +Feedback +Taskexecution +result +Previoustask +Current taskDeviation positive as positive feedback and Deviation negative as negative feedback. Panel A +in Table 1 provides an overview of the main treatment variables. Both feedback variables, with +mean values of 0.426 (0.316) for positive feedback and 0.477 (0.357) for negative feedback, range +from 0 (for neutral feedback) to 2.5 (for increasingly positive/negative feedback). +To measure the effect of feedback on subsequent task execution, we use the jury’s perfor- +mance assessment that the task takers receive for their subsequent performance (hereafter, "Per- +formance") as our outcome variable. An alternative variable to measure subsequent performance +is the mean of the ratings from all (5 or 7) judges. +4 +Empirical strategy +We study how positive and negative feedback affect subsequent performance. To this end, our +identification strategy relies on conditional idiosyncratic variations in the differences between +the jury performance assessment and the discarded ratings. This positive (negative) deviation +is irrelevant to the assessment of the task performance but provides feedback in the form of +additional information about the judges’ general perception of the performance. +The identification strategy presumes that, once we condition on a few observable character- +istics, there are no omitted influences that are correlated with both outcome, i.e., performance +in the task, and treatment, i.e., the positive/negative deviation (feedback for the previous task). +Our approach formalizes to the following linear baseline model: +Yi = α + β+A+ +i + β−A− +i + γXi + ϵi, +where the outcome, Yi, is the performance in the (current) task for individual i. The con- +tinuous treatments A+/− +i +are defined as the positive/negative feedback for the (previous) task, +and β+/− are the coefficients of interest to investigate our hypotheses 1 and 2. Xi contains +(pre-determined) covariates of individual i that we need to control for. ϵi is an idiosyncratic +error term. +To give credence to the unconfoundedness assumption, we address concerns raised in the lit- +erature about potential biases in subjective ratings. First, we consider nationality bias (Heiniger +cation descriptions and results for the robustness of the alternative treatment variable specifications can +be found in Section 5.5. +12 + +& Mercier, 2021; Krumer et al., 2022; Sandberg, 2018; Zitzewitz, 2006), i.e., a judge from the +same country as the task taker rates the compatriot better than other individuals. To account +for potentially more positive ratings from judges who are compatriots, we include a) a binary +variable indicating whether a judge on the panel is a compatriot of the task taker, and b) an +indicator if the individual competes in a home event in Xi.6 To alleviate remaining concerns +about bias based on common nationality, we conduct two further checks. A balancing test in Ta- +ble 8 shows no balancing issues related to compatriot judges. To ensure that the results are not +driven by individuals that are potentially subject to nationality bias, we perform a robustness +check in which the affected task takers are removed from the sample.7 +Second, there is evidence in the literature of an order of action bias (Damisch, Mussweiler, +& Plessner, 2006; Ginsburgh & Van Ours, 2003). Subjective ratings are found to be affected by +the order of task performance, which threatens our identification when some but not all judges +are affected. We account for this by controlling for the order in which individuals perform tasks +(starting order). +Third, more difficult tasks were found to be rewarded with higher scores– +the difficulty bias (Morgan & Rotthoff, 2014). The difficulty of a task in our case is precisely +measurable and predetermined. Specifically, in diving, we control for the difficulty of the jump +(chosen a priori); in ski jumping, we control for the (previous and current) wind and gate, i.e., +the length of the hill–both factors that can influence difficulty and subjective evaluation. +Fourth, there could be reputation bias (Findlay & Ste-Marie, 2004). This bias can lead to +better ratings for well-established individuals who typically have a better reputation. To ensure +conditional independence, we take into account a) individual and individual-by-season fixed +effects and b) current rank in the competition. Fifth, the accuracy of subjective performance +ratings is found to vary for different performance qualities (Heiniger & Mercier, 2021). Therefore, +we include the individual mean and standard deviation of the jury’s performance assessment of +the previous task in Xi. +While not testable, we are confident that the conditional independence assumption is satis- +fied. Still, we offer two types of checks for it. First, in a total of 20 balancing checks in Table 8, +only one statistically significant test indicates a solid balancing among observable characteristics. +Second, with respect to unobservable characteristics, we provide an indirect approach to sup- +6Judges’ decisions regarding possible bias in favor of compatriots might be different in front of a +supportive crowd (Page & Page, 2010; Goller & Krumer, 2020). +7The results for this can be found in Table 11 and hardly differ materially from the main results. +13 + +port the conditional independence assumption by implementing a placebo treatment test. We +replace the treatment variable with a pseudo-treatment variable recorded in the future. The task +performance cannot be influenced by the feedback given in the future of this task. Therefore, if +we observe all confounding influences, the placebo treatment effect should be zero. If we reject +this placebo null hypothesis this points to some unobserved confounding (or other issues like +endogeneity or reverse causality), while not rejecting gives some evidence that the conditional +independence assumption is plausible. Table 7 shows that this placebo test cannot reject our +assumption of unconfoundedness. +To estimate the main effects of interest, we use linear regression and cluster standard errors +on the individual level. In the second step, we apply a method from the causal machine learning +literature. For this research, the importance of investigating potential non-linearities in the effect +lies in the differently observed treatment intensities, i.e., high or low quantified feedback, for +which it is unclear if an estimated constant treatment effect reflects various treatment intensities +properly. +With the non-parametric kernel method for continuous treatment effects introduced by +Kennedy, Ma, McHugh, and Small (2017) we investigate the effects for different intensities +of the treatment. The method builds on two steps. First, a (doubly-robust) pseudo-outcome is +constructed as follows: +ξ(π, µ) = Y − µ(X, A) +π(A|X) +� +π(A|x)dP(x) + +� +µ(x, A)dP(x), +where the nuisance functions π(A|X) and µ(X, A) are estimated using a random forest estimator +(Breiman, 2001). The pseudo-outcome ξ(π, µ) is doubly-robust in the sense that only (at least) +one of the two nuisances needs to be consistent, not both, and is free from confounding influences. +In the second step, the average potential outcome for given treatment levels is estimated using +a non-parametric kernel regression of the pseudo-outcome on the continuous treatment variable: +E(Y a) = E(ξ(π, µ|A = a)). +14 + +5 +Results +5.1 +Main results +Our first main finding is that positive feedback is enhancing (subsequent) performance. Panel +A in Table 2 shows a statistically significant and positive coefficient for positive feedback. The +effect is robust to the inclusion of different sets of covariates. In each specification, the average +effects are statistically significant at the 1% level. Panel B replicates this finding for our second +data set. As our second main finding, we observe that negative feedback causes an effect close +to zero in both panels and all specifications. We do not see any effect of negative feedback on +performance. +Table 2: The effect of feedback on performance – sensitivity to different specifications +Performance +(1) +(2) +(3) +(4) +Panel A: Diving (N=13075) +Positive Feedback +0.242*** +0.208*** +0.115*** +0.100*** +(0.036) +(0.034) +(0.032) +(0.035) +Negative Feedback +0.018 +0.024 +0.001 +0.007 +(0.030) +(0.030) +(0.029) +(0.030) +Panel B: Ski jumping (N=4529) +Positive Feedback +0.201*** +0.180*** +0.145*** +0.107*** +(0.035) +(0.036) +(0.034) +(0.034) +Negative Feedback +-0.063 +-0.055 +-0.049 +-0.026 +(0.043) +(0.041) +(0.037) +(0.041) +Base Covariates +x +x +x +x +All Covariates +x +x +x +Individual Fixed Effect +x +Individual x Season FE +x +Notes: Linear regression. Full regressions in Tables 9 and 10. All regressions contain previous’ +jumps jury assessment (Base Covariates). All Covariates include prev. jumps wind and +gate points and distance (ski jumping) or difficulty (diving). Also, points behind, +compatriot judge, home event, current ranking, SD of previous performance, and start +order. Standard errors are clustered on the individual level. *, **, and *** represents +statistical significance at the 10 %, 5 %, and 1 % level, respectively. +The performance-enhancing impact of positive feedback is rather insensitive to the inclusion +of more covariates and fixed effects. +We start with controlling only for performance in the +previous task in column (1). In column (2) we add several control variables as discussed in +15 + +Section 4. +Columns (3) and (4) add individual fixed effects and individual-by-season fixed +effects to the regressions. Detailed result tables can be found in the appendix in Tables 9 and +10, and for the sake of simplicity, all of the following regressions are based on the specification +used in column (3). +Figure 2: Non-linear estimation of feedback on performance +Notes: Non-parametric kernel regression for different levels of positive (left) and negative (right). +feedback. Expected outcomes (y-axis) and treatment levels (x-axis) are displayed. +Kernel bandwidths are 0.300 (left) and 0.214 (right) and are determined in a data-driven +approach using a cross-validation method. To obtain treatment effects, one might +calculate the difference of the expected outcomes for two treatment levels and divide +this by the difference in the treatment levels (treatment intensity). Diving data. +The broken lines represent the 90% confidence intervals. +Our results show that, on average, positive feedback is enhancing performance. In the fol- +lowing, we go beyond average effects and investigate the effect of positive and negative feedback +for different magnitudes of feedback. Figure 2 provides non-linear estimates of positive and neg- +ative feedback showing the expected outcome (performance) against the extent of the feedback, +i.e., the level of the treatment. The (treatment) effect of different feedback intensities can be +calculated as the difference in expected outcomes for an increase from some treatment level to +another.8 In the graph on the left, the effect of positive feedback is positive throughout all feed- +8For two different treatment levels A = a1 and A = a0, the effect can be calculated as θ(a1, a0) = +E(Y (A=a1))−E(Y (A=a0)) +a1−a0 +. The treatment intensity in this example is a1 − a0, while for a complete picture, +it needs to be clear that the treatment level from which the treatment intensity is evaluated is a0 here. +16 + +Positive feedback +E(Y(a) +7.75 +7.50 +7.25 +7.00 +1 +6.75 +1 +1 +6.50 +1 +0.0 +0.5 +1.0 +1.5 +2.0 +Treatment level A=aNegative feedback +E(Y(a) +7.75 +7.50 +7.25 +7.00 +6.75 +1 +6.50 +1 +0 +2 +Treatment level A=aback intensities, i.e., the expected outcome increases almost steadily as the level of treatment +increases. With negative feedback, on the right side of Figure 2, the effect varies slightly up +and down for different treatment intensities – although the effect does not appear to be different +from zero for any treatment intensity, consistent with the average effect of zero reported in Table +2. For both estimations, we find that the linearity assumption in the regression analyses is a +good approximation for the non-linear effect curves. Still, especially for the higher treatment +intensities the confidence intervals become large and conclusions become imprecise–a fact to +which global linear regression models do not give any hint. +Overall, the results provide support for hypothesis 1: The performance is better after receiv- +ing positive feedback than after receiving neutral feedback. Contrarily, we do not find support +for hypothesis 2, i.e., the performance is not better after receiving negative feedback than after +receiving neutral feedback. In the next section, we test if the positive effect of positive feedback +and the null effect of negative feedback persists in different sub-populations and is generalizable +for diverse personal or situational conditions. +5.2 +Sub-population and context heterogeneity +In the feedback-intervention model of Kluger and DeNisi (1996), as well as, for example, in +the meta-study of Fong et al. (2019) aspects are collected for which the effects of feedback +potentially differ. Personal characteristics, situational aspects, and task characteristics, among +other factors, might shape the reaction of individuals to positive and negative feedback. +A +strength of our unique data set is that it allows us to investigate if we can generalize the results +of our analysis. +Panel A of Table 3 exhibits that positive feedback has a favorable impact irrespective of in- +dividuals’ personal characteristics. We consider three categorizations of the individuals’ cultural +backgrounds. First, we report that the favorable effect of feedback on performance is present +for individuals from WEIRD and non-WEIRD countries. Second, we find a favorable impact +of positive feedback irrespective of the relative cultural distance to the U.S.. Third, individuals +coming from relatively individualistic and relatively collectivistic countries both react favorably +to positive feedback.9 +Other personal characteristics that we investigate are experience and +9We classify (non-)WEIRD countries according our own assessment based on Henrich et al. (2010); +the respective list can be obtained upon request. For cultural distance to the U.S., we use the metrics +provided in Table 1 in the research article by Muthukrishna et al. (2020). +For individualistic and +17 + +gender. +We find a performance-enhancing effect of positive feedback for both the relatively +more and less experienced. Similar to Bear et al. (2017), we also explore whether there are +gender differences in the reaction to feedback. We find that both sexes react favorably to posi- +tive feedback For none of the three different definitions of cultural background, nor gender and +experience, do the two-sample WALD tests show statistically significant differences. This leads +to the conclusion that the effects of feedback are consistent and generalizable across these three +personal characteristics. +Importantly, we find some heterogeneity with respect to the characteristics of the task. +Contested situations offer greater incentives to perform (Goller & Heiniger, 2022), with higher +task focus and more pressure. Panel B of Table 3 shows large and positive effects for positive +feedback in close competitions, but an insignificant effect for situations that are less competitive. +This is in line with the argumentation by Kluger and DeNisi (1996) and our expectations. +Contrary, we find no support for differential effects for the difficulty of the task. Positive feedback +leads to a performance-enhancing impact for easy and hard tasks. +The results of the heterogeneity analysis on the impact of negative feedback are largely in line +with the main finding. The second column of Table 3 shows a null effect of negative feedback for +most subgroups and all contexts. The only exception is the experience of the individuals, where +we find that relatively more experienced individuals improve their performance after receiving +negative feedback. A two-sample Wald test (in square brackets) shows that the difference in +the reaction between the more and less experienced individuals is statistically significant. The +favorable impact of negative ratings for experienced individuals is in line with findings by Eggers +and Suh (2019) on the firm level. +collectivistic countries, we use data from the index created by Hofstede (2011). +18 + +Table 3: Differential effects +Positive Feedback +Negative Feedback +Panel A: Individuals‘ characteristics +WEIRD1 (N=4955) +0.086* (0.048) +0.006 (0.049) +Non–WEIRD (N=8120) +0.135*** (0.043) +0.004 (0.037) +[0.447] +[0.974] +Culturally close to U.S.2 (N=6223) +0.132*** (0.046) +-0.007 (0.047) +Not culturally close to U.S. (N=6852) +0.101** (0.044) +0.008 (0.037) +[0.626] +[0.802] +Individualistic country3 (N=6013) +0.096** (0.047) +0.007 (0.045) +Collectivistic country (N=6872) +0.144*** (0.045) +0.001 (0.040) +[0.461] +[0.921] +More experienced (age ≥ 23y, N=6176) +0.146*** (0.045) +0.076* (0.039) +Less experienced (age < 23y; N=6899) +0.081* (0.047) +-0.062 (0.044) +[0.318] +[0.019] +Female (N=5885) +0.087* (0.047) +-0.028 (0.042) +Male (N=7190) +0.128*** (0.043) +0.018 (0.039) +[0.520] +[0.422] +Panel B: Task characteristics +Tight competition4 (N=5118) +0.173*** (0.056) +-0.033 (0.052) +Non–tight competition (N=7957) +0.064 (0.039) +0.007 (0.037) +[0.110] +[0.531] +Easy task5 (N=7267) +0.154*** (0.043) +-0.027 (0.037) +Hard task (N=5808) +0.086* (0.048) +0.025 (0.044) +[0.291] +[0.366] +Notes: Linear Regression estimates. Diving data. Control variables as in column (3) in Table 2. +Standard errors are clustered on the individual level. *, **, and *** represents statistical +significance at the 10 %, 5 %, and 1 % level, respectively. P-value of WALD test for +equality in square brackets. 1Western, Educated, Industrialized, Rich, Democratic. 2Cultural +closeness is divided at the median level of an index taken Muthukrishna et al. (2020). 3Divided +at median level of an individualism index constructed by Hofstede (2011); (some countries +missing). 4Athlete is within ten points to first place in final, and to the cut-off in preliminary +rounds. 5Easy and hard according to the median chosen difficulty of the (assessed) task. +19 + +5.3 +Repetition and long-term effects +For practitioners, it is crucial to know about the impact of feedback when it is given repeatedly +and about its long-term effect. Fortunately, our data allows for analyzing the impact of feedback +on performance in a repeated setup. +Figure 3 shows that the favorable impact of positive feedback is non-diminishing with repe- +tition. As a benchmark, Baseline shows the average effect of receiving feedback as reported in +Table 2, which is not conditional on further previously received feedback. We find that for those +who have received positive feedback at least one time before, further positive feedback continues +to have a positive impact on their performance. Similarly, we find a positive influence of positive +feedback if the individual has received positive feedback at least two or three times before. +Figure 3: A non-diminishing effect of positive feedback +Notes: Linear regression estimates. Diving data. Specifications as in column (3) in Table 2. +Standard errors are clustered on the individual level. Effect among those that +experienced positive feedback at least one, two, or three times (in the respective +round) before. The whiskers mark the 90 % confidence intervals. +Table 4 shows the non-persistence of the effect of positive feedback on performance. For +reference, column (1) reports the baseline effect for the performance in the task that is conducted +directly after the feedback is received. Columns (2-4) provide estimates for the effect of feedback +on performance in tasks carried out thereafter. +For all follow-up tasks, we find statistically +insignificant effects. This indicates that the favorable short-term effect of positive feedback does +not carry on to future tasks. Negative feedback has no impact, neither on subsequent nor future +tasks. +20 + +Positive feedback before +0 +0.15 +Effect +0.09 +0.03 +0.03 +Baseline +One +Two +ThreeTable 4: A non-persistent effect of feedback on performance +Performance +(1) +(2) +(3) +(4) +Positive feedback +0.115*** +-0.010 +0.073 +-0.062 +(0.032) +(0.047) +(0.050) +(0.061) +Negative feedback +0.001 +-0.049 +0.023 +-0.079 +(0.029) +(0.036) +(0.046) +(0.056) +Periods after feedback: +1 +2 +3 +4 +N +13075 +10130 +7350 +4512 +Notes: Linear Regression on future outcomes. Diving data. Specifications as in column (3) in +Table 2. Standard errors are clustered on the individual level. *, **, and *** represents +statistical significance at the 10 %, 5 %, and 1 % level, respectively. +5.4 +Spillover effects on related tasks +Previously presented evidence shows the favorable effects of positive feedback on the task for +which the feedback was obtained. In practice, individuals might do several tasks simultaneously, +or a task containing different elements, that potentially influence each other. For example, Hecht, +Tafkov, and Towry (2012) show spillover effects of incentive schemes in one task on a related, +simultaneously conducted second task. Our settings allow us to study, both, a single-task and a +multi-task environment. +Panel A presents the results for the single-task setup. As presented previously in Table 2, we +find a performance-enhancing impact of positive feedback and no impact of negative feedback +on performance. The difficulty is fixed ex-ante. That we find no impact of feedback on the +difficulty can be regarded as a placebo outcome test and supports our identification strategy. +Difficulty and performance evaluation jointly determine the combined outcome. Consequently, +we observe a favorable effect of positive feedback on the total score. +Panel B exhibits the results for the multi-task environment. We observe favorable spillover +effects. Receiving positive feedback in Task 1 enhances subsequent performance in Task 1 and +the related Task 2. +Negative feedback has no impact on either of the tasks. +In the setup, +performance in Task 1 and Task 2 are the most important determinants of combined success +and the only ones that can be influenced by the task taker. Consistently, we also find a favorable +influence of positive feedback on the total score. +21 + +Table 5: Spillover effects +Panel A: One isolated task, diving +Task 1: +Multiplier: +Combined: +Performance +Difficulty +Total score +Positive Feedback +0.115*** +-0.002 +1.071*** +(0.032) +(0.006) +(0.324) +Negative Feedback +0.001 +-0.001 +-0.029 +(0.029) +(0.005) +(0.282) +Panel B: Two simultaneous tasks, ski jumping +Task 1: +Task 2: +Combined: +Performance +Distance points +Total score +Positive Feedback +0.145*** +1.692*** +2.126*** +(0.034) +(0.634) +(0.693) +Negative Feedback +-0.049 +0.072 +-0.080 +(0.037) +(0.545) +(0.631) +Notes: Linear Regression estimates. Control variables as in column (3) in Table 2. Feedback was +given previously for Task 1 only. Standard errors are clustered on the individual level. *, ** +, and *** represents statistical significance at the 10 %, 5 %, and 1 % level, respectively. +5.5 +Robustness +To ensure that our results are robust to different specifications we conduct several supplementary +analyses. First, we consider alternative specifications of our key variables. In a first regression, +we take the mean of all (five or seven) judges’ ratings, instead of the performance, i.e., the +mean of the (after discarding the extreme ratings) remaining three ratings, as an alternative +outcome variable. With the treatment, the second key variable is (additionally) constructed +in two different ways. Instead of subtracting the jury’s performance assessment from the most +extreme (positive/negative) discarded rating we deduct (a) the lowest (highest) rating included +in the jury’s performance assessment from the lowest (highest) discarded rating (Deviation +positive/negative+) and (b) the jury’s performance assessment from the mean of the two dis- +carded highest or lowest ratings (Deviation positive/negative++, in diving only). +Table 11 +presents the results for these alternative specifications and shows robust estimates. We conclude +22 + +from this that the result does neither depend on the concrete choice of the treatment variable, +nor on the selection of the outcome variable. +Second, we consider different choices with respect to the sample that is used for the inves- +tigation. Data cleaning might offer some leeway to researchers influencing results. Thus, we +provide additional analyses in Table 11 using (a) the full sample without any data cleaning and +(b) without excluding failed attempts (but excluding boundary values as described in Section +3.1). We find robust results for both supplementary analyses, indicating that our data-cleaning +step does not drive the results. +Third, to prove that nationality bias is not responsible for the effect, i.e., judges favor their +compatriots and potentially influence other judges on the panel, we re-estimate the results +excluding all athletes with a compatriot judge in the panel. If the effect would be driven by +these individuals the results might just be some mechanical effect. Though, the effect is also +found for individuals not sharing nationality with a judge. +6 +Managerial implications and conclusions +Giving feedback is one of the most important tasks of managers. On a typical workday, managers +regularly provide feedback to their teams. Some of this feedback is subconscious, such as facial +expressions or nodding as a sign of appreciation and approval. Other feedback can be formal +and dictated by the institution, as is the case with appraisal interviews. It can be constructive +and substantive. But it can also be purely motivational. Common examples would be phrases +like “Good job!” or “You can do better!” embedded in the context of everyday conversations. +The crucial question is whether such motivational feedback, given consciously by managers, +can serve the goal of increasing the future productivity of workers. For both valences of feedback, +i.e. positive and negative feedback, this question is not trivial. The appreciation that positive +feedback expresses can motivate but also cause employees to rest on their laurels. Negative +feedback can spur on but it can also hurt and discourage. +Our causal analysis indicates that managers can use positive feedback to enhance productiv- +ity. Our results show a favorable impact of positive feedback on (subsequent) performance. The +heterogeneity analysis indicates that this favorable effect of positive feedback can be found for +feedback recipients coming from varying cultural backgrounds, for recipients of both male and +23 + +female gender, and for relatively more and less experienced recipients. We find that the favorable +effect of positive feedback is short-term, repeatable, and with potentially favorable spillover to +related tasks. The favorable impact of positive feedback is robust to the setup in which the +activity is performed and is more pronounced in highly relevant situations. All this makes us +confident that giving positive motivational feedback is a performance-enhancing strategy. +Furthermore, we find no significant impact of negative feedback on performance. This null +effect might explain why managers and other raters are often reluctant to give negative feedback +(Fisher, 1979), a phenomenon termed as leniency bias (Cheng, Hui, & Cascio, 2017) or MUM- +effect (Rosen & Tesser, 1970). While in other contexts the lack of negative ratings is decreasing +efficiency (Cannon & Witherspoon, 2005; Bolton, Kusterer, & Mans, 2019; Keser & Späth, +2021), we report no need to give negative motivational feedback. +Despite the robustness of our results, we acknowledge some limitations of our approach. +First, our sample consists of internationally competing athletes. While their level of profession- +alism and self-discipline might be comparable to those of employees in highly competitive work +environments, top athletes are not representative of the general population. Second, we consider +an environment in which individuals receive feedback from multiple, external sources. Again, +this is more comparable to daily life at large and competitive companies than at small firms. +Third, we analyze a domain in which feedback recipients directly benefit from improvements in +their performance, while feedback providers do not. In other domains, raters might be more +prone to willfully bias their feedback. +Therefore, we suggest that future research could contrast our results to environments, in +which feedback providers benefit from an increased performance more than feedback recipients +do. Employees in such environments might be prone to exploitation when employers use positive +feedback as a substitute for more substantial improvements in the employees’ well-being. Fur- +thermore, future research could analyze the long-term effects of positive and negative feedback. +With this study, we contribute to the literature that provides guidelines for optimal feedback +(Balcazar et al., 1985; Alvero et al., 2001; Sleiman et al., 2020). 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Journal of Economics & Management Strategy, 15(1), +67–99. +31 + +Appendix +Descriptive statistics +Table 6: Full descriptive statistics +Diving +Ski jumping +Mean +Std. dev. +Mean +Std. dev. +Treatments: +Positive feedback (deviation positive) +0.426 +(0.286) +0.316 +(0.262) +Negative feedback (deviation negative) +0.477 +(0.320) +0.357 +(0.290) +Positive feedback+ +0.314 +(0.297) +0.179 +(0.258) +Negative feedback+ +0.363 +(0.328) +0.218 +(0.289) +Future positive feedback +0.439 +(0.301) +Future negative feedback +0.489 +(0.325) +Outcomes: +Performance (rem. 3 judges’ ratings) +7.119 +(1.189) +17.771 +(0.744) +Performance (all 5 / 7 judges’ ratings) +7.110 +(1.182) +17.765 +(0.741) +Score +68.737 +(14.557) +118.647 +(16.204) +Distance +122.608 +(11.837) +Covariates: +Difficulty +3.211 +(0.331) +Compatriot judge +0.248 +0.457 +Home event +0.099 +0.127 +Final +0.291 +Female +0.450 +Age +22.429 +(3.789) +26.836 +(4.949) +Current ranking +8.490 +(9.655) +15.357 +(8.582) +Start order +9.490 +(11.082) +Points behind leader +31.491 +(31.011) +19.247 +(10.132) +In range (within 5 pts. to threshold) +0.264 +Gate points +0.093 +(3.270) +Wind points +-0.291 +(8.225) +Prev. performance +7.270 +(0.958) +17.854 +(0.580) +Prev. SD performance +0.130 +(0.151) +0.157 +(0.159) +Prev. wind points +-1.685 +(8.136) +Prev. gate points +-0.163 +(4.386) +Prev. distance +123.940 +(11.143) +Prev. difficulty +3.166 +(0.317) +N +13075 +4529 +Notes: Mean and standard deviation (in parentheses; for non-binary variables). Some variables +only observed in one of the data sets. +Alternative definition as defined in the main text. +32 + +Placebo and balancing tests +Table 7: Placebo treatment regressions +Judges’ +Judges’ +Judges’ +ratings 3 +ratings 5 +ratings 7 +Score +Future positive feedback +0.028 +0.030 +0.027 +-0.012 +(0.035) +(0.035) +(0.035) +(0.345) +Future negative feedback +-0.045 +-0.043 +-0.041 +-0.437 +(0.035) +(0.035) +(0.035) +(0.318) +N +10256 +10256 +10256 +10256 +Notes: Linear Regression on the outcome mentioned in the column header. 3, 5, and 7 refer +to discarding four, two, or none of the extreme judges’ ratings. Diving data. Pseudo- +treatment is the deviation of next (future) jump. Jumps 2–4/5 only. Specifications as +in column (3) in Table 2. Standard errors are clustered on the individual level. *, **, +and *** represents statistical significance at the 10 %, 5 %, and 1 % level, respectively. +33 + +Table 8: Balancing Tests +Diving +Compatriot judge +Home event +SD prev. perform. +(1) +(2) +(3) +(4) +(5) +(6) +Feedback positive +-0.000 +-0.004 +0.007 +(0.014) +(0.008) +(0.005) +Feedback negative +-0.017 +-0.012 +0.002 +(0.012) +(0.007) +(0.005) +Difficulty +Final +(1) +(2) +(3) +(4) +Feedback positive +-0.005 +-0.018 +(0.007) +(0.013) +Feedback negative +0.006 +-0.017 +(0.006) +(0.013) +Ski jumping +Compatriot judge +Home event +Prev. distance +(1) +(2) +(3) +(4) +(5) +(6) +Feedback positive +-0.045 +-0.023 +0.918 +(0.028) +(0.018) +(0.696) +Feedback negative +-0.010 +-0.048*** +0.345 +(0.022) +(0.017) +(0.674) +Prev. gate +SD prev. perform. +(1) +(2) +(3) +(4) +Feedback positive +-0.304 +0.024 +(0.306) +(0.092) +Feedback negative +0.031 +0.018 +(0.217) +(0.095) +Notes: Linear Regression estimates. Each regression includes athlete fixed-effects. Standard errors +are clustered on the individual level. *, **, and *** represents statistical significance at the +10 %, 5 %, and 1 %, respectively. +34 + +Additional and full results tables +Table 9: Feedback on performance – sensitivity to different specifications, ski jumping +Ski jumping +Performance +(1) +(2) +(3) +(4) +Positive feedback +0.201*** +0.180*** +0.145*** +0.107*** +(0.035) +(0.036) +(0.034) +(0.034) +Negative feedback +-0.063 +-0.055 +-0.049 +-0.026 +(0.043) +(0.041) +(0.037) +(0.041) +Prev. jury assessment +0.593*** +0.465*** +0.402*** +0.329*** +(0.027) +(0.041) +(0.043) +(0.031) +Prev. wind points +0.044*** +0.040*** +0.036*** +(0.002) +(0.002) +(0.002) +Prev. gate points +0.003 +0.002 +0.000 +(0.003) +(0.003) +(0.003) +Prev. distance +0.002*** +0.003*** +0.002** +(0.001) +(0.001) +(0.002) +Wind points +-0.041*** +-0.038*** +-0.036*** +(0.002) +(0.002) +(0.002) +Gate points +-0.020*** +-0.019*** +-0.019*** +(0.003) +(0.002) +(0.003) +Points behind +-0.015*** +-0.016*** +-0.015*** +(0.002) +(0.002) +(0.002) +Compatriot judge +0.021 +0.016 +0.024 +(0.020) +(0.022) +(0.023) +Home event +0.013 +0.028 +0.041 +(0.032) +(0.032) +(0.035) +Start order +0.002 +-0.003 +-0.005* +(0.002) +(0.002) +(0.003) +SD prev. judges’ ratings. +-0.019 +-0.001 +-0.017 +(0.061) +(0.063) +(0.065) +Athlete Fixed Effect +x +Athlete x Season FE +x +N +4529 +4529 +4529 +4529 +Notes: Linear regression. Prev. (= previous) refers to a lagged variable from the previous jump. +SD = standard deviation. Standard errors are clustered on the individual level. *, **, +and *** represents statistical significance at the 10 %, 5 %, and 1 % level, respectively. +35 + +Table 10: Feedback on performance – sensitivity to different specifications, diving +Diving +Performance +(1) +(2) +(3) +(4) +Positive Feedback +0.242*** +0.208*** +0.115*** +0.100*** +(0.036) +(0.034) +(0.032) +(0.035) +Negative Feedback +0.018 +0.024 +0.001 +0.007 +(0.030) +(0.030) +(0.029) +(0.030) +Prev. jury assessment +0.430*** +0.284*** +0.103*** +0.073*** +(0.026) +(0.022) +(0.016) +(0.016) +Prev. difficulty +0.794*** +0.540*** +0.147 +0.228** +(0.079) +(0.087) +(0.091) +(0.100) +SD prev. judges’ ratings +0.095 +0.056 +0.029 +(0.067) +(0.067) +(0.070) +Compatriot judge +-0.015 +-0.024 +-0.016 +(0.024) +(0.022) +(0.025) +Home event +0.129*** +0.164*** +0.196*** +(0.038) +(0.045) +(0.054) +Current ranking +-0.020*** +0.000 +0.011*** +(0.002) +(0.002) +(0.003) +Start order +-0.003*** +-0.006*** +-0.009*** +(0.001) +(0.001) +(0.001) +Points behind +-0.003*** +-0.000 +0.001 +(0.001) +(0.000) +(0.001) +Penalty +-0.288 +-0.362* +-0.310 +(0.187) +(0.187) +(0.200) +Jump and Event Fixed Effect +x +x +x +Athlete Fixed Effect +x +Athlete x Season Fixed Effects +x +N +13075 +13075 +13075 +13075 +Notes: Prev. (= previous) refers to a lagged variable from the previous jump. SD = standard +deviation. Fixed effects for Events are 1m and 3m Springboard and 10m Platform, and the +five (female) or six (male) jumps. Standard errors are clustered on the individual level. +*, **, and *** represents statistical significance at the 10 %, 5 %, and 1 %, respectively. +36 + +Table 11: Robustness Checks +Ski jumping +Diving +Positive +Negative +Positive +Negative +Feedback +Feedback +Baseline results +0.121*** +-0.036 +0.115*** +0.001 +(0.033) +(0.044) +(0.032) +(0.029) +Other outcome variable +0.129*** +-0.070* +0.111*** +0.005 +(all ratings, incl. discarded) +(0.035) +(0.038) +(0.032) +(0.029) +Treatment definition 2 +0.119*** +-0.062 +0.109*** +0.005 +(Discarded vs. last credited) +(0.033) +(0.039) +(0.032) +(0.030) +Treatment definition 3 +0.127*** +0.007 +(Mean discarded vs. mean credited) +(0.043) +(0.039) +Without data cleaning +0.124*** +-0.056 +0.059* +0.017 +(0.044) +(0.047) +(0.035) +(0.031) +Without dropping failed attempts +0.124*** +-0.042 +0.109*** +0.031 +(0.045) +(0.042) +(0.037) +(0.033) +Only athletes not sharing +0.175*** +-0.044 +0.113*** +-0.021 +nationality with a judge +(0.040) +(0.053) +(0.038) +(0.034) +Only jumps with no variance +0.132*** +-0.044 +0.143*** +0.056 +in scoring ratings +(0.043) +(0.051) +(0.043) +(0.038) +Notes: Linear regression. Every line represents two separate regressions, one in each data set. +Specification as in column (3) in Table 2. Standard errors are clustered on the individual +level. *, **, and *** represents statistical significance at the 10 %, 5 %, and 1 %, respectively. +37 + diff --git a/5dFKT4oBgHgl3EQfSi3T/content/tmp_files/load_file.txt b/5dFKT4oBgHgl3EQfSi3T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c02501e52efb46c624f5dbc20b6e1892d087f861 --- /dev/null +++ b/5dFKT4oBgHgl3EQfSi3T/content/tmp_files/load_file.txt @@ -0,0 +1,1791 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf,len=1790 +page_content='‘Good job!’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The impact of positive and negative feedback on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Daniel Goller1,2, Maximilian Späth3* 1 Centre for Research in Economics of Education, University of Bern 2 Swiss Institute for Empirical Economic Research, University of St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='Gallen 3 Department of Economics, University of Potsdam This version: January 30, 2023 Abstract We analyze the causal impact of positive and negative feedback on professional performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We exploit a unique data source in which quasi-random, naturally occurring variations within subjective ratings serve as positive and negative feed- back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The analysis shows that receiving positive feedback has a favorable impact on subsequent performance, while negative feedback does not have an effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' These main results are found in two different environments and for distinct cultural back- grounds, experiences, and gender of the feedback recipients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The findings imply that managers should focus on giving positive motivational feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Keywords: Feedback, Performance, Causal Analysis, Cultural Background ∗Preliminary version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Do not quote or circulate without permission of one of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Comments are very welcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We like to thank Swiss-ski (Swiss ski federation), Michel Roth, David Morris, Alexan- der Mesch, and the DSV (German swimming federation) for valuable insights into the sports contests from the perspective of (former) professional athletes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Also, we thank participants of the ESEA Confer- ence, 2022, and the Berlin School of Economics Workshop, 2022, as well as Enzo Brox, Lisa Bruttel, and Sandro Heiniger for their helpful comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='11776v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='GN] 27 Jan 2023 1 Introduction Providing performance feedback is one of the main tasks of managers and leaders (Morgeson, DeRue, & Karam, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' One important aim of feedback is to create a favorable emotional response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' At best, positive or negative feedback can motivate employees and increase their productivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In the worst case, it leaves the employees frustrated and unproductive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Therefore, the question of how feedback impacts subsequent performance is of tremendous importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Consequently, numerous studies investigating the impact of feedback on creativity (Harrison & Rouse, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Itzchakov & Latham, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Kim & Kim, 2020), the learning process of indi- viduals and firms (Hattie & Timperley, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Lee, Lee, & Kim, 2021) or motivation (Deci & Casico, 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Fong, Patall, Vasquez, & Stautberg, 2019) emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In particular for positive and negative feedback on performance or productivity, studies show the full range from favorable to unfavorable effects (Eggers & Suh, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Kluger & DeNisi, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Podsakoff & Farh, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Sleiman, Sigurjonsdottir, Elnes, Gage, & Gravina, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Waldersee & Luthans, 1994, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The two major difficulties when investigating the impact of feedback on performance are (1) observing truthful and trustworthy feedback in real-incentive situations and (2) quantifying feedback and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' While observational studies typically fail to satisfactorily tackle the second difficulty, experimental studies cannot fulfill the first requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We are not aware of any causal study in which both requirements are met together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' To address this common shortcoming, we exploit a unique setting to estimate the causal effect of positive and negative feedback on subsequent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' For this purpose, we use data from professional sports: diving as the primary data source, and ski jumping for supplementary analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In these sports, individuals’ performance is evaluated subjectively by a jury of seven (or five) experienced judges according to precise rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Each judge independently issues one rating for the task performance (hereafter, "judges rating" or “rating”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Discarding the highest and lowest rating(s), the common assessment of the jury is calculated from the average of the three remaining ratings (hereafter, “jury performance assessment”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='1 Following the definition in Kluger and DeNisi (1996), stating that feedback is information about one’s task performance provided by an external agent, we consider the deviation of the discarded (highest and lowest) ratings from the jury’s performance assessment as feedback on 1Receiving the jury performance assessment can already be seen as a knowledge of results (Kluger & DeNisi, 1996) intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The analysis of this knowledge of results, however, is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 2 task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The discarded ratings are not relevant to the assessment of task performance, but this additional information about judges’ general perceptions of performance provides feed- back that can only work through the motivational channel on subsequent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Kluger and DeNisi (1996) argue that the feedback sign depends on the relation between the performance rating and a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In line with this, discarded ratings define quasi-randomly occurring positive (negative) deviations from the jury performance evaluation that serve as positive (neg- ative) feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' No deviation from the benchmark implies neutral feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We describe the evaluation and feedback process in more detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='2, Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We test several of the propositions from the model of the seminal work by Kluger and DeNisi (1996) within a single framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In our setup, the feedback is truthful, accurately observable, and from an external source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Feedback can impact subsequent performance only through its motivational impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Performance is strongly incentivized and can be precisely quantified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The performance is measured in non-artificial tasks that individuals are not only familiar with but that are routine aspects of their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' What is particularly valuable from a management perspective is that we can investigate the impact of feedback in an international context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Theoretically guided by the feedback intervention model (Kluger & DeNisi, 1996), we inves- tigate the effect of positive and negative feedback on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Further, we investigate the internal and external generalizability of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' To assess internal generalizability, we can use our extensive data to analyze whether situational (or personal) variables and task charac- teristics moderate the effects of the feedback intervention on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The international sample covering female and male individuals from more than 50 nations from 6 continents offer the unique opportunity to analyze feedback effects for different cultural backgrounds and gender within the same framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' To investigate external generalizability, we complement the main findings with a second, independent setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We investigate these aspects using both classical statistical and causal machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' This is followed by analyses examining the feedback interventions’ long-term, repetition, and spill-over effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Our analysis shows a performance-enhancing causal effect of positive feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The favorable effect of positive feedback is found for recipients from different cultural backgrounds, experience levels, and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We observe favorable effects even when individuals repeatedly receive positive feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The impact of positive feedback is stronger when the relevance of the task is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 3 In contrast to all this, negative feedback on average does not have an impact on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Merely, the subgroup of the more experienced individuals benefits from negative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Our findings imply that managers can use positive feedback to enhance the performance of their employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Importantly, positive feedback can be given repeatedly on a regular basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' It has a favorable impact irrespective of several relevant characteristics of the recipient and can be universally applied in an international context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' With our main finding we are in line with the studies conducted by Azmat and Iriberri (2010), Bandiera, Larcinese, and Rasul (2015), Choi, Johnson, Moon, and Oah (2018), and Itzchakov and Latham (2020) for positive feedback and the meta-study by Fong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2019) for negative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We complement decades of research that provides guidelines on how to optimally give feedback (Balcazar, Hopkins, & Suarez, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Alvero, Bucklin, & Austin, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Sleiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 2 Theoretical framing To provide a theoretical foundation for the later empirical analysis, we begin by describing the concept of feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Then, we collect relevant empirical research and form predictions based on propositions stated by Kluger and DeNisi (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='1 The concept of feedback Feedback exists in many forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Kluger and DeNisi (1996) define feedback as "[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='] actions taken by (an) external agent (s) to provide information regarding some aspect (s) of one’s task per- formance" (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 255).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Burgers, Eden, van Engelenburg, and Buningh (2015) distinguish between elaborate and simple feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Elaborate feedback typically includes a lengthy explanation, which provides a guide for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Simple feedback merely gives information, about whether something was done right or wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Burgers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2015) further distinguish between descrip- tive, comparative, and evaluative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Descriptive feedback – sometimes called objective feedback (Johnson, 2013) – merely sums up behavior shown by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Comparative feedback uses the performance of other individuals as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Evaluative feedback provides a judg- ment of the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Villeval (2020) distinguishes between a cognitive and a motivational perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The cognitive perspective rests on the assumption that individuals have imperfect knowledge about their skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Here, feedback serves as a signal used in an information-updating 4 process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The motivational perspective focuses on the impact of feedback on intrinsic motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Individuals might receive feedback from one agent or several agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Stone and Stone (1984) find that receiving feedback from two sources instead of one source increases self-perceived task competence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Related, there is a strand of literature analyzing multi-source feedback (Bailey & Fletcher, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Smither, London, & Reilley, 2005), also called 360 degree feedback (DeNisi & Kluger, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Finally, feedback can be with direct consequences or inconsequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Often feedback comes without direct (monetary) consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Still, research shows that agents also react to irrelevant information (Abeler, Falk, Goette, & Huffman, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Cason & Mui, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The focus of our paper lies on the impact of simple and evaluative feedback on subsequent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The feedback is subjective in the sense that is created by subjective evaluation based on objective guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Our study focuses on the impact of single feedback embedded in a multi-source evaluative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The feedback has no further consequences besides that it can motivate or demotivate the recipient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' One important distinction is between positive and negative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We define positive feedback, sometimes called promotion-orientated feedback (Carpentier & Mageau, 2013), as the expression that the evaluated performance is above a certain reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We define negative feedback, sometimes called change-orientated feedback (Carpentier & Mageau, 2013) or corrective feedback (Waldersee & Luthans, 1994), as the expression that the rated performance is below the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='2 Review and hypotheses In their influential model, Kluger and DeNisi (1996) assume that there are no behavioral effects when there is no discrepancy between the rating and the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Positive feedback increases effort if the agent has the possibility to set new self-goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Likewise, negative feedback leads to an increase in effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Similarly, Villeval (2020) argues that positive and negative feedback fosters motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' On the other hand, positive feedback can lead to a decrease in efforts, when individuals have no possibility to set new goals (Kluger & DeNisi, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Negative feedback can discourage individuals when it threatens the self-perception of their competence (Fong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Some empirical studies show a favorable impact of positive feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2018) find a better performance in a computerized task after purely positive feedback than in a baseline treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Itzchakov and Latham (2020) report better performance in a brainstorming task 5 after positive than after neutral feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Bandiera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2015) report that positive feedback improves the performance of university students and Azmat and Iriberri (2010) that positive relative rank feedback enhances the performance of high school students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Other studies, such as Podsakoff and Farh (1989) reporting no impact of positive feedback on performance in an object-listing task, find no influence of positive feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Waldersee and Luthans (1994) even report an adverse impact of positive feedback on the performance of employees of fast food restaurants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Empirical work on the effect of negative feedback provides an ambiguous picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Several studies show a favorable impact of negative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' As for positive feedback, Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2018) find an improved performance after purely negative feedback in comparison to a baseline treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Azmat and Iriberri (2010) find a favorable effect of negative relative rank feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Itzchakov and Latham (2020) report a positive impact of negative feedback on performance in a brainstorming task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Podsakoff and Farh (1989) report a favorable impact of negative feedback in an object-listing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Waldersee and Luthans (1994) find a performance-enhancing effect of negative feedback for employees of fast food restaurants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Some research, such as the meta-study by Fong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2019), shows no impact of negative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Other studies show an unfavorable impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' For example, Deci and Casico (1972) observe that a negative feedback group shows lower motivation to conduct a puzzle task than a control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' A reason for the ambiguity in reaction to negative feedback might be heterogeneity in the way how individuals update their perception after receiving self-relevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Some research finds that agents do not fully update their self-perception after negative information, while they update their self-perception after observing a positive signal (Eil & Rao, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Kuzmanovic, Jefferson, & Vogeley, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Möbius, Niederle, Niehaus, & Rosenblat, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Sharot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' This would imply to find no reaction to negative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Yet, other studies observe a rational updating of beliefs for positive and negative information (Barron, 2021) or even an overweighting of negative information (Coutts, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Ertac, 2011), leaving this strand of empirical research inconclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We build our hypotheses on the theoretical model by Kluger and DeNisi (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We argue that in the domain of professional performance, there is always the possibility to set more am- bitious goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' This indicates that positive feedback might have a favorable impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 6 Hypothesis 1 - Positive Feedback: The performance is better after receiving positive feedback than after receiving neutral feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We follow Kluger and DeNisi (1996) and Villeval (2020) by assuming that also negative feed- back has a performance-enhancing effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We argue that in the field of professional performance, individuals have a rather stable self-perception of confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Hypothesis 2 - Negative Feedback: The performance is better after receiving negative feedback than after receiving neutral feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' A vital aspect that most empirical studies usually can barely answer is the question of the generalizability of these hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Here, it is useful to distinguish between the two superor- dinate layers of personal and task-specific characteristics by which effects could be moderated (compare Fong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2019), for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' For task characteristics, our hypotheses more readily generalize when individuals’ responses to feedback are inherently similar irrespective of the difficulty and importance of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Difficult and easy tasks might be perceived differently (Moore & Healy, 2008), which can lead to different perceptions of feedback (Pulford & Colman, 1997) and varying subsequent performance (Vancouver & Tischner, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Kluger and DeNisi (1996) argue that the reaction to feedback is stronger the fewer cognitive resources are needed to perform the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Likewise, performance might differ depending on the importance of the task (Goller & Heiniger, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Here, Kluger and DeNisi (1996) argue that the effectiveness of feedback increases the more attention is on the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Guided by the model predictions of Kluger and DeNisi (1996), we do not expect generalizability across task characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Accordingly, we expect stronger feedback effects on performance for (relatively) easier tasks needing fewer cognitive resources and more important tasks that require more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Within the personal domain, three potential moderators seem highly relevant in modern workplaces: cultural background, gender, and experience of the feedback recipients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The litera- ture acknowledges that despite the high relevance of cultural differences in a globalized world, non-WEIRD (not coming from Western, Educated, Industrialized, Rich, and Democratic coun- tries) individuals are largely underrepresented in behavioral research (Henrich, Heine, & Noren- zayan, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' For example, authors postulate differences in self-construals (Markus & Kitayama, 7 1991), in feedback seeking of individuals (Sully De Luque & Sommer, 2000) and in feedback re- action of firms (Rhee, Alexandra, & Powell, 2020) between collectivistic and individualistic cultures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Bear, Cushenbery, London, and Sherman (2017) postulate and Berlin and Dargnies (2016), respectively, Roberts and Nolen-Hoeksema (1994) observe different feedback reactions for women than for men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Eggers and Suh (2019) find that the reaction of organizations to negative feedback depends on the experience in the business area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Kluger and DeNisi (1996) propose differential effects for individuals’ behavioral or psychological traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' More relevant from a managerial perspective is if those potentially moderating traits are associated with directly observable char- acteristics of individuals in a company’s diverse context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We refrain from forming explicit ex- pectations and leave the question of generalizability for different cultural backgrounds, genders, and experience levels exploratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 3 Setting and data We collect data on international competitions of two competitive sports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In the two sports, namely, ski jumping and diving, athletes compete individually in multi-round competitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In each round, the athletes’ task execution is evaluated by multiple professional judges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Besides the similarities, there are several specifics to each of the sports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In diving, athletes acrobatically jump into the water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We use data on individual performances in three different types of competitions: 1m springboard, 3m springboard, and 10m platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The scoring consists of two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' First, each jump is rated by seven judges with respect to the proper execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Each judge can reward up to 10 style points (in increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The two highest and the two lowest judges’ ratings are discarded for the jury performance assessment of the jump, for which the remaining three judges’ ratings are summed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Second, the jury performance assessment is multiplied by the difficulty coefficient, which depends on the complexity of the jump and is assigned to the jump according to the official rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='2 In competitions between women, points are accumulated over five jumps, and in competitions between men, over six jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Depending on the contest there are preliminary rounds and/or semi-finals and the final round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 2See https://resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='fina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='org/fina/document/2021/01/12/916f78f6-2a42-46d6-bea8 e49130211edf/2017-2021_diving_16032018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='pdf for a current version of the rules (last accessed on 01/23/2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 8 In the winter sport of ski jumping, athletes jump on skis after sliding down a ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Scoring consists of four components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' First, athletes receive points for the length of their jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Second, there are compensation points for the force and direction of the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Third, scoring depends on the length of the ramp (gate points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Fourth, athletes receive up to 20 style points (in increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='5) for the flight and landing of the jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The (style) ratings are independently rewarded by five judges according to official rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='3 The worst and the best rating are discarded and the other three are accounted for the athletes’ score of the round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In a typical competition, 50 athletes start in the first round, of which the 30 best reach the final round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' After the final round, both jumps’ total scores are added to determine the winner and the succeeding rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='1 Data sets Table 1: Descriptive statistics Diving Ski jumping Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Panel A: Treatments Positive Feedback (deviation positive) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='426 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='286) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='316 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='262) Negative Feedback (deviation negative) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='477 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='320) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='357 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='290) Panel B: Outcomes Score 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='737 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='557) 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='647 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='204) Performance (rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 3 judges’ ratings) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='119 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='189) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='771 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='744) Performance (all 5 / 7 judges’ ratings) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='110 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='182) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='765 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='741) Panel C: Covariates Compatriot judge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='457 Home event 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='127 Experience (Age in years) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='429 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='789) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='836 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='949) Female 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='450 Difficulty 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='211 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='331) Distance 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='608 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='837) Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Distance 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='940 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='143) Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Difficulty 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='166 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='317) Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Performance 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='270 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='958) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='854 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='580) N 13075 4529 Notes: Mean and standard deviation (in parentheses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' for non-binary variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' = remaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Some variables were only observed in one of the data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Full descriptive statistics in Appendix Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 3See https://assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='fis-ski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='com/image/upload/v1665482445/fis-prod/assets/ICR_Ski _Jumping_2022_marked-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='pdf for a current version of the rules (last accessed on 01/23/2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 9 The main analysis is conducted using data on official diving competitions from 2013 through 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' This includes special events such as World Championships and the Summer Olympics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Except for the first jump, each jump constitutes one observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We exclude observations where the rating points of the current or subsequent jump are at the lower or upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='4 Athletes who stop competing during the contest are excluded, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', due to injury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We conduct the analysis based on 13075 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The data consists of the jumps performed by 434 athletes from 54 countries in Africa, Asia, Europe, North America, Oceania, and South America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' As visible in panel C of Table 1, roughly one-half of the athletes are female and on average 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='4 years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In 25 percent of the cases, at least one of the judges has the same nationality as the task taker and about 10 percent of observations are at a home event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Difficulty and previous difficulty of the jump are on average around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='2, and (current and previous) performance are on average around 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='1 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' For our analysis on ski jumping, we have 4529 observations on events from the 2010/11 through 2016/17 season (based on a collection conducted by Krumer, Otto, and Pawlowski (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Each observation refers to a second jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Athletes who fail to qualify for the second round are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In 13 percent of the cases, athletes perform in their respective country of birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In 45 percent of the cases, one of the judges is of the same nationality as the performing athlete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The average age is about 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='8 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Jumps are on average about 123 meters and (current and previous) performance are on average around 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='7 (see panels B and C of Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 4To put it more concretely: We remove observations that have received an average score of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='5 or higher (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='5 in ski jumping), as well as those with an average score of less than 5 (14 in ski jumping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Furthermore, we remove observations with individual scores of 3 or lower (14 in ski jumping), as these are most likely to be crashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' All of these choices are robust to changes, and we show the robustness of the results to data pre-processing in the results section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='2 Variables Figure 1: Illustration of the evaluation and feedback process Notes: For a current task (on the right), feedback is given for the previous task (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The broken arrows represent our main hypotheses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', the potential influence of feedback on performance in the subsequent task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Task and individual characteristics (dotted square) potentially moderate this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In the case of seven judges, the two highest and lowest ratings are discarded, and only the most extreme ratings are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' See Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='5 for other specifications used in the robustness checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Figure 1 describes the evaluation and feedback process in our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' For the task execution evaluation, each judge in the jury independently gives a numerical rating for the task execution of the task taker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The largest and smallest of those judges’ ratings are discarded and the jury performance assessment is the mean of the remaining (three) judges’ ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The task performance assessment quantifies the task performance result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In our study, we focus on the discarded judges’ ratings that are not regarded for the jury’s performance assessment and can affect subsequent performance only through their motivational impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Our treatment variables are constructed as deviations of the discarded judges’ ratings from the jury performance assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' More concrete, Deviation positive is constructed by sub- tracting the jury performance assessment (the mean of the ratings in absence of the discarded ratings) from the largest discarded judges’ rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Deviation negative is constructed by sub- tracting the smallest discarded judges’ rating from the jury performance assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='5 We define 5Additionally, we construct and test two alternative specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' All specifications can be found in the full descriptive statistics in Appendix Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Especially, for diving, there are two (highest/lowest) judges’ ratings discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The base specification uses the most extreme judges’ ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=" Other specifi- 11 Highest rating Deviation Taskandindividual Positive characteristics Jury= ratings feedback Panel Jury of Y Ratings Judges' performance Judges remaining assessment Deviation (5 or 7) Negative feedback Lowest rating Task Task taker Taskexecution performance Feedback Taskexecution result Previoustask Current taskDeviation positive as positive feedback and Deviation negative as negative feedback." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Panel A in Table 1 provides an overview of the main treatment variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Both feedback variables, with mean values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='426 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='316) for positive feedback and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='477 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='357) for negative feedback, range from 0 (for neutral feedback) to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='5 (for increasingly positive/negative feedback).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' To measure the effect of feedback on subsequent task execution, we use the jury’s perfor- mance assessment that the task takers receive for their subsequent performance (hereafter, "Per- formance") as our outcome variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' An alternative variable to measure subsequent performance is the mean of the ratings from all (5 or 7) judges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 4 Empirical strategy We study how positive and negative feedback affect subsequent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' To this end, our identification strategy relies on conditional idiosyncratic variations in the differences between the jury performance assessment and the discarded ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' This positive (negative) deviation is irrelevant to the assessment of the task performance but provides feedback in the form of additional information about the judges’ general perception of the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The identification strategy presumes that, once we condition on a few observable character- istics, there are no omitted influences that are correlated with both outcome, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', performance in the task, and treatment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', the positive/negative deviation (feedback for the previous task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Our approach formalizes to the following linear baseline model: Yi = α + β+A+ i + β−A− i + γXi + ϵi, where the outcome, Yi, is the performance in the (current) task for individual i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The con- tinuous treatments A+/− i are defined as the positive/negative feedback for the (previous) task, and β+/− are the coefficients of interest to investigate our hypotheses 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Xi contains (pre-determined) covariates of individual i that we need to control for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' ϵi is an idiosyncratic error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' To give credence to the unconfoundedness assumption, we address concerns raised in the lit- erature about potential biases in subjective ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' First, we consider nationality bias (Heiniger cation descriptions and results for the robustness of the alternative treatment variable specifications can be found in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 12 & Mercier, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Krumer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Sandberg, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Zitzewitz, 2006), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', a judge from the same country as the task taker rates the compatriot better than other individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' To account for potentially more positive ratings from judges who are compatriots, we include a) a binary variable indicating whether a judge on the panel is a compatriot of the task taker, and b) an indicator if the individual competes in a home event in Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='6 To alleviate remaining concerns about bias based on common nationality, we conduct two further checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' A balancing test in Ta- ble 8 shows no balancing issues related to compatriot judges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' To ensure that the results are not driven by individuals that are potentially subject to nationality bias, we perform a robustness check in which the affected task takers are removed from the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='7 Second, there is evidence in the literature of an order of action bias (Damisch, Mussweiler, & Plessner, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Ginsburgh & Van Ours, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Subjective ratings are found to be affected by the order of task performance, which threatens our identification when some but not all judges are affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We account for this by controlling for the order in which individuals perform tasks (starting order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Third, more difficult tasks were found to be rewarded with higher scores– the difficulty bias (Morgan & Rotthoff, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The difficulty of a task in our case is precisely measurable and predetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Specifically, in diving, we control for the difficulty of the jump (chosen a priori);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' in ski jumping, we control for the (previous and current) wind and gate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', the length of the hill–both factors that can influence difficulty and subjective evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Fourth, there could be reputation bias (Findlay & Ste-Marie, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' This bias can lead to better ratings for well-established individuals who typically have a better reputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' To ensure conditional independence, we take into account a) individual and individual-by-season fixed effects and b) current rank in the competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Fifth, the accuracy of subjective performance ratings is found to vary for different performance qualities (Heiniger & Mercier, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Therefore, we include the individual mean and standard deviation of the jury’s performance assessment of the previous task in Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' While not testable, we are confident that the conditional independence assumption is satis- fied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Still, we offer two types of checks for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' First, in a total of 20 balancing checks in Table 8, only one statistically significant test indicates a solid balancing among observable characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Second, with respect to unobservable characteristics, we provide an indirect approach to sup- 6Judges’ decisions regarding possible bias in favor of compatriots might be different in front of a supportive crowd (Page & Page, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Goller & Krumer, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 7The results for this can be found in Table 11 and hardly differ materially from the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 13 port the conditional independence assumption by implementing a placebo treatment test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We replace the treatment variable with a pseudo-treatment variable recorded in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The task performance cannot be influenced by the feedback given in the future of this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Therefore, if we observe all confounding influences, the placebo treatment effect should be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' If we reject this placebo null hypothesis this points to some unobserved confounding (or other issues like endogeneity or reverse causality), while not rejecting gives some evidence that the conditional independence assumption is plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Table 7 shows that this placebo test cannot reject our assumption of unconfoundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' To estimate the main effects of interest, we use linear regression and cluster standard errors on the individual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In the second step, we apply a method from the causal machine learning literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' For this research, the importance of investigating potential non-linearities in the effect lies in the differently observed treatment intensities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', high or low quantified feedback, for which it is unclear if an estimated constant treatment effect reflects various treatment intensities properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' With the non-parametric kernel method for continuous treatment effects introduced by Kennedy, Ma, McHugh, and Small (2017) we investigate the effects for different intensities of the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The method builds on two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' First, a (doubly-robust) pseudo-outcome is constructed as follows: ξ(π, µ) = Y − µ(X, A) π(A|X) � π(A|x)dP(x) + � µ(x, A)dP(x), where the nuisance functions π(A|X) and µ(X, A) are estimated using a random forest estimator (Breiman, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The pseudo-outcome ξ(π, µ) is doubly-robust in the sense that only (at least) one of the two nuisances needs to be consistent, not both, and is free from confounding influences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In the second step, the average potential outcome for given treatment levels is estimated using a non-parametric kernel regression of the pseudo-outcome on the continuous treatment variable: E(Y a) = E(ξ(π, µ|A = a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 14 5 Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='1 Main results Our first main finding is that positive feedback is enhancing (subsequent) performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Panel A in Table 2 shows a statistically significant and positive coefficient for positive feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The effect is robust to the inclusion of different sets of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In each specification, the average effects are statistically significant at the 1% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Panel B replicates this finding for our second data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' As our second main finding, we observe that negative feedback causes an effect close to zero in both panels and all specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We do not see any effect of negative feedback on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Table 2: The effect of feedback on performance – sensitivity to different specifications Performance (1) (2) (3) (4) Panel A: Diving (N=13075) Positive Feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='242*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='208*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='115*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='100*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='036) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='034) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='032) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='035) Negative Feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='007 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='030) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='030) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='029) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='030) Panel B: Ski jumping (N=4529) Positive Feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='201*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='180*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='145*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='107*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='035) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='036) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='034) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='034) Negative Feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='026 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='043) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='041) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='037) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='041) Base Covariates x x x x All Covariates x x x Individual Fixed Effect x Individual x Season FE x Notes: Linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Full regressions in Tables 9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' All regressions contain previous’ jumps jury assessment (Base Covariates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' All Covariates include prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' jumps wind and gate points and distance (ski jumping) or difficulty (diving).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Also, points behind, compatriot judge, home event, current ranking, SD of previous performance, and start order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Standard errors are clustered on the individual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' *, **, and *** represents statistical significance at the 10 %, 5 %, and 1 % level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The performance-enhancing impact of positive feedback is rather insensitive to the inclusion of more covariates and fixed effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We start with controlling only for performance in the previous task in column (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In column (2) we add several control variables as discussed in 15 Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Columns (3) and (4) add individual fixed effects and individual-by-season fixed effects to the regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Detailed result tables can be found in the appendix in Tables 9 and 10, and for the sake of simplicity, all of the following regressions are based on the specification used in column (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Figure 2: Non-linear estimation of feedback on performance Notes: Non-parametric kernel regression for different levels of positive (left) and negative (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Expected outcomes (y-axis) and treatment levels (x-axis) are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Kernel bandwidths are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='300 (left) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='214 (right) and are determined in a data-driven approach using a cross-validation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' To obtain treatment effects, one might calculate the difference of the expected outcomes for two treatment levels and divide this by the difference in the treatment levels (treatment intensity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Diving data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The broken lines represent the 90% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Our results show that, on average, positive feedback is enhancing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In the fol- lowing, we go beyond average effects and investigate the effect of positive and negative feedback for different magnitudes of feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Figure 2 provides non-linear estimates of positive and neg- ative feedback showing the expected outcome (performance) against the extent of the feedback, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', the level of the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The (treatment) effect of different feedback intensities can be calculated as the difference in expected outcomes for an increase from some treatment level to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='8 In the graph on the left, the effect of positive feedback is positive throughout all feed- 8For two different treatment levels A = a1 and A = a0, the effect can be calculated as θ(a1, a0) = E(Y (A=a1))−E(Y (A=a0)) a1−a0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The treatment intensity in this example is a1 − a0, while for a complete picture, it needs to be clear that the treatment level from which the treatment intensity is evaluated is a0 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 16 Positive feedback E(Y(a) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='00 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='75 1 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='50 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='0 Treatment level A=aNegative feedback E(Y(a) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='75 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='50 1 0 2 Treatment level A=aback intensities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', the expected outcome increases almost steadily as the level of treatment increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' With negative feedback, on the right side of Figure 2, the effect varies slightly up and down for different treatment intensities – although the effect does not appear to be different from zero for any treatment intensity, consistent with the average effect of zero reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' For both estimations, we find that the linearity assumption in the regression analyses is a good approximation for the non-linear effect curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Still, especially for the higher treatment intensities the confidence intervals become large and conclusions become imprecise–a fact to which global linear regression models do not give any hint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Overall, the results provide support for hypothesis 1: The performance is better after receiv- ing positive feedback than after receiving neutral feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Contrarily, we do not find support for hypothesis 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', the performance is not better after receiving negative feedback than after receiving neutral feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In the next section, we test if the positive effect of positive feedback and the null effect of negative feedback persists in different sub-populations and is generalizable for diverse personal or situational conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='2 Sub-population and context heterogeneity In the feedback-intervention model of Kluger and DeNisi (1996), as well as, for example, in the meta-study of Fong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2019) aspects are collected for which the effects of feedback potentially differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Personal characteristics, situational aspects, and task characteristics, among other factors, might shape the reaction of individuals to positive and negative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' A strength of our unique data set is that it allows us to investigate if we can generalize the results of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Panel A of Table 3 exhibits that positive feedback has a favorable impact irrespective of in- dividuals’ personal characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We consider three categorizations of the individuals’ cultural backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' First, we report that the favorable effect of feedback on performance is present for individuals from WEIRD and non-WEIRD countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Second, we find a favorable impact of positive feedback irrespective of the relative cultural distance to the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='. Third, individuals coming from relatively individualistic and relatively collectivistic countries both react favorably to positive feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='9 Other personal characteristics that we investigate are experience and 9We classify (non-)WEIRD countries according our own assessment based on Henrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' the respective list can be obtained upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' For cultural distance to the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', we use the metrics provided in Table 1 in the research article by Muthukrishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' For individualistic and 17 gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We find a performance-enhancing effect of positive feedback for both the relatively more and less experienced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Similar to Bear et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2017), we also explore whether there are gender differences in the reaction to feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We find that both sexes react favorably to posi- tive feedback For none of the three different definitions of cultural background, nor gender and experience, do the two-sample WALD tests show statistically significant differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' This leads to the conclusion that the effects of feedback are consistent and generalizable across these three personal characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Importantly, we find some heterogeneity with respect to the characteristics of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Contested situations offer greater incentives to perform (Goller & Heiniger, 2022), with higher task focus and more pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Panel B of Table 3 shows large and positive effects for positive feedback in close competitions, but an insignificant effect for situations that are less competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' This is in line with the argumentation by Kluger and DeNisi (1996) and our expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Contrary, we find no support for differential effects for the difficulty of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Positive feedback leads to a performance-enhancing impact for easy and hard tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The results of the heterogeneity analysis on the impact of negative feedback are largely in line with the main finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The second column of Table 3 shows a null effect of negative feedback for most subgroups and all contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The only exception is the experience of the individuals, where we find that relatively more experienced individuals improve their performance after receiving negative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' A two-sample Wald test (in square brackets) shows that the difference in the reaction between the more and less experienced individuals is statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The favorable impact of negative ratings for experienced individuals is in line with findings by Eggers and Suh (2019) on the firm level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' collectivistic countries, we use data from the index created by Hofstede (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 18 Table 3: Differential effects Positive Feedback Negative Feedback Panel A: Individuals‘ characteristics WEIRD1 (N=4955) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='086* (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='048) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='006 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='049) Non–WEIRD (N=8120) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='135*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='043) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='004 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='037) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='447] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='974] Culturally close to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='2 (N=6223) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='132*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='046) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='007 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='047) Not culturally close to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (N=6852) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='101** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='044) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='008 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='037) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='626] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='802] Individualistic country3 (N=6013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='096** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='047) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='007 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='045) Collectivistic country (N=6872) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='144*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='045) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='040) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='461] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='921] More experienced (age ≥ 23y, N=6176) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='146*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='045) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='076* (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='039) Less experienced (age < 23y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' N=6899) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='081* (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='047) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='044) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='318] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='019] Female (N=5885) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='087* (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='047) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='028 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='042) Male (N=7190) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='128*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='043) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='018 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='039) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='520] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='422] Panel B: Task characteristics Tight competition4 (N=5118) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='173*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='056) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='033 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='052) Non–tight competition (N=7957) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='064 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='039) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='007 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='037) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='110] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='531] Easy task5 (N=7267) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='154*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='043) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='027 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='037) Hard task (N=5808) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='086* (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='048) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='025 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='044) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='291] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='366] Notes: Linear Regression estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Diving data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Control variables as in column (3) in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Standard errors are clustered on the individual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' *, **, and *** represents statistical significance at the 10 %, 5 %, and 1 % level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' P-value of WALD test for equality in square brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 1Western, Educated, Industrialized, Rich, Democratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 2Cultural closeness is divided at the median level of an index taken Muthukrishna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 3Divided at median level of an individualism index constructed by Hofstede (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (some countries missing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 4Athlete is within ten points to first place in final, and to the cut-off in preliminary rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 5Easy and hard according to the median chosen difficulty of the (assessed) task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='3 Repetition and long-term effects For practitioners, it is crucial to know about the impact of feedback when it is given repeatedly and about its long-term effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Fortunately, our data allows for analyzing the impact of feedback on performance in a repeated setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Figure 3 shows that the favorable impact of positive feedback is non-diminishing with repe- tition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' As a benchmark, Baseline shows the average effect of receiving feedback as reported in Table 2, which is not conditional on further previously received feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We find that for those who have received positive feedback at least one time before, further positive feedback continues to have a positive impact on their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Similarly, we find a positive influence of positive feedback if the individual has received positive feedback at least two or three times before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Figure 3: A non-diminishing effect of positive feedback Notes: Linear regression estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Diving data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Specifications as in column (3) in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Standard errors are clustered on the individual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Effect among those that experienced positive feedback at least one, two, or three times (in the respective round) before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The whiskers mark the 90 % confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Table 4 shows the non-persistence of the effect of positive feedback on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' For reference, column (1) reports the baseline effect for the performance in the task that is conducted directly after the feedback is received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Columns (2-4) provide estimates for the effect of feedback on performance in tasks carried out thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' For all follow-up tasks, we find statistically insignificant effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' This indicates that the favorable short-term effect of positive feedback does not carry on to future tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Negative feedback has no impact, neither on subsequent nor future tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 20 Positive feedback before 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='15 Effect 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='03 Baseline One Two ThreeTable 4: A non-persistent effect of feedback on performance Performance (1) (2) (3) (4) Positive feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='115*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='032) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='047) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='050) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='061) Negative feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='079 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='029) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='036) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='046) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='056) Periods after feedback: 1 2 3 4 N 13075 10130 7350 4512 Notes: Linear Regression on future outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Diving data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Specifications as in column (3) in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Standard errors are clustered on the individual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' *, **, and *** represents statistical significance at the 10 %, 5 %, and 1 % level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='4 Spillover effects on related tasks Previously presented evidence shows the favorable effects of positive feedback on the task for which the feedback was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In practice, individuals might do several tasks simultaneously, or a task containing different elements, that potentially influence each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' For example, Hecht, Tafkov, and Towry (2012) show spillover effects of incentive schemes in one task on a related, simultaneously conducted second task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Our settings allow us to study, both, a single-task and a multi-task environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Panel A presents the results for the single-task setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' As presented previously in Table 2, we find a performance-enhancing impact of positive feedback and no impact of negative feedback on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The difficulty is fixed ex-ante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' That we find no impact of feedback on the difficulty can be regarded as a placebo outcome test and supports our identification strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Difficulty and performance evaluation jointly determine the combined outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Consequently, we observe a favorable effect of positive feedback on the total score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Panel B exhibits the results for the multi-task environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We observe favorable spillover effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Receiving positive feedback in Task 1 enhances subsequent performance in Task 1 and the related Task 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Negative feedback has no impact on either of the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In the setup, performance in Task 1 and Task 2 are the most important determinants of combined success and the only ones that can be influenced by the task taker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Consistently, we also find a favorable influence of positive feedback on the total score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 21 Table 5: Spillover effects Panel A: One isolated task, diving Task 1: Multiplier: Combined: Performance Difficulty Total score Positive Feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='115*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='071*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='032) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='006) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='324) Negative Feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='029 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='029) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='005) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='282) Panel B: Two simultaneous tasks, ski jumping Task 1: Task 2: Combined: Performance Distance points Total score Positive Feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='145*** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='692*** 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='126*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='034) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='634) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='693) Negative Feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='080 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='037) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='545) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='631) Notes: Linear Regression estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Control variables as in column (3) in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Feedback was given previously for Task 1 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Standard errors are clustered on the individual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' *, ** , and *** represents statistical significance at the 10 %, 5 %, and 1 % level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='5 Robustness To ensure that our results are robust to different specifications we conduct several supplementary analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' First, we consider alternative specifications of our key variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In a first regression, we take the mean of all (five or seven) judges’ ratings, instead of the performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', the mean of the (after discarding the extreme ratings) remaining three ratings, as an alternative outcome variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' With the treatment, the second key variable is (additionally) constructed in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Instead of subtracting the jury’s performance assessment from the most extreme (positive/negative) discarded rating we deduct (a) the lowest (highest) rating included in the jury’s performance assessment from the lowest (highest) discarded rating (Deviation positive/negative+) and (b) the jury’s performance assessment from the mean of the two dis- carded highest or lowest ratings (Deviation positive/negative++, in diving only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Table 11 presents the results for these alternative specifications and shows robust estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We conclude 22 from this that the result does neither depend on the concrete choice of the treatment variable, nor on the selection of the outcome variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Second, we consider different choices with respect to the sample that is used for the inves- tigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Data cleaning might offer some leeway to researchers influencing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Thus, we provide additional analyses in Table 11 using (a) the full sample without any data cleaning and (b) without excluding failed attempts (but excluding boundary values as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We find robust results for both supplementary analyses, indicating that our data-cleaning step does not drive the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Third, to prove that nationality bias is not responsible for the effect, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', judges favor their compatriots and potentially influence other judges on the panel, we re-estimate the results excluding all athletes with a compatriot judge in the panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' If the effect would be driven by these individuals the results might just be some mechanical effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Though, the effect is also found for individuals not sharing nationality with a judge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 6 Managerial implications and conclusions Giving feedback is one of the most important tasks of managers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' On a typical workday, managers regularly provide feedback to their teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Some of this feedback is subconscious, such as facial expressions or nodding as a sign of appreciation and approval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Other feedback can be formal and dictated by the institution, as is the case with appraisal interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' It can be constructive and substantive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' But it can also be purely motivational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Common examples would be phrases like “Good job!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' or “You can do better!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' embedded in the context of everyday conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The crucial question is whether such motivational feedback, given consciously by managers, can serve the goal of increasing the future productivity of workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' For both valences of feedback, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' positive and negative feedback, this question is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The appreciation that positive feedback expresses can motivate but also cause employees to rest on their laurels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Negative feedback can spur on but it can also hurt and discourage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Our causal analysis indicates that managers can use positive feedback to enhance productiv- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Our results show a favorable impact of positive feedback on (subsequent) performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The heterogeneity analysis indicates that this favorable effect of positive feedback can be found for feedback recipients coming from varying cultural backgrounds, for recipients of both male and 23 female gender, and for relatively more and less experienced recipients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' We find that the favorable effect of positive feedback is short-term, repeatable, and with potentially favorable spillover to related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The favorable impact of positive feedback is robust to the setup in which the activity is performed and is more pronounced in highly relevant situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' All this makes us confident that giving positive motivational feedback is a performance-enhancing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Furthermore, we find no significant impact of negative feedback on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' This null effect might explain why managers and other raters are often reluctant to give negative feedback (Fisher, 1979), a phenomenon termed as leniency bias (Cheng, Hui, & Cascio, 2017) or MUM- effect (Rosen & Tesser, 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' While in other contexts the lack of negative ratings is decreasing efficiency (Cannon & Witherspoon, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Bolton, Kusterer, & Mans, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Keser & Späth, 2021), we report no need to give negative motivational feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Despite the robustness of our results, we acknowledge some limitations of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' First, our sample consists of internationally competing athletes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' While their level of profession- alism and self-discipline might be comparable to those of employees in highly competitive work environments, top athletes are not representative of the general population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Second, we consider an environment in which individuals receive feedback from multiple, external sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Again, this is more comparable to daily life at large and competitive companies than at small firms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Third, we analyze a domain in which feedback recipients directly benefit from improvements in their performance, while feedback providers do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In other domains, raters might be more prone to willfully bias their feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Therefore, we suggest that future research could contrast our results to environments, in which feedback providers benefit from an increased performance more than feedback recipients do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Employees in such environments might be prone to exploitation when employers use positive feedback as a substitute for more substantial improvements in the employees’ well-being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Fur- thermore, future research could analyze the long-term effects of positive and negative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' With this study, we contribute to the literature that provides guidelines for optimal feedback (Balcazar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Alvero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Sleiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Our causal analysis shows that positive feedback 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='1744-6570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='514 _1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='x Stone, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', & Stone, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The effects of multiple sources of perfor- mance feedback and feedback favorability on self-perceived task competence and perceived feedback accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Journal of Management, 10(3), 371–378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='1177/014920638401000311 Sully De Luque, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', & Sommer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The impact of culture on feedback- seeking behavior: An integrated model and propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Academy of Management Review, 25(4), 829–849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='5465/amr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='3707736 Vancouver, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', & Tischner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The effect of feedback sign on task perfor- mance depends on self-concept discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The Journal of Applied Psychology, 89(6), 1092–1098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='1037/0021-9010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='1092 Villeval, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Performance feedback and peer effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' In K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Zimmermann (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' ), Handbook of labor, human resources and population economics (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 1–38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Cham: Springer International Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='1007/978-3-319-57365-6_126-1 Waldersee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=', & Luthans, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' The impact of positive and corrective feedback on customer service performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Journal of Organizational Behavior, 15(1), 83–95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='1002/job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='4030150109 Zitzewitz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Nationalism in winter sports judging and its lessons for organi- zational decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Journal of Economics & Management Strategy, 15(1), 67–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 31 Appendix Descriptive statistics Table 6: Full descriptive statistics Diving Ski jumping Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Mean Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Treatments: Positive feedback (deviation positive) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='426 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='286) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='316 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='262) Negative feedback (deviation negative) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='477 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='320) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='357 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='290) Positive feedback+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='314 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='297) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='179 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='258) Negative feedback+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='363 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='328) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='218 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='289) Future positive feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='439 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='301) Future negative feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='489 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='325) Outcomes: Performance (rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 3 judges’ ratings) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='119 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='189) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='771 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='744) Performance (all 5 / 7 judges’ ratings) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='110 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='182) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='765 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='741) Score 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='737 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='557) 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='647 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='204) Distance 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='608 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='837) Covariates: Difficulty 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='211 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='331) Compatriot judge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='457 Home event 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='127 Final 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='291 Female 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='450 Age 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='429 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='789) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='836 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='949) Current ranking 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='490 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='655) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='357 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='582) Start order 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='490 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='082) Points behind leader 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='491 (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='011) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='247 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='132) In range (within 5 pts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' to threshold) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='264 Gate points 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='093 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='270) Wind points 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='291 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='225) Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' performance 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='270 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='958) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='854 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='580) Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' SD performance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='130 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='151) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='157 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='159) Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' wind points 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='685 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='136) Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' gate points 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='163 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='386) Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' distance 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='940 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='143) Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' difficulty 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='166 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='317) N 13075 4529 Notes: Mean and standard deviation (in parentheses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' for non-binary variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Some variables only observed in one of the data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' +Alternative definition as defined in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 32 Placebo and balancing tests Table 7: Placebo treatment regressions Judges’ Judges’ Judges’ ratings 3 ratings 5 ratings 7 Score Future positive feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='012 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='035) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='035) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='035) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='345) Future negative feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='437 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='035) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='035) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='035) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='318) N 10256 10256 10256 10256 Notes: Linear Regression on the outcome mentioned in the column header.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 3, 5, and 7 refer to discarding four, two, or none of the extreme judges’ ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Diving data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Pseudo- treatment is the deviation of next (future) jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Jumps 2–4/5 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Specifications as in column (3) in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Standard errors are clustered on the individual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' *, **, and *** represents statistical significance at the 10 %, 5 %, and 1 % level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 33 Table 8: Balancing Tests Diving Compatriot judge Home event SD prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (1) (2) (3) (4) (5) (6) Feedback positive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='007 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='014) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='008) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='005) Feedback negative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='002 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='012) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='007) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='005) Difficulty Final (1) (2) (3) (4) Feedback positive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='018 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='007) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='013) Feedback negative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='017 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='006) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='013) Ski jumping Compatriot judge Home event Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' distance (1) (2) (3) (4) (5) (6) Feedback positive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='918 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='028) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='018) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='696) Feedback negative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='048*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='345 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='022) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='017) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='674) Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' gate SD prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (1) (2) (3) (4) Feedback positive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='024 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='306) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='092) Feedback negative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='018 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='217) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='095) Notes: Linear Regression estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Each regression includes athlete fixed-effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Standard errors are clustered on the individual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' *, **, and *** represents statistical significance at the 10 %, 5 %, and 1 %, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 34 Additional and full results tables Table 9: Feedback on performance – sensitivity to different specifications, ski jumping Ski jumping Performance (1) (2) (3) (4) Positive feedback 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='201*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='180*** 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='003) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='003) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='003) Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='002*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='003*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='002** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='001) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='001) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='002) Wind points 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='041*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='038*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='036*** (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} 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Home event 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='041 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='032) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='032) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='035) Start order 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} 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FE x N 4529 4529 4529 4529 Notes: Linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (= previous) refers to a lagged variable from the previous jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' SD = standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Standard errors are clustered on the individual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' *, **, and *** represents statistical significance at the 10 %, 5 %, and 1 % level, respectively.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='030) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='030) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='029) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='030) Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' jury assessment 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='430*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='001) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='000) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='001) Penalty 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='288 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='362* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='310 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='187) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='187) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='200) Jump and Event Fixed Effect x x x Athlete Fixed Effect x Athlete x Season Fixed Effects x N 13075 13075 13075 13075 Notes: Prev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' (= previous) refers to a lagged variable from the previous jump.' metadata={'source': 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Positive Negative Feedback Feedback Baseline results 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='121*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='115*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='033) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='044) (0.' metadata={'source': 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definition 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='127*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='007 (Mean discarded vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' mean credited) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='043) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='039) Without data cleaning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content='124*** 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' Standard errors are clustered on the individual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' *, **, and *** represents statistical significance at the 10 %, 5 %, and 1 %, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} +page_content=' 37' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQfSi3T/content/2301.11776v1.pdf'} diff --git a/5tE2T4oBgHgl3EQfOwZB/content/tmp_files/2301.03751v1.pdf.txt b/5tE2T4oBgHgl3EQfOwZB/content/tmp_files/2301.03751v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b11b3adc029332206da199e5885028c17c45d416 --- /dev/null +++ b/5tE2T4oBgHgl3EQfOwZB/content/tmp_files/2301.03751v1.pdf.txt @@ -0,0 +1,1141 @@ +Generative Emotional AI for Speech Emotion Recognition: +The Case for Synthetic Emotional Speech Augmentation +Abdullah Shahida, Siddique Latifb,∗ and Junaid Qadirc +aInformation Technology University (ITU),Punjab, Pakistan +bUniversity of Southern Queensland, Australia +cQatar University, Doha, Qatar +A R T I C L E I N F O +Keywords: +Tacotron, WaveRNN, speech synthesis, +text-to-speech, emotional speech syn- +thesis, speech emotion recognition +A B S T R A C T +Despite advances in deep learning, current state-of-the-art speech emotion recognition (SER) systems +still have poor performance due to a lack of speech emotion datasets. This paper proposes augmenting +SER systems with synthetic emotional speech generated by an end-to-end text-to-speech (TTS) system +based on an extended Tacotron architecture. The proposed TTS system includes encoders for speaker +and emotion embeddings, a sequence-to-sequence text generator for creating Mel-spectrograms, and a +WaveRNN to generate audio from the Mel-spectrograms. Extensive experiments show that the quality +of the generated emotional speech can significantly improve SER performance on multiple datasets, +as demonstrated by a higher mean opinion score (MOS) compared to the baseline. The generated +samples were also effective at augmenting SER performance. +1. Introduction +Speech emotion recognition (SER) is a rapidly growing +field with many applications in fields such as healthcare, cus- +tomer service, media, education, and forensics. While deep +learning (DL) has shown promise in developing SER sys- +tems, their performance is still limited by the scarcity of +emotion datasets [20, 24]. Existing SER corpora are small +since the process of creating emotional data is costly and +time-consuming, as multiple annotators have to manually +listen to and annotate the material [26, 31]. To increase data +size, some studies have used multiple corpora, but the num- +ber of standard benchmark datasets is also limited, hindering +progress in SER systems [19]. +Researchers have long been interested in creating natural- +sounding TTS systems. TTS technology has come a long +way from early TTS systems that often used pre-recorded +waveforms pieced together based on input text [11]. Such +systems were prone to boundary artefact issues and statis- +tical techniques were later developed to generate smoothed +audio features for the vocoder to synthesise speech [37, 44]. +More recently, end-to-end neural network-based approaches +have been proposed that can synthesise more natural-sounding +human speech [3, 34]. Current state-of-the-art TTS systems +are trained using DL algorithms in an end-to-end fashion, +with popular models including Tacotron [41], Deepvoice [3], +Fastspeech [33, 34], Fastpitch [18], to name a few. +Unlike traditional systems, end-to-end TTS models can +learn to generate a spectrogram directly from text without +any complex pre-processing. These models, however, are +currently only able to synthesise natural speech. Using gen- +erative DL techniques such as generative adversarial net- +works (GANs) [8] for emotional speech synthesis is also +challenging, as it requires a large amount of time-aligned +data of a single speaker speaking the same content in dif- +∗Corresponding author +ORCID(s): +ferent emotions and complex equations to guide the model +in converting emotions using audio features. Some studies +have achieved promising results in single-speaker emotional +speech synthesis using TTS models [17], but the quality of +synthetic speech in augmenting SER has not been evaluated. +In this paper, we propose a method for augmenting SER +systems using an emotional text-to-speech (TTS) system and +make two main contributions. Firstly, we develop an end-to- +end multi-speaker emotional TTS system that does not re- +quire any alignment of audio files for emotion conversion +or complex pre-processing of input data. Inspired by the +success of end-to-end TTS models, we adopt a similar ar- +chitecture to Tacotron. We propose to use a condition en- +coder to control the speakers’ voices and emotions in the +output speech. We generate speaker voice feature vectors +using the encoder network. These feature vectors are modu- +lated with one of the encoded emotional feature representa- +tions. These modulated feature vectors are used to condition +the Tacotron to synthesise speech in different speaker voices +and emotions. Subjective evaluation tasks show that our pro- +posed model improves controllability and successfully syn- +thesises emotional speech. Secondly, we use the synthesised +emotional speech to augment an SER system and conduct +multiple experiments to evaluate the generated data quan- +titatively. Results show that the synthesised data can help +improve SER performance in both within-corpus and cross- +corpus settings. +The rest of the paper is organised as follows. In Section +2, we briefly introduce the related work to change different +features of audio. The model’s architecture, loss functions, +and flow of our architecture are described in Section 3. The +details of the dataset and experimental condition in which +we trained our model and hyper-parameters are provided in +Section 4. We report our results in Section 5. Finally, this +paper is concluded in Section 6. +A Shahid et al.: Preprint submitted to Elsevier +Page 1 of 9 +arXiv:2301.03751v1 [cs.SD] 10 Jan 2023 + +Multi-Speaker Emotional Speech Synthesis +2. Previous Work +In this section, we review the literature that has emerged +around (1) the use of Tacotron for TTS, and for (2) emotional +speech synthesis, and (3) the process of augmenting SER. +2.1. Tacotron Based TTS Systems +Many recent studies have focused on modifying the Tacotron +model in order to better control the output of TTS systems. +For instance, [13] presented a Tacotron-based model that +synthesises multi-speaker speech by conditioning the Tacotron +on the speaker’s voice embedding, which was generated from +a speaker verification model [40]. [42] introduced a Tacotron +variant that can change the speaking style, by learning dif- +ferent styles and saving them as vectors or tokens. These +tokens are obtained by clustering similar accents and repre- +senting each cluster with an average. During synthesis, the +Tacotron is conditioned on one of these tokens to produce +speech with a specific style. [35] presented a multi-speaker +Tacotron that can change accents (e.g., American, Indian, +British). Their model uses two encoder networks with the +Tacotron and requires two audio samples (one for the ac- +cent and one for the speaker’s voice) as input to generate +the desired output. [36] proposed a Tacotron model that is +trained with encoded output audio from a variational autoen- +coder as input. This not only improves the multi-speaker +performance of Tacotron but also allows for control over the +energy of the generated audio through the mean-variance +property of the variational autoencoder. [43] developed a +Tacotron model that can learn more complex vocalisations +by using the self-attention mechanism in Tacotron to learn +complex dependencies related to pitch in different accents. +They claim that their model outperforms traditional end-to- +end approaches for languages with more pitch-dependent ac- +cents, such as Japanese. Our proposed model also gener- +ates speech in a multi-speaker setting and includes additional +control over the emotions in the output. +2.2. Tacotron Based Emotional TTS Systems +Several previous works have attempted to generate emo- +tional speech using TTS systems. For example, [38] devel- +oped an emotion control method for a TTS system based on +the GST-Tacotron network [35], and demonstrated its effec- +tiveness in synthesising emotional speech in a single-speaker +setting in Korean. [27] also evaluated a Tacotron-based emo- +tional speech synthesizer in Korean, and found improvements +in the quality of the generated speech for a single speaker. +Other studies, such as [15, 17], have also proposed meth- +ods for controlling emotional speech synthesis, but these ap- +proaches only synthesise emotional speech in a single speaker’s +voice. In contrast, our proposed method achieves control +over emotional speech synthesis for multi-speaker TTS and +we also evaluate the quality of the synthesised data to aug- +ment the SER system. +2.3. Augmenting Techniques for SER +Speed perturbation [16] is a popular data augmentation +technique that has been widely studied in different contexts +[2, 23]. It has been found to improve speech emotion recog- +nition (SER) performance by creating copies of input data +with different speed effects. Mixup [45] is another data aug- +mentation technique that generates augmented samples as a +linear combination of original samples from the input data. +Several studies have demonstrated the effectiveness of mixup +in SER, including Latif et al. [25], who used the technique to +augment an SER system and achieve improved performance +and robustness. A recent method called SpecAugment [30], +originally proposed for automatic speech recognition, has +also been applied to SER [4]. In this study, the authors aug- +mented the SER system with duplicate samples by a factor +of two and found that SpecAugment improved model per- +formance. Other studies [2, 21, 23] have also achieved im- +proved performance by using input perturbation-based data +augmentation techniques to increase the training data. +Further research is required to explore data-driven ap- +proaches to increase the training data for SER. In this paper, +we propose to explore TTS based data augmentation method +where we explored different variations in the training data +by changing the speaker and gender voices in different emo- +tions. +3. Proposed Framework +We propose to generate synthetic speech using a Tacotron- +based emotional TTS system. We use synthetic speech data +to augment the speech emotion classifier. The details of both +emotional TTS and classifier are presented next. +3.1. Emotional Speech Synthesis +Our model consists of an encoder which conditions Tacotron +(as depicted in Figure 1) to alter the speaker’s voice and emo- +tion in the output. Tacotron generates a Mel-spectrogram +from a given text and embedding vector, while a Wave-RNN- +based vocoder is used to generate an audio signal from the +Mel-spectrogram +3.1.1. Condition Encoder +We propose using a condition encoder to create an em- +bedding that represents both speaker identity and emotion. +To do this, we use a speaker identification model presented +in [40], which creates a fixed-dimensional embedding, known +as a d-vector [10, 39], using a sequence of Mel-spectrograms +computed from a speech signal of arbitrary length. We train +this model using an end-to-end speaker verification loss that +maximises the cosine similarity between utterances from the +same speaker while minimising the cosine similarity between +utterances from different speakers. We fine-tune this net- +work on an emotional corpus to create an emotional embed- +ding as well. Thus, the condition encoder is optimised to +maximise the cosine similarity between embeddings of the +same speaker with different emotions and to minimise the +similarity between different emotions and different speak- +ers. In this way, the model learns to generate a feature vector +that contains both emotion and speaker identity information. +The speaker’s voice audio and emotion audio are embedded +A Shahid et al.: Preprint submitted to Elsevier +Page 2 of 9 + +Multi-Speaker Emotional Speech Synthesis +Figure 1: Architectural flow diagram. The reference speaker’s voice is first encoded and then modulated to desired emotion as +described in the model schema. The output is then passed to the Tacotron decoder with the text embedding to synthesise the +Mel-spectrogram. +using the condition encoder and combined to generate a fi- +nal embedding, which is used to condition the synthesizer to +output speech with the selected emotion and speaker’s voice. +For each unique emotion of every speaker in dataset, a +centroid 푐푘 is calculated by taking the average of embedding +for each unique emotion of every unique speaker. Loss for +an embedding 푒푖 when the embedding and the centroid 푐푘 +have the same speaker and emotion is calculated as: +(푒푖, 푐푘) = −1 × 휎(cos(푒푖, 푐푘)) +(1) +When 푒푖 have different emotion or different speaker for cen- +troid 푐푘 then loss is calculated as:- +(푒푖, 푐푘) = 휎(cos(푒푖, 푐푘)) +(2) +퐺(푆) = +∑ +푖,푘 +퐿(푒푖, 푐푘) +(3) +Equation 1 maximises the cosine similarity between em- +beddings for the same speaker voice and same emotion. Equa- +tion 2 represents the cosine similarity between embedding +and centroid when they have different speaker voices or dif- +ferent emotions or both. Equation 3 represents the final loss +over every embedding, which is calculated as the sum of the +loss for every embedding with every centroid. +The condition encoder consists of three LSTM layers +with 768 cells each, and a final 256-length fully connected +layer. The input to the model is the Mel-spectrogram gener- +ated from a speech utterance of a reference speaker’s audio +sample, and its output is an embedding vector of size 256 +which represents the speaker’s identity. After training the +model, we use it to extract speaker and emotional informa- +tion from a given audio. To separate the emotion from the +speaker’s voice, we generate vectors that only contain emo- +tional information by using the trained condition encoder to +generate embedding vectors for both the neutral and emo- +tional voices of the same speaker. The neutral embedding +vector is then subtracted from the emotional ones using Equa- +tion (4), resulting in a vector that only contains emotional +information. This vector can be used at inference time to +control the emotion of the synthesised audio. +푒푚푏em = (푒푚푏en − 푒푚푏neu) +(4) +Where 푒푚푏en represents the embedding with emotion and +voice information generated from the emotional voice of a +speaker; 푒푚푏neu is generated from neutral audio of the same +speaker, and 푒푚푏em represents the embedding that only con- +tains emotional information. During inference, reference au- +dio embedding (voice in which we want our output sample +to be synthesised) and emotional embedding are added to +generate a final embedding vector. +푒푚푏final = 푒푚푏ref + 푒푚푏em +(5) +Finally, the modulated embedding vector and text are fed +to Tacotron, which generates the Mel-spectrograms. These +Mel-spectrograms are converted to the time domain using a +vocoder, resulting in an audio signal. +3.1.2. Synthesizer architecture +The synthesizer is a variation of Tacotron [41], which +is a sequence-to-sequence model that generates output one +frame at a time based on the input. In addition, we condition +this synthesizer on an embedding vector generated by the +condition encoder, which contains information about the de- +sired output emotion and the speaker’s voice. The condition +embedding is concatenated with the text embedding of the +synthesizer and then passed through a decoder to synthesise +A Shahid et al.: Preprint submitted to Elsevier +Page 3 of 9 + +Audio +Reference audio +Emotional embedding +Speaker +Vocoder +Encoder +Concatenation +Condition encoder +Mel +Spectrogram +Text +Concatenation +Attention +Decoder +embedding +Synthesizer +Character SeguenceMulti-Speaker Emotional Speech Synthesis +the output Mel-spectrogram. The synthesizer was trained on +80-channel Mel-spectrograms with a window size of 50 ms +and a hop size of 12.5 ms. The synthesizer encodes the input +characters into a hidden representation using three convolu- +tion layers, which learn longer-term context like an n-gram. +The output of these convolution layers is passed to a single +bi-directional LSTM layer with 256 units, which learns time +dependencies from these n-gram-like features. The LSTM +layer returns an encoded vector that fully represents the in- +put text sequence. This vector is concatenated with a vector +of emotional and speaker embeddings from the encoder. +It is worth noting that at this point, the encoder has been +trained and its weights are not updated. The combined text, +speaker, and emotion embedding is passed to the decoder to +generate a Mel-spectrogram. The decoder architecture in- +cludes a location-sensitive attention mechanism that trans- +forms the input embedding into a fixed-length vector. The +output frame from the previous step is passed through two +fully connected layers and concatenated with the embedding +vector to ensure that sequences are generated without any +time artefacts. This vector is then passed through two LSTM +layers, and a linear transformation is applied to generate the +next frame of the Mel-spectrogram. The output from this +LSTM is also projected down to a single scalar, which serves +as a stop token and indicates when to stop generating further +frames. Once the Mel-spectrogram has been generated, it +is passed through a 5-layer convolution network called the +PostNet to improve overall reconstruction. +3.1.3. Vocoder +Traditionally, the Griffin-Lim algorithm [9] was used to +generate time-domain audio from a spectrogram, but it was +slow and the output speech lacked naturalness. To address +this, we use a vocoder based on the WaveRNN architecture +[14], which is a faster and more powerful recurrent network +for sequential modelling of high-fidelity audio. It employs +residual convolutions and GRU layers to generate a time- +domain audio signal frame by frame from a Mel-spectrogram. +3.2. Emotion Classifier +To evaluate the synthesised emotions, we trained a deep +neural network (DNN) for SER. We implemented a convo- +lutional neural network (CNN)-based classifier that consists +of a convolutional layer, a batch normalisation layer, and a +dense layer before the softmax layer. Mel-frequency cepstral +coefficients (MFCCs) are used as the input to the classifier. +The CNN layers learn high-level features from the input fea- +tures, which are then transformed by the dense layer into a +more discriminative space for better emotion classification +after passing through the normalisation layer. +4. Experimental Protocol +This section describes the details of the dataset, input +feature, and model training. +4.1. Datasets +We used the Librispeech dataset [29] to train our TTS +model. It consists of 1000 hours of speech data from various +speakers, sampled at 16 kHz. For the emotion embeddings, +we used the Emotional Voices Database (EVD) [1] and the +Toronto Emotional Speech Set (TESS) [7], which contain six +different speakers reading different sentences with different +emotions. We conducted multiple experiments to evaluate +the performance of our model. For emotion classification +experiments, we used the Ryerson Audio-Visual Database +of Emotional Speech and Song (RAVDESS) [28] and TESS. +For cross-corpus emotion classification, we used the CREMA- +D [6], SAVEE [12], EmoDB [5], and synthesised audio. The +details of these datasets are presented in Table 1. We used +one speaker from Librispeech, as well as all the speakers +from EVD and TESS with two samples that were not in- +cluded in the training set, to determine the mean opinion +score. For emotion classification experiments, we use speaker- +independent emotion classification. We randomly select 70% +of CREMA-D for training, 10% for validation, and 20% for +testing. The full corpora including RAVDESS and EmoDB +were used as the test set in the emotion classification experi- +ments, and the SAVEE dataset was used as the test set in the +cross-corpus emotion classification experiments. +4.2. Input Features +Tacotron takes text strings as input, which are sequences +of characters. Each character is encoded into a one-hot en- +coded vector and embedded in a continuous vector. The +other input to Tacotron is a condition embedding vector that +contains speaker and emotion information. This vector is +obtained from an encoder, which takes speaker audio as in- +put and converts it into Mel-frequency cepstral coefficients +(MFCCs). These MFCCs have 40 log filter banks, 80 frames, +and no overlapping window. To generate t-distributed stochas- +tic neighbour embedding (t-SNE) plots of synthesised audio, +we encoded our synthesised audio using the model presented +in [13]. The input to this model is also MFCCs with 40 log +filter banks, 80 frames, and no overlapping window, result- +ing in an 80x40-dimensional feature vector. This model is +also used in evaluating the equal error rate (EER) in speaker +verification. In emotion classification and cross-corpus emo- +tion classification, we use MFCCs with 40 log filter banks +and a hop size of 64 milliseconds. The MFCC array is trans- +posed and the arithmetic mean is calculated across its hori- +zontal axis as in a previous work [32]. +4.3. Speech Synthesis Models Training +First, the encoder is trained on the Librispeech dataset +to learn to generate a speaker embedding that is distinct for +each speaker. It takes a Mel-spectrogram as input and out- +puts an embedding vector of size 256. From these embed- +ding vectors, a similarity matrix is constructed such that each +column contains an embedding vector for a unique speaker, +and cosine similarity is maximised in all cells of the columns +and minimised in all cells of the rows. Cosine similarity is +maximised along the columns because they contain audio +embeddings for the same person, whereas it is minimised +A Shahid et al.: Preprint submitted to Elsevier +Page 4 of 9 + +Multi-Speaker Emotional Speech Synthesis +Table 1 +Description of all the considered datasets. +Name +Number of +Speakers +Number of +Utterances +CREMA-D +91 +7,442 +EmoDB +10 +535 +EVD +5 +7,590 +Librispeech +2484 +281,241 +REVDESS +24 +7,356 +SAVEE +4 +480 +TESS +2 +2,800 +along the rows because they contain audio embeddings for +different people. In this way, the embeddings of the same +people are similar and those of different people are differ- +ent. +After training the encoder on the Librispeech data, it is +fine-tuned on the EVD and TESS datasets to generate dis- +tinct embedding vectors for different emotions. This time, +a similarity matrix is constructed such that a column con- +tains embedding vectors generated for a single emotion for +the same speaker, and other emotions are placed in other +columns. This is done for all speakers, and then cosine sim- +ilarity is maximised along a column and minimised across +columns. This is done to increase the distance between dif- +ferent emotions of the same person, so cosine similarity is +minimised by adding it across columns rather than within +the same columns. We used a batch size of 30 and a learn- +ing rate of 10−4. +During training, the synthesizer model is first trained on +the Librispeech data so that it can learn to generate audio +of different speakers from a diverse range of text. This is +because the EVD and TESS datasets combined only have six +speakers. Once the synthesizer is trained enough that it can +generate audio resembling the reference speaker, we fine- +tune it to generate different emotional Mel-spectrograms by +training it on the EVD and TESS datasets. We use a learning +rate of 103 that exponentially decays to 10−5, and a batch size +of 30 for training the synthesizer. The Adam optimiser with +훽1 = 0.9, 훽2 = 0.999, and 휖 = 10−6 is used as the optimiser. +The teacher forcing ratio is set to 1 (meaning the original +previous sequence is shown to the model for prediction of +the next sequence). The mean squared error is minimised +for the predicted Mel-spectrogram. +5. Results +In this section, we evaluate the performance of our pro- +posed model in terms of the similarity of the synthesized +speakers and the granularity of synthesized emotions. +5.1. Evaluating Synthetic Speech Quality +To evaluate the quality of synthetic speech, we conducted +multiple experiments. The details of these experiments are +presented below. +Table 2 +Speaker verification EERs of different synthesizers. +# of samples +EER +Emotion + voice conversion TTS +100 +0.16 +Baseline Emotion conversion TTS +100 +0.24 +Voice conversion TTS +100 +0.10 +Real audios +100 +0.04 +Table 3 +Mean Opinion Score (MOS) with 95% confidence interval. +Emotion +Angry +Happy +Sad +Neutral +Overall +Recorded +4.6 +4.50 +4.50 +4.60 +4.55 +Baseline +2.80 +3.10 +2.70 +4.20 +3.20 +Proposed +3.60 +3.70 +3.80 +4.10 +3.80 +5.1.1. Speaker Verification +We evaluated the speaker similarity of synthesised au- +dios with real speech using speaker verification and mea- +sured the equal error rate (EER) following [13]. The EER +is used to measure the performance of a speaker verification +system by comparing the false reject rate (FRR) and false ac- +cept rate (FAR) at different sensitivity levels. The EER is the +point at which the FRR and FAR are equal. To calculate the +EER, we used 100 audio samples, 40 of which were synthe- +sised. We enrolled only synthesised speakers in the system +and calculated the EER. We achieved an EER of 0.10% by +performing voice conversion using a multi-speaker Tacotron +model [13]. We also generated emotional audio samples us- +ing a base model, and the speaker verification model gave an +EER of 0.24% on these synthesised audios. In contrast, we +achieved an EER of 0.16% when using the proposed model +for both emotion and voice conversion. The EER on real +samples using the approach in [13] was 0.04%. We have +compared the EER of these models in Table 2. +5.1.2. Listening Experiments +We performed mean opinion score (MOS) evaluations +to measure the quality of synthesised speech. We asked sub- +jects with post-graduate exposure to give a score after listen- +ing to the audio based on the following standard: 1 = Bad; +2 = Poor; 3 = Fair; 4 = Good; and 5 = Excellent. The re- +sults, shown in Table 3, indicate that the proposed model +can synthesise high-quality emotional speech compared to +the baseline model. The proposed model significantly im- +proves the MOS score for emotions including angry, sad, +and happy compared to the baseline. However, it achieves +slightly lower MOS scores for natural speech compared to +the baseline. This may be because the baseline model is +specifically designed to generate natural speech and there- +fore performs better for neutral speech. Nevertheless, our +proposed model performs well for all emotions. Readers can +listen to samples of the generated speech at this URL1. +1https://emotaco.github.io/Emotional_Tacotron/ +A Shahid et al.: Preprint submitted to Elsevier +Page 5 of 9 + +Multi-Speaker Emotional Speech Synthesis +Figure 2: Comparison of target and synthesized Mel-spectrograms for various emotions in Male and Female audios. +5.1.3. Speaker and Emotion Visualisation +During this experiment, we did not use teacher forcing +and generated audio as described in the inference part. The +synthesised Mel-spectrograms for different emotions by the +baseline and proposed models were plotted in Figure 2, and +the results were compared with the target Mel-spectrograms. +In contrast to the baseline, our proposed model did not smooth +the generated Mel-spectrograms that help produce a better +quality of emotional speech using WaveRNN vocoder. +For the purpose of evaluation, we present the t-SNE plot, +which was generated by embedding vectors generated from +synthesised output samples using a speaker verification model +as the encoder. Note that the speaker encoder was not trained +with the synthesizer, so it is not optimised for synthesizer +output. We generated t-SNE plots for emotional audio syn- +thesised using the model from the base papers and compared +the results with the proposed model. These t-SNE plots for +synthesised speech in both male and female voices are shown +in Figure 3 and 4, respectively. These plots demonstrate that +our model is able to synthesise distinct emotions compared +to the base model. It can be observed that different emotions +are separated and similar emotions are clustered together, in- +dicating similarity between emotions. +Since the angry emotion has more expression compared +to the sad and happy emotions, which are tone variations, +the cluster of angry emotions is farther from the happy emo- +tions. We also visualise the t-SNE plot of multiple speakers +in neutral speech using our proposed model in Figure 5. It +shows distinct clusters for different speakers indicating that +the model is able to learn the multiple speaker embeddings +effectively. +. +5.2. Augmenting Speech Emotion Recognition +(SER) +In this section, we used the synthetic speech to augment +the SER system. We performed our evaluations using corpus +Figure 3: Comparison of t-SNE plots of male audio for various +emotions using baseline and our proposed model shows that +our model demonstrates better emotion performance. +and cross-corpus settings. Results for these experiments are +presented next. +5.2.1. Within Corpus Evaluations +We used the RAVDESS and TESS datasets for evalua- +tions. We combined both datasets and then randomly split +the data into a ratio of 70:10:20 for train, validation, and +test sets, respectively. We trained the model for 45 epochs. +We compared the results for speaker recognition on real and +synthesised speech in Figure 6. We achieved an accuracy +of 80% for synthesised speech, while the accuracy for the +real speech test set was 92.4%. This demonstrates that our +model can synthesise the emotional characteristics of out- +put speech. We also augmented the classifier with synthetic +data and performed training using both real and synthesised +speech data. We achieved an accuracy of 94.6%, which is +better compared to the classifier trained on real data alone. +This experiment shows that our model can also be used to +A Shahid et al.: Preprint submitted to Elsevier +Page 6 of 9 + +Female-happy +Female-angry +Female-sad + Male-happy +Male-angry +Male-sad +Target +20 +20 +20 +20 +20 +导 +40 +40 +40 +60 +50 +60 +25 +50 +75 +100125 +25 +50 +75 +100 +125 +50 +100 +25 +50 +75 +100 +125 +25 +50 +75 +100 +125 +50 +100 +Proposed +20 +20 +20 +20 +20 +20 +导 +40 +40 +40 +60 +60 +60 +100125 +100 +50 +100 +25 +50 +100 +125 +25 +50 +75 +100 +125 +50 +100 +25 +50 +75 +25 +Baseline +20 +20 +2 +2 +40 +40 +40 +40 +40 +60 +50 +60 +60 +25 +50 +100 +125 +05 +50 +100 +125 + 50 +100 +0 +25 +75 +100125 +0 +25 +50 +75 +100 +125 +0 +50 +100Proposed +Baseline +30 +30 +20 +20 +10 +10 +0 +0 +10 +-10 +20 +20 +30 +30 +20 +20 +20 +D +20Multi-Speaker Emotional Speech Synthesis +Figure 4: Comparison of t-SNE plots of female audio for var- +ious emotions using baseline and our proposed model shows +that our model demonstrates better emotion performance. +Figure 5: The t-SNE plot for speaker voice of synthesised re- +sults shows that individual speakers’ voices are distinctly clus- +tered together. +generate additional audio data which can be used to augment +speaker recognition systems to improve their performance. +Figure 6: Bar plot which shows that our synthesized audio’s +emotion and real audio emotions are almost similarly classified +by the classification model. +We have also plotted confusion matrices in Figure 7 for +emotion classification on real audio, synthetic audio, and a +combination of real and synthetic data in the training set. +The confusion matrix shows that the model augmented with +synthetic data is able to better classify speech emotions. The +accuracy of other emotions has also been improved, but the +most significant improvement can be seen in the classifica- +tion of happy emotions. +5.2.2. Cross-Corpus Corpus Evaluations +We also evaluated the effect of augmenting with syn- +thetic data by performing cross-corpus emotion classifica- +tion. To do this, we implemented a classifier consisting of an +LSTM layer, three dense layers, and a softmax layer for emo- +tion classification. We also used two dropout layers between +dense layers to learn more generalised representations. We +selected the architecture of the model based on previous re- +search findings [22, 23]. We trained the classifier on MFCC +features extracted from the input audio. The model was trained +with a sparse categorical cross-entropy loss and Adam op- +timiser for 100 epochs. The model was trained using the +CREMA-D dataset and the CREMA-D dataset augmented +with synthetic data and was evaluated on the CREMA-D, +SAVEE, and EMODB datasets. The results, shown in Figure +8, demonstrate that adding synthesised data increases accu- +racy not only on the SAVEE and EMODB datasets without +fine-tuning the model but also on the CREMA-D test set as +well. +5.2.3. Changing Gender and Speaker Distributions +In this experiment, we compare the results of data aug- +mentation with new speaker voices that are not present in +the given corpus. For instance, the SAVEE corpus has four +male speakers, and synthetic data can be created either in the +voices of these four male speakers or in the voices of addi- +tional male and female speakers to bring diversity to the data +and augment speech emotion classification. We present the +results in Table 4. We compared the results with the baseline +model, which was trained without any augmentation, and +also with the application of speed perturbation to the train- +ing data. We followed [19] and created two copies of aug- +mented samples using the speed perturbation data augmenta- +tion technique. We found that augmenting the data with dif- +ferent speaker voices helps improve performance compared +to the baseline and the widely used data augmentation tech- +nique of speed perturbation. +6. Conclusions +This paper proposes to utilise an emotional text-to-speech +(TTS) system to augment a speech emotion recognition (SER) +system. We present a Tacotron-based multi-speaker emo- +tional TTS system for synthetic speech generation in dif- +ferent speaker voices and use it for data augmentation in +speech emotion recognition to improve performance. The +results showed that the proposed TTS system can generate +high-quality emotionally discriminative samples. When we +augment the SER system with these augmented samples, we +find that using synthetic data in different emotional voices +A Shahid et al.: Preprint submitted to Elsevier +Page 7 of 9 + +Proposed +Baseline +30 +30 +20 +20 +10 +10 +D +10 +-10 +20 +-20 +-30 +-30 +20 +20 +0100 +50 +0 +50 +100 +100 +50 +50 +10080 +ccuracy +60 +40 +20 +0 +Real +Synthetic +Real + +Synthetic +augmentationMulti-Speaker Emotional Speech Synthesis +Figure 7: Confusion matrix for the test set of real, synthetic, and combined synthetic and real audio. The addition of synthetic +data improves emotion classification. +Table 4 +Results using different distributions of synthetic data for speakers and gender +Dataset +Accuracy (%) +Baseline +Speed perturbation +augmentation +Male spakers +synthetic data +Female speakers +synthetic data +Both female and +male synthetic data +SAVEE +65.4 +66.8 +68.2 +69.4 +72.3 +CREMA-D +68.3 +70.1 +72.7 +72.9 +74.3 +Figure 8: Test results in cross-corpus setting, which shows +improvements when the model is augmented with synthetic +data. +can help improve performance compared to the widely used +speech data augmentation technique in SER. 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Statistical parametric speech +synthesis. speech communication 51, 1039–1064. +[45] Zhang, H., Cisse, M., Dauphin, Y.N., Lopez-Paz, D., 2018. mixup: +Beyond empirical risk minimization, in: International Conference on +Learning Representations. +A Shahid et al.: Preprint submitted to Elsevier +Page 9 of 9 + diff --git a/5tE2T4oBgHgl3EQfOwZB/content/tmp_files/load_file.txt b/5tE2T4oBgHgl3EQfOwZB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6023da5c6e24310abeda51fd57ede662456384d6 --- /dev/null +++ b/5tE2T4oBgHgl3EQfOwZB/content/tmp_files/load_file.txt @@ -0,0 +1,951 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf,len=950 +page_content='Generative Emotional AI for Speech Emotion Recognition: The Case for Synthetic Emotional Speech Augmentation Abdullah Shahida,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Siddique Latifb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='∗ and Junaid Qadirc aInformation Technology University (ITU),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='Punjab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Pakistan bUniversity of Southern Queensland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Australia cQatar University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Doha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Qatar A R T I C L E I N F O Keywords: Tacotron,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' WaveRNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' speech synthesis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' text-to-speech,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' emotional speech syn- thesis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' speech emotion recognition A B S T R A C T Despite advances in deep learning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' current state-of-the-art speech emotion recognition (SER) systems still have poor performance due to a lack of speech emotion datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' This paper proposes augmenting SER systems with synthetic emotional speech generated by an end-to-end text-to-speech (TTS) system based on an extended Tacotron architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The proposed TTS system includes encoders for speaker and emotion embeddings, a sequence-to-sequence text generator for creating Mel-spectrograms, and a WaveRNN to generate audio from the Mel-spectrograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Extensive experiments show that the quality of the generated emotional speech can significantly improve SER performance on multiple datasets, as demonstrated by a higher mean opinion score (MOS) compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The generated samples were also effective at augmenting SER performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Introduction Speech emotion recognition (SER) is a rapidly growing field with many applications in fields such as healthcare, cus- tomer service, media, education, and forensics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' While deep learning (DL) has shown promise in developing SER sys- tems, their performance is still limited by the scarcity of emotion datasets [20, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Existing SER corpora are small since the process of creating emotional data is costly and time-consuming, as multiple annotators have to manually listen to and annotate the material [26, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' To increase data size, some studies have used multiple corpora, but the num- ber of standard benchmark datasets is also limited, hindering progress in SER systems [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Researchers have long been interested in creating natural- sounding TTS systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' TTS technology has come a long way from early TTS systems that often used pre-recorded waveforms pieced together based on input text [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Such systems were prone to boundary artefact issues and statis- tical techniques were later developed to generate smoothed audio features for the vocoder to synthesise speech [37, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' More recently, end-to-end neural network-based approaches have been proposed that can synthesise more natural-sounding human speech [3, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Current state-of-the-art TTS systems are trained using DL algorithms in an end-to-end fashion, with popular models including Tacotron [41], Deepvoice [3], Fastspeech [33, 34], Fastpitch [18], to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Unlike traditional systems, end-to-end TTS models can learn to generate a spectrogram directly from text without any complex pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' These models, however, are currently only able to synthesise natural speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Using gen- erative DL techniques such as generative adversarial net- works (GANs) [8] for emotional speech synthesis is also challenging, as it requires a large amount of time-aligned data of a single speaker speaking the same content in dif- ∗Corresponding author ORCID(s): ferent emotions and complex equations to guide the model in converting emotions using audio features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Some studies have achieved promising results in single-speaker emotional speech synthesis using TTS models [17], but the quality of synthetic speech in augmenting SER has not been evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' In this paper, we propose a method for augmenting SER systems using an emotional text-to-speech (TTS) system and make two main contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Firstly, we develop an end-to- end multi-speaker emotional TTS system that does not re- quire any alignment of audio files for emotion conversion or complex pre-processing of input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Inspired by the success of end-to-end TTS models, we adopt a similar ar- chitecture to Tacotron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We propose to use a condition en- coder to control the speakers’ voices and emotions in the output speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We generate speaker voice feature vectors using the encoder network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' These feature vectors are modu- lated with one of the encoded emotional feature representa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' These modulated feature vectors are used to condition the Tacotron to synthesise speech in different speaker voices and emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Subjective evaluation tasks show that our pro- posed model improves controllability and successfully syn- thesises emotional speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Secondly, we use the synthesised emotional speech to augment an SER system and conduct multiple experiments to evaluate the generated data quan- titatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Results show that the synthesised data can help improve SER performance in both within-corpus and cross- corpus settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The rest of the paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' In Section 2, we briefly introduce the related work to change different features of audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The model’s architecture, loss functions, and flow of our architecture are described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The details of the dataset and experimental condition in which we trained our model and hyper-parameters are provided in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We report our results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Finally, this paper is concluded in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' A Shahid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 1 of 9 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='03751v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='SD] 10 Jan 2023 Multi-Speaker Emotional Speech Synthesis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Previous Work In this section, we review the literature that has emerged around (1) the use of Tacotron for TTS, and for (2) emotional speech synthesis, and (3) the process of augmenting SER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Tacotron Based TTS Systems Many recent studies have focused on modifying the Tacotron model in order to better control the output of TTS systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' For instance, [13] presented a Tacotron-based model that synthesises multi-speaker speech by conditioning the Tacotron on the speaker’s voice embedding, which was generated from a speaker verification model [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' [42] introduced a Tacotron variant that can change the speaking style, by learning dif- ferent styles and saving them as vectors or tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' These tokens are obtained by clustering similar accents and repre- senting each cluster with an average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' During synthesis, the Tacotron is conditioned on one of these tokens to produce speech with a specific style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' [35] presented a multi-speaker Tacotron that can change accents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=', American, Indian, British).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Their model uses two encoder networks with the Tacotron and requires two audio samples (one for the ac- cent and one for the speaker’s voice) as input to generate the desired output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' [36] proposed a Tacotron model that is trained with encoded output audio from a variational autoen- coder as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' This not only improves the multi-speaker performance of Tacotron but also allows for control over the energy of the generated audio through the mean-variance property of the variational autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' [43] developed a Tacotron model that can learn more complex vocalisations by using the self-attention mechanism in Tacotron to learn complex dependencies related to pitch in different accents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' They claim that their model outperforms traditional end-to- end approaches for languages with more pitch-dependent ac- cents, such as Japanese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Our proposed model also gener- ates speech in a multi-speaker setting and includes additional control over the emotions in the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Tacotron Based Emotional TTS Systems Several previous works have attempted to generate emo- tional speech using TTS systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' For example, [38] devel- oped an emotion control method for a TTS system based on the GST-Tacotron network [35], and demonstrated its effec- tiveness in synthesising emotional speech in a single-speaker setting in Korean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' [27] also evaluated a Tacotron-based emo- tional speech synthesizer in Korean, and found improvements in the quality of the generated speech for a single speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Other studies, such as [15, 17], have also proposed meth- ods for controlling emotional speech synthesis, but these ap- proaches only synthesise emotional speech in a single speaker’s voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' In contrast, our proposed method achieves control over emotional speech synthesis for multi-speaker TTS and we also evaluate the quality of the synthesised data to aug- ment the SER system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Augmenting Techniques for SER Speed perturbation [16] is a popular data augmentation technique that has been widely studied in different contexts [2, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' It has been found to improve speech emotion recog- nition (SER) performance by creating copies of input data with different speed effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Mixup [45] is another data aug- mentation technique that generates augmented samples as a linear combination of original samples from the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Several studies have demonstrated the effectiveness of mixup in SER, including Latif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' [25], who used the technique to augment an SER system and achieve improved performance and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' A recent method called SpecAugment [30], originally proposed for automatic speech recognition, has also been applied to SER [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' In this study, the authors aug- mented the SER system with duplicate samples by a factor of two and found that SpecAugment improved model per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Other studies [2, 21, 23] have also achieved im- proved performance by using input perturbation-based data augmentation techniques to increase the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Further research is required to explore data-driven ap- proaches to increase the training data for SER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' In this paper, we propose to explore TTS based data augmentation method where we explored different variations in the training data by changing the speaker and gender voices in different emo- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Proposed Framework We propose to generate synthetic speech using a Tacotron- based emotional TTS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We use synthetic speech data to augment the speech emotion classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The details of both emotional TTS and classifier are presented next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Emotional Speech Synthesis Our model consists of an encoder which conditions Tacotron (as depicted in Figure 1) to alter the speaker’s voice and emo- tion in the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Tacotron generates a Mel-spectrogram from a given text and embedding vector, while a Wave-RNN- based vocoder is used to generate an audio signal from the Mel-spectrogram 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Condition Encoder We propose using a condition encoder to create an em- bedding that represents both speaker identity and emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' To do this, we use a speaker identification model presented in [40], which creates a fixed-dimensional embedding, known as a d-vector [10, 39], using a sequence of Mel-spectrograms computed from a speech signal of arbitrary length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We train this model using an end-to-end speaker verification loss that maximises the cosine similarity between utterances from the same speaker while minimising the cosine similarity between utterances from different speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We fine-tune this net- work on an emotional corpus to create an emotional embed- ding as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Thus, the condition encoder is optimised to maximise the cosine similarity between embeddings of the same speaker with different emotions and to minimise the similarity between different emotions and different speak- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' In this way, the model learns to generate a feature vector that contains both emotion and speaker identity information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The speaker’s voice audio and emotion audio are embedded A Shahid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 2 of 9 Multi-Speaker Emotional Speech Synthesis Figure 1: Architectural flow diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The reference speaker’s voice is first encoded and then modulated to desired emotion as described in the model schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The output is then passed to the Tacotron decoder with the text embedding to synthesise the Mel-spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' using the condition encoder and combined to generate a fi- nal embedding, which is used to condition the synthesizer to output speech with the selected emotion and speaker’s voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' For each unique emotion of every speaker in dataset, a centroid 푐푘 is calculated by taking the average of embedding for each unique emotion of every unique speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Loss for an embedding 푒푖 when the embedding and the centroid 푐푘 have the same speaker and emotion is calculated as: \ue238(푒푖, 푐푘) = −1 × 휎(cos(푒푖, 푐푘)) (1) When 푒푖 have different emotion or different speaker for cen- troid 푐푘 then loss is calculated as:- \ue238(푒푖, 푐푘) = 휎(cos(푒푖, 푐푘)) (2) \ue238퐺(푆) = ∑ 푖,푘 퐿(푒푖, 푐푘) (3) Equation 1 maximises the cosine similarity between em- beddings for the same speaker voice and same emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Equa- tion 2 represents the cosine similarity between embedding and centroid when they have different speaker voices or dif- ferent emotions or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Equation 3 represents the final loss over every embedding, which is calculated as the sum of the loss for every embedding with every centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The condition encoder consists of three LSTM layers with 768 cells each, and a final 256-length fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The input to the model is the Mel-spectrogram gener- ated from a speech utterance of a reference speaker’s audio sample, and its output is an embedding vector of size 256 which represents the speaker’s identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' After training the model, we use it to extract speaker and emotional informa- tion from a given audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' To separate the emotion from the speaker’s voice, we generate vectors that only contain emo- tional information by using the trained condition encoder to generate embedding vectors for both the neutral and emo- tional voices of the same speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The neutral embedding vector is then subtracted from the emotional ones using Equa- tion (4), resulting in a vector that only contains emotional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' This vector can be used at inference time to control the emotion of the synthesised audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 푒푚푏em = (푒푚푏en − 푒푚푏neu) (4) Where 푒푚푏en represents the embedding with emotion and voice information generated from the emotional voice of a speaker;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 푒푚푏neu is generated from neutral audio of the same speaker, and 푒푚푏em represents the embedding that only con- tains emotional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' During inference, reference au- dio embedding (voice in which we want our output sample to be synthesised) and emotional embedding are added to generate a final embedding vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 푒푚푏final = 푒푚푏ref + 푒푚푏em (5) Finally, the modulated embedding vector and text are fed to Tacotron, which generates the Mel-spectrograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' These Mel-spectrograms are converted to the time domain using a vocoder, resulting in an audio signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Synthesizer architecture The synthesizer is a variation of Tacotron [41], which is a sequence-to-sequence model that generates output one frame at a time based on the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' In addition, we condition this synthesizer on an embedding vector generated by the condition encoder, which contains information about the de- sired output emotion and the speaker’s voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The condition embedding is concatenated with the text embedding of the synthesizer and then passed through a decoder to synthesise A Shahid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 3 of 9 Audio Reference audio Emotional embedding Speaker Vocoder Encoder Concatenation Condition encoder Mel Spectrogram Text Concatenation Attention Decoder embedding Synthesizer Character SeguenceMulti-Speaker Emotional Speech Synthesis the output Mel-spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The synthesizer was trained on 80-channel Mel-spectrograms with a window size of 50 ms and a hop size of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='5 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The synthesizer encodes the input characters into a hidden representation using three convolu- tion layers, which learn longer-term context like an n-gram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The output of these convolution layers is passed to a single bi-directional LSTM layer with 256 units, which learns time dependencies from these n-gram-like features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The LSTM layer returns an encoded vector that fully represents the in- put text sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' This vector is concatenated with a vector of emotional and speaker embeddings from the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' It is worth noting that at this point, the encoder has been trained and its weights are not updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The combined text, speaker, and emotion embedding is passed to the decoder to generate a Mel-spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The decoder architecture in- cludes a location-sensitive attention mechanism that trans- forms the input embedding into a fixed-length vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The output frame from the previous step is passed through two fully connected layers and concatenated with the embedding vector to ensure that sequences are generated without any time artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' This vector is then passed through two LSTM layers, and a linear transformation is applied to generate the next frame of the Mel-spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The output from this LSTM is also projected down to a single scalar, which serves as a stop token and indicates when to stop generating further frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Once the Mel-spectrogram has been generated, it is passed through a 5-layer convolution network called the PostNet to improve overall reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Vocoder Traditionally, the Griffin-Lim algorithm [9] was used to generate time-domain audio from a spectrogram, but it was slow and the output speech lacked naturalness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' To address this, we use a vocoder based on the WaveRNN architecture [14], which is a faster and more powerful recurrent network for sequential modelling of high-fidelity audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' It employs residual convolutions and GRU layers to generate a time- domain audio signal frame by frame from a Mel-spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Emotion Classifier To evaluate the synthesised emotions, we trained a deep neural network (DNN) for SER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We implemented a convo- lutional neural network (CNN)-based classifier that consists of a convolutional layer, a batch normalisation layer, and a dense layer before the softmax layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Mel-frequency cepstral coefficients (MFCCs) are used as the input to the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The CNN layers learn high-level features from the input fea- tures, which are then transformed by the dense layer into a more discriminative space for better emotion classification after passing through the normalisation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Experimental Protocol This section describes the details of the dataset, input feature, and model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Datasets We used the Librispeech dataset [29] to train our TTS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' It consists of 1000 hours of speech data from various speakers, sampled at 16 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' For the emotion embeddings, we used the Emotional Voices Database (EVD) [1] and the Toronto Emotional Speech Set (TESS) [7], which contain six different speakers reading different sentences with different emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We conducted multiple experiments to evaluate the performance of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' For emotion classification experiments, we used the Ryerson Audio-Visual Database of Emotional Speech and Song (RAVDESS) [28] and TESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' For cross-corpus emotion classification, we used the CREMA- D [6], SAVEE [12], EmoDB [5], and synthesised audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The details of these datasets are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We used one speaker from Librispeech, as well as all the speakers from EVD and TESS with two samples that were not in- cluded in the training set, to determine the mean opinion score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' For emotion classification experiments, we use speaker- independent emotion classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We randomly select 70% of CREMA-D for training, 10% for validation, and 20% for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The full corpora including RAVDESS and EmoDB were used as the test set in the emotion classification experi- ments, and the SAVEE dataset was used as the test set in the cross-corpus emotion classification experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Input Features Tacotron takes text strings as input, which are sequences of characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Each character is encoded into a one-hot en- coded vector and embedded in a continuous vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The other input to Tacotron is a condition embedding vector that contains speaker and emotion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' This vector is obtained from an encoder, which takes speaker audio as in- put and converts it into Mel-frequency cepstral coefficients (MFCCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' These MFCCs have 40 log filter banks, 80 frames, and no overlapping window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' To generate t-distributed stochas- tic neighbour embedding (t-SNE) plots of synthesised audio, we encoded our synthesised audio using the model presented in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The input to this model is also MFCCs with 40 log filter banks, 80 frames, and no overlapping window, result- ing in an 80x40-dimensional feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' This model is also used in evaluating the equal error rate (EER) in speaker verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' In emotion classification and cross-corpus emo- tion classification, we use MFCCs with 40 log filter banks and a hop size of 64 milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The MFCC array is trans- posed and the arithmetic mean is calculated across its hori- zontal axis as in a previous work [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Speech Synthesis Models Training First, the encoder is trained on the Librispeech dataset to learn to generate a speaker embedding that is distinct for each speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' It takes a Mel-spectrogram as input and out- puts an embedding vector of size 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' From these embed- ding vectors, a similarity matrix is constructed such that each column contains an embedding vector for a unique speaker, and cosine similarity is maximised in all cells of the columns and minimised in all cells of the rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Cosine similarity is maximised along the columns because they contain audio embeddings for the same person, whereas it is minimised A Shahid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 4 of 9 Multi-Speaker Emotional Speech Synthesis Table 1 Description of all the considered datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Name Number of Speakers Number of Utterances CREMA-D 91 7,442 EmoDB 10 535 EVD 5 7,590 Librispeech 2484 281,241 REVDESS 24 7,356 SAVEE 4 480 TESS 2 2,800 along the rows because they contain audio embeddings for different people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' In this way, the embeddings of the same people are similar and those of different people are differ- ent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' After training the encoder on the Librispeech data, it is fine-tuned on the EVD and TESS datasets to generate dis- tinct embedding vectors for different emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' This time, a similarity matrix is constructed such that a column con- tains embedding vectors generated for a single emotion for the same speaker, and other emotions are placed in other columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' This is done for all speakers, and then cosine sim- ilarity is maximised along a column and minimised across columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' This is done to increase the distance between dif- ferent emotions of the same person, so cosine similarity is minimised by adding it across columns rather than within the same columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We used a batch size of 30 and a learn- ing rate of 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' During training, the synthesizer model is first trained on the Librispeech data so that it can learn to generate audio of different speakers from a diverse range of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' This is because the EVD and TESS datasets combined only have six speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Once the synthesizer is trained enough that it can generate audio resembling the reference speaker, we fine- tune it to generate different emotional Mel-spectrograms by training it on the EVD and TESS datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We use a learning rate of 103 that exponentially decays to 10−5, and a batch size of 30 for training the synthesizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The Adam optimiser with 훽1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='9, 훽2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='999, and 휖 = 10−6 is used as the optimiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The teacher forcing ratio is set to 1 (meaning the original previous sequence is shown to the model for prediction of the next sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The mean squared error is minimised for the predicted Mel-spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Results In this section, we evaluate the performance of our pro- posed model in terms of the similarity of the synthesized speakers and the granularity of synthesized emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Evaluating Synthetic Speech Quality To evaluate the quality of synthetic speech, we conducted multiple experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The details of these experiments are presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Table 2 Speaker verification EERs of different synthesizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' # of samples EER Emotion + voice conversion TTS 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='16 Baseline Emotion conversion TTS 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='24 Voice conversion TTS 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='10 Real audios 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='04 Table 3 Mean Opinion Score (MOS) with 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Emotion Angry Happy Sad Neutral Overall Recorded 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='55 Baseline 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='20 Proposed 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='70 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='80 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='80 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Speaker Verification We evaluated the speaker similarity of synthesised au- dios with real speech using speaker verification and mea- sured the equal error rate (EER) following [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The EER is used to measure the performance of a speaker verification system by comparing the false reject rate (FRR) and false ac- cept rate (FAR) at different sensitivity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The EER is the point at which the FRR and FAR are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' To calculate the EER, we used 100 audio samples, 40 of which were synthe- sised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We enrolled only synthesised speakers in the system and calculated the EER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We achieved an EER of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='10% by performing voice conversion using a multi-speaker Tacotron model [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We also generated emotional audio samples us- ing a base model, and the speaker verification model gave an EER of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='24% on these synthesised audios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' In contrast, we achieved an EER of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='16% when using the proposed model for both emotion and voice conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The EER on real samples using the approach in [13] was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='04%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We have compared the EER of these models in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Listening Experiments We performed mean opinion score (MOS) evaluations to measure the quality of synthesised speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We asked sub- jects with post-graduate exposure to give a score after listen- ing to the audio based on the following standard: 1 = Bad;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 2 = Poor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 3 = Fair;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 4 = Good;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' and 5 = Excellent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The re- sults, shown in Table 3, indicate that the proposed model can synthesise high-quality emotional speech compared to the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The proposed model significantly im- proves the MOS score for emotions including angry, sad, and happy compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' However, it achieves slightly lower MOS scores for natural speech compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' This may be because the baseline model is specifically designed to generate natural speech and there- fore performs better for neutral speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Nevertheless, our proposed model performs well for all emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Readers can listen to samples of the generated speech at this URL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 1https://emotaco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='io/Emotional_Tacotron/ A Shahid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 5 of 9 Multi-Speaker Emotional Speech Synthesis Figure 2: Comparison of target and synthesized Mel-spectrograms for various emotions in Male and Female audios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Speaker and Emotion Visualisation During this experiment, we did not use teacher forcing and generated audio as described in the inference part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The synthesised Mel-spectrograms for different emotions by the baseline and proposed models were plotted in Figure 2, and the results were compared with the target Mel-spectrograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' In contrast to the baseline, our proposed model did not smooth the generated Mel-spectrograms that help produce a better quality of emotional speech using WaveRNN vocoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' For the purpose of evaluation, we present the t-SNE plot, which was generated by embedding vectors generated from synthesised output samples using a speaker verification model as the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Note that the speaker encoder was not trained with the synthesizer, so it is not optimised for synthesizer output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We generated t-SNE plots for emotional audio syn- thesised using the model from the base papers and compared the results with the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' These t-SNE plots for synthesised speech in both male and female voices are shown in Figure 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' These plots demonstrate that our model is able to synthesise distinct emotions compared to the base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' It can be observed that different emotions are separated and similar emotions are clustered together, in- dicating similarity between emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Since the angry emotion has more expression compared to the sad and happy emotions, which are tone variations, the cluster of angry emotions is farther from the happy emo- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We also visualise the t-SNE plot of multiple speakers in neutral speech using our proposed model in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' It shows distinct clusters for different speakers indicating that the model is able to learn the multiple speaker embeddings effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Augmenting Speech Emotion Recognition (SER) In this section, we used the synthetic speech to augment the SER system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We performed our evaluations using corpus Figure 3: Comparison of t-SNE plots of male audio for various emotions using baseline and our proposed model shows that our model demonstrates better emotion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' and cross-corpus settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Results for these experiments are presented next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Within Corpus Evaluations We used the RAVDESS and TESS datasets for evalua- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We combined both datasets and then randomly split the data into a ratio of 70:10:20 for train, validation, and test sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We trained the model for 45 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We compared the results for speaker recognition on real and synthesised speech in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We achieved an accuracy of 80% for synthesised speech, while the accuracy for the real speech test set was 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' This demonstrates that our model can synthesise the emotional characteristics of out- put speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We also augmented the classifier with synthetic data and performed training using both real and synthesised speech data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We achieved an accuracy of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='6%, which is better compared to the classifier trained on real data alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' This experiment shows that our model can also be used to A Shahid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' : Preprint submitted to Elsevier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='Page 6 of 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='Female-happy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='Female-angry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='Female-sad ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='Male-happy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='Male-angry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='Male-sad ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='20 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='20Multi-Speaker Emotional Speech Synthesis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='Figure 4: Comparison of t-SNE plots of female audio for var- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='ious emotions using baseline and our proposed model shows ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='that our model demonstrates better emotion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Figure 5: The t-SNE plot for speaker voice of synthesised re- sults shows that individual speakers’ voices are distinctly clus- tered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' generate additional audio data which can be used to augment speaker recognition systems to improve their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Figure 6: Bar plot which shows that our synthesized audio’s emotion and real audio emotions are almost similarly classified by the classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We have also plotted confusion matrices in Figure 7 for emotion classification on real audio, synthetic audio, and a combination of real and synthetic data in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The confusion matrix shows that the model augmented with synthetic data is able to better classify speech emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The accuracy of other emotions has also been improved, but the most significant improvement can be seen in the classifica- tion of happy emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Cross-Corpus Corpus Evaluations We also evaluated the effect of augmenting with syn- thetic data by performing cross-corpus emotion classifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' To do this, we implemented a classifier consisting of an LSTM layer, three dense layers, and a softmax layer for emo- tion classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We also used two dropout layers between dense layers to learn more generalised representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We selected the architecture of the model based on previous re- search findings [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We trained the classifier on MFCC features extracted from the input audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The model was trained with a sparse categorical cross-entropy loss and Adam op- timiser for 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The model was trained using the CREMA-D dataset and the CREMA-D dataset augmented with synthetic data and was evaluated on the CREMA-D, SAVEE, and EMODB datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The results, shown in Figure 8, demonstrate that adding synthesised data increases accu- racy not only on the SAVEE and EMODB datasets without fine-tuning the model but also on the CREMA-D test set as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Changing Gender and Speaker Distributions In this experiment, we compare the results of data aug- mentation with new speaker voices that are not present in the given corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' For instance, the SAVEE corpus has four male speakers, and synthetic data can be created either in the voices of these four male speakers or in the voices of addi- tional male and female speakers to bring diversity to the data and augment speech emotion classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We present the results in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We compared the results with the baseline model, which was trained without any augmentation, and also with the application of speed perturbation to the train- ing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We followed [19] and created two copies of aug- mented samples using the speed perturbation data augmenta- tion technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We found that augmenting the data with dif- ferent speaker voices helps improve performance compared to the baseline and the widely used data augmentation tech- nique of speed perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Conclusions This paper proposes to utilise an emotional text-to-speech (TTS) system to augment a speech emotion recognition (SER) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' We present a Tacotron-based multi-speaker emo- tional TTS system for synthetic speech generation in dif- ferent speaker voices and use it for data augmentation in speech emotion recognition to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The results showed that the proposed TTS system can generate high-quality emotionally discriminative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' When we augment the SER system with these augmented samples, we find that using synthetic data in different emotional voices A Shahid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 7 of 9 Proposed Baseline 30 30 20 20 10 10 D 10 10 20 20 30 30 20 20 0100 50 0 50 100 100 50 50 10080 ccuracy 60 40 20 0 Real Synthetic Real + Synthetic augmentationMulti-Speaker Emotional Speech Synthesis Figure 7: Confusion matrix for the test set of real, synthetic, and combined synthetic and real audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' The addition of synthetic data improves emotion classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Table 4 Results using different distributions of synthetic data for speakers and gender Dataset Accuracy (%) Baseline Speed perturbation augmentation Male spakers synthetic data Female speakers synthetic data Both female and male synthetic data SAVEE 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='8 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='2 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='3 CREMA-D 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='7 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='9 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='3 Figure 8: Test results in cross-corpus setting, which shows improvements when the model is augmented with synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' can help improve performance compared to the widely used speech data augmentation technique in SER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' Our future work will focus on investigating the learning of a unified embed- ding for controlling style and emotions for all people, regard- less of age, background, and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=' References [1] Adigwe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=', Tits, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content=', Haddad, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE2T4oBgHgl3EQfOwZB/content/2301.03751v1.pdf'} +page_content='E.' metadata={'source': 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0000000000000000000000000000000000000000..02a37a45134f3cb0433aea180a850c7edad8bbec --- /dev/null +++ b/6dE2T4oBgHgl3EQfPAZa/content/tmp_files/2301.03754v1.pdf.txt @@ -0,0 +1,1095 @@ +1 + +Inert gas as electronic impurity in semiconductors: The case for +active infrared absorption in silicon +Nian-Ke Chen1,#, Yu-Chen Gao1,#, Ji-Hong Zhao1,*, Chun-Hao Li1, Qi-Dai Chen1, +Hong-Bo Sun2,*, Shengbai Zhang3,*, and Xian-Bin Li1,* + +1State Key Laboratory of Integrated Optoelectronics, College of Electronic Science and +Engineering, Jilin University, Changchun 130012, China +2State Key Lab of Precision Measurement Technology and Instruments, Department of +Precision Instrument, Tsinghua University, Beijing 100084, China +3Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic +Institute, Troy, New York 12180, USA +Corresponding +authors: +lixianbin@jlu.edu.cn, +or +zhaojihong@jlu.edu.cn, +hbsun@tsinghua.edu.cn, or zhangs9@rpi.edu + +Abstract +Inert (noble gas) elements are extremely inactive to surrounding chemical environment +and are frequently employed as protective gas in various semiconductor fabrication +processes. In this work, we surprisingly discover that high doses of argon up to 1017- +1020 cm-3 can be measured in silicon exposed by laser pulses even after 1300 days. First- +principles calculations and molecular dynamics identify a unique argon-locking- +vacancy (ALV) defect atomic model in silicon. The ALV defect is dynamically robust +in contrast to the frequently moving pure Si vacancy. While argon is chemically inert, +it readily modulates defect states of the occupied vacancy via steric repulsion and +rattling motions, leading to significant band splitting within bandgap and thus strong +infrared absorptions. Moreover, the repulsion between substitutional argon and +dangling bonds results in shallow donors which explains the confusion of enhanced n- +type carriers in experiments. The work paves a way of using noble gas element to +produce active infrared absorption source for the non-heteroepitaxy photonic detectors +directly on silicon wafer at infrared communication wavelength. + +2 + +Silicon (Si) based optoelectronic devices is at the heart of optoelectronic industry owing +to their ability for Si integration. Among them, photodetectors working at infrared (IR) +communication wavelength (λ) of 1.31/1.55 μm are indispensable. However, due to the +well-known problem of low absorption at λ ≥ 1.1 μm, corresponding to the Si bandgap, +Si is powerless in communication applications. Often, a different semiconductor with a +suitable bandgap is heterogeneously grown on Si. However, issues with heteroepitaxy +such as lattice mismatch can reduce or even degrade the performance of the detectors +[1]. Another way is to introduce IR absorption sources in Si. For example, gap states +can be created inside Si by chalcogenide dopants with the help of ultrafast laser pulses +to result in doped black silicon [2-6]. It has a strong IR absorption at λ = 1.31/1.55 μm. +However, such IR sources are often not stable enough for applications. For example, +the IR absorption at 1.31/1.55 μm in black silicon can be significantly reduced by +annealing at 775 K for half an hour [4]. + +In 2018, Zhao et al. reported a form of black Si, which was fabricated by nanosecond +laser pulses without any intentional element dopant except for a protective Ar gas [7,8]. +It was quite unexpected that the photodiode fabricated based on such a black Si has a +high and stable photoresponsivity of 260 mA/W at 5 V at λ = 1.31 μm [8], which paves +the way for practical sensing by a Si detector at the IR communication wavelength. +Argon is a noble gas widely used as a protective gas in the electronic industry. As a +matter of fact, the name of argon is derived from a Greek word that means lazy or +inactive. Due to its fully occupied valence band electronic shell with eight electrons, +there is little chemical reaction between argon and other elements. As such, it is also +expected that the Ar gas has no effect on the property of semiconductors. + +In this work, we report the observation of very high concentration of Ar (1017-1020 cm- +3) in ultrafast laser-modified Si using the secondary-ion mass spectrometry (SIMS) +measurement. First-principles calculations and molecular dynamic studies reveal the +unique atomic and electronic properties of the Ar-doped Si to result in an unexpected +and strong IR absorptions. While the pure Si vacancy can produce dangling-bond state + +3 + +within bandgap, it is movable and unstable. In contrast, Ar atom can lock the Si vacancy +to form a dynamically stable defect complex even up to 900 K. Thanks to its full +electronic shell, Ar protects the dangling electrons of vacancy and retains its gap states. +Moreover, rattling motion and Coulomb repulsion of Ar atoms can lead to an enhanced +structural distortion and the further splitting of defect energy levels within the bandgap. +Unexpectedly, the repulsion between substitutional Ar and dangling electrons makes +the defect a shallow donor, which explains the confusion of laser irradiation induced n- +type doping effect. As a result, the inert Ar atom in fact acts as an electronic impurity +and offers active and robust sub-bandgap IR absorption source for Si photodetector. +This solves the long-term difficulty of high photoresponsivity of Si based detectors at +IR communication wavelength. The physics behind shed new light on a general strategy +of employing inert elements to raise performances of semiconductor devices. + +The concentrations of Ar atoms are measured by dynamic secondary ion mass +spectrometer (D-SIMS) in laser modified Si samples in argon protecting atmosphere. +The fabrication of such samples were reported in our previous work [8]. The D-SIMS +instrument is equipped with a Cameca IMS-4F device using 8 keV Cs+ primary beam. +Density-functional theory (DFT) calculations are performed using the VASP code +[9,10], where the projector-augmented wave (PAW) pseudo potential and generalized- +gradient approximation (GGA) exchange-correlation functional developed by Perdew, +Burke and Ernzerh are adopted [11-13]. A Si supercell that contains 216 atoms are used +to describe defect effects. The energy cutoff for plane-wave expansion is 380 eV. The +3×3×3 Monkhorst-Pack grids are used as Brillouin-zone sampling for static energy and +property calculations, while the Γ point is used for structural relaxation and molecular +dynamic (MD) simulations. The band structures of supercells are unfolded by the +modified VaspBandUnfolding package [14]. Energy barriers are calculated using the +climbing image nudged elastic band (c-NEB) method [15,16]. The structures and +charge density are visualized by the VESTA code [17]. The positions of vacancy defect +are determined by the Wigner-Seitz method in the OVITO code [18]. + + +4 + +To figure out the actual role of Ar atoms in laser modified Si, we carefully analyze the +dose of Ar by SIMS measurements in this work. Figure 1(a) displays Ar concentration +for such a typical Si sample, which was fabricated previously by nanosecond laser in +Ar atmosphere [8]. It is unexpected that even the sample has been made over 1300 days, +a very high dose of Ar atoms can be detected as from ~ 4×1021 cm-3 at the surface to ~ +4×1017 cm-3 at 2 µm below the surface. We reevaluate the specific detectivity (D*) of +the photodetector based on the Si sample [8] and compare it to those of non-silicon +photodetectors at the IR wavelength of λ = 1.31 μm [19]. The 1.31-µm wavelength, +whose photonic energy is below the bandgap of Si, is out of the detecting scope of +detectors based on intrinsic Si. A higher D* reflects a higher signal-to-noise ratio of +detectors. For example, Fig. 1(b) shows that the D* of the laser modified Si in Ar +atmosphere here working at 295 K (1011 cmHz1/2W-1) is not only higher than those of +PbSe and InAs working at the same temperature but also higher than or close to those +of InAs and InSb working at a much lower temperature of 193 K or 77 K. All of these +indicate the inclusion of Ar in Si could potentially offer a highly effective IR absorption +source for detectors. + +To uncover the microscopic picture and critical role of Ar in Si samples, we carry out +first-principles calculations. In fact, the formation energies (ΔHf) of Ar defects in +crystalline Si are very large. Table S1 in Supplemental Material summarizes the +calculated ΔHf of several defects in Si, which agrees with previous reports [20,21]. +The calculated ΔHf of interstitial and substitutional Ar defects are as large as 6.05-7.19 +eV while the ΔHf of a silicon vacancy (VSi) is about 3.67 eV. Despite of the high ΔHf, +the Ar-related defects can still be formed under laser irradiations. A key reason is that +the ΔHf can be substantially reduced when Si is melted by laser irradiations, see Fig. +1(c). Also, compared with interstitial Ar defects, substitutional Ar defects should be +dominate because an interstitial Ar and a VSi will be annihilated into a substitutional Ar +once they encounter during the annealing process. This annihilation is obviously energy +favorable. Moreover, the ΔHf of substitutional Ar defects can further be lowered by +their accumulation [see Fig. 1(d)] because such an accumulation reduces the number of + +5 + +dangling bonds. + + +FIG. 1. (a) Concentration of Ar in the nanosecond-laser modified silicon sample +measured by the secondary ion mass spectroscopy. (b) Comparison of specific +detectivity (D*) between the black silicon detector [8] and other reported infrared +detectors at 1.31 µm [19]. (c) Formation energies of a substitutional Ar defect (ArSi) at +various temperatures. The energy of a high-temperature state is calculated by the +average free energy of the last 10 ps frames of a 20-ps NVT MD simulation. (d) +Formation energies of the accumulated substitutional Ar defects in Si. The formation +energies of the multi-substitutional defects are averaged by the number of Ar atoms. + +Figure 2 elucidates atomic and electronic structures after an Ar atom is introduced into +Si. In order to study Ar induced defect states in Si, a large supercell with 216 atoms is +employed. Due to Brillouin Zone folding in the supercell, band structures are calculated +and further reanalyzed by the effective band unfolding method [22]. The unfolded band +structure in Fig. 2(d) reproduces the correct E-k dispersion for the intrinsic Si despite +of underestimating the bandgap by the GGA-DFT method. + +Naturally, an Ar atom filling in a Si lattice vacancy (VSi) could be an ideal candidate +for the IR absorption. Because an intrinsic vacancy has the ability of offering defect + +1E22 +(a) +3 +Concentration (Atoms/cm +1E13 +(b) +1.31 μum +InGaAs(PV,295K) +PbS(PC,295K) +InAs(PV,193K) +PbS(PC,77K) +-Black Si in this work O (PV, 295K) +nslaserpulse +1E21 +1E12 +InSb(PV,77K) +InSb(PC,77K) +InAs(PC,77K) +lens +Si substrate +PbS(PC,193K) +1E11 +1E20 +■PbSe(PC,295K) +■ InAs(PV,295K) +1E10 +1E19 +1E9 +1E18 +Ar ( +Intrinsic absorption +1E8 +impurityabsorption +0.0 +0.5 +1.0 +1.5 +2.0 +Detectors +Depth (μm) +(c) +6 +(d) +6 +H (eV per Ar) + (eV per Ar) +4 +melting states +5 +H. +0 +4 +2Ar,si +3Ar3si +4Ar4si +OK +2000 K 2500K 3000 K +Temperature +Substutional Ar in Si6 + +states within the bandgap. Fig. 2(a) displays the local tetrahedral motif in intrinsic Si. +When a Si atom is removed to form a VSi, four dangling bonds are produced, see Fig. +2(b). Here, the four atoms around the vacancy are noted as atoms 1, 2, 3, 4 (Si1, Si2, Si3, +Si4), respectively. Li,j is the distance between any two of them. If no any atomic +relaxation, the vacancy with all the same Li,j = 3.87 Å holds an ideal Td local symmetry. +However, due to the well-known Jahn-Teller distortion effect, the four atoms are +relaxed closer to the center of the vacancy with L1,2 (3.03 Å) ≈ L3,4 (3.12 Å) and other +Li,j = 3.54 Å. The symmetry is thus lowered to a near D2d symmetry (~D2d) shown in +Fig. 2(b), which is consistent with a previous result [23]. Accordingly, the energy levels +of defect states can split into two parts: two occupied levels in the lower part within the +bandgap while two unoccupied levels in the upper part within the bandgap, see Fig. 2(e) +of the band structure and its schematic drawing [Fig. 2(h)] for Si with a ~D2d vacancy. + +After Ar atoms are introduced into Si which is confirmed by our SIMS measurement, +the local structures of VSi will be changed. For example, as shown in Fig. 2(c), the +atomic distortion of VSi can be further enhanced to hold a C2v symmetry due to a Ar +atom substitutionally filled in VSi (ArSi). In this case, the Ar atom is closer to Si3 and +Si4, which makes Si3 and Si4 move away from each other. Meanwhile, Si1 and Si2 move +closer to each other. Therefore, L1,2 and L3,4 are changed to 3.46 Å and 5.40 Å, +respectively. The ~120°bond angles of Si3 and Si4 indicate the bonding type is changed +to be sp2-hybridization like while the sp3-hybridization like bonding still remains for +Si1 and Si2. Our calculations found that the filling of Ar into VSi costs about 2.56-eV +extra energy. Such large an interaction energy and the large structural distortion suggest +that Ar should have a significant repulsive interaction with surrounding dangling bonds +of Si atoms, i.e., the steric repulsion effect. The steric-repulsive induced distortion +(SRD) also leads to a larger bonding hierarchy and changes the defect states of VSi. +Accordingly, comparing to the case of VSi, defect levels of ArSi are further splitted, see +Figs. 2(f) and (i): the energies of highest occupied defect orbital (HODO), lowest +unoccupied defect orbital (LUDO) and LUDO+1 is raised and closer to conduction +band while the energy of HODO-1 lowers into the valence band. + +7 + + +FIG. 2. Local atomic structures of (a) Si, (b) VSi and (c) ArSi with the steric-repulsive +distortion (SRD) holding a C2v symmetry . The arrows indicate the directions of atomic +relaxation referred to their original positions. (d)-(f) correspondingly show the unfolded +band structures of the three supercell models. The size of the scatters represents the +atomic weight. The weights of Si1,2,3,4 (green and red scatters) are amplified by a factor +of 10. The green and red scatters indicate the contributions from Si1,2 and Si3,4, +respectively. (g)-(i) show corresponding schematic band structures. Ef is set as 0 eV. + +In fact, the local atomic configuration of ArSi defect with SRD is not unique. It can also +have other structures with a C3v or ~Td symmetry, see Figs. S1(e) and (f) in +Supplemental Material. In both cases, the four dangling Si atoms (Si1,2,3,4) are repulsed +by the Ar and thus tend to have a sp2-like bonding characteristic due to the bond angle +of ~120°. In the two cases, not only the defect levels but also the spin states are splitted, +see Figs. S2 and S3 in Supplemental Material. Three of the four electrons from dangling +bonds occupy three spin-up levels while one occupies a spin-down level. + +(a) +Si +(b) +V. +(c) +Ar..SRD-C +Si +Energy [eV] +Energy [eV] +Energy [eV] +Q +1 +-1 +2 +2 +2 +-3 +3 +xu +K +W +X +X U +K +L +W +X +xU +K +L +L +W +X +(g) +(h) +(0) +Conduction Band +ConductionBand +Conduction Band +ValenceBand +Valence Band +Valence Band8 + + +In fact, the three configurations of ArSi with C2v, C3v and ~Td SRDs almost have the +same formation energy with little differnece of 0.01-0.03 eV (see Table S1 in +Supplemental Material). The energy barrier of the tansition between C2v and C3v (C3v +and ~Td) configurations calculated by the NEB method is also as small as 0.015 (0.013) +eV, indicating readily transitions to each other. Defect levels always preserve inside the +bandgap after inert Ar doping in VSi, despite of these different Ar configurations with +close eneriges. Moreover, the SRD effect still stands in the accumulated substitutional +Ar defects (3Ar3Si and 4Ar4Si) and thus the splitted defect states also exist (see Fig. S4 +in Supplemental Material). In addition, a bond-centered (B-C.) site interstitial Ar defect +also has two dangling bonds while the tetrahedral/hexagonal-site interstitial Ar defects +have no dangling bonds, see Figs. S1(a)-(c) in Supplemental Material. As such, the B- +C. interstitial Ar also has two defect levels inside the bandgap, see Fig. S5 in +Supplemental Material. All these defect states no doubt will benefit a robust sub- +bandgap IR absorption in Si. + +Next, electronic bonding mechanisms of ArSi are analyzed to figure out the mechanism +of the Ar doping induced change of defect states. Taking the C2v configuration as an +example, the charge density difference (CDD) analyses [24] in Figs. 3(a)-(c) show no +transferred or shared electrons between Ar and its surrounding Si due to the chemical +inertia of Ar. Therefor, the interaction between Ar and the dangling Si atoms should be +Coulomb repulsion effect which agrees with the SRD of the local structure. In fact, +although Ar atom does not offer any defect levels but indirectly modulate defect states +of a Si vacancy via the repulsion. Figures 3(d)-(h) elucidate the spatial distributions of +defect states. The updated sp2-like configuration of Si3,4 [see the bond angles of 121° in +Fig. 3(a)] may release a dangling pz orbital from the orginal sp3 orbital. Obviously, the +LUDO+1 and the HODO are the unoccupied and occupied pz-like orbitals of Si3,4, +respectively, see Figs. 3(e) and (g). Figure 3(f) shows the LUDO corresponding to the +unoccupied sp3-like orbital of Si1,2. Since Si1 and Si2 has also a certain interaction due +to a charge accumulation between them shown in Fig. 3(b), their occupied sp3-like state + +9 + +further moves below the valence band maximum. We indeed find the state by projecting +the orbital-decomposed partial charge density inside valence band [see Fig. 3(h)]. Such +a defect state will move back into bandgap in the case of C3v and ~Td SRDs due to the +absence of such interaction between dangling Si atoms (see Figs. S2 and S3 in +Supplemental Material). + + +FIG. 3. (a) The charge density difference (CDD) of ArSi with the C2v SRD. The value +of isosurface is 0.015 e/a03, a0 is the Bohr radius. (b) and (c) show the respective (11̅0) +and (110) slices of the CDD. The unit of the color bar is e/a03. (d) Schematic defect +levels of ArSi with C2v SRD. (e)-(h) The corresponding orbital-decomposed partial +charge density projected to real space. The values of isosurface are 0.005 e/a03 for (e)- +(g) and 0.0015 e/a03 for (h). + +To directly demonstrate the influence of the Ar-related defect on the IR absorption at λ += 1.31/1.55 μm, the aobsorption coefficients are calculated [see Fig. 4(a)]. Here, the +meta-GGA method using the modified Becke-Johnson (MBJ) exchange potential +[25,26] is adopted to correct the bandgap underestimated by the traditional PBE +functional (see Fig. S6 in Supplemental Material for the effect of correction). Obviously, + +a +≥ 0.015 +0.006 +105° +0.000 +121 +121° +-0.006 +<-0.015 +e +(d) +Conduction +Band +Pz +(g) +(h) +sp +Valence +Band10 + +a substantial enhancement of the sub-bandgap IR absorption is achieved by ArSi in great +contrast to the case without defect (Si) that displays almost no absoprtion. Moreover, +the substitutioanl defects also have stronger abosorptions compared with the interstitial +Ar and VSi, owing to the repulsion induced splitting of defect levels. + + +FIG. 4. (a) Calculated absorption coefficient of Si, VSi, B-C. interstitial Ar and ArSi with +the C2v and C3v SRDs. (b)-(d) The ΔHf of neutral and charged states of the ArSi with +C2v SRD, the ArSi with C3v SRD and the B-C. interstitial Ar defect, respectively. + +In fact, the repulsion effect of Ar on dangling electrons not only changes the atomic +structure and the position of defect levels but also significantly affects the ionization +process. When the Ar is filled into the vacancy, the strong repulsive interaction between +Ar and dangling electrons should make the electrons be ionized easily. Figures 4(b)-(d) +show the calculated chemical potential dependent ΔHf of neutral and charged states of +the Ar defects [27,28]. According to the position of transition levels (where the ΔHf of +neutral and charged states equal), it is surprising that the substitutional Ar defects are +shallow donors but deep acceptors. In contrast, the B-C. interstitial Ar still has deep +donor and acceptor levels. The shallow donor effect of ArSi defects can explain the + +Absorption Coefficient (cm +6000 +(a) +by meta-GGA +Ar.. SRD-C +3V +4000 +Ar.. SRD-C +2000 +B-C. interstitial +Si +0 +1.2 +1.3 +1.4 +1.5 +1.6 +Wavelength (μm) +6.8 +(b) +(c) + Ar SRD-C +(d)B-C. interstitial +Ar SRD-C +IS +2v +1 +S +3v +-1 +6.4 +(eV) +0 +0 +△H. +0 +6.0 ++1 ++1 ++1 +0.0 +0.3 +0.6 0.0 +0.3 +0.6 0.0 +0.3 +0.6 +E.- E +(eV) +E.-E +(eV) +E.-E +(eV) +f +VBV +VBM +VBV11 + +confusion that the fabricated sample shows a substantially enhanced n-type carrier +concentration [8]. In other words, the substitutional Ar can act as an electronic impurity +despite of its inert chemical property. + + + +FIG. 5. The projected trajectories of the local structure of (a) ArSi and (b) VSi during 60- +ps MD simulations at 500 K. The colored dots represent positions of ArSi and Vsi while +the grey dots indicate the ones of Si atoms with time evolution. The energy barriers for +a diffusion of (c) ArSi and (d) VSi calculated by c-NEB method. (e) The MSD of the VSi +during the MD simulation. Inset in (e) shows the schematic trajectory of diffusion of a +pure VSi in the simulation. + +It worth noting that VSi has the lowest ΔHf (3.67 eV) among the defects in Si and it +can also offer a significant sub-bandgap IR absorption. As mentioned above, the +absorption of ArSi is in fact originated from defect states of dangling bonds around the +Ar filled vacancy. However, for practical device applications, IR absorption sources +should be reliable enough. Therefore, two ab initio MD simulations of 60 ps at 500 K, + +ps +(a) +(b) +60 +(010) +(100) +(100) +40 +20 +0 +(001) +(001) +Si +Si +Z +0.3 +Ar +V +Energy (eV) +(c) +(d) +Si +Si +0.2 +0.1 +0 +0.0 +Configurations +Configurations +40 +(e) +V +MSD (A^ +Si +4 +20 +2 +6 +8 +1. +0 +0 +10 +20 +30 +40 +50 +60 +Time (ps)12 + +which is significantly higher than room temperature at which detector devices work, +are performed to compare the structural stability between ArSi and pure VSi. Figures 5(a) +and (b) show the projected atomic trajectories with time for the local structures of ArSi +and the VSi, respectively. During the MD simulations, the instant sites of the vacancy +are determined by the Wigner-Seitz method as implemented in the OVITO code [18]. +It is clear that the Ar atom in ArSi is in form of a kind of rattling motion and moves +much more intensely than normal Si atoms behave in lattices. The result is consistent +with the negligible energy barriers among different SRD configurations of a ArSi. +Despite of such intense motions, ArSi stays in a same lattice firmly during the whole +MD simulation. Note that the multi-substitutional Ar defects (e.g., 4Ar4Si) also stable +during a 500 K MD simulation (see Fig. S7 in Supplemental Material). In a great +contrast, the pure VSi shows an easy diffusion among different lattices. Next, the c-NEB +analyses further evaluate barriers of atomic diffusions in Si. First, the energy barrier of +a single VSi diffusion can be as low as 0.26 eV, see Fig. 5(d). The probability of its +diffusion behavior at 500 K could be simply estimated by Arrhenius equation: 𝑃 = 𝑣 ∙ +exp (− +𝐸𝑎 +𝑘𝐵𝑇), where kB is the Boltzmann constant, T is the temperature, Ea is the energy +barrier of diffusion and v can be regarded as atomic vibration frequency. Using the +frequency of highest optical phonon of crystalline Si, i.e., ~15 THz [29], the estimated +characteristic time for one diffusion event is 𝜏 = +1 +𝑃 ≈ 30 ps for the VSi. Figure 5(e) +shows the mean square displacement (MSD) of VSi during the 500-K MD simulations. +The pure VSi can move in different lattices as many as 8 times within 60 ps, which is +close to the estimated characteristic time. That indicates in a practical Si sample the +pure VSi can readily diffuses among the lattices at the raised temperature and will be +eliminated when it gets to a grain boundary or a surface by an annealing process. As +such, the significant sub-bandgap absorption by VSi as shown in Fig. 4(f) cannot readily +happens. Second, in a great contrast, the energy barrier for an Ar diffusion out of a +vacancy (i.e., an Ar exchanges its site with an adjacent Si atom) is as high as 1.8 eV, +see Fig. 5(c). We have performed c-NEB calculations with other two different paths for +Ar diffusions and the barriers are almost the same (~1.8 eV), see Fig. S8 in + +13 + +Supplemental Material. The atomic pictures of the ArSi diffusions are presented in Figs. +S9-S11 in Supplemental Material. According to the Arrhenius equation, the +characteristic time for the Ar diffusion is at least ~ 70000 s, indicating a robust stability +of ArSi. Moreover, in another 60-ps MD in Fig. S12 of Supplemental Material, the +diffusion of ArSi is still absent at a much higher temperature of 900 K, which is usually +as an annealing temperature for fabrications of black Si detectors. Therefore, ArSi can +be regarded as a kind of Ar locking vacancy (ALV) defect. + +Finally, we discuss benefits of the ArSi or ALV doping for IR absorptions in Si detectors. +Firstly, the steric repulsion induced band splitting makes the ArSi defect readily +contribute multi defect states and thus offer effective sub-bandgap IR sources, which is +impossible for intrinsic Si. Secondly, due to Coulomb repulsion of the fully occupuied +shell of Ar, local configurations of the ALV can be dynamically adjusted by the Ar atom +at a raised temperature or even room temperature, see Fig. S13 in Supplemental +Material, which leads to a broad IR absorption band. Thirdly, Ar acts like an electronic +impurity with shallow donor defect levels. Then, the N+ layer can be formed which is +the key to construct the N+-N- junction in the device of IR detector [8]. + +Conclusion +In summary, we detect an unexpected high dose of inert Ar (with 1017-1020 cm-3) by +SIMS measurements in laser modified Si samples at Ar protective gas even after more +than 1300 days from when it was fabricated. First-principles calculations and molecular +dynamics simulations demonstrate a mechanism of Ar locking vacancy (ALV or ArSi) +happening in Ar doped silicon. While the pure vacancy in silicon can readily diffuse at +500 K, the ALV defect is dynamically robust at the same condition even no direct +chemical bonding connection between Ar and its neighboring Si atoms. Despite of the +chemical inert property of Ar, it can still act as an electronic impurity via strong +Coulomb repulsion effect which leads to significant splitting of defect levels within +bandgap, and thus have a strong sub-bandgap IR absorption in Si. It is an impossible +task for intrinsic Si. Moreover, the repulsion between Ar and dangling electrons leads + +14 + +to a shallow donor effect, which explains the confusion of n-type doping effect of laser +irradiation observed in previous experiments. It is reasonable to expect that such an +inert element induced IR absorption mechanism may also happen in other +semiconductors. Our work opens up a new door of using inert element doping +engineering to develop high performance IR Si detector, which is urgently required in +current Si based integrated optoelectronics. + +Acknowledgements +Work in China was supported by the National Natural Science Foundation of China +(Grants No. 62275098, No. 12274180, No. 12274172) and the Fundamental Research +Funds for the Central Universities. S.Z. was supported by the US Department of Energy +under Award No. DE-SC0002623. We sincerely thank Prof. Q.Z. at USTC for his +supports on band-unfolding analyses. 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Hussein, Thermal characterization of nanoscale phononic +crystals using supercell lattice dynamics, AIP Adv. 1, 041701 (2011). + + +1 + +Supplemental Material for +“Inert gas as electronic impurity in semiconductors: +The case for active infrared absorption in silicon” + +Nian-Ke Chen1,#, Yu-Chen Gao1,#, Ji-Hong Zhao1,*, Chun-Hao Li1, Qi-Dai Chen1, +Hong-Bo Sun2,*, Shengbai Zhang3,*, and Xian-Bin Li1,* + +1State Key Laboratory of Integrated Optoelectronics, College of Electronic Science +and Engineering, Jilin University, Changchun 130012, China +2State Key Lab of Precision Measurement Technology and Instruments, Department of +Precision Instrument, Tsinghua University, Beijing 100084, China +3Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic +Institute, Troy, New York 12180, USA + +Corresponding +authors: +lixianbin@jlu.edu.cn, +or +zhaojihong@jlu.edu.cn, +hbsun@tsinghua.edu.cn, or zhangs9@rpi.edu + + +2 + +CONTENTS: +Table S1. Formation energies of different defects in crystalline Si. +Fig. S1. Atomic structures of different Ar defects in crystalline Si. +Fig. S2. Band structure of ArSi defect with the configuration of C3v symmetry. +Fig. S3. Band structure of ArSi defect with the configuration of ~Td symmetry. +Fig. S4. Band structures of the 3Ar3Si and 4Ar4Si defects. +Fig. S5. Band structures of the interstitial Ar defects. +Fig. S6 Comparison of the density of states calculated by traditional PBE functional +and the meta-GGA method. +Fig. S7. A transient structure of 4Ar4Si and the projected trajectories of the local +structure of the 4Ar4Si during the 60-ps MD simulations at 500 K. +Fig. S8. Energy barriers for the diffusion of ArSi defect calculated by c-NEB method. +Fig.S9-S11. Atomic pictures of ArSi diffusion corresponding to the c-NEB calculations +in Fig. 5(c), Fig. S8(a) and Fig. S8(b), respectively. +Fig. S12-S13. The projected trajectories of the local structure of ArSi during the 60-ps +MD simulations at 900 K and 300 K, respectively. + + +3 + +TABLE S1. Formation energies of the defects in crystalline Si. The unit is eV. VSi is a +silicon vacancy. Tetrahedral (Tetra.), hexagonal (Hex.) and bond-centered (B-C.) +represent three different sites of the interstitial Ar defects [see Fig. S1(a)-(c) for the +atomic pictures]. C2v, C3v and ~Td represent three different configurations of the +substitutional Ar defects as described in the main text [see Fig. S1(d)-(f) for the +atomic +pictures]. +2Ar2Si, +3Ar3Si +and +4Ar4Si +represent +the +accumulated +multi-substitutional Ar defects [see Fig. S1(g)-(i) for the atomic pictures]. The +formation energies of multi-substitutional Ar defects are averaged by the number of +Ar atoms. Under the jellium approximation, the formation energy of defect w with +charge q can be calculated by the following equation: +𝛥𝐻𝑓(𝑞, 𝑤) = 𝐸𝑡𝑜𝑡(𝑞, 𝑤) − 𝐸𝑡𝑜𝑡(ℎ𝑜𝑠𝑡) + ∑ 𝑛𝑖𝜇𝑖 +𝑖 ++ 𝑞(𝜀𝑉𝐵𝑀 + 𝜀𝐹) +where 𝐸𝑡𝑜𝑡(𝑞, 𝑤) is the total energy of the supercell with defects, 𝐸𝑡𝑜𝑡(ℎ𝑜𝑠𝑡) is the +energy without the defect (i.e., bulk Si), 𝑛𝑖 is the number of atoms being exchanged +during the formation of the defect, 𝜇𝑖 is the atomic chemical potential of an element +(here we use the 𝜇𝑖 of bulk and isolated atom for Si and Ar, respectively), and 𝜀𝐹 is +the Fermi energy relative to the valence band maximum, 𝜀𝑉𝐵𝑀. +VSi +Interstitial +Substitutional +Multi-substitutional +Tetra. +Hex. +B-C. +C2v +C3v +~Td +2Ar2Si 3Ar3Si 4Ar4Si +3.69 +6.05 +7.19 +6.10 +6.25 +6.24 +6.27 +4.97 +4.52 +4.28 + + + +4 + + +FIG. S1. (a)-(c) Local atomic structures of three interstitial Ar defects. (d)-(f) Local +atomic structures of ArSi with the steric-repulsive distortion (SRD) of C2v, C3v and ~Td +symmetry. (g)-(i) The atomic structures of accumulated multi-Ar doping defects +including two-Ar-atoms substitution (2Ar2Si), three-Ar-atoms substitution (3Ar3Si) and +the four-Ar-atoms substitution (4Ar4Si). + + + +(a) Tetra. interstitial +(b) Hex. interstitial +(c) B-C. interstitial +(d) +Ar.. +c +(e) +Ar.. +C +(f) +Ar. +Si +2v +Si +3v +120° +120° +105 +~120° +115° +121 +121 +120° +~120° +120% +~120° +2Ar +(h) +3Ar +(i) +4Ar +(g) +2Si +3Si +4Si5 + + +FIG. S2. Unfolded band structures of ArSi with the SRD-C3v configuration [see Fig. +S1(e) for the atomic structure] holding (a) spin-up and (b) spin-down polarizations. +The green and red scatters indicate the contributions from Si1 and Si2,3,4, respectively. +Lower panels show corresponding schematic band structures. Ef is set as 0 eV. + + + +(a) Ars, SRD-C3spin个 +(b) Ars, SRD-C3 spin +2 +1 +1 +0 +0 +Energy [eV] +[eV] +Energy +2 +2 +-3 +3 +XU +K +L +W +X +XU +K +L +W +X +Conduction Band +Conduction Band +Valence Band +Valence Band6 + + +FIG. S3. The unfolded band structure of the ArSi with the ~Td configuration [see Fig. +S1(f) for the atomic structure], which is very similar to the ArSi SRD-C3v +configuration. The states contributed by the defective atoms are highlighted by red +color. + + +FIG. S4. The unfolded band structures of the (a) 3Ar3Si [see Fig. S1(h) for the atomic +structure] and (b) 4Ar4Si [see Fig. S1(i) for the atomic structure], respectively. The +states contributed by the defective atoms are highlighted by red color. + +a +SRD-~T +spin个 +(b) +Ar.. +SRD-~TspinV +2 +7 +0 +0 +Energy [ev] +1 +T +2 +2 +3 +3 +.4 +5 +5 +X U +K +W +X +x U +K +W +X(a) +3Ar +(b) +4Ar +3Si +4Si +2 +2 +1 +1 +Energy[eV] +Energy [eV] +0 +0 +1 +2 +-3 +3 +XU +K +厂 +W +X +XU +K +W +X7 + + + +FIG. S5. The unfolded band structure of the (a) Tetra.-site [see Fig. S1(a) for the +atomic picture] and (b) B-C.-site [see Fig. S1(c) for the atomic picture] interstitial Ar +defects. The states contributed by the defective atoms are highlighted by red color. +The defect states exist in the supercell with a B-C. interstitial Ar defect because the Ar +atom in this case breaks a Si-Si bond. + + +FIG. S6. Comparison of the density of states (DOS) of calculated by traditional PBE +functional and the meta-GGA method using the modified Becke-Johnson (MBJ) +exchange potential. The bandgap underestimated by PBE is corrected by the +meta-GGA method while the defect states still retain. + + +(a) Tetra. interstitia +(b) +B-C. interstitial +2 +Energy [eV] +Energy [eV] +1 +1 +2 +2 +-3 +3 +. +X +K +W +X +XU +K +L +W +X300 +(a) +Si +PBE +(b) +PBE +Ar..SRD-C +PBE +metaGGA +metaGGA +2 +metaGGA +200 +DOS +100 +0 +-3 +-2 +-1 +0 +2 +3 +.3 +-2 +1 +0 +2 +3 +2 +0 +3 +E- E, (eV) +E- E. (ev) +E-E.(eV)8 + + +FIG. S7. (a) A snapshot of the atomic structure of the 4Ar4Si after the 60-ps MD +simulations at 500 K. (b) The projected trajectories of the local structure of the 4Ar4Si +during the 60-ps MD simulations at 500 K. The color coding is the same with that in +the main text. + + + +FIG. S8. The energy barriers for diffusion of ArSi calculated by c-NEB method along +different paths. The atomic pictures of different paths are shown in Figs. S9-S11 +below. + + +(a) 4Aras; after 60-ps MD at 500 K +60 ps +(b) +(010) +(100) +40 +20 +(001) +02 +2 +(a) +(b) +1 +0 +0 +Configurations +Configurations9 + + +FIG. S9. The picture of ArSi diffusion corresponding to the c-NEB calculation of the +path in Fig. 5(c) of the main text. The color coding is the same with that in the main +text. The yellow label indicates the Si atom who exchanged its position with Ar atom +during the diffusion. (a) and (b) show the views along [010] and [100] directions, +respectively. + + +FIG. S10. The picture of ArSi diffusion corresponding to the c-NEB calculation of the +path in Fig. S8(a). The color coding is the same with that in the main text. The yellow +label indicates the Si atom who exchanged its position with Ar atom during the +diffusion. (a) and (b) show the views along [010] and [100] directions, respectively. + +aa) +b10 + + + +FIG. S11. The picture of ArSi diffusion corresponding to the c-NEB calculation of the +path in Fig. S8(b). The color coding is the same with that in the main text. The yellow +label indicates the Si atom who exchanged its position with Ar atom during the +diffusion. (a) and (b) show the views along [010] and [100] directions, respectively. + + +FIG. S12. The projected trajectories of the local structure of ArSi during the 60-ps MD +simulations at 900 K. The color coding is the same with that in the main text. + +a +b60ps +(010) +(100) +40 +20 +(001) +011 + + + +FIG. S13. The projected trajectories of the local structure of ArSi during the 60-ps MD +simulations at 300 K. The color coding is the same with that in the main text. + + + +60ps +(010) +(100) +40 +20 +(001) +0 \ No newline at end of file diff --git a/6dE2T4oBgHgl3EQfPAZa/content/tmp_files/load_file.txt b/6dE2T4oBgHgl3EQfPAZa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..55c402440cb564e894890573a948401d1dba11e3 --- /dev/null +++ b/6dE2T4oBgHgl3EQfPAZa/content/tmp_files/load_file.txt @@ -0,0 +1,794 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf,len=793 +page_content='1 Inert gas as electronic impurity in semiconductors: The case for active infrared absorption in silicon Nian-Ke Chen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='#,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Yu-Chen Gao1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='#,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Ji-Hong Zhao1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Chun-Hao Li1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Qi-Dai Chen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Hong-Bo Sun2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Shengbai Zhang3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' and Xian-Bin Li1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='* 1State Key Laboratory of Integrated Optoelectronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' College of Electronic Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Jilin University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Changchun 130012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' China 2State Key Lab of Precision Measurement Technology and Instruments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Department of Precision Instrument,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Beijing 100084,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' China 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Rensselaer Polytechnic Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Troy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' New York 12180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' USA Corresponding authors: lixianbin@jlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='cn, or zhaojihong@jlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='cn, hbsun@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='cn, or zhangs9@rpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='edu Abstract Inert (noble gas) elements are extremely inactive to surrounding chemical environment and are frequently employed as protective gas in various semiconductor fabrication processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In this work, we surprisingly discover that high doses of argon up to 1017- 1020 cm-3 can be measured in silicon exposed by laser pulses even after 1300 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' First- principles calculations and molecular dynamics identify a unique argon-locking- vacancy (ALV) defect atomic model in silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The ALV defect is dynamically robust in contrast to the frequently moving pure Si vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' While argon is chemically inert, it readily modulates defect states of the occupied vacancy via steric repulsion and rattling motions, leading to significant band splitting within bandgap and thus strong infrared absorptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Moreover, the repulsion between substitutional argon and dangling bonds results in shallow donors which explains the confusion of enhanced n- type carriers in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The work paves a way of using noble gas element to produce active infrared absorption source for the non-heteroepitaxy photonic detectors directly on silicon wafer at infrared communication wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 2 Silicon (Si) based optoelectronic devices is at the heart of optoelectronic industry owing to their ability for Si integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Among them, photodetectors working at infrared (IR) communication wavelength (λ) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='31/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='55 μm are indispensable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' However, due to the well-known problem of low absorption at λ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='1 μm, corresponding to the Si bandgap, Si is powerless in communication applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Often, a different semiconductor with a suitable bandgap is heterogeneously grown on Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' However, issues with heteroepitaxy such as lattice mismatch can reduce or even degrade the performance of the detectors [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Another way is to introduce IR absorption sources in Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' For example, gap states can be created inside Si by chalcogenide dopants with the help of ultrafast laser pulses to result in doped black silicon [2-6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' It has a strong IR absorption at λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='31/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='55 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' However, such IR sources are often not stable enough for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' For example, the IR absorption at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='31/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='55 μm in black silicon can be significantly reduced by annealing at 775 K for half an hour [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In 2018, Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' reported a form of black Si, which was fabricated by nanosecond laser pulses without any intentional element dopant except for a protective Ar gas [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' It was quite unexpected that the photodiode fabricated based on such a black Si has a high and stable photoresponsivity of 260 mA/W at 5 V at λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='31 μm [8], which paves the way for practical sensing by a Si detector at the IR communication wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Argon is a noble gas widely used as a protective gas in the electronic industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' As a matter of fact, the name of argon is derived from a Greek word that means lazy or inactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Due to its fully occupied valence band electronic shell with eight electrons, there is little chemical reaction between argon and other elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' As such, it is also expected that the Ar gas has no effect on the property of semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In this work, we report the observation of very high concentration of Ar (1017-1020 cm- 3) in ultrafast laser-modified Si using the secondary-ion mass spectrometry (SIMS) measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' First-principles calculations and molecular dynamic studies reveal the unique atomic and electronic properties of the Ar-doped Si to result in an unexpected and strong IR absorptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' While the pure Si vacancy can produce dangling-bond state 3 within bandgap, it is movable and unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In contrast, Ar atom can lock the Si vacancy to form a dynamically stable defect complex even up to 900 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Thanks to its full electronic shell, Ar protects the dangling electrons of vacancy and retains its gap states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Moreover, rattling motion and Coulomb repulsion of Ar atoms can lead to an enhanced structural distortion and the further splitting of defect energy levels within the bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Unexpectedly, the repulsion between substitutional Ar and dangling electrons makes the defect a shallow donor, which explains the confusion of laser irradiation induced n- type doping effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' As a result, the inert Ar atom in fact acts as an electronic impurity and offers active and robust sub-bandgap IR absorption source for Si photodetector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' This solves the long-term difficulty of high photoresponsivity of Si based detectors at IR communication wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The physics behind shed new light on a general strategy of employing inert elements to raise performances of semiconductor devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The concentrations of Ar atoms are measured by dynamic secondary ion mass spectrometer (D-SIMS) in laser modified Si samples in argon protecting atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The fabrication of such samples were reported in our previous work [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The D-SIMS instrument is equipped with a Cameca IMS-4F device using 8 keV Cs+ primary beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Density-functional theory (DFT) calculations are performed using the VASP code [9,10], where the projector-augmented wave (PAW) pseudo potential and generalized- gradient approximation (GGA) exchange-correlation functional developed by Perdew, Burke and Ernzerh are adopted [11-13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' A Si supercell that contains 216 atoms are used to describe defect effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The energy cutoff for plane-wave expansion is 380 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The 3×3×3 Monkhorst-Pack grids are used as Brillouin-zone sampling for static energy and property calculations, while the Γ point is used for structural relaxation and molecular dynamic (MD) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The band structures of supercells are unfolded by the modified VaspBandUnfolding package [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Energy barriers are calculated using the climbing image nudged elastic band (c-NEB) method [15,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The structures and charge density are visualized by the VESTA code [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The positions of vacancy defect are determined by the Wigner-Seitz method in the OVITO code [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 4 To figure out the actual role of Ar atoms in laser modified Si, we carefully analyze the dose of Ar by SIMS measurements in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Figure 1(a) displays Ar concentration for such a typical Si sample, which was fabricated previously by nanosecond laser in Ar atmosphere [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' It is unexpected that even the sample has been made over 1300 days, a very high dose of Ar atoms can be detected as from ~ 4×1021 cm-3 at the surface to ~ 4×1017 cm-3 at 2 µm below the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' We reevaluate the specific detectivity (D*) of the photodetector based on the Si sample [8] and compare it to those of non-silicon photodetectors at the IR wavelength of λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='31 μm [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='31-µm wavelength, whose photonic energy is below the bandgap of Si, is out of the detecting scope of detectors based on intrinsic Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' A higher D* reflects a higher signal-to-noise ratio of detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' For example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 1(b) shows that the D* of the laser modified Si in Ar atmosphere here working at 295 K (1011 cmHz1/2W-1) is not only higher than those of PbSe and InAs working at the same temperature but also higher than or close to those of InAs and InSb working at a much lower temperature of 193 K or 77 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' All of these indicate the inclusion of Ar in Si could potentially offer a highly effective IR absorption source for detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' To uncover the microscopic picture and critical role of Ar in Si samples, we carry out first-principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In fact, the formation energies (ΔHf) of Ar defects in crystalline Si are very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Table S1 in Supplemental Material summarizes the calculated ΔHf of several defects in Si, which agrees with previous reports [20,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The calculated ΔHf of interstitial and substitutional Ar defects are as large as 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='05-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='19 eV while the ΔHf of a silicon vacancy (VSi) is about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='67 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Despite of the high ΔHf, the Ar-related defects can still be formed under laser irradiations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' A key reason is that the ΔHf can be substantially reduced when Si is melted by laser irradiations, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Also, compared with interstitial Ar defects, substitutional Ar defects should be dominate because an interstitial Ar and a VSi will be annihilated into a substitutional Ar once they encounter during the annealing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' This annihilation is obviously energy favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Moreover, the ΔHf of substitutional Ar defects can further be lowered by their accumulation [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 1(d)] because such an accumulation reduces the number of 5 dangling bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (a) Concentration of Ar in the nanosecond-laser modified silicon sample measured by the secondary ion mass spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (b) Comparison of specific detectivity (D*) between the black silicon detector [8] and other reported infrared detectors at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='31 µm [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (c) Formation energies of a substitutional Ar defect (ArSi) at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The energy of a high-temperature state is calculated by the average free energy of the last 10 ps frames of a 20-ps NVT MD simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (d) Formation energies of the accumulated substitutional Ar defects in Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The formation energies of the multi-substitutional defects are averaged by the number of Ar atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Figure 2 elucidates atomic and electronic structures after an Ar atom is introduced into Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In order to study Ar induced defect states in Si, a large supercell with 216 atoms is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Due to Brillouin Zone folding in the supercell, band structures are calculated and further reanalyzed by the effective band unfolding method [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The unfolded band structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 2(d) reproduces the correct E-k dispersion for the intrinsic Si despite of underestimating the bandgap by the GGA-DFT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Naturally, an Ar atom filling in a Si lattice vacancy (VSi) could be an ideal candidate for the IR absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Because an intrinsic vacancy has the ability of offering defect 1E22 (a) 3 Concentration (Atoms/cm 1E13 (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='31 μum InGaAs(PV,295K) PbS(PC,295K) InAs(PV,193K) PbS(PC,77K) -Black Si in this work O (PV, 295K) nslaserpulse 1E21 1E12 InSb(PV,77K) InSb(PC,77K) InAs(PC,77K) lens Si substrate PbS(PC,193K) 1E11 1E20 ■PbSe(PC,295K) ■ InAs(PV,295K) 1E10 1E19 1E9 1E18 Ar ( Intrinsic absorption 1E8 impurityabsorption 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='0 Detectors Depth (μm) (c) 6 (d) 6 H (eV per Ar) (eV per Ar) 4 melting states 5 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 0 4 2Ar,si 3Ar3si 4Ar4si OK 2000 K 2500K 3000 K Temperature Substutional Ar in Si6 states within the bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 2(a) displays the local tetrahedral motif in intrinsic Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' When a Si atom is removed to form a VSi, four dangling bonds are produced, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Here, the four atoms around the vacancy are noted as atoms 1, 2, 3, 4 (Si1, Si2, Si3, Si4), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Li,j is the distance between any two of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' If no any atomic relaxation, the vacancy with all the same Li,j = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='87 Å holds an ideal Td local symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' However, due to the well-known Jahn-Teller distortion effect, the four atoms are relaxed closer to the center of the vacancy with L1,2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='03 Å) ≈ L3,4 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='12 Å) and other Li,j = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='54 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The symmetry is thus lowered to a near D2d symmetry (~D2d) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 2(b), which is consistent with a previous result [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Accordingly, the energy levels of defect states can split into two parts: two occupied levels in the lower part within the bandgap while two unoccupied levels in the upper part within the bandgap, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 2(e) of the band structure and its schematic drawing [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 2(h)] for Si with a ~D2d vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' After Ar atoms are introduced into Si which is confirmed by our SIMS measurement, the local structures of VSi will be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' For example, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 2(c), the atomic distortion of VSi can be further enhanced to hold a C2v symmetry due to a Ar atom substitutionally filled in VSi (ArSi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In this case, the Ar atom is closer to Si3 and Si4, which makes Si3 and Si4 move away from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Meanwhile, Si1 and Si2 move closer to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Therefore, L1,2 and L3,4 are changed to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='46 Å and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='40 Å, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The ~120°bond angles of Si3 and Si4 indicate the bonding type is changed to be sp2-hybridization like while the sp3-hybridization like bonding still remains for Si1 and Si2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Our calculations found that the filling of Ar into VSi costs about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='56-eV extra energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Such large an interaction energy and the large structural distortion suggest that Ar should have a significant repulsive interaction with surrounding dangling bonds of Si atoms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=', the steric repulsion effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The steric-repulsive induced distortion (SRD) also leads to a larger bonding hierarchy and changes the defect states of VSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Accordingly, comparing to the case of VSi, defect levels of ArSi are further splitted, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 2(f) and (i): the energies of highest occupied defect orbital (HODO), lowest unoccupied defect orbital (LUDO) and LUDO+1 is raised and closer to conduction band while the energy of HODO-1 lowers into the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Local atomic structures of (a) Si, (b) VSi and (c) ArSi with the steric-repulsive distortion (SRD) holding a C2v symmetry .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The arrows indicate the directions of atomic relaxation referred to their original positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (d)-(f) correspondingly show the unfolded band structures of the three supercell models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The size of the scatters represents the atomic weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The weights of Si1,2,3,4 (green and red scatters) are amplified by a factor of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The green and red scatters indicate the contributions from Si1,2 and Si3,4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (g)-(i) show corresponding schematic band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Ef is set as 0 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In fact, the local atomic configuration of ArSi defect with SRD is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' It can also have other structures with a C3v or ~Td symmetry, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S1(e) and (f) in Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In both cases, the four dangling Si atoms (Si1,2,3,4) are repulsed by the Ar and thus tend to have a sp2-like bonding characteristic due to the bond angle of ~120°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In the two cases, not only the defect levels but also the spin states are splitted, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S2 and S3 in Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Three of the four electrons from dangling bonds occupy three spin-up levels while one occupies a spin-down level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (a) Si (b) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (c) Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='.SRD C Si Energy [eV] Energy [eV] Energy [eV] Q 1 1 2 2 2 3 3 xu K W X X U K L W X xU K L L W X (g) (h) (0) Conduction Band ConductionBand Conduction Band ValenceBand Valence Band Valence Band8 In fact, the three configurations of ArSi with C2v, C3v and ~Td SRDs almost have the same formation energy with little differnece of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='01-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='03 eV (see Table S1 in Supplemental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The energy barrier of the tansition between C2v and C3v (C3v and ~Td) configurations calculated by the NEB method is also as small as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='015 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='013) eV, indicating readily transitions to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Defect levels always preserve inside the bandgap after inert Ar doping in VSi, despite of these different Ar configurations with close eneriges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Moreover, the SRD effect still stands in the accumulated substitutional Ar defects (3Ar3Si and 4Ar4Si) and thus the splitted defect states also exist (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S4 in Supplemental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In addition, a bond-centered (B-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=') site interstitial Ar defect also has two dangling bonds while the tetrahedral/hexagonal-site interstitial Ar defects have no dangling bonds, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S1(a)-(c) in Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' As such, the B- C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' interstitial Ar also has two defect levels inside the bandgap, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S5 in Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' All these defect states no doubt will benefit a robust sub- bandgap IR absorption in Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Next, electronic bonding mechanisms of ArSi are analyzed to figure out the mechanism of the Ar doping induced change of defect states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Taking the C2v configuration as an example, the charge density difference (CDD) analyses [24] in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 3(a)-(c) show no transferred or shared electrons between Ar and its surrounding Si due to the chemical inertia of Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Therefor, the interaction between Ar and the dangling Si atoms should be Coulomb repulsion effect which agrees with the SRD of the local structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In fact, although Ar atom does not offer any defect levels but indirectly modulate defect states of a Si vacancy via the repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Figures 3(d)-(h) elucidate the spatial distributions of defect states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The updated sp2-like configuration of Si3,4 [see the bond angles of 121° in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 3(a)] may release a dangling pz orbital from the orginal sp3 orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Obviously, the LUDO+1 and the HODO are the unoccupied and occupied pz-like orbitals of Si3,4, respectively, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 3(e) and (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Figure 3(f) shows the LUDO corresponding to the unoccupied sp3-like orbital of Si1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Since Si1 and Si2 has also a certain interaction due to a charge accumulation between them shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 3(b), their occupied sp3-like state 9 further moves below the valence band maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' We indeed find the state by projecting the orbital-decomposed partial charge density inside valence band [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 3(h)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Such a defect state will move back into bandgap in the case of C3v and ~Td SRDs due to the absence of such interaction between dangling Si atoms (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S2 and S3 in Supplemental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (a) The charge density difference (CDD) of ArSi with the C2v SRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The value of isosurface is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='015 e/a03, a0 is the Bohr radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (b) and (c) show the respective (11̅0) and (110) slices of the CDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The unit of the color bar is e/a03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (d) Schematic defect levels of ArSi with C2v SRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (e)-(h) The corresponding orbital-decomposed partial charge density projected to real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The values of isosurface are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='005 e/a03 for (e)- (g) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='0015 e/a03 for (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' To directly demonstrate the influence of the Ar-related defect on the IR absorption at λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='31/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='55 μm, the aobsorption coefficients are calculated [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 4(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Here, the meta-GGA method using the modified Becke-Johnson (MBJ) exchange potential [25,26] is adopted to correct the bandgap underestimated by the traditional PBE functional (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S6 in Supplemental Material for the effect of correction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Obviously, a ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='006 105° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='000 121 121° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='006 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='015 e (d) Conduction Band Pz (g) (h) sp Valence Band10 a substantial enhancement of the sub-bandgap IR absorption is achieved by ArSi in great contrast to the case without defect (Si) that displays almost no absoprtion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Moreover, the substitutioanl defects also have stronger abosorptions compared with the interstitial Ar and VSi, owing to the repulsion induced splitting of defect levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (a) Calculated absorption coefficient of Si, VSi, B-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' interstitial Ar and ArSi with the C2v and C3v SRDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (b)-(d) The ΔHf of neutral and charged states of the ArSi with C2v SRD, the ArSi with C3v SRD and the B-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' interstitial Ar defect, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In fact, the repulsion effect of Ar on dangling electrons not only changes the atomic structure and the position of defect levels but also significantly affects the ionization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' When the Ar is filled into the vacancy, the strong repulsive interaction between Ar and dangling electrons should make the electrons be ionized easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Figures 4(b)-(d) show the calculated chemical potential dependent ΔHf of neutral and charged states of the Ar defects [27,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' According to the position of transition levels (where the ΔHf of neutral and charged states equal), it is surprising that the substitutional Ar defects are shallow donors but deep acceptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In contrast, the B-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' interstitial Ar still has deep donor and acceptor levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The shallow donor effect of ArSi defects can explain the Absorption Coefficient (cm 6000 (a) by meta GGA Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='. SRD C 3V 4000 Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='. SRD C 2000 B C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' interstitial Si 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='6 Wavelength (μm) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='8 (b) (c) Ar SRD C (d)B C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' interstitial Ar SRD C IS 2v 1 S 3v 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='4 (eV) 0 0 △H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='0 +1 +1 +1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' E (eV) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' E (eV) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' E (eV) f VBV VBM VBV11 confusion that the fabricated sample shows a substantially enhanced n-type carrier concentration [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In other words, the substitutional Ar can act as an electronic impurity despite of its inert chemical property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The projected trajectories of the local structure of (a) ArSi and (b) VSi during 60- ps MD simulations at 500 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The colored dots represent positions of ArSi and Vsi while the grey dots indicate the ones of Si atoms with time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The energy barriers for a diffusion of (c) ArSi and (d) VSi calculated by c-NEB method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (e) The MSD of the VSi during the MD simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Inset in (e) shows the schematic trajectory of diffusion of a pure VSi in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' It worth noting that VSi has the lowest ΔHf (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='67 eV) among the defects in Si and it can also offer a significant sub-bandgap IR absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' As mentioned above, the absorption of ArSi is in fact originated from defect states of dangling bonds around the Ar filled vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' However, for practical device applications, IR absorption sources should be reliable enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Therefore, two ab initio MD simulations of 60 ps at 500 K, ps (a) (b) 60 (010) (100) (100) 40 20 0 (001) (001) Si Si Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='3 Ar V Energy (eV) (c) (d) Si Si 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='0 Configurations Configurations 40 (e) V MSD (A^ Si 4 20 2 6 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 0 0 10 20 30 40 50 60 Time (ps)12 which is significantly higher than room temperature at which detector devices work, are performed to compare the structural stability between ArSi and pure VSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Figures 5(a) and (b) show the projected atomic trajectories with time for the local structures of ArSi and the VSi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' During the MD simulations, the instant sites of the vacancy are determined by the Wigner-Seitz method as implemented in the OVITO code [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' It is clear that the Ar atom in ArSi is in form of a kind of rattling motion and moves much more intensely than normal Si atoms behave in lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The result is consistent with the negligible energy barriers among different SRD configurations of a ArSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Despite of such intense motions, ArSi stays in a same lattice firmly during the whole MD simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Note that the multi-substitutional Ar defects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=', 4Ar4Si) also stable during a 500 K MD simulation (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S7 in Supplemental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' In a great contrast, the pure VSi shows an easy diffusion among different lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Next, the c-NEB analyses further evaluate barriers of atomic diffusions in Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' First, the energy barrier of a single VSi diffusion can be as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='26 eV, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The probability of its diffusion behavior at 500 K could be simply estimated by Arrhenius equation: 𝑃 = 𝑣 ∙ exp (− 𝐸𝑎 𝑘𝐵𝑇), where kB is the Boltzmann constant, T is the temperature, Ea is the energy barrier of diffusion and v can be regarded as atomic vibration frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Using the frequency of highest optical phonon of crystalline Si, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=', ~15 THz [29], the estimated characteristic time for one diffusion event is 𝜏 = 1 𝑃 ≈ 30 ps for the VSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Figure 5(e) shows the mean square displacement (MSD) of VSi during the 500-K MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The pure VSi can move in different lattices as many as 8 times within 60 ps, which is close to the estimated characteristic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' That indicates in a practical Si sample the pure VSi can readily diffuses among the lattices at the raised temperature and will be eliminated when it gets to a grain boundary or a surface by an annealing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' As such, the significant sub-bandgap absorption by VSi as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 4(f) cannot readily happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Second, in a great contrast, the energy barrier for an Ar diffusion out of a vacancy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=', an Ar exchanges its site with an adjacent Si atom) is as high as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='8 eV, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' We have performed c-NEB calculations with other two different paths for Ar diffusions and the barriers are almost the same (~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='8 eV), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S8 in 13 Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The atomic pictures of the ArSi diffusions are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S9-S11 in Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' According to the Arrhenius equation, the characteristic time for the Ar diffusion is at least ~ 70000 s, indicating a robust stability of ArSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Moreover, in another 60-ps MD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S12 of Supplemental Material, the diffusion of ArSi is still absent at a much higher temperature of 900 K, which is usually as an annealing temperature for fabrications of black Si detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Therefore, ArSi can be regarded as a kind of Ar locking vacancy (ALV) defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Finally, we discuss benefits of the ArSi or ALV doping for IR absorptions in Si detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Firstly, the steric repulsion induced band splitting makes the ArSi defect readily contribute multi defect states and thus offer effective sub-bandgap IR sources, which is impossible for intrinsic Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Secondly, due to Coulomb repulsion of the fully occupuied shell of Ar, local configurations of the ALV can be dynamically adjusted by the Ar atom at a raised temperature or even room temperature, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S13 in Supplemental Material, which leads to a broad IR absorption band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Thirdly, Ar acts like an electronic impurity with shallow donor defect levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Then, the N+ layer can be formed which is the key to construct the N+-N- junction in the device of IR detector [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Conclusion In summary, we detect an unexpected high dose of inert Ar (with 1017-1020 cm-3) by SIMS measurements in laser modified Si samples at Ar protective gas even after more than 1300 days from when it was fabricated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' First-principles calculations and molecular dynamics simulations demonstrate a mechanism of Ar locking vacancy (ALV or ArSi) happening in Ar doped silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' While the pure vacancy in silicon can readily diffuse at 500 K, the ALV defect is dynamically robust at the same condition even no direct chemical bonding connection between Ar and its neighboring Si atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Despite of the chemical inert property of Ar, it can still act as an electronic impurity via strong Coulomb repulsion effect which leads to significant splitting of defect levels within bandgap, and thus have a strong sub-bandgap IR absorption in Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' It is an impossible task for intrinsic Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Moreover, the repulsion between Ar and dangling electrons leads 14 to a shallow donor effect, which explains the confusion of n-type doping effect of laser irradiation observed in previous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' It is reasonable to expect that such an inert element induced IR absorption mechanism may also happen in other semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Our work opens up a new door of using inert element doping engineering to develop high performance IR Si detector, which is urgently required in current Si based integrated optoelectronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Acknowledgements Work in China was supported by the National Natural Science Foundation of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 62275098, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 12274180, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 12274172) and the Fundamental Research Funds for the Central Universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' was supported by the US Department of Energy under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' DE-SC0002623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' We sincerely thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' at USTC for his supports on band-unfolding analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The High-Performance Computing Center (HPCC) at Jilin University for computational resources is also acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' References [1] A.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='#,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Ji Hong Zhao1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Chun Hao Li1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Qi Dai Chen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Hong Bo Sun2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Shengbai Zhang3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' and Xian Bin Li1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='* 1State Key Laboratory of Integrated Optoelectronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' College of Electronic Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Jilin University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Changchun 130012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' China 2State Key Lab of Precision Measurement Technology and Instruments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Department of Precision Instrument,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Beijing 100084,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' China 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Rensselaer Polytechnic Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Troy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' New York 12180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' USA Corresponding authors: lixianbin@jlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='cn, or zhaojihong@jlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='cn, hbsun@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='cn, or zhangs9@rpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='edu 2 CONTENTS: Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Formation energies of different defects in crystalline Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Atomic structures of different Ar defects in crystalline Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Band structure of ArSi defect with the configuration of C3v symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Band structure of ArSi defect with the configuration of ~Td symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Band structures of the 3Ar3Si and 4Ar4Si defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Band structures of the interstitial Ar defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S6 Comparison of the density of states calculated by traditional PBE functional and the meta-GGA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' A transient structure of 4Ar4Si and the projected trajectories of the local structure of the 4Ar4Si during the 60-ps MD simulations at 500 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Energy barriers for the diffusion of ArSi defect calculated by c-NEB method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='S9-S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Atomic pictures of ArSi diffusion corresponding to the c-NEB calculations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 5(c), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S8(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S8(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S12-S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The projected trajectories of the local structure of ArSi during the 60-ps MD simulations at 900 K and 300 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 3 TABLE S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Formation energies of the defects in crystalline Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The unit is eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' VSi is a silicon vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Tetrahedral (Tetra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' ), hexagonal (Hex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=') and bond-centered (B-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=') represent three different sites of the interstitial Ar defects [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S1(a)-(c) for the atomic pictures].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' C2v, C3v and ~Td represent three different configurations of the substitutional Ar defects as described in the main text [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S1(d)-(f) for the atomic pictures].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 2Ar2Si, 3Ar3Si and 4Ar4Si represent the accumulated multi-substitutional Ar defects [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S1(g)-(i) for the atomic pictures].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The formation energies of multi-substitutional Ar defects are averaged by the number of Ar atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Under the jellium approximation, the formation energy of defect w with charge q can be calculated by the following equation: 𝛥𝐻𝑓(𝑞, 𝑤) = 𝐸𝑡𝑜𝑡(𝑞, 𝑤) − 𝐸𝑡𝑜𝑡(ℎ𝑜𝑠𝑡) + ∑ 𝑛𝑖𝜇𝑖 𝑖 + 𝑞(𝜀𝑉𝐵𝑀 + 𝜀𝐹) where 𝐸𝑡𝑜𝑡(𝑞, 𝑤) is the total energy of the supercell with defects, 𝐸𝑡𝑜𝑡(ℎ𝑜𝑠𝑡) is the energy without the defect (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=', bulk Si), 𝑛𝑖 is the number of atoms being exchanged during the formation of the defect, 𝜇𝑖 is the atomic chemical potential of an element (here we use the 𝜇𝑖 of bulk and isolated atom for Si and Ar, respectively), and 𝜀𝐹 is the Fermi energy relative to the valence band maximum, 𝜀𝑉𝐵𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' VSi Interstitial Substitutional Multi-substitutional Tetra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Hex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' B-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' C2v C3v ~Td 2Ar2Si 3Ar3Si 4Ar4Si 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='69 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='97 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='52 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='28 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (a)-(c) Local atomic structures of three interstitial Ar defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (d)-(f) Local atomic structures of ArSi with the steric-repulsive distortion (SRD) of C2v, C3v and ~Td symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (g)-(i) The atomic structures of accumulated multi-Ar doping defects including two-Ar-atoms substitution (2Ar2Si), three-Ar-atoms substitution (3Ar3Si) and the four-Ar-atoms substitution (4Ar4Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (a) Tetra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' interstitial (b) Hex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' interstitial (c) B C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' interstitial (d) Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='. c (e) Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='. C (f) Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Si 2v Si 3v 120° 120° 105 ~120° 115° 121 121 120° ~120° 120% ~120° 2Ar (h) 3Ar (i) 4Ar (g) 2Si 3Si 4Si5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Unfolded band structures of ArSi with the SRD-C3v configuration [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S1(e) for the atomic structure] holding (a) spin-up and (b) spin-down polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The green and red scatters indicate the contributions from Si1 and Si2,3,4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Lower panels show corresponding schematic band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Ef is set as 0 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (a) Ars, SRD C3spin个 (b) Ars, SRD C3 spin 2 1 1 0 0 Energy [eV] [eV] Energy 2 2 3 3 XU K L W X XU K L W X Conduction Band Conduction Band Valence Band Valence Band6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The unfolded band structure of the ArSi with the ~Td configuration [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S1(f) for the atomic structure], which is very similar to the ArSi SRD-C3v configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The states contributed by the defective atoms are highlighted by red color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The unfolded band structures of the (a) 3Ar3Si [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S1(h) for the atomic structure] and (b) 4Ar4Si [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S1(i) for the atomic structure], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The states contributed by the defective atoms are highlighted by red color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' a SRD ~T spin个 (b) Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='. SRD ~TspinV 2 7 0 0 Energy [ev] 1 T 2 2 3 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='4 5 5 X U K W X x U K W X(a) 3Ar (b) 4Ar 3Si 4Si 2 2 1 1 Energy[eV] Energy [eV] 0 0 1 2 3 3 XU K 厂 W X XU K W X7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The unfolded band structure of the (a) Tetra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='-site [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S1(a) for the atomic picture] and (b) B-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='-site [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S1(c) for the atomic picture] interstitial Ar defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The states contributed by the defective atoms are highlighted by red color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The defect states exist in the supercell with a B-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' interstitial Ar defect because the Ar atom in this case breaks a Si-Si bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' Comparison of the density of states (DOS) of calculated by traditional PBE functional and the meta-GGA method using the modified Becke-Johnson (MBJ) exchange potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The bandgap underestimated by PBE is corrected by the meta-GGA method while the defect states still retain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (a) Tetra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' interstitia (b) B C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' interstitial 2 Energy [eV] Energy [eV] 1 1 2 2 3 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' X K W X XU K L W X300 (a) Si PBE (b) PBE Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='.SRD C PBE metaGGA metaGGA 2 metaGGA 200 DOS 100 0 3 2 1 0 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='3 2 1 0 2 3 2 0 3 E E, (eV) E E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (ev) E E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content='(eV)8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (a) A snapshot of the atomic structure of the 4Ar4Si after the 60-ps MD simulations at 500 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (b) The projected trajectories of the local structure of the 4Ar4Si during the 60-ps MD simulations at 500 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The color coding is the same with that in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The energy barriers for diffusion of ArSi calculated by c-NEB method along different paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The atomic pictures of different paths are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S9-S11 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (a) 4Aras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' after 60-ps MD at 500 K 60 ps (b) (010) (100) 40 20 (001) 02 2 (a) (b) 1 0 0 Configurations Configurations9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The picture of ArSi diffusion corresponding to the c-NEB calculation of the path in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 5(c) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The color coding is the same with that in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The yellow label indicates the Si atom who exchanged its position with Ar atom during the diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (a) and (b) show the views along [010] and [100] directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The picture of ArSi diffusion corresponding to the c-NEB calculation of the path in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The color coding is the same with that in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The yellow label indicates the Si atom who exchanged its position with Ar atom during the diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (a) and (b) show the views along [010] and [100] directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' aa) b10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The picture of ArSi diffusion corresponding to the c-NEB calculation of the path in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The color coding is the same with that in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The yellow label indicates the Si atom who exchanged its position with Ar atom during the diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' (a) and (b) show the views along [010] and [100] directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The projected trajectories of the local structure of ArSi during the 60-ps MD simulations at 900 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The color coding is the same with that in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' a b60ps (010) (100) 40 20 (001) 011 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The projected trajectories of the local structure of ArSi during the 60-ps MD simulations at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' The color coding is the same with that in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} +page_content=' 60ps (010) (100) 40 20 (001) 0' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfPAZa/content/2301.03754v1.pdf'} diff --git a/79AyT4oBgHgl3EQfc_dz/content/tmp_files/2301.00293v1.pdf.txt b/79AyT4oBgHgl3EQfc_dz/content/tmp_files/2301.00293v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa5bf5d397b2019c0f75d487ad0bd8962350a131 --- /dev/null +++ b/79AyT4oBgHgl3EQfc_dz/content/tmp_files/2301.00293v1.pdf.txt @@ -0,0 +1,1187 @@ +arXiv:2301.00293v1 [astro-ph.EP] 31 Dec 2022 +Orbital Migration of Protoplanets in a Marginally Gravitationally +Unstable Disk. II. Migration, Merging, and Ejection +Alan P. Boss +Earth & Planets Laboratory, Carnegie Institution for Science, 5241 Broad Branch Road, +NW, Washington, DC 20015-1305 +aboss@carnegiescience.edu +ABSTRACT +Protoplanets formed in a marginally gravitationally unstable (MGU) disk by +either core accretion or disk instability will be subject to dynamical interactions +with massive spiral arms, possibly resulting in inward or outward orbital migra- +tion, mergers with each other, or even outright ejection from the protoplanetary +system. The latter process has been hypothesized as a possible formation sce- +nario for the unexpectedly high frequency of unbound gas giant exoplanets (free +floating planets, FFP). Previous calculations with the EDTONS fixed grid three +dimensional (3D) hydrodynamics code found that protoplanets with masses from +0.01 M⊕ to 3 MJup could undergo chaotic orbital evolutions in MGU disks for +∼ 1000 yrs without undergoing monotonic inward or outward migration. Here +the Enzo 2.5 adaptive mesh refinement (AMR) 3D hydrodynamics code is used +to follow the formation and orbital evolution of protoplanets in MGU disks for +up to 2000 yrs. The Enzo results confirm the basic disk fragmentation results +of the EDTONS code, as well as the absence of monotonic inward or outward +orbital migration. In addition, Enzo allows protoplanet mergers to occur, unlike +EDTONS, resulting in a significant decrease in the number of protoplanets that +survive for 1000 to 2000 yrs in the Enzo models. These models also imply that +gas giants should be ejected frequently in MGU disks that fragment into large +numbers of protoplanets, supporting ejection as a possible source mechanism for +the observed FFPs. +Subject headings: planets and satellites: formation — protoplanetary disks +1. +Introduction +Exoplanet demographics provide one of the ultimate arbiters of theories of exoplanet +formation and evolution. Nielsen et al. (2019) used the GPI exoplanet survey to search + +– 2 – +for planets with masses between 2 and 13 MJup and semimajor axes between 3 and 100 au, +finding that the peak occurrence distance of giant planets was in the range of 1 to 10 au. +Fulton et al. (2021) found the same peak occurrence distance of 1 to 10 au for the California +Legacy Doppler velocities survey. +Vigan et al. +(2021) showed that the VLT SPHERES +direct imaging survey of 150 stars detected 13 sub-stellar companions with masses between +1 and 75 MJup and semimajor axes between 5 and 300 au, finding that both core accretion +(CA; Mizuno 1980) and disk instability (DI; Boss 1997) appeared necessary to explain the +detections for the FGK stars in their sample. +Gas giant planets with orbital distances as large as 980 au have been discovered and +studied (Wu et al. 2022). Forming giant exoplanets at such large distances by CA within +the ∼ 1 Myr lifetimes of the gaseous portion of protoplanetary disks is challenging (e.g., +Chambers 2021), if not impossible. DI has the advantage of forming dense, self-gravitating +clumps in a few orbital periods, relaxing the disk lifetime constraint for forming wide-orbit +gas giants in situ (e.g., Boss 2011). Evidence for a gas giant protoplanet embedded in a spiral +arm 93 au from AB Aurigae has been interpreted as an example of gas giant planet formation +by DI (Currie et al. 2022; cf. Cadman et al. 2021; Zhou et al. 2022). DI has also been +proposed as the source of the ∼ 10MJup exoplanet that orbits ∼ 560 au from the massive +binary b Centauri (Janson et al. 2021). Goda & Matsuo (2019) examined the demographics +of 485 planetary systems and concluded that a hybrid theory of planet formation, involving +both CA and DI, was needed to explain the exoplanet detections. +Miret-Roig et al. (2022) used a direct imaging survey coupled with Gaia and Hipparchos +astrometry to search for unbound gas giant exoplanets in the Upper Scorpius and Ophiuchus +young stellar association. Their survey yielded between 70 and 170 free floating planets +(FFP), considerably more than might be expected to form as the tail end of the star formation +process of molecular cloud core collapse and fragmentation, and suggested that ejection +from unstable planetary systems might make a major contribution during the first 10 Myr. +Gravitational microlensing has also found an abundance of likely FFPs, though these could +also simply be bound planets with orbital distances greater than about 10 au (Mr´oz et al. +2020; Ryu et al. 2021). Vorobyov (2016) performed numerical simulations that supported +the hypothesis that FFPs might be the result of planets ejected from massive MGU disks. +While exoplanet demographics reveal orbital characteristics at the present epoch, unless +exoplanets do not undergo significant orbital evolution or migration following their formation, +the present epoch orbital parameters are of limited usefulness in constraining their initial +orbital distances. CA is the favored mechanism closer to the host star, as a result of shorter +orbital periods, higher gas disk temperatures, and higher surface densities of solids, to name +a few factors, while DI may be more effective at larger distances in suitably massive and cool + +– 3 – +protoplanetary disks. For either CA or DI, a key question then becomes the extent to which +protoplanets might migrate away from their birth orbits to their present epoch orbits. +As noted by Boss (2013), CA and DI both require giant protoplanets to form in the +presence of disk gas. Theoretical work on protoplanetary orbital migration (e.g., Kley & +Nelson 2012) usually focuses on protoplanets in disks where the disk mass is low enough +that the disk self-gravity can be neglected, greatly simplifying the analysis. Protoplanet +evolution in MGU disk models has been calculated by Boss (2005), Baruteau et al. (2011), +and Michael et al. (2011). These studies each considered quite different initial conditions +and found a wide range of outcomes, ranging from large-scale inward orbital migration to +relatively little orbital migration. Boss (2013) studied the evolution of protoplanets formed +by either CA and DI in MGU disks, noting that while a MGU disk is essential for formation +by DI, even a giant planet formed by CA in a quiescent, non-MGU disk can experience a +later phase of MGU disk interactions during the periodic FU Orionis outbursts experienced +by young solar-type protostars, which are thought to involve a phase of disk gravitational +instability that dumps disk mass onto the protostar (e.g., Zhu et al. 2010; Kuffmeier et al. +2018). Dunhill (2018) similarly suggested that giant planets formed by CA might undergo +orbital migration during FU Orionis outbursts. +The Boss (2005, 2013) models were performed using the EDTONS three dimensional +radiative hydrodynamics code, with a spherical coordinate grid that was fixed at moderate +spatial resolution throughout the MGU disk evolutions. Virtual protoplanets were introduced +at the beginning of each model to represent protoplanets as point sources of gravity, able +to interact gravitationally with the disk and with each other and to accrete mass from the +disk by Bondi-Hoyle accretion. Boss (2013) found that protoplanets with initial masses in +the range from 0.01 M⊕ to 3 MJup and initial orbital distances of 6 to 12 au in a MGU disk +around a solar-mass protostar underwent chaotic orbital evolutions for ∼ 1000 yr without +undergoing the monotonic inward or outward migration that typically characterizes the Type +I or Type II behavior of non-self-gravitating disk models (e.g., Kley & Nelson 2012). +The present models of protoplanet orbital evolution employ the Enzo 2.5 hydrodynamics +code. Enzo is also a three dimensional (3D) code and uses Adaptive Mesh Refinement (AMR) +in Cartesian coordinates to ensure that sharp gradients in fluid quantities such as shock fronts +can be handled accurately. Enzo is able to replace exceptionally dense disk clumps with sink +particles representing newly formed (by DI) protoplanets, which thereafter interact with each +other and the disk while accreting disk gas, as do the virtual protoplanets in the Boss (2013) +models. We thus seek here to use a completely different 3D hydro code to learn more about +the possible outcomes for protoplanet orbital evolution in MGU disks, and to compare the +results with the latest advances in exoplanet demographics. + +– 4 – +2. +Numerical Hydrodynamics Code +As noted by Boss & Keiser (2013), the Enzo 2.5 AMR code performs hydrodynamics +(HD) using any one of three different algorithms (Collins et al. 2010; Bryan et al. 2014): (1) +the piecewise parabolic method (PPM) of Colella & Woodward (1984), (2) the ZEUS method +of Stone & Norman (1992), or (3) a Runge–Kutta third-order-based MUSCL (“monotone +upstream-centered schemes for conservation laws”) algorithm based on the Godunov (1959) +shock-handling HD method. Enzo is designed for handling strong shock fronts by solving +the Riemann problem (e.g., Godunov 1959) for discontinuous solutions of a fluid quantity +that should be conserved. The PPM option was used in the current models as a result of the +testing on mass and angular momentum conservation performed with Enzo 2.0 by Boss & +Keiser (2013), who found that PPM was better able to conserve mass and angular momentum +during the collapse of a rotating isothermal cloud core (Boss & Bodenheimer 1979) than +either ZEUS or MUSCL. Enzo is designed for parallel processing on high performance clusters +(HPC), but when run on a single, dedicated 32-core node of the Carnegie memex HPC, a +typical model still required 7 months of continuous computation to evolve for ∼ 103 yrs of +model time. +The Enzo 2.5 models were initialized on a 3D Cartesian grid with 32 top grid points +in each direction. +A maximum of 7 levels of refinement was used, with a factor of two +refinement occurring for each level, so that the maximum possible effective grid resolution +was 27 = 128 times higher than the initial resolution of 323, i.e., 40963. The models with 7 +levels needed an increase in the number of cell buffer zones (NumberBufferZones) to 3 from +the default value of 1, which was used for the lower levels of refinement, in order to maintain +reasonable time steps. Grid refinement was performed whenever necessary to ensure that +the Jeans length constraint (e.g., Truelove et al. 1997; Boss et al. 2000) was satisfied by a +factor of 4 for cells with a density at least eight times the initial density. Periodic boundary +conditions were applied on each face of the grid cubic box, with each side either 60 au or +120 au in length. A point source of external gravity was used to represent a 1 M⊕ protostar +at the center of the grid. The maximum number of Green’s functions used to calculate the +gravitational potential was 10. The time step typically used was 0.15 of the limiting Courant +time step. The results were analyzed with the yt astrophysical analysis and visualization +toolkit (Turk et al. 2011). +Following Boss & Keiser (2014), we used the Enzo 2.2 sink particle coding described by +Wang et al. (2010). Sink particles are created in grid cells that have already been refined +to the maximum extent permitted by the specified number of levels of grid refinement, +but where the gas density still exceeds that consistent with the Jeans length criterion for +avoiding spurious fragmentation (Truelove et al. 1997; Boss et al. 2000). As described by + +– 5 – +Boss & Keiser (2014), sink particles accrete gas from their host cells at the modified Bondi- +Hoyle accretion rate proposed by Ruffert (1994). Two parameters control the conditions +under which sink particles can be merged together: the merging mass (SinkMergeMass) and +the merging distance (SinkMergeDistance). The former of these two parameters is used to +divide the sink particles into either large or small particles. Particles with less mass than +SinkMergeMass are first subjected to being combined with any large particles that are located +within the SinkMergeDistance. Any surviving small particles after this first step are then +merged with any other small particles within the SinkMergeDistance. The merging process +is performed in such a way as to ensure conservation of mass and linear momentum. Boss & +Keiser (2014) found that their results for collapse and fragmentation of magnetic molecular +cloud cores were not particularly sensitive to the choice of these two key parameters with +regard to the tendency of the cores to undergo fragmentation into multiple protostar systems. +The current paper uses the Wang et al. (2010) sink particle coding with the SinkMergeMass +set equal to 0.01 MJup and the SinkMergeDistance set equal to 0.1 au, appropriate values +for studying gas giant protoplanets in a 120 au-size region. Sink creation was only allowed +for cells with densities exceeding the values listed in Table 1 (DensThresh in code units in +the sink maker.C subroutine). These densities were chosen to be low enough that sinks do +form in the models, as the point of the present models was to study the orbital evolution +of sink particles representing protoplanets in MGU disks rather than to study the precise +physics of DI-induced fragmentation and clump formation in such disks (e.g., Boss 2021a,b). +The sink particles used in the Enzo models are similar to the virtual protoplanets (VPs) +used in the EDTONS models: both are introduced in regions of density maxima and are +intended to represent gravitationally bound clumps of disk gas that will contract to form +gaseous protoplanets, as they orbit in the disk around the central protostar, interacting +gravitationally with each other and the disk gas, even as they accrete more disk gas. There +are several differences, however. Sink particles are created automatically by Enzo following +the criteria noted above, sink particles with close encounters can be merged together, and +sink particles that encounter a grid boundary reappear on the opposite boundary as a result +of the periodic boundary conditions. VPs, on the other hand, are inserted when a density +maximum exceeds the Jeans length or Toomre length criteria (Nelson 2006; Boss 2021a,b) for +the current grid spatial resolution. VPs may undergo close encounters with each other but +do not suffer mergers. VPs that strike either the inner or outer grid boundary are removed +from the calculation. +While it would be desirable to compare flux-limited diffusion (FLD) approximation +radiative hydrodynamic models from the EDTONS code with FLD radiative hydrodynamic +models calculated by Enzo, the FLD routines available in Enzo are limited to non-local +thermodynamic equilibrium (non-LTE), as Enzo was developed primarily for cosmological + +– 6 – +simulations, whereas EDTONS assumes LTE. As a result, we are limited to using a simpler +approach to handling the disk thermodynamics with the Enzo code. Boss (1998) showed that +disk fragmentation could occur for strongly gravitationally unstable disks with either locally +isothermal or locally adiabatic thermodynamics, using disk gas adiabatic exponents ranging +from γ = 1 (purely isothermal) to γ = 7/5, which is appropriate for molecular hydrogen. +Given that disks are subject to compressional heating, γ = 1 is not strictly correct, and given +that disks that are optically thick at their midplanes can cool from their surfaces, γ = 7/5 +is not strictly correct either. The physically correct behavior presumably lies somewhere in +the middle of these two extremes. +Radiative cooling in optically thin regions was employed in the Enzo models, with a +critical density for cooling of 10−13 g cm−3; regions with densities above this critical value +had the cooling rate decreased proportionally. +This critical density was chosen because +that is the disk midplane density where the dust grain opacity produces optical depths of +order unity (e.g., Boss 1986). The cooling rates were modified from the default values in +cool rates.in to rates consistent with molecular line cooling in optically thin regions (Boss +et al. +2010; Neufeld & Kaufman 1993). +Because Enzo PPM hydrodynamics involves a +Riemann solver that cannot be purely isothermal, i.e., γ cannot equal unity, the adiabatic +index for the disk gas was taken to be γ = 1.001, appropriate for a nearly isothermal, but +still adiabatic equation of state for an ideal gas. Test runs were computed for 100 yrs of +evolution with both γ = 7/5 and γ = 5/3, but in both cases Enzo produced midplane +disk temperatures that were over 104 K, whereas the initial disk had a maximum midplane +temperature of 1500 K. The test runs with γ = 1.001 produced the expected maximum +temperatures of ∼ 1500 K, and hence γ = 1.001 was adopted for the models presented here. +The resulting temperature distributions were also affected by the assumption of radiative +cooling; spiral features in the midplane temperature distribution accompanied spiral features +in the midplane density distribution, as is to be expected. Finally, the mean molecular weight +of the gas was effectively taken to be µ = 2.4, appropriate for a solar composition mixture +of molecular hydrogen and helium. +3. +Initial Conditions +Table 1 lists the models with variations in the number of levels of grid refinement, +the outer disk and envelope temperatures, initial minimum value of the Toomre (1964) Q +parameter, disk radius, calculational grid box size, and critical density for sink particle +creation. A 60 au box size was used for the 20 au and 30 au radius disks, while a 120 au box +size was used for 60 au radius disks, in order to give the disks sufficient room to evolve and + +– 7 – +expand by the outward transport of angular momentum through gravitational interactions +with the spiral arms and clumps. In the number of levels column, 34 means the model was +initially run with 3 levels and then a fourth level of refinement was added. +The initial disks are based on the model HR disk from Boss (2001), with an outer disk +temperature of 40 K and and disk envelope temperature of 50 K, which has been used as a +standard initial model for many of the author’s disk instability models (e.g., Boss 2021a,b). +Model HR has an initial minimum Toomre Q ≈ 1.3, implying marginal stability to the +growth of rings and spiral arms. The model HR initial disk has a mass of 0.091 M⊙ within +an inner radius of 4 au and an outer radius of 20 au and orbits a 1 M⊙ central protostar. +The Enzo models have have masses of 0.102 M⊙ for 20 au outer radius disks, slightly higher +than in model HR because the Enzo models extend inward to 1 au, 0.142 M⊙ for 30 au outer +radius disks, and 0.210 M⊙ for 60 au outer radius disks. The same disk density power-law-like +Keplerian structure as in Boss (2001) is used for all of the models, with the structure being +terminated at 20 au, 30 au, or 60 au. Figures 1 and 2 show cross sections of the initial disk +density distribution for the 20 au disks, both parallel and perpendicular (i.e., disk midplane) +to the disk rotation axis. +4. +Results +Figure 3 shows the intermediate results for two of the four models that have the identical +initial disk configuration (20 au radius) as the Boss (2001) model HR, depicted at the same +time (190 yrs of evolution) as the same initial disk model (fldA) in Boss (2021b, cf. Figure +2a). Figure 3 shows that both of these models (3-1K-20 and 6-1K-20) rapidly evolved into +a configuration of multiple spiral arms interspersed with dense clumps, as expected for a +marginally gravitationally unstable disk. +Also as expected, the degree of fragmentation +and clump formation increases as the numerical grid resolution increases from 3 to 6 levels. +When sink particles are allowed to form, the number of sink particles similarly increases as +the resolution is improved. While the background disk looks quite similar for model 3-1K-20 +with or without sink particles (Figure 3a,c), there is a clear difference in the case of model +6-1K-20 (Figure 3b,d), where the background disk has become perturbed into a prolate +configuration due to the formation of a massive (∼ 20MJup) secondary companion (at one +o’clock), with its own circumplanetary disk and tertiary companion, whose combined tidal +forces have evidently distorted the disk’s overall appearance. Model fldA of Boss (2021b) +had fragmented into a five clumps and three virtual protoplanets (i.e., sink particles) by +189 yrs, for a total of eight, considerably more than formed in the present model 3-1K-20, +but not as many as in model 6-1K-20, suggesting that even with the quadrupled spatial + +– 8 – +resolution of the Boss (2021b) EDTONS models, the adaptive mesh refinement feature of +Enzo results in significantly improved numerical spatial resolution of the disk instability and +fragmentation. Confirmation of the formation of long-lived fragments in the model HR disk +(Boss 2001, 2021b) with the completely different hydrodynamical method used here provides +strong support for the viability of the disk instability mechanism for the formation of gas +giant protoplanets and higher mass companions. +Figure 4 displays the results after 2000 yrs for the Enzo models in Figure 3. +The +EDTONS model fldA in Boss (2021b) was stopped after only 189 yrs of evolution, but even +still required over 4 years of computation on a single core of a node on memex, a Carnegie +Institution computer cluster. EDTONS is based on code initially written in the late 1970s +and is not parallelized. Enzo, in contrast, is a modern code designed to run on parallel +processing systems like memex, and as a result the Enzo models can be computed much +farther in time. Even still, model 3-1K-20 required one week to run for 2000 yrs of model +time on a dedicated single memex node with 28 cores, while model 6-1K-20 required one +year to run 2000 yrs on a dedicated 28-core node. Three dimensional hydrodynamics at high +spatial resolution is computationally expensive, even when a parallelized code is employed. +Figure 4 shows that the evolution of these two models diverged considerably following +the early fragmentation phase depicted in Figure 3. The two sink particles evident in Figure +4a have masses of ∼ 2MJup and ∼ 0.6MJup, with a total gas disk mass of ∼ 99MJup, while +the 13-odd sink particles in Figure 4b have masses ranging from ∼ 0.2MJup to ∼ 23MJup, +for a total sink particle mass of ∼ 96MJup, leaving a disk gas mass of only ∼ 5MJup. Clearly, +the final disk mass in model 3-1K-20 far outweighs the mass of the sink particles, and as +a result the particles are unable to open gaps in the disk, though the disk has expanded +outward to a radius of about 30 au as a result of the transport of disk mass and angular +momentum outward, caused by the strong spiral arms evident in Figure 3a,c. The fact that +sink particle formation has been so efficient in model 6-1K-20, with the total particle mass +some 20 times larger than the disk mass, means that the particles rule the evolution and +are able to clear out a distinct inner gap, centered on about 5 au (Figure 4b). In model +6-1K-20, the sink particles gained the bulk of the disk’s mass and angular momentum, so +that the disk is not able to expand beyond its initial radius of 20 au. Three sink particles +were accelerated to speeds high enough at their orbital location to be ejected altogether from +the system, but because of the periodic boundary conditions imposed on the calculations +by the Enzo self-gravity solver, these ejectable particles were returned to the system and +underwent further interactions with the sink particles and disk gas. +Table 2 gives the maximum number of sink particles formed for all the models, as well +as the number surviving at the end of the run. Table 2 shows that the maximum number of + +– 9 – +sinks formed decreases as the initial disk gas temperature is increased, as this results in an +increase in the Toomre minimum Q value (Table 1), i.e., in greater stability to the growth of +rings and spiral arms, and hence to fragmentation and sink particle formation. By the time +that the disk temperature is increased to 160 K, disk fragmentation is completely stifled in +the Enzo models, consistent with the flux-limited diffusion approximation models of Boss +(2021b), where fragmentation ceased for a minimum Toomre Q greater than 2.2. +Models 6-2K-30 and 7-2K-30 did not form sink particles, unlike the otherwise identical, +but lower resolutions models 3-2K-30, 4-2K-30, and 5-2K-30, because this sequence used a +fixed critical density for sink particle formation of 10−9 g cm−3. That choice meant that +the dense clumps formed in the two higher resolution models could always be resolved with +more grid levels and finer spatial resolution, thereby preventing the clumps from exceeding +the critical density required for sink particle formation, at least during the limited amount +of model time that the 6- and 7-level models were able to be evolved (326 and 340 yrs, +respectively). +Small time steps prevented these two models for being evolved farther in +time. Figure 5 shows these two models at their final times, showing that the spiral arms and +nascent clumps become more distinct as the number of grid levels is increased, as expected +when approaching the continuum limit of infinite spatial resolution. +Table 2 also lists the number of sink particles mergers, where Nmerged−sinks = Nmax−sinks− +Nfinal−sinks, and the number of times that a sink particle would have been ejected if periodic +boundary conditions were not required. Table 2 shows that mergers of sink particles are +quite common in all of the models that formed sink particles, and evidently are responsi- +ble for much of the gain in mass of the particles, along with the ongoing accretion of disk +gas, given that the number of mergers is usually comparable to, or far greater than, the final +number of sink particles. The value Nescaped−sinks can be quite large due to the sink particles’ +inability to escape the system; often the same particle bounces in and out in orbital radius +and achieves escape velocity multiple times. Achieving the escape velocity usually occurs for +particle orbital distances of 30 au to 40 au, but can also occur from 10 au to 20 au in the +more unstable disks (e.g., 5-1K-20, 6-1K-20). The large numbers of escape episodes in the +latter two models are clearly solely a result of the periodic boundary conditions, but they +do indicate that ejected protoplanets are to be expected as a natural outcome of a phase of +gas disk gravitational instability. Table 2 suggests that such a phase of protoplanetary disk +evolution should result in the ejection of several gas giant protoplanets. +Figure 6 presents all of the sink particle masses and distances from the central star +at the final times for the models. These distances correspond to observed separations in +the absence of any other knowledge of the orbital parameters, i.e., the semimajor axis and +eccentricity, The final masses range from ∼ 0.1MJup to ∼ 100MJup, i.e., sub-Jupiters to + +– 10 – +brown dwarfs and late M dwarf stars. Separations range from inside 1 au to over 30 au. +Ejected particles would be at much larger distances, were ejection permitted. +Figure 6 shows that the black dots, representing the 20 au radius disks, tend to have +higher masses (> 10MJup) inside 10 au than the blue dots, representing the 60 au radius +disks, which tend to have lower masses (< 1MJup) inside 10 au. This outcome is the result +of the 20 au radius disks all starting their evolutions from considerably more gravitationally +unstable initial states, i.e., Toomre Qminimum = 1.3 than the 60 au radius disks, with initial +Toomre Qminimum = 1.9 or 2.2. The 20 au radius models thus generally form more massive +sink particles, as would be expected. +Figure 7 shows the sink particle masses as a function of the orbital semimajor axis at the +final times for the models, while Figure 8 depicts these properties for the known exoplanets +on the same scales. Figure 3b shows that fragmenting dense clumps appear between about +5 au and 20 au, which is the same distance range as most of the sink particles in Figure 7; +only a few have migrated inside 1 au, and only a few orbit beyond about 20 au. Clearly +the present models produce a goodly number of cool gas giants and brown dwarfs, but do +not lend support for the formation and inward migration of the numerous hot and warm +exoplanets evident in Figure 8: little evidence for monotonic inward orbital migration is +seen. This result is consistent with the EDTONS models of Boss (2013). +Finally, Figure 9 shows the sink particle masses as a function of the orbital eccentricity +at the final times for the models, while Figure 10 depicts these properties for the known +exoplanets on the same scales. The present models show that the processes studied here of +fragmentation, mergers, chaotic orbits, and ejections result in the observed wide range of +eccentricities, though not the presumably tidally damped, near-zero eccentricities of the hot +Jupiters. +5. +Discussion +Drass et al. (2016) showed that the initial mass function in the Orion nebula cloud has +two peaks, one at 0.25 M⊙ and another at 0.025 M⊙, and suggested that the latter peak was +composed of brown dwarfs and isolated planetary-mass objects that had been ejected from +circumstellar disks or multiple star systems. The large number of attempted ejections in the +Enzo models that are listed in Table 2 fully support this hypothesis. +Feng et al. (2022) combined high-precision Doppler velocity data with Gaia and Hippar- +cos astrometry to constrain the masses and orbital parameters of 167 sub-stellar companions +to nearby stars. Their Figure 3 shows that these 167 companions fully populate a parameter + +– 11 – +space ranging from semimajor axes of ∼ 2 au to ∼ 40 au, with masses from ∼ 4MJup to +∼ 100MJup, much like the upper right quadrant of Figure 7. Their Figure 3 also shows orbital +eccentricities varying from 0 to 0.75, again in basic agreement with the range evident in the +present models in Figure 9. These Enzo models suggest a unified formation mechanism of +the 167 sub-stellar companions studied by Feng et al. (2022): fragmentation of MGU disks. +Galvagni et al. (2012) used a smoothed particle hydrodynamics (SPH) code to study +clumps formed at ∼ 100 au in a MGU disk, finding that the clumps could contract and heat +up enough to begin molecular hydrogen dissociation, resulting in a dynamical collapse phase +that can ensure their survival to tidal forces. Their results showed that this collapse phase +could occur within ∼ 103 yrs, shorter than the evolution times of the models considered here +(Table 2), justifying the replacement of dense clumps with Enzo sink particles or EDTONS +virtual protoplanets (e.g., Boss 2005, 2013). +Lichtenberg & Schleicher (2015) used Enzo to study fragments formed by the disk in- +stability process in isothermal disks, but did not employ sink particles or radiative transfer +effects, finding that the clumps formed were all lost by inward migration combined with the +tidal force of the protostar. Stamatellos (2015) used a SPH code to study disks with radii +of 100 au and high Toomre Q values. Planets inserted at 50 au either migrated inward or +outward over 2 × 104 yrs, depending on whether they were allowed to gain mass or not, +respectively. +Hall et al. (2017) used an SPH code to study the identification and interactions of disk +fragments composed of clumps of SPH particles that formed from the fragmentation of a +0.25M⊙ disk of radius 100 au around a 1M⊙ protostar. Their models showed that fragment- +fragment interactions early in the evolutions led to scattering of fragments to larger semi- +major axes, as large as 250 au, and to increased eccentricities, as high as 0.7. While the +periodic boundary conditions used in the present models preclude an assessment of the final +semi-major axes after close encounters, the fact that the sink particle velocities were often +sufficiently high to predict ejection from the system is consistent with the Hall et al. (2017) +results showing efficient scattering outward (cf., Table 2). The eccentricity pumping found +by Hall et al. (2017) is similarly consistent with that found in the present models (cf. Figure +9). +Hall et al. (2017) also studied tidal downsizing and disruption of fragments that ven- +tured too close to the tidal forces of the central protostar, finding that more clumps were +destroyed by tidal disruption than by disappearing in a merger event. Tidal downsizing was +proposed by Nayakshin (2010, 2017) as a means for forming inner rocky worlds from gas +giants formed in a disk instability, following the formation of rocky inner cores by the sed- +imentation of dust grains and pebbles to the center of the giant gaseous protoplanet (Boss + +– 12 – +1997). Tidal downsizing remains as a creative means to form inner rocky worlds as a result +of a gravitationally unstable gas disk. The present sink particle models, as well as the virtual +protoplanet models of the EDTONS code, do not allow tidal downsizing to occur, though +implicitly the loss of virtual protoplanets that hit the inner disk boundary in EDTONS code +calculations could be considered the equivalent of the loss of gas giant protoplanets by tidal +disruption. Modeling the interior structure and thermal evolution of slowly contracting gas +giant protoplanets is a future challenge for these types of models, and tidal disruption could +result in the loss of sink particles that pass close to the central protostar, though it can be +seen in Figure 6 that few sink particles passed inside 1 au. +Fletcher et al. (2019) performed a code comparison study of the orbital migration of +protoplanets inserted at 120 au in disks of 300 au radius, finding that protoplanets of 2 +MJup migrated inward to ∼ 40 au to ∼ 60 au within ∼ 104 yr. These code comparisons +differ considerably from the present models, as only single protoplanets were injected, the +disks used a γ = 7/5 adiabatic index, and the disks were gravitationally stable everywhere, +with Toomre Q ≥ 2. As a result, the evolutions did not undergo the chaotic evolutions of +the present models, where the MGU disk produces strong spiral arms that interact with the +numerous protoplanets that formed near the outset. +Finally, Rowther & Meru (2020) used a SPH code to study planet survival in self- +gravitating disks. They found that a fixed-mass planet with a range of masses would migrate +inward in the cool outer regions of their disks, but that this migration was halted once the +planet reached the warm inner disk. In their models, a single planet at a time is embedded +in a disk with a mild spiral arm structure. Compared to the multiple clumps, sink particles, +and strong spiral arms that form and interact in the present models (e.g., Figure 3), it is +clear that the Rowther & Meru (2020) planets do not undergo the chaotic orbital motions +experienced by the Enzo models here (or the EDTONS models of Boss 2013), which prevent +monotonic orbital migration. +6. +Conclusions +The use of a completely different three dimensional hydrodynamical code (Enzo 2.5), +with a completely different method for handling nascent protoplanets (sink particles), has +produced results in good agreement with those obtained by the EDTONS code and the +virtual protoplanet method (Boss 2005). Both codes agree that with high spatial resolution, +the standard model HR disk (Boss 2001) fragments rapidly into multiple dense clumps and +strong spiral arms. Both codes agree that when these clumps are replaced with particles that +can accrete mass from the disk, the particles grow in mass and can orbit chaotically for 1000 + +– 13 – +yrs to 2000 yrs without suffering monotonic inward or outward orbital migration. In addition, +the Enzo models show that the protoplanets have a high probability of close encounters with +each other, leading either to mergers, or to being ejected from the protoplanetary system. +Comparisons with the observational data on exoplanet demographics and FFPs suggest that +gas disk gravitational instabilities have an important role to play in explaining the formation +of sub-stellar companions with a wide range of masses and orbital distances. +I thank Sean Raymond for discussions about FFPs and Floyd Fayton for his invaluable +assistance with the memex cluster. I also thank the reviewer for providing several suggestions +for improving the manuscript. 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With six levels of refinement, the inner 1 au is better resolved, but otherwise +the initial disk is identical. + +Density +10~10 +10~12 +10~14 +1015 +10~16 +1017 +11.0 +1013 +3 +2 +(AU) +0 +X +-10 +-20 +0 +-10 +-20 +-30 +y (AU)– 20 – +! +" +# +$ +Fig. 3.— Log density cross-section in the disk midplane (z = 0) after 190 yr of evolution for +model 3-1K-20 without (a) and with (c) sink particles, and model 6-1K-20 without (b) and +with (d) sink particles. + +30 +104 +20 +103 +10 +102 +(nv) +10 +0 +100 +-10 +10-1 +-20 +10-2 +-20 +-10 +0 +10 +20 +30 +X (AU)30 +109 +20 +10-10 +1011 +10 +10-12 +(nv) +0 +1013 +Density +10~14 +-10 +1015 +10~16 +-20 +10-17 +-30 +-30 +-10 +20 +30 +10-18 +-20 +0 +10 +X (AU)30 +10-7 +20 +10-9 +10 +10-1L +(AU) +0 +Density +10-13 +-10 +10-15 +-20 +21-01 +3030 +20 +-10 +0 +10 +20 +30 +X (AU)30 +104 +20 +103 +10 +102 +10 +(AU) +0 +-10 +10-1 +20 +102 +-30 +10-3 +-30 +-20 +-10 +0 +10 +20 +30 +X (AU)– 21 – +! +" +# +$ +! +Fig. 4.— Log density cross-section in the disk midplane (z = 0) after 2000 yr of evolution +for (a) model 3-1K-20 and (b) model 6-1K-20, both with sink particles. + +30 +30 +a +103 +103 +20 +20 +102 +102 +10 +10 +10' +y (AU) +y (AU) +101 +0 +0 +10° +-10 +-10 +100 +10 +20 +102 +20 +10-1 +-30 +10-3 +3030 +-30 +-20 +-10 +0 +10 +20 +30 +-20 +-10 +0 +10 +20 +30 +x (AU) +X (AU)– 22 – +! +" +# +$ +! +Fig. 5.— Log density cross-section in the disk midplane (z = 0) for (a) model 6-2K-30 and +(b) model 7-2K-30 after 326 yr and 340 yr of evolution, respectively. + +30 +30 +10-10 +20 +20 +10-10 +10 +10 +10-12 +10-12 +(nv) +(nv) +0 +-10 +-10 +10-16 +10-16 +20 +20 +-30 +30 +20 +-10 +0 +10 +20 +30 +-20 +-10 +0 +10 +20 +30 +X (AU) +X (AU)– 23 – +Fig. 6.— Sink particle masses and distances from central star at the final times for the +models. These distances correspond to observed separations in the absence of knowledge of +the orbital parameters, i.e., the semi-major axis and eccentricity. Black dots are for models +that started with 20 au radius disks, red dots are for 30 au disks, and blue dots are for 60 +au radius disks (see Table 1). + +– 24 – +Fig. 7.— Sink particle masses as a function of the orbital semimajor axis at the final times +for the models. Black dots are for models that started with 20 au radius disks, red dots are +for 30 au disks, and blue dots are for 60 au radius disks (see Table 1). + +– 25 – +Fig. 8.— Exoplanet masses as a function of orbital semimajor axis from the Extrasolar +Planets Encyclopaedia (exoplanet.eu) as of 24 August 2022. + +1ie+2 +. +exoplanet.eu, +2e+1 +. +. +. +Semi-Major Axis (AU) +. +2e+0 +. +. +. +. +8 +. +8 +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +: +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +. +！ +. +. +. +1e+2 +1e+1 +Planetary Mass (Mjup)– 26 – +Fig. 9.— Sink particle masses as a function of the orbital eccentricity at the final times for +the models. Black dots are for models that started with 20 au radius disks, red dots are for +30 au disks, and blue dots are for 60 au radius disks (see Table 1). + +– 27 – +Fig. 10.— Exoplanet masses as a function of orbital eccentricity from the Extrasolar Planets +Encyclopaedia (exoplanet.eu) as of 24 August 2022. + +Orbital Eccentricity +0.5 +！ +1e+2 +1e+1 ++1 +3 +Planetary Mass (Mjup) \ No newline at end of file diff --git a/79AyT4oBgHgl3EQfc_dz/content/tmp_files/load_file.txt b/79AyT4oBgHgl3EQfc_dz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3b2f0cb0a2990e5244e0cbed1139c28286030516 --- /dev/null +++ b/79AyT4oBgHgl3EQfc_dz/content/tmp_files/load_file.txt @@ -0,0 +1,785 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf,len=784 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='00293v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='EP] 31 Dec 2022 Orbital Migration of Protoplanets in a Marginally Gravitationally Unstable Disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Migration, Merging, and Ejection Alan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Boss Earth & Planets Laboratory, Carnegie Institution for Science, 5241 Broad Branch Road, NW, Washington, DC 20015-1305 aboss@carnegiescience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='edu ABSTRACT Protoplanets formed in a marginally gravitationally unstable (MGU) disk by either core accretion or disk instability will be subject to dynamical interactions with massive spiral arms, possibly resulting in inward or outward orbital migra- tion, mergers with each other, or even outright ejection from the protoplanetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The latter process has been hypothesized as a possible formation sce- nario for the unexpectedly high frequency of unbound gas giant exoplanets (free floating planets, FFP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Previous calculations with the EDTONS fixed grid three dimensional (3D) hydrodynamics code found that protoplanets with masses from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='01 M⊕ to 3 MJup could undergo chaotic orbital evolutions in MGU disks for ∼ 1000 yrs without undergoing monotonic inward or outward migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Here the Enzo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='5 adaptive mesh refinement (AMR) 3D hydrodynamics code is used to follow the formation and orbital evolution of protoplanets in MGU disks for up to 2000 yrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The Enzo results confirm the basic disk fragmentation results of the EDTONS code, as well as the absence of monotonic inward or outward orbital migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' In addition, Enzo allows protoplanet mergers to occur, unlike EDTONS, resulting in a significant decrease in the number of protoplanets that survive for 1000 to 2000 yrs in the Enzo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' These models also imply that gas giants should be ejected frequently in MGU disks that fragment into large numbers of protoplanets, supporting ejection as a possible source mechanism for the observed FFPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Subject headings: planets and satellites: formation — protoplanetary disks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Introduction Exoplanet demographics provide one of the ultimate arbiters of theories of exoplanet formation and evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2019) used the GPI exoplanet survey to search – 2 – for planets with masses between 2 and 13 MJup and semimajor axes between 3 and 100 au, finding that the peak occurrence distance of giant planets was in the range of 1 to 10 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2021) found the same peak occurrence distance of 1 to 10 au for the California Legacy Doppler velocities survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Vigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2021) showed that the VLT SPHERES direct imaging survey of 150 stars detected 13 sub-stellar companions with masses between 1 and 75 MJup and semimajor axes between 5 and 300 au, finding that both core accretion (CA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Mizuno 1980) and disk instability (DI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Boss 1997) appeared necessary to explain the detections for the FGK stars in their sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Gas giant planets with orbital distances as large as 980 au have been discovered and studied (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Forming giant exoplanets at such large distances by CA within the ∼ 1 Myr lifetimes of the gaseous portion of protoplanetary disks is challenging (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Chambers 2021), if not impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' DI has the advantage of forming dense, self-gravitating clumps in a few orbital periods, relaxing the disk lifetime constraint for forming wide-orbit gas giants in situ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Boss 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Evidence for a gas giant protoplanet embedded in a spiral arm 93 au from AB Aurigae has been interpreted as an example of gas giant planet formation by DI (Currie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Cadman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' DI has also been proposed as the source of the ∼ 10MJup exoplanet that orbits ∼ 560 au from the massive binary b Centauri (Janson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Goda & Matsuo (2019) examined the demographics of 485 planetary systems and concluded that a hybrid theory of planet formation, involving both CA and DI, was needed to explain the exoplanet detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Miret-Roig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2022) used a direct imaging survey coupled with Gaia and Hipparchos astrometry to search for unbound gas giant exoplanets in the Upper Scorpius and Ophiuchus young stellar association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Their survey yielded between 70 and 170 free floating planets (FFP), considerably more than might be expected to form as the tail end of the star formation process of molecular cloud core collapse and fragmentation, and suggested that ejection from unstable planetary systems might make a major contribution during the first 10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Gravitational microlensing has also found an abundance of likely FFPs, though these could also simply be bound planets with orbital distances greater than about 10 au (Mr´oz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Ryu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Vorobyov (2016) performed numerical simulations that supported the hypothesis that FFPs might be the result of planets ejected from massive MGU disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' While exoplanet demographics reveal orbital characteristics at the present epoch, unless exoplanets do not undergo significant orbital evolution or migration following their formation, the present epoch orbital parameters are of limited usefulness in constraining their initial orbital distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' CA is the favored mechanism closer to the host star, as a result of shorter orbital periods, higher gas disk temperatures, and higher surface densities of solids, to name a few factors, while DI may be more effective at larger distances in suitably massive and cool – 3 – protoplanetary disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' For either CA or DI, a key question then becomes the extent to which protoplanets might migrate away from their birth orbits to their present epoch orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' As noted by Boss (2013), CA and DI both require giant protoplanets to form in the presence of disk gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Theoretical work on protoplanetary orbital migration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Kley & Nelson 2012) usually focuses on protoplanets in disks where the disk mass is low enough that the disk self-gravity can be neglected, greatly simplifying the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Protoplanet evolution in MGU disk models has been calculated by Boss (2005), Baruteau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2011), and Michael et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' These studies each considered quite different initial conditions and found a wide range of outcomes, ranging from large-scale inward orbital migration to relatively little orbital migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Boss (2013) studied the evolution of protoplanets formed by either CA and DI in MGU disks, noting that while a MGU disk is essential for formation by DI, even a giant planet formed by CA in a quiescent, non-MGU disk can experience a later phase of MGU disk interactions during the periodic FU Orionis outbursts experienced by young solar-type protostars, which are thought to involve a phase of disk gravitational instability that dumps disk mass onto the protostar (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Kuffmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Dunhill (2018) similarly suggested that giant planets formed by CA might undergo orbital migration during FU Orionis outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The Boss (2005, 2013) models were performed using the EDTONS three dimensional radiative hydrodynamics code, with a spherical coordinate grid that was fixed at moderate spatial resolution throughout the MGU disk evolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Virtual protoplanets were introduced at the beginning of each model to represent protoplanets as point sources of gravity, able to interact gravitationally with the disk and with each other and to accrete mass from the disk by Bondi-Hoyle accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Boss (2013) found that protoplanets with initial masses in the range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='01 M⊕ to 3 MJup and initial orbital distances of 6 to 12 au in a MGU disk around a solar-mass protostar underwent chaotic orbital evolutions for ∼ 1000 yr without undergoing the monotonic inward or outward migration that typically characterizes the Type I or Type II behavior of non-self-gravitating disk models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Kley & Nelson 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The present models of protoplanet orbital evolution employ the Enzo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='5 hydrodynamics code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Enzo is also a three dimensional (3D) code and uses Adaptive Mesh Refinement (AMR) in Cartesian coordinates to ensure that sharp gradients in fluid quantities such as shock fronts can be handled accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Enzo is able to replace exceptionally dense disk clumps with sink particles representing newly formed (by DI) protoplanets, which thereafter interact with each other and the disk while accreting disk gas, as do the virtual protoplanets in the Boss (2013) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' We thus seek here to use a completely different 3D hydro code to learn more about the possible outcomes for protoplanet orbital evolution in MGU disks, and to compare the results with the latest advances in exoplanet demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' – 4 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Numerical Hydrodynamics Code As noted by Boss & Keiser (2013), the Enzo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='5 AMR code performs hydrodynamics (HD) using any one of three different algorithms (Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Bryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2014): (1) the piecewise parabolic method (PPM) of Colella & Woodward (1984), (2) the ZEUS method of Stone & Norman (1992), or (3) a Runge–Kutta third-order-based MUSCL (“monotone upstream-centered schemes for conservation laws”) algorithm based on the Godunov (1959) shock-handling HD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Enzo is designed for handling strong shock fronts by solving the Riemann problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Godunov 1959) for discontinuous solutions of a fluid quantity that should be conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The PPM option was used in the current models as a result of the testing on mass and angular momentum conservation performed with Enzo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='0 by Boss & Keiser (2013), who found that PPM was better able to conserve mass and angular momentum during the collapse of a rotating isothermal cloud core (Boss & Bodenheimer 1979) than either ZEUS or MUSCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Enzo is designed for parallel processing on high performance clusters (HPC), but when run on a single, dedicated 32-core node of the Carnegie memex HPC, a typical model still required 7 months of continuous computation to evolve for ∼ 103 yrs of model time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The Enzo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='5 models were initialized on a 3D Cartesian grid with 32 top grid points in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' A maximum of 7 levels of refinement was used, with a factor of two refinement occurring for each level, so that the maximum possible effective grid resolution was 27 = 128 times higher than the initial resolution of 323, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', 40963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The models with 7 levels needed an increase in the number of cell buffer zones (NumberBufferZones) to 3 from the default value of 1, which was used for the lower levels of refinement, in order to maintain reasonable time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Grid refinement was performed whenever necessary to ensure that the Jeans length constraint (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Truelove et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Boss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2000) was satisfied by a factor of 4 for cells with a density at least eight times the initial density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Periodic boundary conditions were applied on each face of the grid cubic box, with each side either 60 au or 120 au in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' A point source of external gravity was used to represent a 1 M⊕ protostar at the center of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The maximum number of Green’s functions used to calculate the gravitational potential was 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The time step typically used was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='15 of the limiting Courant time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The results were analyzed with the yt astrophysical analysis and visualization toolkit (Turk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Following Boss & Keiser (2014), we used the Enzo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='2 sink particle coding described by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Sink particles are created in grid cells that have already been refined to the maximum extent permitted by the specified number of levels of grid refinement, but where the gas density still exceeds that consistent with the Jeans length criterion for avoiding spurious fragmentation (Truelove et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Boss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' As described by – 5 – Boss & Keiser (2014), sink particles accrete gas from their host cells at the modified Bondi- Hoyle accretion rate proposed by Ruffert (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Two parameters control the conditions under which sink particles can be merged together: the merging mass (SinkMergeMass) and the merging distance (SinkMergeDistance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The former of these two parameters is used to divide the sink particles into either large or small particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Particles with less mass than SinkMergeMass are first subjected to being combined with any large particles that are located within the SinkMergeDistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Any surviving small particles after this first step are then merged with any other small particles within the SinkMergeDistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The merging process is performed in such a way as to ensure conservation of mass and linear momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Boss & Keiser (2014) found that their results for collapse and fragmentation of magnetic molecular cloud cores were not particularly sensitive to the choice of these two key parameters with regard to the tendency of the cores to undergo fragmentation into multiple protostar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The current paper uses the Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2010) sink particle coding with the SinkMergeMass set equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='01 MJup and the SinkMergeDistance set equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='1 au, appropriate values for studying gas giant protoplanets in a 120 au-size region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Sink creation was only allowed for cells with densities exceeding the values listed in Table 1 (DensThresh in code units in the sink maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='C subroutine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' These densities were chosen to be low enough that sinks do form in the models, as the point of the present models was to study the orbital evolution of sink particles representing protoplanets in MGU disks rather than to study the precise physics of DI-induced fragmentation and clump formation in such disks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Boss 2021a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The sink particles used in the Enzo models are similar to the virtual protoplanets (VPs) used in the EDTONS models: both are introduced in regions of density maxima and are intended to represent gravitationally bound clumps of disk gas that will contract to form gaseous protoplanets, as they orbit in the disk around the central protostar, interacting gravitationally with each other and the disk gas, even as they accrete more disk gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' There are several differences, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Sink particles are created automatically by Enzo following the criteria noted above, sink particles with close encounters can be merged together, and sink particles that encounter a grid boundary reappear on the opposite boundary as a result of the periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' VPs, on the other hand, are inserted when a density maximum exceeds the Jeans length or Toomre length criteria (Nelson 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Boss 2021a,b) for the current grid spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' VPs may undergo close encounters with each other but do not suffer mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' VPs that strike either the inner or outer grid boundary are removed from the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' While it would be desirable to compare flux-limited diffusion (FLD) approximation radiative hydrodynamic models from the EDTONS code with FLD radiative hydrodynamic models calculated by Enzo, the FLD routines available in Enzo are limited to non-local thermodynamic equilibrium (non-LTE), as Enzo was developed primarily for cosmological – 6 – simulations, whereas EDTONS assumes LTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' As a result, we are limited to using a simpler approach to handling the disk thermodynamics with the Enzo code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Boss (1998) showed that disk fragmentation could occur for strongly gravitationally unstable disks with either locally isothermal or locally adiabatic thermodynamics, using disk gas adiabatic exponents ranging from γ = 1 (purely isothermal) to γ = 7/5, which is appropriate for molecular hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Given that disks are subject to compressional heating, γ = 1 is not strictly correct, and given that disks that are optically thick at their midplanes can cool from their surfaces, γ = 7/5 is not strictly correct either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The physically correct behavior presumably lies somewhere in the middle of these two extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Radiative cooling in optically thin regions was employed in the Enzo models, with a critical density for cooling of 10−13 g cm−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' regions with densities above this critical value had the cooling rate decreased proportionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' This critical density was chosen because that is the disk midplane density where the dust grain opacity produces optical depths of order unity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Boss 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The cooling rates were modified from the default values in cool rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='in to rates consistent with molecular line cooling in optically thin regions (Boss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Neufeld & Kaufman 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Because Enzo PPM hydrodynamics involves a Riemann solver that cannot be purely isothermal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', γ cannot equal unity, the adiabatic index for the disk gas was taken to be γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='001, appropriate for a nearly isothermal, but still adiabatic equation of state for an ideal gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Test runs were computed for 100 yrs of evolution with both γ = 7/5 and γ = 5/3, but in both cases Enzo produced midplane disk temperatures that were over 104 K, whereas the initial disk had a maximum midplane temperature of 1500 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The test runs with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='001 produced the expected maximum temperatures of ∼ 1500 K, and hence γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='001 was adopted for the models presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The resulting temperature distributions were also affected by the assumption of radiative cooling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' spiral features in the midplane temperature distribution accompanied spiral features in the midplane density distribution, as is to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Finally, the mean molecular weight of the gas was effectively taken to be µ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='4, appropriate for a solar composition mixture of molecular hydrogen and helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Initial Conditions Table 1 lists the models with variations in the number of levels of grid refinement, the outer disk and envelope temperatures, initial minimum value of the Toomre (1964) Q parameter, disk radius, calculational grid box size, and critical density for sink particle creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' A 60 au box size was used for the 20 au and 30 au radius disks, while a 120 au box size was used for 60 au radius disks, in order to give the disks sufficient room to evolve and – 7 – expand by the outward transport of angular momentum through gravitational interactions with the spiral arms and clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' In the number of levels column, 34 means the model was initially run with 3 levels and then a fourth level of refinement was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The initial disks are based on the model HR disk from Boss (2001), with an outer disk temperature of 40 K and and disk envelope temperature of 50 K, which has been used as a standard initial model for many of the author’s disk instability models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Boss 2021a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Model HR has an initial minimum Toomre Q ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='3, implying marginal stability to the growth of rings and spiral arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The model HR initial disk has a mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='091 M⊙ within an inner radius of 4 au and an outer radius of 20 au and orbits a 1 M⊙ central protostar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The Enzo models have have masses of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='102 M⊙ for 20 au outer radius disks, slightly higher than in model HR because the Enzo models extend inward to 1 au, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='142 M⊙ for 30 au outer radius disks, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='210 M⊙ for 60 au outer radius disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The same disk density power-law-like Keplerian structure as in Boss (2001) is used for all of the models, with the structure being terminated at 20 au, 30 au, or 60 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Figures 1 and 2 show cross sections of the initial disk density distribution for the 20 au disks, both parallel and perpendicular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', disk midplane) to the disk rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Results Figure 3 shows the intermediate results for two of the four models that have the identical initial disk configuration (20 au radius) as the Boss (2001) model HR, depicted at the same time (190 yrs of evolution) as the same initial disk model (fldA) in Boss (2021b, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Figure 3 shows that both of these models (3-1K-20 and 6-1K-20) rapidly evolved into a configuration of multiple spiral arms interspersed with dense clumps, as expected for a marginally gravitationally unstable disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Also as expected, the degree of fragmentation and clump formation increases as the numerical grid resolution increases from 3 to 6 levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' When sink particles are allowed to form, the number of sink particles similarly increases as the resolution is improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' While the background disk looks quite similar for model 3-1K-20 with or without sink particles (Figure 3a,c), there is a clear difference in the case of model 6-1K-20 (Figure 3b,d), where the background disk has become perturbed into a prolate configuration due to the formation of a massive (∼ 20MJup) secondary companion (at one o’clock), with its own circumplanetary disk and tertiary companion, whose combined tidal forces have evidently distorted the disk’s overall appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Model fldA of Boss (2021b) had fragmented into a five clumps and three virtual protoplanets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', sink particles) by 189 yrs, for a total of eight, considerably more than formed in the present model 3-1K-20, but not as many as in model 6-1K-20, suggesting that even with the quadrupled spatial – 8 – resolution of the Boss (2021b) EDTONS models, the adaptive mesh refinement feature of Enzo results in significantly improved numerical spatial resolution of the disk instability and fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Confirmation of the formation of long-lived fragments in the model HR disk (Boss 2001, 2021b) with the completely different hydrodynamical method used here provides strong support for the viability of the disk instability mechanism for the formation of gas giant protoplanets and higher mass companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Figure 4 displays the results after 2000 yrs for the Enzo models in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The EDTONS model fldA in Boss (2021b) was stopped after only 189 yrs of evolution, but even still required over 4 years of computation on a single core of a node on memex, a Carnegie Institution computer cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' EDTONS is based on code initially written in the late 1970s and is not parallelized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Enzo, in contrast, is a modern code designed to run on parallel processing systems like memex, and as a result the Enzo models can be computed much farther in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Even still, model 3-1K-20 required one week to run for 2000 yrs of model time on a dedicated single memex node with 28 cores, while model 6-1K-20 required one year to run 2000 yrs on a dedicated 28-core node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Three dimensional hydrodynamics at high spatial resolution is computationally expensive, even when a parallelized code is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Figure 4 shows that the evolution of these two models diverged considerably following the early fragmentation phase depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The two sink particles evident in Figure 4a have masses of ∼ 2MJup and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='6MJup, with a total gas disk mass of ∼ 99MJup, while the 13-odd sink particles in Figure 4b have masses ranging from ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='2MJup to ∼ 23MJup, for a total sink particle mass of ∼ 96MJup, leaving a disk gas mass of only ∼ 5MJup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Clearly, the final disk mass in model 3-1K-20 far outweighs the mass of the sink particles, and as a result the particles are unable to open gaps in the disk, though the disk has expanded outward to a radius of about 30 au as a result of the transport of disk mass and angular momentum outward, caused by the strong spiral arms evident in Figure 3a,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The fact that sink particle formation has been so efficient in model 6-1K-20, with the total particle mass some 20 times larger than the disk mass, means that the particles rule the evolution and are able to clear out a distinct inner gap, centered on about 5 au (Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' In model 6-1K-20, the sink particles gained the bulk of the disk’s mass and angular momentum, so that the disk is not able to expand beyond its initial radius of 20 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Three sink particles were accelerated to speeds high enough at their orbital location to be ejected altogether from the system, but because of the periodic boundary conditions imposed on the calculations by the Enzo self-gravity solver, these ejectable particles were returned to the system and underwent further interactions with the sink particles and disk gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Table 2 gives the maximum number of sink particles formed for all the models, as well as the number surviving at the end of the run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Table 2 shows that the maximum number of – 9 – sinks formed decreases as the initial disk gas temperature is increased, as this results in an increase in the Toomre minimum Q value (Table 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', in greater stability to the growth of rings and spiral arms, and hence to fragmentation and sink particle formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' By the time that the disk temperature is increased to 160 K, disk fragmentation is completely stifled in the Enzo models, consistent with the flux-limited diffusion approximation models of Boss (2021b), where fragmentation ceased for a minimum Toomre Q greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Models 6-2K-30 and 7-2K-30 did not form sink particles, unlike the otherwise identical, but lower resolutions models 3-2K-30, 4-2K-30, and 5-2K-30, because this sequence used a fixed critical density for sink particle formation of 10−9 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' That choice meant that the dense clumps formed in the two higher resolution models could always be resolved with more grid levels and finer spatial resolution, thereby preventing the clumps from exceeding the critical density required for sink particle formation, at least during the limited amount of model time that the 6- and 7-level models were able to be evolved (326 and 340 yrs, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Small time steps prevented these two models for being evolved farther in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Figure 5 shows these two models at their final times, showing that the spiral arms and nascent clumps become more distinct as the number of grid levels is increased, as expected when approaching the continuum limit of infinite spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Table 2 also lists the number of sink particles mergers, where Nmerged−sinks = Nmax−sinks− Nfinal−sinks, and the number of times that a sink particle would have been ejected if periodic boundary conditions were not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Table 2 shows that mergers of sink particles are quite common in all of the models that formed sink particles, and evidently are responsi- ble for much of the gain in mass of the particles, along with the ongoing accretion of disk gas, given that the number of mergers is usually comparable to, or far greater than, the final number of sink particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The value Nescaped−sinks can be quite large due to the sink particles’ inability to escape the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' often the same particle bounces in and out in orbital radius and achieves escape velocity multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Achieving the escape velocity usually occurs for particle orbital distances of 30 au to 40 au, but can also occur from 10 au to 20 au in the more unstable disks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', 5-1K-20, 6-1K-20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The large numbers of escape episodes in the latter two models are clearly solely a result of the periodic boundary conditions, but they do indicate that ejected protoplanets are to be expected as a natural outcome of a phase of gas disk gravitational instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Table 2 suggests that such a phase of protoplanetary disk evolution should result in the ejection of several gas giant protoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Figure 6 presents all of the sink particle masses and distances from the central star at the final times for the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' These distances correspond to observed separations in the absence of any other knowledge of the orbital parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', the semimajor axis and eccentricity, The final masses range from ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='1MJup to ∼ 100MJup, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', sub-Jupiters to – 10 – brown dwarfs and late M dwarf stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Separations range from inside 1 au to over 30 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Ejected particles would be at much larger distances, were ejection permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Figure 6 shows that the black dots, representing the 20 au radius disks, tend to have higher masses (> 10MJup) inside 10 au than the blue dots, representing the 60 au radius disks, which tend to have lower masses (< 1MJup) inside 10 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' This outcome is the result of the 20 au radius disks all starting their evolutions from considerably more gravitationally unstable initial states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Toomre Qminimum = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='3 than the 60 au radius disks, with initial Toomre Qminimum = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='9 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The 20 au radius models thus generally form more massive sink particles, as would be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Figure 7 shows the sink particle masses as a function of the orbital semimajor axis at the final times for the models, while Figure 8 depicts these properties for the known exoplanets on the same scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Figure 3b shows that fragmenting dense clumps appear between about 5 au and 20 au, which is the same distance range as most of the sink particles in Figure 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' only a few have migrated inside 1 au, and only a few orbit beyond about 20 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Clearly the present models produce a goodly number of cool gas giants and brown dwarfs, but do not lend support for the formation and inward migration of the numerous hot and warm exoplanets evident in Figure 8: little evidence for monotonic inward orbital migration is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' This result is consistent with the EDTONS models of Boss (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Finally, Figure 9 shows the sink particle masses as a function of the orbital eccentricity at the final times for the models, while Figure 10 depicts these properties for the known exoplanets on the same scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The present models show that the processes studied here of fragmentation, mergers, chaotic orbits, and ejections result in the observed wide range of eccentricities, though not the presumably tidally damped, near-zero eccentricities of the hot Jupiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Discussion Drass et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2016) showed that the initial mass function in the Orion nebula cloud has two peaks, one at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='25 M⊙ and another at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='025 M⊙, and suggested that the latter peak was composed of brown dwarfs and isolated planetary-mass objects that had been ejected from circumstellar disks or multiple star systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The large number of attempted ejections in the Enzo models that are listed in Table 2 fully support this hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2022) combined high-precision Doppler velocity data with Gaia and Hippar- cos astrometry to constrain the masses and orbital parameters of 167 sub-stellar companions to nearby stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Their Figure 3 shows that these 167 companions fully populate a parameter – 11 – space ranging from semimajor axes of ∼ 2 au to ∼ 40 au, with masses from ∼ 4MJup to ∼ 100MJup, much like the upper right quadrant of Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Their Figure 3 also shows orbital eccentricities varying from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='75, again in basic agreement with the range evident in the present models in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' These Enzo models suggest a unified formation mechanism of the 167 sub-stellar companions studied by Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2022): fragmentation of MGU disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Galvagni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2012) used a smoothed particle hydrodynamics (SPH) code to study clumps formed at ∼ 100 au in a MGU disk, finding that the clumps could contract and heat up enough to begin molecular hydrogen dissociation, resulting in a dynamical collapse phase that can ensure their survival to tidal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Their results showed that this collapse phase could occur within ∼ 103 yrs, shorter than the evolution times of the models considered here (Table 2), justifying the replacement of dense clumps with Enzo sink particles or EDTONS virtual protoplanets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Boss 2005, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Lichtenberg & Schleicher (2015) used Enzo to study fragments formed by the disk in- stability process in isothermal disks, but did not employ sink particles or radiative transfer effects, finding that the clumps formed were all lost by inward migration combined with the tidal force of the protostar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Stamatellos (2015) used a SPH code to study disks with radii of 100 au and high Toomre Q values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Planets inserted at 50 au either migrated inward or outward over 2 × 104 yrs, depending on whether they were allowed to gain mass or not, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2017) used an SPH code to study the identification and interactions of disk fragments composed of clumps of SPH particles that formed from the fragmentation of a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='25M⊙ disk of radius 100 au around a 1M⊙ protostar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Their models showed that fragment- fragment interactions early in the evolutions led to scattering of fragments to larger semi- major axes, as large as 250 au, and to increased eccentricities, as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' While the periodic boundary conditions used in the present models preclude an assessment of the final semi-major axes after close encounters, the fact that the sink particle velocities were often sufficiently high to predict ejection from the system is consistent with the Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2017) results showing efficient scattering outward (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The eccentricity pumping found by Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2017) is similarly consistent with that found in the present models (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2017) also studied tidal downsizing and disruption of fragments that ven- tured too close to the tidal forces of the central protostar, finding that more clumps were destroyed by tidal disruption than by disappearing in a merger event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Tidal downsizing was proposed by Nayakshin (2010, 2017) as a means for forming inner rocky worlds from gas giants formed in a disk instability, following the formation of rocky inner cores by the sed- imentation of dust grains and pebbles to the center of the giant gaseous protoplanet (Boss – 12 – 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Tidal downsizing remains as a creative means to form inner rocky worlds as a result of a gravitationally unstable gas disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The present sink particle models, as well as the virtual protoplanet models of the EDTONS code, do not allow tidal downsizing to occur, though implicitly the loss of virtual protoplanets that hit the inner disk boundary in EDTONS code calculations could be considered the equivalent of the loss of gas giant protoplanets by tidal disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Modeling the interior structure and thermal evolution of slowly contracting gas giant protoplanets is a future challenge for these types of models, and tidal disruption could result in the loss of sink particles that pass close to the central protostar, though it can be seen in Figure 6 that few sink particles passed inside 1 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' (2019) performed a code comparison study of the orbital migration of protoplanets inserted at 120 au in disks of 300 au radius, finding that protoplanets of 2 MJup migrated inward to ∼ 40 au to ∼ 60 au within ∼ 104 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' These code comparisons differ considerably from the present models, as only single protoplanets were injected, the disks used a γ = 7/5 adiabatic index, and the disks were gravitationally stable everywhere, with Toomre Q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' As a result, the evolutions did not undergo the chaotic evolutions of the present models, where the MGU disk produces strong spiral arms that interact with the numerous protoplanets that formed near the outset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Finally, Rowther & Meru (2020) used a SPH code to study planet survival in self- gravitating disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' They found that a fixed-mass planet with a range of masses would migrate inward in the cool outer regions of their disks, but that this migration was halted once the planet reached the warm inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' In their models, a single planet at a time is embedded in a disk with a mild spiral arm structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Compared to the multiple clumps, sink particles, and strong spiral arms that form and interact in the present models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Figure 3), it is clear that the Rowther & Meru (2020) planets do not undergo the chaotic orbital motions experienced by the Enzo models here (or the EDTONS models of Boss 2013), which prevent monotonic orbital migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Conclusions The use of a completely different three dimensional hydrodynamical code (Enzo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='5), with a completely different method for handling nascent protoplanets (sink particles), has produced results in good agreement with those obtained by the EDTONS code and the virtual protoplanet method (Boss 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Both codes agree that with high spatial resolution, the standard model HR disk (Boss 2001) fragments rapidly into multiple dense clumps and strong spiral arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Both codes agree that when these clumps are replaced with particles that can accrete mass from the disk, the particles grow in mass and can orbit chaotically for 1000 – 13 – yrs to 2000 yrs without suffering monotonic inward or outward orbital migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' In addition, the Enzo models show that the protoplanets have a high probability of close encounters with each other, leading either to mergers, or to being ejected from the protoplanetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Comparisons with the observational data on exoplanet demographics and FFPs suggest that gas disk gravitational instabilities have an important role to play in explaining the formation of sub-stellar companions with a wide range of masses and orbital distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' I thank Sean Raymond for discussions about FFPs and Floyd Fayton for his invaluable assistance with the memex cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' I also thank the reviewer for providing several suggestions for improving the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The computations were performed on the Carnegie Institu- tion memex computer cluster (hpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='carnegiescience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='edu) with the support of the Carnegie Scientific Computing Committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' The computations were performed using the Enzo code originally developed by the Laboratory for Computational Astrophysics at the University of California San Diego and now available at https://enzo-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' REFERENCES Baruteau, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', Meri, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', & Paadekooper, S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' – 16 – Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Initial conditions for the models with varied maximum number of refinement levels, initial outer disk and envelope temperatures (K), initial minimum Toomre Q, outer disk radii (au), box size (au), and critical density needed for sink particle formation (cgs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Model Nlevels Tdisk Tenvelope Qminimum Rdisk Sbox ρcrit−sinks 3-1K-20 3 40 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} 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4-3K-60-11 4 120 120 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='2 60 120 10−11 3-4K-60-10 3 160 120 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='5 60 120 10−10 3-4K-60-11 3 160 120 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='5 60 120 10−11 – 17 – Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Results for the models, showing the maximum number of sinks formed, final number of sinks, number of times a sink reached escape velocity, number of sinks lost to mergers, and final time (yrs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='— Initial log density cross-section in a vertical section (x = 0) showing the entire computational grid with a maximum of three levels of refinement for the 20 au outer disk radius models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Density 01- 10°12 C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='01 b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='01 ST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='O1 LI- 3 2 (nv) 0 y 01- 20 3 10 20 30 z (AU)– 19 – Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='— Initial log density cross-section in the disk midplane (z = 0) showing the entire computational grid with a maximum of three levels of refinement for the 20 au outer disk radius models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' With six levels of refinement, the inner 1 au is better resolved, but otherwise the initial disk is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Density 10~10 10~12 10~14 1015 10~16 1017 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='0 1013 3 2 (AU) 0 X 10 20 0 10 20 30 y (AU)– 20 – !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' " # $ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='— Log density cross-section in the disk midplane (z = 0) after 190 yr of evolution for model 3-1K-20 without (a) and with (c) sink particles, and model 6-1K-20 without (b) and with (d) sink particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 30 104 20 103 10 102 (nv) 10 0 100 10 10-1 20 10-2 20 10 0 10 20 30 X (AU)30 109 20 10-10 1011 10 10-12 (nv) 0 1013 Density 10~14 10 1015 10~16 20 10-17 30 30 10 20 30 10-18 20 0 10 X (AU)30 10-7 20 10-9 10 10-1L (AU) 0 Density 10-13 10 10-15 20 21-01 3030 20 10 0 10 20 30 X (AU)30 104 20 103 10 102 10 (AU) 0 10 10-1 20 102 30 10-3 30 20 10 0 10 20 30 X (AU)– 21 – !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' " # $ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='— Log density cross-section in the disk midplane (z = 0) after 2000 yr of evolution for (a) model 3-1K-20 and (b) model 6-1K-20, both with sink particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=" 30 30 a 103 103 20 20 102 102 10 10 10' y (AU) y (AU) 101 0 0 10° 10 10 100 10 20 102 20 10-1 30 10-3 3030 30 20 10 0 10 20 30 20 10 0 10 20 30 x (AU) X (AU)– 22 – !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' " # $ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='— Log density cross-section in the disk midplane (z = 0) for (a) model 6-2K-30 and (b) model 7-2K-30 after 326 yr and 340 yr of evolution, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 30 30 10-10 20 20 10-10 10 10 10-12 10-12 (nv) (nv) 0 10 10 10-16 10-16 20 20 30 30 20 10 0 10 20 30 20 10 0 10 20 30 X (AU) X (AU)– 23 – Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='— Sink particle masses and distances from central star at the final times for the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' These distances correspond to observed separations in the absence of knowledge of the orbital parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=', the semi-major axis and eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Black dots are for models that started with 20 au radius disks, red dots are for 30 au disks, and blue dots are for 60 au radius disks (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' – 24 – Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='— Sink particle masses as a function of the orbital semimajor axis at the final times for the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Black dots are for models that started with 20 au radius disks, red dots are for 30 au disks, and blue dots are for 60 au radius disks (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' – 25 – Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='— Exoplanet masses as a function of orbital semimajor axis from the Extrasolar Planets Encyclopaedia (exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='eu) as of 24 August 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 1ie+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='eu, 2e+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Semi-Major Axis (AU) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 2e+0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 8 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 1e+2 1e+1 Planetary Mass (Mjup)– 26 – Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='— Sink particle masses as a function of the orbital eccentricity at the final times for the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Black dots are for models that started with 20 au radius disks, red dots are for 30 au disks, and blue dots are for 60 au radius disks (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' – 27 – Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='— Exoplanet masses as a function of orbital eccentricity from the Extrasolar Planets Encyclopaedia (exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='eu) as of 24 August 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' Orbital Eccentricity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content='5 ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} +page_content=' 1e+2 1e+1 +1 3 Planetary Mass (Mjup)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AyT4oBgHgl3EQfc_dz/content/2301.00293v1.pdf'} diff --git a/7tE1T4oBgHgl3EQfnQST/content/tmp_files/2301.03307v1.pdf.txt b/7tE1T4oBgHgl3EQfnQST/content/tmp_files/2301.03307v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ca3e19e2bcbeb858820a887402bb8d15582dd784 --- /dev/null +++ b/7tE1T4oBgHgl3EQfnQST/content/tmp_files/2301.03307v1.pdf.txt @@ -0,0 +1,3162 @@ +MNRAS 000, 1–20 (2022) +Preprint 10 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Isolating the Extreme Debris Disk Signature - Explorations of Eccentric +Extreme Debris Disks Formed by Giant Impacts +Thomas Lewis,★ Lewis Watt, and Zoë M. Leinhardt +School of Physics, University of Bristol, H. H. Wills Physics Laboratory, Tyndall Avenue, Bristol, BS8 1TL, UK +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +In this work we used 𝑁-body simulations and a radiative transfer package to model the evolution of eccentric debris disks +produced by giant impacts between planetary embryos. This included how the morphology and infrared emission of these disks +varied with embryo eccentricity and collision true anomaly. We found that eccentric disks inherit the eccentric properties of +the centre of mass orbit of the two colliding embryos. However, the orientation of the collision with the respect to this orbit +plays a key role in determining how closely the disk material resembles the centre of mass orbit. Additionally, we found that +increased eccentricity acted to suppress the formation of certain short-term variations in the disk emission depending on the +collision position. These short-term variations have been associated with an observational phenomenon called extreme debris +disks. Short-term variability has been suggested as a potential signature for giant impacts. +Key words: circumstellar matter – planets and satellites: formation – method: numerical +1 INTRODUCTION +Debris disks are one of the most useful observational features of +solar systems, as they encode a large amount of information on the +evolution of a system and are fairly easy to observe around other stars. +Additionally, debris disks appear to be quite common with examples +in our own Solar System in the form of the Asteroid Belt and the +Kuiper Belt, as well as in many other systems (e.g. Hughes et al. +2018). +Traditional debris disks are belts of material orbiting around stars. +This material is composed of particles of a range of sizes from +larger planetesimals to smaller dust particles. Debris disks are most +commonly detected through observation of excess infrared emission +from a star. The dust grains in debris disks are heated by their host star +and re-radiate energy in the infrared. This dust emission is visible in +the spectral energy density (SED) profile of the star as a small bump +in the IR wavelengths when compared to a pure stellar blackbody. +Debris disks are often characterised using fractional luminosity, 𝑓 = +𝐿𝑑𝑖𝑠𝑘/𝐿∗, which compares the disk luminosity (𝐿𝑑𝑖𝑠𝑘) to that of the +host star (𝐿∗) (e.g. Wyatt 2008). Typically, the fractional luminosity +of a debris disk is 𝑓 < 10−3 (Meng et al. 2015). +Most debris disks that have been observed are so-called exo-Kuiper +belts with inner radii of tens or hundreds of au and similar magnitudes +in width, analogous to the Kuiper belt (located between ∼30 au and +∼50 au, Trujillo & Brown 2001). A prototypical example of an exo- +Kuiper belt is the debris disk around Vega which was first reported +in Aumann et al. (1984) and has an inner radius of 86 au and extends +out to hundreds of au (Su et al. 2005). +Debris disks are thought to be the remnants of structures called +protoplanetary disks. These structures are collections of gas and +★ E-mail: tom.lewis@bristol.ac.uk +dust orbiting around young, newly-formed stars from which planets +and planetesimals are formed. Protoplanetary disks lose mass and +dissipate over time through accretion, leaving a dusty debris disk as a +remnant. Debris disks are therefore generally an order of magnitude +fainter than protoplanetary disks which have fractional luminosity +values of at least 𝑓 ≈ 10−2. Additionally, as a consequence of their +origins, debris disks contain little to no gas and exist around more +mature stars with ages ≳10 Myrs (Alexander et al. 2014). Finally, +debris disks also typically have very low optical depth in optical +wavelengths when compared to protoplanetary disks (e.g. Hughes +et al. 2018). These four factors – luminosity, system age, gas content, +and optical depth, help observationally distinguish debris disks from +protoplanetary disks. +1.1 Extreme Debris Disks +The standard model for debris disks assumes a steady-state collisional +cascade which gradually grinds down large 1-100 km planetesimals +into smaller and smaller particles (Wyatt & Dent 2002; Quillen et al. +2007; Wyatt 2008). Smaller particles are able to re-emit absorbed +radiation in the infrared much more efficiently than larger particles, +meaning collisional grinding helps to create a dust population which +is observable through its IR emission. Sub-micron dust particles +are small enough to be affected by radiation pressure from the host +star, so most simple models assume these particles are blown out +of the debris disk. The balance of collisional grinding and radiation +pressure blow out leads to a stable dust population in the disk. This +implies the observed fractional luminosity should be constant over +orbital timescales (Wyatt et al. 2011). However, it is important to note +that this steady-state is only maintained while there is sufficient mass +in the large planetesimal population to keep a roughly consistent +collision rate. Once mass runs out at the top of the distribution the +© 2022 The Authors +arXiv:2301.03307v1 [astro-ph.EP] 9 Jan 2023 + +2 +T. Lewis et al. +Figure 1. The IR excess from the debris disk of ID8 detected by the Spitzer +Space Telescope at 3.6 (blue dots) and 4.5 (red crosses) 𝜇m. The error bars +(1𝜎) on both sets of data points represent the uncertainty in excess flux. Data +from Su et al. (2019) reproduced here with permission. +amount of dust produced decreases leading to a drop in fractional +luminosity over several hundreds of Myrs (Wyatt 2008). +However, some debris disks do not seem to be sustained by the +traditional steady-state collisional cascade. This sub-class of debris +disks are much brighter than traditional debris disks and often highly +variable, so are usually referred to as extreme debris disks (EDDs). +EDDs have average fractional luminosities in excess of 10−2 (Meng +et al. 2015), but this value can vary significantly. +Two examples of observed EDDs are ID8 (Meng et al. 2014) and +HD23514 (Meng et al. 2015). Both of these disks display variability +in their infrared output. ID8 showed a rapid increase in infrared +luminosity (in the 3.6𝜇m and 4.5𝜇m wavebands) of roughly 40% at +the start of 2013. This was followed by a gradual decay in output +throughout the rest of the year. ID8 also displays short-term, quasi- +periodic variability overlaid on the longer-term decay trend Su et al. +(2019). Observational data of the excess flux from ID8 over 2012 +and 2013 is shown in Fig. 1. +HD23514 shows a similar decay trend to ID8 in the 3.6𝜇m and +4.5𝜇m wavebands without any significant short-term variability. +These two types of variability both occur on timescales of years +and decades which is much faster than the Myr evolution timescales +associated with dust generated by a collisional cascade Wyatt et al. +(2007). Additionally, they are both found around fairly young stars +with ages of ∼35 Myr and ∼120 Myr respectively. At these ages (>10 +Myr) protoplanetary disks will likely have been cleared of gas Math- +ews et al. (2011), implying the unusual brightness and variability +is not related to collisional activity during earliest stages of planet +formation (Meng et al. 2012). +The unusual brightness and variability of EDDs, exemplified by +ID8 and HD23514, cannot be explained via the traditional, steady- +state model alone. This is because the short-term and medium-term +variations in luminosity of disks like ID8 are far too rapid to be +attributed to the dust produced by slow, collisional grinding. In- +stead, other processes have been suggested to account for these ob- +servations, including dynamical instabilities (Bonsor et al. 2013) and +comets scattering into the inner regions of the system (Marino et al. +2016; Nesvorný et al. 2010; Bonsor et al. 2012). In Moór et al. (2021) +a variety of the possible explanations for EDDs are explored and eval- +uated. One of these explanations that has received significant interest +in recent years has been giant impacts (Watt et al. 2021; Jackson et al. +2014; Wyatt & Jackson 2016; Wyatt et al. 2017). Giant impacts are a +type of collision between large terrestrial bodies such as planets and +planetary embryos. Whilst there is no standard definition of the giant +in giant impacts, in this work we will assume this refers to rocky +planetesimals with a diameter >1500 km Carter et al. (2020). +Giant impacts are highly energetic interactions which can partly +melt and vapourise the surface of the colliding embryos. The ejected +vapour cools and condenses after the collision to form a cloud of +small dust particles in the mm-cm range (Johnson & Melosh 2012). +This cloud of dust would be detectable in the infrared almost immedi- +ately after impact due to the small particle size (Jackson et al. 2014). +The sudden appearance of this vapour population could explain the +rapid increase in luminosity of EDDs like ID8. The decay trend of +ID8 and HD23514 could be attributed to the transient nature of the +vapour condensate. Dust particles of this size are small enough to be +significantly affected by radiation pressure and Poynting-Robertson +drag which dissipates the disk and decreases the total disk luminosity +over time. The ejected melt material will also cool and solidify into a +population of planetesimals which can undergo the traditional colli- +sion cascade. This fresh population could eventually produce visible +dust once the collisional cascade has reached steady-state (which +could take many thousands of orbits). Giant impacts could therefore +produce enough ejected material to form a new transient debris disk +which would be observable. +Numerical simulations have suggested that giant impacts are com- +mon in the late stages of terrestrial planet formation (Chambers & +Wetherill 1998; Agnor et al. 1999; Chambers 2001; Quintana et al. +2016) and could play a key role in planet formation. Evidence of +possible giant impacts can be seen across our own Solar System, +including the formation of the Moon (Canup & Asphaug 2001; Hart- +mann 2014), the size and location of Mercury (Benz et al. 1988), the +origin of the Pluto-Charon system (McKinnon 1989; Canup 2010), +and collisional family around Haumea (Leinhardt et al. 2010). This +does provide some evidence that giant impacts would be occurring +at the right time in stellar evolution to cause the observed EDDs. +The observation of EDDs has led several authors to a focus on giant +impacts as a possible explanation. +1.2 Previous Work on Modelling Extreme Debris Disks +Jackson et al. (2014) and Jackson & Wyatt (2012) modelled the +dynamical evolution of debris disks produced by planetary collisions. +One of the key conclusions from both of these projects was that the +morphology of the disk is primarily shaped by the collision point. +This is the point in space at which the giant impact occurs. At the +moment of collision all of the source particles which will make up +the disk are located at this point. During the collision the particles +each receive a velocity kick which places the dust on a distribution +of defined orbits. Over time the dust clump will shear out due to +differences in their respective orbits. However, the collision point +remains a fixed point on all of their orbits. In other words, all particles +must pass through the collision point. In reality, the collision point +will not be a single point, but a small volume which depends on the +size of the colliding objects. Jackson et al. (2014) assumed a single +point for simplicity. +In addition to the collision point, there is also the anti-collision +line. This is a radial line on the opposite side of the star to the collision +point and in the plane of progenitor embryo which all particles will +cross at some point in their orbit. Jackson et al. (2014) found that +this confluence of orbital paths leads to an asymmetry in the disk +structure with an over-density of material in these two regions which +increases the perceived optical depth. This asymmetry effect should +create a quasi-periodic variation in the luminosity of the disk on +orbital timescales, although the observability of this variation would +MNRAS 000, 1–20 (2022) + +20122013 +3.6μm +3.0 +? +4.5μm +2.5 +Excess Flux (mjy) +2.0 +1.5 +1.0 +0.5 +56100 +56200 +56300 +56400 +56500 +Barycentric Modified Julian DateEccentricity and Extreme Debris Disks +3 +depend on viewingangle.Disk asymmetry hasbeenused as apossible +explanation for the short-term variability observed in ID8, as well +as the observable characteristics of EDDs more generally. Jackson +et al. (2014) showed that this asymmetry eventually smears out as the +particle orbits precess over many orbits. Typically, the asymmetric +phase lasts around 1000 orbits, so the observable lifetime largely +depends on the semi-major axis of the original planetary embryo. +Watt et al. (2021) followed on from this work but took a different +approach by simulating the entire process from collision to debris +disk evolution, as well as the expected infrared emission of the de- +bris post-impact. To simulate collisions between planetary embryos +they used a modified version of an SPH (smooth particle hydrody- +namics) code called GADGET-2 (Springel 2005; Carter 2022). This +code was originally developed to model cosmological events, such +as galaxy cluster formations, however it has been re-purposed to be +used in many other astrophysical contexts, including planet forma- +tion and planetary collisions. The modified version of GADGET-2 +allows the use of tabulated equations of state (EOS) to determine +the thermodynamic state of the particles (Ćuk & Stewart 2012). The +planetary embryos were initialised with an iron core and forsterite +mantle using ANEOS equations of state for these two materials (Mar- +cus et al. 2009; Melosh 2007; Carter et al. 2019). The embryos were +equilibrated as in previous work (Denman et al. 2020; Carter et al. +2020). +Performing these simulations required an understanding of the +state and composition of the mass ejected from the embryo colli- +sion. As mentioned earlier, numerical simulations have shown that +giant impacts with sufficient energy can produce a vapour conden- +sate cloud with particles in mm-cm range (Johnson & Melosh 2012), +as well as a more standard population of planetesimals. We refer to +these two populations of ejecta as the vapour condensate and boulder +populations respectively. +The boulder population is formed from material that has been +melted by the giant impact and then re-solidified into planetesimals. +The boulder population would generally contain large km-sized plan- +etesimals which grind down through a collision cascade until they +reach a steady-state with a fixed size distribution. This size distribu- +tion is usually assumed to resemble a power law based on observa- +tions of debris disks. In this way the disk formed from the boulder +population is similar to a traditional debris disk. +Conversely, the vapour condensate population forms directly from +material vapourised in the collision and is thought to be composed of +much smaller particles, generally in the mm/cm range depending on +impact velocity and impactor size (Johnson & Melosh 2012). Dust +particles in this size range are able to absorb and re-emit in the in- +frared much more efficiently than larger particles. This implies that +the vapour condensate population would be visible to observers al- +most immediately after the collision. The boulder population would +eventually becomevisible in theinfrared, butwould takemuch longer, +as it would need time for the large planetesimals to grind down into +sufficiently small dust. This difference in formation would also likely +have an effect on the lifetime on these two populations. Assuming we +consider the two populations completely separately, the vapour con- +densate population has no larger planetesimals to replenish its dust +leading to a shorter overall lifetime when compared to the boulder +population. +Given this dichotomy in the ejected mass, an important aspect +of the collision simulation was determining the fraction of mass in +liquid and vapour post-impact, as this dictated the relative ratio of the +boulder and vapour condensate populations respectively. Watt et al. +(2021) assumed that the supercritical ejecta cooled isentropically +until the triple point temperature was reached inside a liquid-vapour +dome determined from the material equation of state. The vapour +fraction of each SPH particle was then calculated using the lever rule. +This allowed them to determine the mass of the vapour condensate +population. +Watt et al. (2021) assumed that the vapour condensate population +would be visible immediately after the collision and therefore sim- +ulated the infrared emission from this dusty debris while ignoring +the boulder population. They found that in certain circumstances the +infrared emission of the vapour disk could exhibit short-term vari- +ations on orbital timescales, similar to the variations observed in +ID8 and P1121. This was assumed to be related the disk asymmetry +highlighted in Jackson & Wyatt (2012). Increased optical depth at +the collision point and anti-collision line led to a drop in the ob- +served emission. However, the appearance of these variations was +highly dependent on the parameters of the collision, in particular the +orientation with respect to the orbital path of the centre of mass of +the two colliding embryos. Collisions which predominately launched +ejecta perpendicular to the centre of mass orbit produced disks with +variability while collisions which launched ejecta parallel to the cen- +tre of mass orbit did not. Variability is a good indicator that dust +has been generated by an impact rather than some other mechanism. +Any factor which suppresses variability would make detection and +characterisation of giant impacts less likely. +The question of the link between giant impacts and EDDs is an +important one, because there is a fundamental tension between our +assumptions and the observations. Giant impacts are thought to be +common during the later stages of planet formation. They are as- +sumed to occur during a separate stage of solar system formation af- +ter the conclusion of the oligarchic stage (Kenyon & Bromley 2006). +If giant impacts can lead to EDDs and if giant impacts are common, +why do we not observe more EDDs? Depending on the definition of +EDD, the number of observed EDDs is around a few dozen at the +time of publication (Moór et al. 2021; Melis et al. 2012; Meng et al. +2012; Kennedy et al. 2017; Rieke et al. 2021). This implies there +could be something wrong with our assumptions about the regular- +ity of giant impacts or perhaps something about the way EDDs are +formed which makes them difficult to detect and observe. +EDDs give us a vital observational foothold when trying to under- +stand planet formation in other solar systems and provide evidence +for different planet formation models. Despite their rarity, EDDs can +play a key role in our understanding of planet formation. +Through this investigation we hoped to more fully understand the +factors which suppress EDD variability and observability. +1.3 Aims +In this project we expanded upon the work first outlined in Watt et al. +(2021). Watt et al. (2021) found that simulated collisions between +planetary embryos on circular orbits could produce debris disks with +distinct, short-term variability in their infrared output, similar in na- +ture to observations of EDDs. They also found that the presence of +this variability was highly dependent on the specific parameters of +the collision, such as impact parameter, impact speed, and collision +orientation. This result implied a potentially narrow parameter space +over which EDDs could be observable, leaving the vast majority of +debris disks created by giant impacts observationally indistinguish- +able from traditional debris disks. +However, not all planetary collisions are likely to occur on per- +fectly circular orbits. Instead, we would expect the population of +embryos to exist on orbits with a range of eccentricities. Eccentricity +would likely change the morphology of the resultant debris disk and +affect its infrared emission. In addition, an embryo on an eccentric +MNRAS 000, 1–20 (2022) + +4 +T. Lewis et al. +orbit will have an instantaneous orbital velocity that varies depend- +ing on its position in the orbit. The embryo orbital velocity at the +moment before the collision would likely affect the velocity distribu- +tion of the ejected material which would change the morphology of +the resultant debris disk and again affect the infrared emission. The +combined impact of these effects is unclear, but it is important to un- +derstand whether the parameter space which can generate observable +variability is as narrow as Watt et al. (2021) concludes when consid- +ering more realistic orbits. The main aim of this work was therefore +to investigate how embryo eccentricity and collision position affects +the observability of these short-term variations in disk flux. +In section 2 we outline the steps we performed to simulate embryo +collisions and subsequent disk evolution. We then detail the analysis +we performed on the simulation data in order to compare the disks +produced by different parameters. In section 3, we examine how the +morphology and observability of the simulated disks changed with +eccentricity and collision position. We then discuss how these results +compare to other simulated and observed data and the implications +on explanations for the origin of EDDs. Finally, in section 4 we +summarise the work and suggest areas of future exploration. +2 METHODS +Our numerical campaign was broken down into several steps which +we will cover briefly in this section. +2.1 Modelling the Collisions +The first step was modelling the collision between the planetary +embryos. Watt et al. (2021) ran a large array of collision scenarios +covering a range of impact speeds and impact parameters. However, +we focussed on a single collision simulation between two 0.1 Earth- +mass embryos (containing 4 × 104 SPH particles) with an impact +velocity of 10 km s−1. Index 8 of Table A1 in Watt et al. (2021) gives +the full details of this collision. The general simulation setup, includ- +ing embryo composition and equations of state used, are provided in +section 1.2 of this paper. Watt et al. (2021) demonstrated that this +particular collision generated the most distinct short-term variations +in infrared emission and provided a good starting point to study the +effect of orbital eccentricity and collision position on variability. +We used this single SPH simulation to generate three contrasting +baseline cases. As mentioned earlier in section 1.2, Watt et al. (2021) +found that infrared variability was highly dependent on whether the +collision occurs parallel or perpendicular to the orbital path of the +centre of mass of the two embryos. We therefore rotated the initial +simulation data by 90◦ to ensure we investigated both the perpendic- +ular and parallel cases. The impact parameters of both of these cases +are summarised in Table 1. +The Parallel configuration in Table 1 was a set of parameters +which Watt et al. (2021) showed generated observable short-term +variations in the simulated disk emission for a circular orbit, whereas +the Perpendicular configuration did not. Collisions that are parallel +to the orbital path of the centre of mass of the two embryos are +denoted by 𝜃 = 0◦ while collisions that occur perpendicular to that +path are denoted by 𝜃 = 90◦. The orientations of these two collisions +are shown in Fig. 2. +We also performed analysis of a third orientation where the col- +lision occurs perpendicular to both the preceding cases. Both the +Parallel and Perpendicular collisions still occurred within the orbital +plane of the original centre of mass. However, there is another possi- +ble orientation where the velocities of the embryos are perpendicular +Table 1. The two collision configurations used throughout this work. The +columns are as follows: 𝑀𝑒𝑚𝑏 is the mass of the projectile and target plan- +etary embryos, 𝑣 is the relative impact velocity between the two embryos, +𝑏 is the impact parameter, 𝑎 is the semi-major axis of the target embryo, +and 𝜃 is the collision orientation. The collision orientation tracks how the +collision occurs with respect to the orbital path of the centre of mass of the +two embryos. In this work we consider parallel and perpendicular collision +orientations along with a range of eccentricities and collision positions. All +simulations are summarised in Tables A1 and A2 of the online supplementary +material. +Config +𝑀𝑒𝑚𝑏 +𝑣 +𝑏 +𝑎 +𝜃 +In plane? +Parallel +0.1𝑀⊕ +10 km s−1 +0 +1 au +0◦ +In +Perpendicular +0.1𝑀⊕ +10 km s−1 +0 +1 au +0◦ +In +Perpendicular* +0.1𝑀⊕ +10 km s−1 +0 +1 au +90◦ +Out +(a) Parallel collision (𝜃=0◦) +(b) Perpendicular collision (𝜃=90◦) +Figure 2. A simple cartoon diagram to demonstrate the two collision cases +studied in this work. The large black circles represent the planetary embryos +involved in the collision. The dashed black line represents the orbital path of +the centre of mass of the two embryos. The thick black arrow indicates the +orbital velocity direction of this centre of mass. The grey clouds and arrows +show the direction in which material is preferentially ejected in each collision +case. The red arrows indicate the relative velocities of the embryos - i.e., the +collision orientation. +to the orbital plane. This case was labelled with Perpendicular* in +Table 1 and was not studied by Watt et al. (2021). +A brief summary of the basic processes of the collision modelling +performed by Watt et al. (2021) is found in section 1.2. For a more +comprehensive explanation see the full details in Watt et al. (2021). +2.2 Evolving the Particles through Time +In order to evolve the ejecta for several orbits after the collision, the +SPH simulation data was handed over to an 𝑁-body integrator. The +output SPH particle data we used had been modified by Watt et al. +(2021) following the procedure outlined in their work, producing +the vapour condensate population and the two largest remnants of +the collision. This simulation had orbital parameters matching our +Parallel and Perpendicular configurations from Table 1. The largest +(0.146𝑀⊕) and second largest (0.001𝑀⊕) remnants account for the +majority of the mass in the boulder population, so were also included +as they could have a noticeable effect on the evolution of the system. +The largest remnants were identified by examining the kinetic and +gravitational potential energies of the SPH particles to determine +which particles are bound and which are unbound. This was an itera- +tive process which identified the particle with the lowest gravitational +potential energy as the seed particle for the largest remnant. Other +particles were then added to this remnant if their kinetic energy was +less than their potential energy in the centre of mass frame of the +remnant. The process was repeated for the second largest remnant +ignoring the largest remnant particles. +In addition to the extraction of the two largest remnants and the +MNRAS 000, 1–20 (2022) + +Eccentricity and Extreme Debris Disks +5 +vapour mass, the vapour condensate population was upscaled using a +process which maintained the velocity distribution and the total mass +of the original SPH particles. This procedure was originally outlined +by Watt et al. (2021) and was done to shift the resolution of the +simulation to focus on the vapour condensate population, improving +the granularity of the simulation when resolving the more complex +gravitational interaction of these particles. The two largest remnants +of the impact were converted directly to 𝑁-body particles, as they +were expected to be single gravitationally coherent object rather than +a distribution of small dust particles. At the end of this processing we +had a set of ∼100,000 particles with individual position and velocity +data which matched the distribution of ejecta after the SPH collision. +We evolved this system of particles using the leapfrog integrator as +described in Watt et al. (2021). +We ran 84 𝑁-body simulations using the same SPH data output +from the process described above, but in each run we varied the centre +of mass orbital eccentricity and the true anomaly of the collision to +see how the debris disk morphology and output flux changed. All of +these simulation runs are summarised in Table A1 and Table A2 in the +online supplementary material. The ’Sim.’ value in these tables will +be used to refer to individual runs throughout the rest of this work. +We evolved the system in each case for 20 orbits of the pre-collision +centre of mass orbit. +It is important to make clear that particles used in these 𝑁-body +simulations are tracers for the mass distribution of the system. In +reality, given the average mass of the particles used in the 𝑁-body +simulation each particle would be roughly 1 km in radius (assum- +ing a particle density of 3 g cm−3). However, an individual vapour +condensate particle is likely to be between a few microns and a few +millimetres in radius (Johnson & Melosh 2012). In these simulations +the 𝑁-body particles are being used as "super-particles" to represent +a distribution of dust particles and track the spatial distribution of +the disk mass rather than the positions of individual particles in a +disk. Additionally, the 𝑁-body particles only interacted gravitation- +ally with the star and the two largest remnants. +2.3 Simulating Disk Emission +We also simulated the total flux emitted by the dust particles in the +disk over time. We used a radiative transfer code package called +RADMC-3D (Dullemond et al. 2012). RADMC-3D takes as input +a cubic grid containing particle densities and one or more energy +sources, usually a single star. It then uses this data to generate syn- +thetic images/spectra. +In our case we generated synthetic images at 0.1 orbit intervals +using the following process - we set up a 3 au cubic grid and binned +the particles into the grid cells based on their current 𝑁-body posi- +tions. In total there were 301 cells along each axis of the grid, giving +a total of 3013 cubic cells. We input this particle density grid into +RADMC-3D alongside a solar-type star as the single energy source +for the system - simulated using a 5700K blackbody. This involved +specifying stellar mass, stellar radius, position, and stellar spectrum. +Additionally, the dust population was assumed to exist in a fixed +power law size distribution with particles sizes ranging from 1mm +to 1𝜇m. The dust opacities were determined from the opacity tool +developed to determine the DIANA standard opacities (Toon & Ack- +erman 1981; Dorschner et al. 1995; Min et al. 2005; Woitke 2015). +Finally, we just needed to configure RADMC-3D to produce images +at some common observation wavelengths: 3.6𝜇m, 4.5𝜇m, 10𝜇m +and 24𝜇m, and select the camera angle. The camera angle defines +from what direction the image is generated. In this work we limited +our observations to the three fundamental planes of the disk. We call +these planes x-y, x-z, and y-z. In the x-y plane the disk is face-on while +in the y-z and x-z planes the disk is edge-on. The collision point and +collision line for all disks are aligned in the y-z plane at (0,0). +RADMC-3D produces observation images which gives the flux +emitted by the disk as viewed from a specific direction. This is meant +to simulate how the object would look when observed. In order to +capture the total flux of the disk at a particular timestep, we summed +the flux values from each pixel in the observation image. We used +RADMC-3D to simulate the disk flux for the entire 20 orbits of the +𝑁-body simulations. +3 RESULTS AND DISCUSSION +In total, we ran 84 𝑁-body simulations using the output of a sin- +gle SPH simulation to generate three different collision scenarios +(Table 1). Using these three configurations as base cases, we varied +the centre of mass orbital eccentricity between e=0.0 and e=0.8. In +addition, we varied the position of the collision along the centre of +mass orbit with the true anomaly value, 𝜈, used to track this collision +point. A value of 𝜈 = 0.0𝜋 represented a collision at the periapsis of +an eccentric orbit while a value of 𝜈 = 1.0𝜋 represented a collision +at the apoapsis of an eccentric orbit. We varied the collision point +between 𝜈 = 0.0𝜋 and 𝜈 = 1.0𝜋. +These parameter limits were chosen because they represented the +extremes of the parameter space. A maximum eccentricity of 𝑒 = 0.8 +was chosen, because we expect the number of planetary embryos on +orbits with eccentricity greater than this value to be quite low based +on numerical planet formation simulations of runaway and oligarchic +growth, as well as pebble accretion models (Chambers & Wetherill +1998; Izidoro et al. 2014; Levison et al. 2015; Raymond & Izidoro +2017; Matsumura et al. 2017; Izidoro & Raymond 2018). +All of the simulation runs are noted in Table A1 and A2 of the +online supplementary material with their collision parameters and an +associated simulation number. We will use these simulation numbers +throughout this section to refer to the different simulations. Note that +Sim. 0 and Sim. 35 are the 𝑁-body simulations shown in Watt et al. +(2021). +3.1 The Morphology of Circular Disks +Firstly, we examined how the morphology of the giant impact ejecta +evolves through time on a circular orbit. Simulations from Jackson +et al. (2014), Wyatt & Jackson (2016), and Watt et al. (2021) show that +immediately after the collision the ejecta is clumped together without +any discernible structure, however after several orbits of the centre +of mass the material shears out to form a clear disk. We find that +this is true throughout all of our simulations, but the precise shape +and structure of the disk varies greatly as we adjust the collision +parameters. +The most distinct morphological difference in all of our simu- +lations was found between disks created by Parallel collisions and +disks created by Perpendicular collisions. This is illustrated for cir- +cular orbits in Fig. 3 which shows the spatial evolution of a debris +disk from a Parallel collision and a Perpendicular collision on a +circular orbit. The snapshots in this figure range from just a few days +after the collision to 8 orbits/years after the collision. +Both disks shown in Fig. 3, start with a clump of material at the col- +lision point with some anisotropy, but evolve in very different ways. +The disk from the Perpendicular collision (bottom row) produces +spiral arm structures which eventually evolve into concentric rings +that spread out across a broad range of semi-major axes. On the other +MNRAS 000, 1–20 (2022) + +6 +T. Lewis et al. +Figure 3. Evolution of the morphology of two debris disk created from a giant impact of two different configurations over eight orbits. The top row (a) shows +the evolution of a debris disk from Parallel collision (Sim. 0 from Table A1 in the online supplementary material). The bottom row (b) shows the evolution of a +debris disk from a Perpendicular collision (Sim. 35). The inset figures in the bottom left of the first timestep shows the collision orientation with respect to the +centre of mass orbit. Brighter colours indicate a greater density of material. The anisotropic distribution of the dust at t=0.05 in both plots shows the effect of +changing the collision orientation. The two plots at the end of (a) and (b) show the view of each disk in the x-z and y-z planes at the final timestep. +hand, the Parallel collision produces a largely contiguous disk with +few distinguishable rings. The clear difference between disks pro- +duced by Parallel and Perpendicular collisions is consistent across +parameter space explored in this work, although eccentricity and +collision position do have an impact on morphology. +The difference in the morphology of disks created by Parallel and +Perpendicular collisions can be explained by understanding how the +collision affects the distribution of semi-major axis. When a collision +occurs parallel to the orbital path of the centre of mass of the two +embryos (see Fig. 2a), material is preferentially ejected in a direction +perpendicular to the orbital path. The direction of this velocity kick +does not greatly change the orbital path of the ejected particles, +leading to a tight distribution of semi-major axes and a clumpier +disk. On the other hand, when a collision occurs perpendicular to the +centre of mass orbital path (see Fig. 2b), material is preferentially +ejected along the orbital path of the target. This causes a greater +change in the velocities of the ejected particles, leading to a broader +distribution of semi-major axes and a more extended disk. This can +be thought of similarly to prograde/retrograde burn by a satellite +versus a radial/anti-radial burn. +On the end of Fig. 3, we have also included the view looking +towards the x-z and y-z planes. The y-z plane on both disks has a +typical flared-out pattern similar to a bow tie. This is because in this +view we are looking directly at the collision point. This is the pinch +point through which all particles must pass at some time. Either side +of the collision point the particle orbits flare out slightly as they all +follow their slightly different orbit inclinations. In the x-z view, we +see two denser regions at either end of the disk, marking the collision +point (right) and anti-collision line (left). In the Parallel collision +case (see (a) in Fig. 3), the anti-collision line region is fairly compact +similar to the collision point, however in the Perpendicular case (see +(b) in Fig. 3), this is closer to a series of dense regions in a line. This +matches what we would expect from the x-y morphology. +Using RADMC-3D to calculate the radiance of these disks over +time shows the expected effect on observation of these two different +morphologies. In Fig. 4a we see clear periodic variation in the radi- +ance of the Parallel collision simulation. Dips occur every half-orbit +which coincides with the collision point and the anti-collision line, +since most of the material tracks closely with the largest remnant of +the collision. This is what would be expected from the increase in +density and optical depth which occurs at these points (see Fig. 3). +Increased optical depth means the total flux visible to an observer +decreases. This is compared to the Perpendicular collision case (Fig. +4b) where we see no distinct periodic variation. As highlighted be- +fore and in Watt et al. (2021), the difference between these two is +likely a result of the different distributions of semi-major axis in each +case. The Parallel case has a tighter distribution of semi-major axis +which creates dense regions of material at the collision point and +anti-collision line. The Perpendicular case has a wider distribution +of semi-major axis which reduces the density of these regions. +3.2 The Morphology of Eccentric Disks +As mentioned in section 2, we ran a number of 𝑁-body simulations +where the eccentricity of centre of mass orbit and collision position +along the centre of mass orbit were varied between 𝑒 = 0.0 and +𝑒 = 0.8 and 𝜈 = 0.0𝜋 and 𝜈 = 1.0𝜋 respectively, where 𝜈 is the +true anomaly of the collision position with 𝜈 = 0.0𝜋 representing +a collision at periapsis and 𝜈 = 1.0𝜋 representing a collision at +apoapsis. We wanted to understand how changes in eccentricity of +the centre of mass affected the structure of the debris disk. +Fig. 5 and Fig. 6 show how the morphology of the debris disks +change with eccentricity and collision position. All disks are plotted +exactly 10 orbits after the collision. This was chosen because at this +point the structure of the disk had stabilised. Additionally, at this +time the largest remnant of the collision and most of the other disk +material was passing through the collision point (marked by white +arrows in each plot). +We find the same dichotomy in morphology exists between Paral- +lel and Perpendicular collisions in eccentric orbits as with circular +orbits. Debris disks from Parallel collisions are more tightly bound +whereas those from Perpendicular collisions are extended over a +greater range of semi-major axes. Additionally, these debris disks +broadly inherit the eccentric characteristics of the centre of mass +orbit with the number density of the disk tracing the original embryo +orbit (Fig. 5). +We also see the same periodic increase in density at the collision +point which is responsible for the short-term variations shown in Fig. +MNRAS 000, 1–20 (2022) + +(a) +25.0 +12.5 +t=0.05 +t=1.25 +=2.0 +t=8.0 +t=8.0 +0.0 +(b) +(n) +8.0 +0.4 +-1 +0.0 +t=0.05 +t=1.25 +t=2.0 +=8.0 +t=8.0 +0.4 +0 +2 +0 +2 +X (AU) +n) AxEccentricity and Extreme Debris Disks +7 +(a) Infrared emission of a simulated debris disk produced by a Parallel colli- +sion. This is from Sim. 0 in Table A1 in the online supplementary material. +(b) Infrared emission of a simulated debris disk produced by a Perpendicular +collision. This is from Sim. 31 in Table A1 in the online supplementary +material. +Figure 4. The total infrared emission of a simulated extreme debris produced +by two different types of collision orientation. This was simulated using the +RADMC-3D package. +4. This region of increased density travels around the disk tracking +with the largest remnant. A gap is also visible in the denser region +which is likely a result of the largest remnant of the collision scatter- +ing surrounding material as it approaches the narrow orbital space of +the collision point. The gap travels around the disk in the middle of +the dense region, but it becomes most distinct as the largest remnant +passes through the collision point. +The eccentricity of the centre of mass also has an effect on the +spread of semi-major axis values in the disk. For the collisions at +periapsis in both the Parallel and Perpendicular collision cases, in- +creasing eccentricity leads to a greater spread of semi-major axis +values, although this effect is much more pronounced in the Per- +pendicular case. The reverse effect occurs for collisions at apoapsis +where the disks are more extended at low eccentricities and become +more tightly bound as eccentricity is increased. This can be explained +with reference to Oberth Effect in astronautics (Oberth 2014). Accel- +erating a body while it is at its maximum orbital velocity at periapsis +will increase the semi-major of the orbit more efficiently, pushing +the apoapsis of the body further from the star. Decelerating the body +at periapsis will have the opposite effect, circularising the orbit and +decreasing the semi-major axis. The strength of this effect roughly +scales with eccentricity. In the case of a Perpendicular collision, ma- +terial is preferentially kicked parallel or anti-parallel to the velocity +of the centre of mass of the two embryos creating a larger range of +semi-major axis values amongst the ejected particle population. In +the Parallel collision case material is preferentially kicked perpen- +dicular to this velocity, meaning the effect is much less pronounced. +Accelerating a body while it is travelling more slowly at apoapsis +will give a much smaller boost to the semi-major axis, so instead we +see a more constrained disk across all eccentricities. +Changing the position of the collision along the orbit has a clear +effect on the morphology of the disks. For circular orbit and low +eccentricity cases shifting the collision position simply rotates the +entire density pattern of the disk when comparing to 𝜈 = 0.0𝜋 case, +but otherwise the morphology remains similar. For example, in row +(a) of Fig. 5 and Fig. 6 the 𝜈 = 0.5𝜋 case (Sim. 2 and Sim. 37) is +essentially the 𝜈 = 0.0𝜋 (Sim. 0 and Sim. 35) case rotated by 90◦. +The higher density region created by the collision point is obviously +in a different spatial position and disk expansion direction has also +changed. +As we increase eccentricity in the middle two intermediate cases +(𝜈 = 0.5𝜋 and 𝜈 = 0.81𝜋), the collision point moves closer to the +apparent periapsis of the eccentric disk. The higher density region +caused by the collision point also shifts accordingly with the collision +point. This is to be expected when the true anomaly is fixed and +eccentricity of an orbit is increased. We also see in these two middle +cases at high eccentricity that the expansion direction of the disk is +no longer on the opposite side of the disk to the collision point, as +the periapsis and the collision point no longer align. This can be seen +most clearly in the third column of Fig. 6. +We do not see a shift in the apoapsis and periapsis collision cases +(outer columns of Fig. 5 and Fig. 6). Both of the higher density +regions remain very close to the apoapsis and periapsis respectively. +An interesting consequence of increasing eccentricity for collisions +at apoapsis is that the periapsis of the centre of mass orbit is brought +closer to the host star. Velocity kicks from the collision can then shift +the periapsis of individual ejected particles even closer to the star. In +the most extreme case (bottom right of Fig. 6) there is a build-up of +material on the host star. In reality, this material would be accreted +onto the surface of the star, but it does serve to highlight how close +material is getting. As will be shown in the next section, this has an +effect on the IR output of the disk. +Finally, we also looked at how eccentricity and collision position +affect the vertical structure of the disks. The inset plots in Fig. 5 and +Fig. 6 show views towards the x-z and y-z planes. As with circular +orbits, there is a bow tie structure in the y-z plane for collisions at +apoapsis and periapsis (first and last columns of Fig. 5 and Fig. 6). +This structure gets flattened as eccentricity is increased in the peri- +apsis case, but increases in the height in the apoapsis case. This is +discussed more quantitatively in section 3.4. In the x-z plane there is +an oval structure with dense regions at each disk ansae for apoapsis +and periapsis collisions. This is an effect of the viewing angle as the +line of sight through each disk ansae will have a greater column den- +sity than the rest of the disk. Similarly to the y-z view, this structure +flattens with increasing eccentricity in the periapsis collision case +leading to a more homogeneous density distribution along the mid- +plane of the disk. Conversely in the apoapsis case, the disk becomes +more extended in x-z view, but the disk ansae are still distinctly dense +regions. In the 𝜈 = 0.5𝜋 case, the structures in the x-z and y-z are +swapped due to the rotated density structure. In the 𝜈 = 0.81𝜋 case +we see a twisted bow tie shape in both of x-z and y-z planes due to the +off-centre location of the collision point from these viewing angles. +3.3 The Morphology Disks from Out-of-plane Collisions +We covered a smaller subset of parameters for collisions in the Per- +pendicular* orientation from Table 1. These results are summarised +in Fig. 7. +As with all other simulations, the Perpendicular* disks resemble +the centre of mass orbit of their progenitor embryos. The morphology +pattern of Perpendicular* collisions across the parameter space is +broadly similar to the Perpendicular collisions. We found the dust +sheared out quickly, spreading across a large range of semi-major axes +in a set of concentric rings. Increasing eccentricity also had a similar +MNRAS 000, 1–20 (2022) + +0.50 +0.45 +0.40 +(A) +Flux +0.35 +0.30 +0.25 +24.0μm +10.0μm +0.20 +6 +8 +10 +Orbits since collisionhw +0.45 +0.40 +(K[) +Flux +0.35 +0.30 +24.0μm +0.25 +10.0μm +5+ +8 +10 +Orbits since collision8 +T. Lewis et al. +Figure 5. Grid of simulated debris disks produced by collisions parallel to the orbital path of the centre of mass of the two colliders. The grid shows the effect +of varying centre of mass eccentricity and position of the collision along the orbital path. Colour in these plots is used to indicate 𝑁 -body particle density. +All plots are taken from the same timestep which is 10 orbits after the collision and shows the disk in the x-y plane. The columns in this figure show different +positions along the centre of mass orbital path at which a collision has occurred. The 𝜈 value at the top of the figure tracks the true anomaly. 𝜈 = 0.0𝜋 denotes +a collision at periapsis while 𝜈 = 1.0𝜋 denotes a collision at apoapsis. The rows represent changing eccentricity with (a) e=0.0, (b) 0.2, (c) 0.4, (d) 0.6, and +(e) 0.8, respectively. The white triangles on each plot point to the spatially fixed collision point. The two inset figures show the same disk from the x-z and y-z +planes. +MNRAS 000, 1–20 (2022) + +V=0.0㎡l +0.3 +V=0.5l +V=0.81π +0.3 +V=1.0 +0.3 +0.0 +0.0 +(a) +-0.3 +-0.3 +0.3 +0.3 +0.3 +8.3 +0.3 +% +N +N +N +-0.3 +0.3 +0.3 +0.3 +0.3 +8. +8.3 +% +V=0.0㎡l +N +V=0.5l +V=0.81π +N +V=1.0π +N +(b) +0.3 +0.3 +0.3 +0.3 +0.3 +0.3 +0.3 +0.3 +0.0 +0.0 +0.0 +0.0 +-0.3 +0.3 +03 +V=0.0㎡l +0.3 +V=0.5l +0.3 +V=0.81π +0.3 +V=1.0π +0.3 +0.0 +0.0 +N +0.0 +(c) +-0.3 +0.3 +-0.3 +0.3 +0.3 +0.3 +0.3 +0.3 +0.0 +Q.0 +0.0 +0.0 +0.3 +0.3 +V=0.0㎡ +0.3 +V=0.5π +0.3 +V=0.81π +0.3 +V=1.0 +0.3 +N +0.0 +N +0.0 +N +0.0 +N +0.0 +(d) +-0.3 +0.3 +0.3 +0.3 +一 +0.3 +8.9 +0.3 +0.3 +0.0 +0.9 +2.0 +V=0.0㎡l +V=0.5m +0.3 +V=0.81π +0.3 +V=1.0π +0.3 +1.5 +0.0 +-0. +-0.9 +0.0 +(e) +-0.3 +0.3 +1.0 +X +X +0.5 +y (AU) +0.0 +0.5 +1.0 +1.5 +0.3 +0.3 +0.3 +0.3 +0.0 +0.0 +0.0 +N +0.0 +-0.3 +-0.3 +-0.3 +-1 +X (AU)Eccentricity and Extreme Debris Disks +9 +Figure 6. Similar to Fig. 5 but the collisions occur perpendicular to the centre of mass orbit. All other information about Fig 5 is valid for this figure. +MNRAS 000, 1–20 (2022) + +V=0.0㎡l +0.3 +V=0.5l +V=0.81π +0.3 +V=1.0 +8.6 +N +0.0 +0.0 +N +0.3 +N +(a) +-0.3 +-0.3 +-0.3 +0.3 +0.3 +0.3 +0.3 +N +0.0 +N +0.0 +N +0.3 +0.3 +0.3 +0.3 +0.3 +8.: +V=0.81π +8.3 +% +V=0.0㎡l +N +V=0.5l +N +N +V=1.0π +(b) +-0.3 +0.3 +0.3 +0.3 +0.3 +0.3 +0.3 +0.0 +0.0 +0.0 +0.0 +0.3 +0.3 +V=0.0㎡l +0.3 +V=0.5l +0.3 +V=0.81π +0.3 +V=1.0m +0.3 +N +0.0 +N +0.0 +N +0.0 +0.0 +(c) +0.3 +-0.3 +0.3 +0.3 +0.3 +0.3 +0.3 +0.3 +0.0 +0.0 +0.0 +0.0 +0.3 +0.3 +0.3 +V=0.0l +0.3 +V=0.5π +0.3 +V=0.81π +0.3 +V=1.0 +0.3 +N +0.0 +N +0.0 +N +0.0 +N +0.0 +(d) +-0.3 +0.3 +0.3 +0.3 +X +0.3 +0.3 +0.3 +0.3 +0.0 +0.0 +0.0 +0.3 +-0.3 +2.0 +V=0.0ml +. +V=0.5ml +0.3 +V=0.81π +0.3 +V=1.0元 +0.3 +1.5 +N +0.0 +0.0 +N +0.0 +(e) +0.3 +-0.3 +1.0 +X +X +X +0.5 +y (AU) +0.0 +0.5 +1.0 +1.5 +0.3 +0.3 +0.3 +0.3 +N +0.0 +0.0 +0.0 +N +0.0 +-0.3 +-0.3 +0.3 +-0.3 +-1 +1 +X (AU)10 +T. Lewis et al. +Figure 7. Grid of simulated debris disks produced by collisions perpendicular +to the orbital path of the centre of mass of the two colliders and perpendicular +to the orbital plane. Colour in these plots is used to indicate 𝑁 -body particle +density. All plots are taken from the same timestep which is 10 orbits after +the collision and shows the disk in the x-y plane. Columns in this figure show +different positions along the centre of mass orbital path at which a collision +has occurred. The 𝜈 value at the top of the figure tracks the true anomaly. +𝜈 = 0.0𝜋 denotes a collision at periapsis while 𝜈 = 1.0𝜋 denotes a collision +at apoapsis. The rows represent changing eccentricity with (a) e=0.0, (b) 0.4, +and (c) 0.8, respectively. The white triangles on each plot point to the spatially +fixed collision point. The two inset figures show the same disk from the x-z +and y-z planes. +effect on the spatial distribution of the rings as in the Perpendicular +case. For collisions at periapsis, increasing eccentricity boosted the +range of semi-major axes while the opposite occurred for collisions +at apoapsis. The main difference between the Perpendicular and +Perpendicular* cases was that the rings appeared much less cleanly +defined than in the Perpendicular case. This effect is likely a result of +more particles receiving both a perpendicular and parallel velocity +kick component in the collision compared to the Perpendicular case. +The additional parallel kick component could help to offset particle +orbits slightly, creating rings that are more indistinct. +Disks in this case are also much thinner in the z-axis than in the +other orientations, with average scale heights typically a tenth the +size of their corresponding Parallel and Perpendicular disks. This +can be seen in the x-z and y-z inset figures in Fig. 7. Flatter disks were +expected given that the material was preferentially ejected into the +orbital (x-y) plane, so the velocity kick components in the z-direction +are minimised. +Figure 8. The average scale height of a disk over the first 10 orbits of Sim. +0 in Table A1 in the online supplementary material. The red dotted line is a +horizontal linear fit to the data, representing the average scale height across +time. +3.4 Scale Height +In order to move beyond simple qualitative descriptions of the simu- +lated disks, we quantified the average scale height of each simulated +disk over time. This allowed us to compare the average height of +different debris disks through time. Initially, we tried using a simple +exponential fit of the particle density against height above orbital +plane, however, the density-height profile did not seem to follow an +exponential decay in every timestep. Instead, we went for a more +general approach and calculated the 36.7th percentile of the particle +density. While this may not be exactly equivalent to the scale height, +it can provide a rough estimate that can be used for comparative +purposes. +Fig. 8 and Fig. 9 summarise this scale height information. +Fig. 8 shows an example of how scale height varies over the first 10 +orbits of Sim. 0. The most obvious trend in this data is the periodic +dips in scale height over the course of a single orbit. This reinforces +our conclusions about the origin of the short-term variations in the +simulated emission shown in section 3.5. The collision point and +anti-collision line are spatially fixed regions in the x-y plane. This +can be seen in the bow tie shape of the disks in the y-z plots of Fig. +5 and Fig. 6. The x-y plane in all of the simulations is defined by +the orbital plane of the centre of mass of the two colliding embryos. +As the largest remnant and a large proportion of the disk material +passes through the pinch points at the collision point and along the +anti-collision line, the average scale height drops because the x-y +plane defines the zero point of the height scale. This drop in average +scale height implies an increase in density at the pinch points which +creates the short-term variations seen in Fig. 4a. +The second trend seen in Fig. 8 is the attenuation of the magnitude +of the short-term variations over time. This is likely a result of mate- +rial being sheared out over time leading to less pronounced clumping +at the collision point and anti-collision line so less variation in the +average scale height of the disk. +To gain an understanding of how eccentricity and collision position +affects scale height we plotted scale height against these parameters. +The results of this are shown in Fig. 9. One of the primary results +from this is that the average scale height decreases with eccentric- +ity when the collision occurs at the periapsis of the centre of mass +orbit, whereas the average scale height increases with eccentricity +when the collision occurs at apoapsis. This is likely related to the +variation in orbital velocity at different points in an eccentric orbit. +MNRAS 000, 1–20 (2022) + +0.03:0 + Height (AU) +0.025 +0.020 +0.0L5 +0.010 +0.005 +2 +t +6 +8 +Timestep0.3 +0.3 +V=0.0 +V=1.0π +0.0 +N +0.0 +(a) +0.3 +0.3 +0.3 +0.3 +N +0.0 +N +0.0 +0.3 +0.3 +EO +0.3 +V=0.0爪 +V=1.0π +N +0.0 +N +0.0 +(c) +0.3 +0.3 +X +0.3 +0.3 +N +0.0 +0.0 +0.3 +0.3 +2.0 +0.3 +0.3 +V=0.0㎡ +V=1.0πl +1.5 +0.0 +N +0.0 +(e) +0.3 +0.3 +-1 +1.0 +0.5 +Y(AU) +0.0 +0.5 +-1.0 +0.3 +0.3 +1.5 +N +0.0 +N +0.0 +-0.3 +0.3 +2.0 +2.0 +1.5 +o'T- +0.0 +0.5 +1.0 +1.5 +2.0 +X (AU)Eccentricity and Extreme Debris Disks +11 +Figure 9. Average scale height variation of simulated debris disks with eccentricity, collision orientation, and collision position. The solid line shows scale height +variation for collisions occurring parallel to the orbital path of the centre of mass of the two colliders while the dotted line shows this value for Perpendicular +collisions. The columns show different collision positions around the orbit. +The velocity of a body in an eccentric orbit is at its maximum at +periapsis and at its minimum at apoapsis. This effect scales with ec- +centricity, meaning increasing eccentricity will increase the velocity +at periapsis, but decrease the velocity at apoapsis. The relative colli- +sion velocity between the projectile and the target embryos is fixed +at 10 km s−1 across all simulations as shown in Table 1. The average +scale height of the disk should be dependent on the distribution of +particle inclination in the disk. A wider inclination distribution leads +to a greater average disk scale height. The inclination of a particle +is much easier to change when its orbital velocity is lower, so this is +why we find a higher average scale height when the centre of mass +of the two embryos is travelling more slowly. +This pattern is reinforced when looking at collision points between +apoapsis and periapsis. For example, 𝜈 = 0.5𝜋 (a collision occurring +halfway between apoapsis and periapsis) results in an average scale +height does that not vary significantly with eccentricity. Following +the reasoning outlined above, this implies that the orbital velocity of +the centre of mass at the point of collision does not vary significantly +with eccentricity. When 𝜈 = 0.81𝜋 we see that average scale height +increases with eccentricity. This again matches our expectations as +the orbital velocity of the centre of mass at the point of collision will +decrease as eccentricity is increased. +Fig. 9 shows the scale height trends for both Parallel (solid blue +line) and Perpendicular collisions (dotted orange line). These lines +follow very closely across all 𝜈 and eccentricity values, implying that +the overall trend in average scale height is related to orbital speed +and collision position rather than collision orientation. +3.5 Infrared Emission from Extreme Disks +The figures in this section contain grids of light curves at various +wavelengths generated by radiative transfer (section 2.3) which show +how the dust emission of the simulated disks varies over the first +10 orbits after the embryo collision. Similarly to Fig. 5 and Fig. +6, these figures show the parameter space we covered during our +investigation. The eccentricity of the centre of mass in the collision +increases as you move down the rows while the true anomaly of the +collision changes from periapsis to apoapsis as you move from left +to right. For example, in Fig. 10, the light curve in the second row +of the final column corresponds to a collision occurring at apoapsis +(𝜈 = 1.0𝜋) on an orbit with eccentricity of 0.2. +Fig. 10 shows the light curves for our simulated collisions occur- +ring parallel to the orbital path of the centre of mass. Short-term +variations are found across all collision positions and eccentricities, +however the periodicity and magnitude of these variations changes +as we traverse the parameter space. +At low eccentricity in Parallel collisions the dust emission dips +at half-integer orbit intervals, so there are two dips in emission per +orbit. As mentioned in section 3.4, this variation can be related to +the average scale height of the disk. Fig. 8 and the light curves in +the top left of Fig. 10 are generated from the same simulation and +demonstrate dips at similar half-integer intervals. +We also find that as eccentricity is increased one of these emission +dips is suppressed. For example, with a Parallel collision at periapsis +(𝜈 = 0.0𝜋, first column in Fig. 10), the dip that occurs on each full +orbit (vertical dotted lines in Fig. 10) is increasingly suppressed with +increasing eccentricity. This continues until at e = 0.6 there is only a +single dip detectable per orbit. Conversely, when the collision occurs +at apoapsis (𝜈 = 1.0𝜋, final column in Fig. 10), the half-integer dip +is suppressed as eccentricity is increased, leaving only the dip that +occurs on each orbit. Additionally, increased eccentricity also results +in a general increase in the magnitude of dips across all collision +positions. +It is not entirely clear what is causing these effects, but one of the +most likely explanations is that ever-changing distance between the +dust and the star in an eccentric orbit creates a periodic variation in +the disk emission. The amount of flux emitted by dust in a debris +disk is dependent on the amount of energy absorbed from the star +which in turn is dependent on the distance to the star. In circular +orbits the distance from the star to an orbiting body does not change +with time, however in eccentric orbits this distance is continually +changing, as a body completes an orbit. This means the amount of +stellar flux received by the dust and therefore the temperature of the +dust continually changes as well. The temperature of the dust should +peak at orbital periapsis and reach a minimum at apoapsis. Hotter +dust will emit more total energy and preferentially emit in shorter +wavelengths. Fig. 5 and Fig. 6 revealed that a denser region of dust is +co-located with the largest remnant as it orbits the star. This is why +MNRAS 000, 1–20 (2022) + +(a) v=0.0 +(b) v=0.25π +(c) v= 0.5π +(d)v=0.81π +(e)v=1.0n +0.035 +(ne) +0.030 +0.025 +0.020 +0.015 +0.010 +Parallel +0.005 +Perpendicular +0.0 +0.2 +0.4 +0.6 +0.8 +Eecentricity12 +T. Lewis et al. +Figure 10. Grid of IR emission in the x-y plane for debris disks produced by collisions parallel to the centre of mass of the two colliders. The first 0.5 orbits for +all simulations have been cropped for clarity (during this period flux density increases rapidly as the disk expands). The grid shows the effect of varying centre +of mass eccentricity, position of the collision along the orbit, and the observation wavelength. The columns indicate different collision positions along the orbit, +𝜈 = 0.0𝜋 denotes a collision at periapsis while 𝜈 = 1.0𝜋 denotes one at apoapsis. The rows represent different eccentricities: (a) e=0.0 (Sim. 0, 2, 3, 4 in Table +A1, left to right); (b) e=0.2 (Sim. 15, 17, 18, 19 in Table A1, left to right); (c) e=0.4 (Sim. 20, 22, 23, 24 in Table A1, left to right); (d) e=0.6 (Sim. 25, 27, 28, 29 +in Table A1, left to right); (e) e=0.8 (Sim. 30, 32, 33, 34 in Table A1, left to right). The different observation wavelengths are denoted by different line colours - +24𝜇m: blue, 10𝜇m: orange, 4.5𝜇m: green, and 3.6𝜇m: red. +a peak is seen in dust emission as the largest remnant passes through +the periapsis of eccentric orbits. +The strength of this dust temperature variation effect would in- +crease with eccentricity, as the periapsis gets closer to the star (mov- +ing down the columns in Fig. 6 and Fig. 5). It is possible as centre of +mass eccentricity is increased the impact of this effect overrides the +impact of any optical depth variation. To investigate this we plotted +the average dust temperature for a set of eccentricities directly. Fig. +11 shows how the temperature varies across different eccentricities +for a Parallel collision orientation. Average dust temperature was +broadly flat in the circular case but had strong peaks in the highly +eccentric cases which coincided with the largest remnant passing +through periapsis (see bottom right of Fig. 10). This suggests tem- +perature variation is a major driver of variability in the most eccentric +Parallel disks. +Fig. 12 shows the light curves for our simulated collisions occur- +ring perpendicular to the orbital path of the target embryo. This grid +covers a subset of the total parameter space, because all of the light +curves from Perpendicular collisions look broadly similar across +most of the parameter space we studied. These light curves are sta- +ble with time and do not display much variability. This reinforces +the conclusion from Watt et al. (2021) that collisions occurring per- +pendicular to the centre of mass orbit do not result in disks with +short-term variations in their emission, at least for the collision con- +figurations shown in Table 1. As mentioned earlier, the suggested +explanation for this result is that collisions perpendicular to the cen- +tre of mass orbit preferentially eject material along the orbital path +(see Fig. 2). This means, on average, that the direction of the velocity +kick given to the ejected material from the collision is more likely to +be parallel or anti-parallel to the original orbital velocity of the centre +of mass. This leads to a faster shearing out of the dust and prevents a +build-up of dust density at the collision point and anti-collision line. +MNRAS 000, 1–20 (2022) + +=0.5斤 +=0.81 +V=1.0 +1.25 +a +24.0μm +4.5μm +1.00 +10.0μm +3.6μm +0.75 +Flux +0.50 +0.25 +0.00 +1.25 +1.00 +0.75 +Flux +0.50 +0.25 +DD'0 +1.25 +c) +DD'T +0.75 +0.50 +0.25 +0.00 +1.25 +DD'T +0.75 +Flux +0.50 +0.25 +0.00 +1.25 +e +DD'T +0.75 +Flux +ANNAAAA +0.50 +0.25 +DDO +2 +6 +10 +2 +4 +6 +8 +10 +2 +女 +6 +8 +10 +2 +6 +8 +10 +Orbital Periods +Orbital Periods +Orbital Periods +Orbital PeriodsEccentricity and Extreme Debris Disks +13 +Figure 11. Average temperature across the entire disk for a set of Parallel +collisions occurring at apoapsis (right-most column of Fig. 10, Sim. 4, Sim. +24, and Sim. 34 in Table A1). +This increase in density and optical depth causes drops in observed +emission, so preventing this build-up removes a source of variability. +The only notable exceptions to this result are the light curves +in the bottom right of Fig. 12. These light curves are the result +of a collision occurring at the apoapsis of a highly eccentric orbit +(e=0.8). In these light curves there is some significant variance in +emission, but not the consistent, periodic variation in Fig. 10. We +can attempt to understand this by looking at the morphology of +the highly eccentric disk in Fig. 6 (bottom right). This disk has +the smallest average disk periapsis compared to other disks in this +figure. Particles making closer approaches to the host star will be +heated to higher dust temperatures and preferentially emit in shorter +wavelengths while at the periapsis of their orbit. This explanation is +supported by Fig. 13 which shows how the average dust temperature +varies over the timeframe shown in Fig. 12. +Average dust temperature peaks at every half-integer orbit for the +first few orbits, aligning with the peaks in emission seen in the bottom +right of Fig. 12. The collision in this case occurred at apoapsis which +implies the emission and temperature peak when the largest remnant +and most of the other disk material is passing through the periapsis +of their orbit. This conclusion is further supported by the relative +flux levels of the different wavelengths in Fig. 12. The average disk +temperature rises over the first 5 orbits before flattening out which +broadly mirrors the ratio. In general, the shorter wavelengths are +much stronger compared to the other collisions in this figure. The +initial periodic variability tends to peter out after a few orbits - likely +due to the rapid Keplerian shearing of the disk. The velocity kicks +from the collision set all dust particles on slightly different orbital +trajectories. Over several orbits the material becomes increasingly +out-of-sync with the original clump of material co-located with the +largest remnant until dust is spread more evenly around the disk +and there is no periodic variation. This shearing effect may also +account for the shorter wavelengths peaking slightly earlier in the +first few orbits of collision shown in the bottom right of Fig. 12. The +velocity kicks from the collision will alter the orbit of some amount of +material, so that it makes a closer approach to the star. This material +will be slightly out-of-sync with the main bulk of material around the +largest remnant, so the shorter wavelength peak from this material +will occur at a slightly different time to the main peak tracked by +10𝜇m. +Another noteworthy observation in both Fig. 10 and Fig. 12 is +that the 24𝜇m and 10𝜇m flux density lines begin to converge as +eccentricity is increased. In some highly eccentric cases the 10𝜇m +line actually exceeds the 24𝜇m line (Row (b) in Fig. 12 and rows (d) +and (e) in Fig. 10). This behaviour is consistent across all collision +positions we simulated. In addition, the 3.6𝜇m and 4.5𝜇m lines are +both essentially zero across most of the parameter space, except for +the most extreme eccentricities (Row (b) in Fig. 12 and row (d) and +(e) in Fig. 10) where these flux densities increase. This result again +supports the idea that the short-term variations at higher eccentric- +ities are driven by distance variation. As eccentricity increases, the +average periapsis of the particles gets closer to the star. Dust particles +approaching closer to the star will be heated to a higher equilibrium +temperature and preferentially radiate in shorter wavelengths. +An important caveat to these results is that in our 𝑁-body simu- +lation particles are only removed when they enter the stellar radius. +This is roughly 0.005 au assuming the host star has the same radius +as the Sun. In reality, dust particles are likely to be sublimated at +a much larger semi-major axis. Some of the material that builds up +close to the star, as shown in the bottom right of Fig. 6, may be re- +moved by this effect which could subdue the temperature variability +somewhat. The persistence of this material may particularly affect the +shorter observation wavelengths (4.5𝜇m and 3.6𝜇m) which do not +seem to drop as expected when the largest remnant passes through +apoapsis. Material building up close to the star (but outside the stel- +lar radius) would help to keep shorter wavelengths more stable than +longer wavelengths. +The full grid of simulated IR emission for disks produced by Per- +pendicular collisions can be found in Fig. B1 in the online supple- +mentary material. Additionally, a set of animated movies showing the +evolution of various disks are included in the online supplementary +material. +3.5.1 Infrared Emission from Out-of-plane Collisions +We also simulated the IR emission from our Perpendicular* col- +lisions where the velocities of the colliding embryos were perpen- +dicular to the centre of mass orbital path and orbital plane. These +collisions preferentially ejected material into the orbital plane of +the disk. Fig. 14 shows the light curves for our simulated collisions +occurring perpendicular to the orbital path of the target embryo. +As might be expected from the morphology pattern discussed in +Section 3.3, Fig. 14 shows a very similar pattern to Fig. 12, with +broadly flat disk emission across most of the parameter space except +for the highly eccentric collision at apoapsis (bottom right) where the +variability was driven by dust temperature variation. The peaks in this +case seem to be more distinct and more clearly defined for longer than +the Perpendicular case. In this orientation some particles are ejected +perpendicular to orbital path which may increase the amount of time +it takes for the disk to shear out completely. The lower shearing rate +may help the disk stay more coherent for slightly longer, making +the temperature-induced variation appear more distinct. As with the +Perpendicular case, we also see the shorter wavelength flux peaking +slightly before the 10𝜇m flux due to Keplerian shear. +3.5.2 Observing Emission from the x-z and y-z Planes +Up until this point we have focussed on observing the emission of +the disk when looking down at the x-y plane. In other words, we have +observed these disks face-on. This is useful as a starting point for +discussions of morphology and observability, but in reality, we are +likely to see EDDs from various viewing angles. +For Perpendicular collisions there is a clear lack of short-term +MNRAS 000, 1–20 (2022) + +e=0.8 +400 +e=0.4 +(K) +e=0.0 + temperature +350 +300 +lisk +250 +abei +Avel +200 +150 +6 +10 +Orbital Periods14 +T. Lewis et al. +Figure 12. Same as Fig. 10 except for simulated debris disks from collisions perpendicular to the orbital path of the centre of mass of the two colliders. The +rows represent different eccentricities: (a) e=0.0 (Sim. 35 and 39 in Table A1, left to right); (b) e=0.8 (Sim. 65 and 69 in Table A2, left to right). +Figure 13. Average temperature across the entire disk for an e=0.8 Perpen- +dicular collision occurring at apoapsis (bottom right corner of Fig. 12, Sim. +69 in Table A2). +variability in the IR emission of the resulting disks when looking at +the x-z and y-z planes. For example, the grid in Fig. 15 shows a subset +of the simulated IR emission for Perpendicular collisions viewed in +the x-z plane. Similarly, Fig. B2 in the online supplementary material +shows the simulated IR emission for Perpendicular collisions viewed +in the y-z plane. The lack of variability in x-z and y-z is consistent +with the view from the x-y plane (see Fig. 12 and Fig. B1 in the online +supplementary material) and implies short-term, periodic variability +is a good observable indicator of giant impact collision orientation. +The exceptions to this rule are the cases with extreme eccentricity +and collision positions closer to apoapsis (bottom right of Fig. 15) +where there is some variability on orbital timescales. Similar to the +x-y view, this is likely a result of oscillating dust temperature as +the largest remnant moves from apoapsis to periapsis and back to +apoapsis over the course of a single orbit. This variation would be +reflected in the IR emission. +As with the x-y results, more complex variability is seen in disks +produced by Parallel collisions when viewed in the x-z and y-z planes +(Fig. B3 and Fig. B4 in the online supplementary material). At these +orientations disk ansae, a product of viewing angle (rather than a mor- +phological feature), should create additional variability. Disk ansae +are the two extreme points at the far end of the disk when viewed +edge-on (see Fig. 16). The apparent column density should increase +at these two ansae points when the largest remnant of the collision +passes through them. This should create a drop in total emission sim- +ilar in nature to the collision point/anti-collision line effect. However, +the disk ansae effect will be obscured at certain viewing angles where +the disk ansae and collision point/anti-collision line align from the +perspective of the observer. Su et al. (2019) attribute some of the +observed variability in ID8 to the disk ansae. +To understand this further take the example of a circular disk shown +in the top left of Fig. 5. A clearer view of the x-z and y-z orientations +is shown at the end of the top row of Fig. 3. In the x-z view the +collision point and anti-collision line coincide with disk ansae at +either end of the disk. This implies there should only be two dips +per orbit corresponding to the collision point and anti-collision line. +This is exactly what is seen in Fig. 17 where the IR emission from +this disk viewed from x-z and y-z planes are compared. The magenta +line in this figure dips at each integer and half-integer orbit which +is what would be expected from the collision point/anti-collision +line effect. Contrasting this with the y-z view where the morphology +resembles a bowtie shape. In this case the disk ansae and collision +point/anti-collision line should be distinct from the observer’s line +of sight with the ansae points found either edge of the disk and +the collision point/anti-collision line found at the pinch point of the +bowtie. With this orientation four distinct dips in emission should +MNRAS 000, 1–20 (2022) + +. +V=1.0 +1.50 +(e) +24.0μm +1.25 +10.0μm +(K) +1.00 +4.5μm +Flux +0.75 +3.6μm +0.50 +0.25 +0.00 +1.50 +1.25 +(b) +(KI) +1.00 +0.75 +Flux +0.50 +0.25 +0.00 +2 +6 +8 +6 +8 +10 +Orbital Peniods +Orbital Peniods(>) : +350 +temperature +325 +300 +275 +: disk +250 +Average I +225 +200 +175 +4 +6 +8 +10 +Orbital PeriodsEccentricity and Extreme Debris Disks +15 +Figure 14. Same as Fig. 12 except for simulated debris disks from collisions perpendicular to both the orbital path and orbital plane of the centre of mass of the +two colliders. The rows represent different eccentricities: (a) e=0.0 (Sim. 78 and 79 in Table A2, left to right); (b) e=0.4 (Sim. 80 and 81 in Table A2, left to +right); (c) e=0.8 (Sim. 82 and 83 in Table A2, left to right). +be seen over the course of an orbit as the largest remnant passes +through the collision point, the first disk ansa, the anti-collision line, +and finally the second disk ansa. However, looking at the orange line +in Fig. 17, although there might be some dips in the first two orbits +which could align the disk ansa, broadly there is conspicuous lack +of consistent periodic variability. The dips corresponding to the two +ansae points could be suppressed due to the flaring of the bowtie +shape. Differences in the orbital inclination of the different particles +creates a disk that flares out at the ansae points. This flaring could be +enough to reduce the column density and optical depth at the ansae +points, eliminating any drop in emission. Another possibility, which +would explain the apparent lack of dips from the collision point and +anti-collision line, is that from this line of sight the disk optical +depth is much more consistent over time leading to little variation +in emission. There is however some limited evidence that disk ansae +could induce or support some variability in disk emission. The full +light curve grids covering the entire parameter space in the x-z and y-z +planes are shown in Fig. B3 and Fig. B4 in the online supplementary +material. Comparing Fig. B3 and Fig. B4 we can see that two dips per +orbit is a persistent trend as eccentricity is increased in the X-Z case +(where the disk ansae and collision-point/anti-collision line align). +This suggests that line of sight ansae effects reinforcing the collision +point/anti-collision line effect. +Much like the emission in the x-y plane, the magnitude and pe- +riodicity of the emission in the x-z and y-z planes is dependent on +the centre of mass eccentricity and collision position along the ec- +centric orbit. In general, the x-z case is qualitatively similar across +our parameter space to the x-y case, with the magnitude of variations +increasing with eccentricity for apoapsis collisions, but decreasing +for collisions at periapsis. Also, some dips appear to be suppressed +with increasing eccentricity which appears to align with the expla- +nation from the x-y case that the major driver of flux variation in +more eccentric disks is the distance to the host star as opposed to +changes in optical depth. This can be seen in Fig. 18 which shows +the disk emission in x-z for a subset of the collision parameters. In +general, we find reduced variability in the y-z case. As with all other +cases where we find fairly quiescent disks, the exceptions to this are +apoapsis collisions at higher eccentricities which have variability that +increases in magnitude with increasing eccentricity. +MNRAS 000, 1–20 (2022) + +=. +=1.D +24.0μm +4.5μm +1.0 +(a) +10.0μm +3.6μm +0.8 +0.6 +Flux +0.4 +0.2 +0.0 +1.0 +(b) +0.8 +(KI) +0.6 +0.4 +0.2 +0.0 +1.0 +(c) +0.8 +0.6 +Flux +0.4 +0.2 +0.0 +2 +8 +6 +8 +10 +Orbital Periods +Orbital Periods16 +T. Lewis et al. +Figure 15. Same as Fig. 12 but for a collision occurring perpendicular to the centre of mass orbit and viewed from the x-z plane. The rows represent different +eccentricities: (a) e=0.0; (b) e=0.8. +Figure 16. A diagram of a debris disk from a collision at the apoapsis of an +eccentric orbit. Observing the disk edge-on towards the periapsis (as shown +in the inset diagram in the top left) creates two ansae, A and B, which would +be the most extreme points at either end of the disk. +3.6 Comparing with other Published Results +Giant impact-produced debris disks have been modelled before, most +notably in Jackson et al. (2014). In that work they used analytical +models to predict the dynamical evolution of material released by +a giant collision. Their results for a collision on a circular orbit are +qualitatively similar to ours with a clump of material released imme- +diately after the collision shearing out into a coiled spiral pattern. We +also recreate their collision point and anti-collision line asymmetry. +Jackson et al. (2014) also modelled disks produced from eccentric +orbits which included varying the eccentricity and the position of +the collision. They found broadly similar patterns to our results, +including the disk being centred on an elliptical orbit rather than a +circular one and additional asymmetry when the collision point is +moved away from either apoapsis or periapsis. One of the main points +they focus on is the interaction between apoapsis of the eccentric disk +Figure 17. The 24𝜇m flux evolution of the debris disk from Sim. 0 in Table +A1 in the y-z (orange) and x-z (magenta) planes. The collision which produces +this disk occurs on a circular orbit and is orientated parallel to the centre of +mass orbit. +and collision point. They argue that dust particles will spend more +time at the apoapsis of their orbit than the periapsis which creates a +higher density region at the apoapsis. Additionally, there is the higher +density region around the collision point. The interaction between +these two dense regions can either be constructive or destructive. +This description seems to fit the results shown in Fig. 10. For col- +lisions occurring at apoapsis and higher eccentricities, the RADMC +light curves show a large fall in flux on each integer orbit. This is +because the dense regions are aligned, creating an ultra-dense region +at the apoapsis. The opposite is true for a collision at periapsis. The +two dense regions are at opposite ends of the orbit, so we find a drop +in flux at half-integer orbits when the bulk of material is passing +through the apoapsis. Material spends much less time at the periap- +sis so the effect of the collision point is attenuated as eccentricity +MNRAS 000, 1–20 (2022) + +V=1.D +1.0 +(a) +24.0μm +0.8 +10.0μm +4.5μm +0.6 +Flux +3.6μm +0.4 +0.2 +:8 +(b) +0.8 +(AI) +0.6 +Flux +0.4 +0.2 +0.0 +4 +6 +8 +2 +4 +6 +8 +Orbital Periods +Orbital PeriodsA +B +Collision point +Diskansaepoints +B +Viewing +direction +Anti-collision liney-z +X-Z +0.35 +Flux (ly) @ 24μm +0.30 +0.25 +0.20 +0.15 +2 +4 +6 +B +10 +Orbits since collisionEccentricity and Extreme Debris Disks +17 +Figure 18. Same as Fig. 10 but for a collision occurring parallel to the centre of mass orbit and viewed from the x-z plane. The rows represent different +eccentricities: (a) e=0.0; (b) e=0.8. +increases. We have also added an understanding of how temperature +variation is also a factor in this dynamic. +The point where our work significantly differs is that Jackson et al. +(2014) assumed an isotropic velocity distribution post-impact. As +covered in previous sections, we have found that velocity distribution +plays a significant role in determining both the disk morphology and +flux. The presence of short-term variability is strongly tied to the +initial dust distribution, so accounting for anisotropy is vital. +3.7 Comparison with Observed Debris Disks +It is also important to consider how our simulated results compare +to observational instances of EDDs. Direct imaging of debris disks +is quite difficult, so instead we will focus on the excess IR emission +from the disk as the comparison value. In particular, the average +fractional luminosity of the disk and the variability of the emission. +As of the date of publication, there are tens of observed examples of +EDDs (Moór et al. 2021). The most well-studied examples of EDDs +currently are ID8 and P1121. +3.7.1 ID8 and P1121 +ID8 is a young solar-type star in NGC2547 which displays strong +infrared excess, implying the presence of a dusty debris disk. The +average fractional luminosity of ID8 is 3.2 × 10−2 (Olofsson et al. +2012). +Yearly variation in the 24𝜇𝑚 IR excess of ID8 was first observed +in Meng et al. (2012). Periodicity analysis in Meng et al. (2014) +revealed two significant periods in the IR excess emission of ID8: +𝑃1 = 25.4 ± 1.1 days and 𝑃2 = 34.0 ± 1.5 days. These two periods +were explained as the combined influence of two orbital effects. The +collision-point collision-line optical depth asymmetry and the disk +ansae viewpoint flux drop for edge-on disks. The first effect was +described early in this work and results from a confluence of particle +orbital paths at the collision point and anti-collision line. The second +effect is a result of the viewing angle. If ID8 is edge-on or nearly edge- +on the two disk ansae would appear to have greater optical depth than +the rest of the disk. Assuming a roughly sinusoidal variation to both +of these effects and fitting these to the photometric measurements of +the disks gives a rough peak-to-peak amplitude of 6×10−3 fractional +luminosity. +Meng et al. (2014) estimated the semi-major axis of the debris +disk in ID8 from periodicity analysis to be roughly 0.33 AU. This is +roughly consistent with analysis performed by Olofsson et al. (2012). +This implies the ID8 disk is slightly smaller than most of the ones +simulated in this work. +P1121 is another solar-type star which has displayed high levels of +IR excess with variability (Su et al. 2019). This star is roughly 120 +Myrs old, so as with ID8 it is in the age range where planet formation +is ongoing. +Observations by Su et al. (2019) revealed a long-term flux decay +in the IR excess of P1121 with a decay timescale of 𝑡0 = 310 ± 60 +days. Additionally, they found short-term variability in this emission +with a period of 16.7 days and an amplitude of ±0.08 mJy. Su et al. +(2019) considers a several explanations for this variability, including +a giant-impact produced cloud of debris. As discussed earlier this +explanation implies a dip in disk emission as the largest collision +remnant passes through the collision point and anti-collision line, +increasing dust density in these two regions. As with Meng et al. +(2014) and ID8, Su et al. (2019) also considers the effect of the +viewing angle on disk emission which can create apparent increases +in optical depth at the disk ansae. Combining these effects should lead +to emission dips at every half-integer or quarter integer depending +on the viewing angle. This gives a true orbital period of 33.4 days for +MNRAS 000, 1–20 (2022) + +V=1.0T +24.0μm +4.5μm +0.8 (a) +10.0μm +3.6μm +0.6 +(A) +0.4 +0.2 +0.0 +(q) +0.6 +0.4 +0.2 +0.0 +2 +6 +8 +10 +6 +8 +10 +Orbital Periods +Orbital Periods18 +T. Lewis et al. +a face-on disk and 66.8 days for an edge-on disk and implies that the +collision point would be at 0.2 au or 0.32 au from P1121 depending +on the orientation. +In our results for edge-on disks we do not see the additional dips +from the disk ansae as suggested by Meng et al. (2014) and Su et al. +(2019). The effect of the disk ansae would only be revealed when +looking down at the collision point and anti-collision line, so only +certain viewing angles would have the possibility of seeing an ansae +effect. However, even when viewed the correct orientation we do not +see any additional dips. This could be explained by the flaring out of +the disk at the ansae point due to the range of orbital inclinations of +the disk particles. This may help to reduce the optical depth at the +ansae points and avoid a noticeable dip in flux. +3.7.2 V488 Persei +Beyond ID8 and P1121, there are growing numbers of examples of +extremely variable debris disks. Recent results by Rieke et al. (2021) +have highlighted a particularly acute example of this type of disk +around V488 Persei. V488 Persei is an 80 Myr-old star (Soderblom +et al. 2014) which places it in a similar age range to ID8 and P1121 +and, as with those stars, at an age where planet formation is assumed +to be ongoing. +Observations of the infrared excess of V488 Persei over a number +of years revealed a relatively quiescent phase, followed by a major +uplift in emission in 2019. During the quiescent phase the disk is +still extremely variable with excess infrared emission varying be- +tween 30% to 60% of the peak signal at 3.6𝜇m and 60% to 75% for +4.5𝜇m. This variability is possibly periodic on the timescale of a few +months, but this is still uncertain. Rieke et al. (2021) used the Debris +Disk Simulator (Wolf & Hillenbrand 2005) to fit a simple three- +component, optically thin debris disk model consisting of an inner +disk at 0.3-0.35 AU, an outer disk at 25-45 AU, and a distribution +of micron-sized grains extended inward from 0.3 AU. This model +estimated the total fractional luminosity for the inner component at +∼ 3.6% and the outer component between 10-16%. The estimated +fractional luminosities ( 𝑓 ≫ 10−3) and variability of this disk marks +it clearly as a strong EDD candidate. Rieke et al. (2021) suggest that +such high fractional luminosity and variability implies a very dense +and collisionally active disk. They propose that a massive planet or +brown dwarf has perturbed the inner disk, creating an exceptionally +collisionally active disk with high levels of dust production. They +estimated the amount of dust mass required to produce the major +boost in infrared flux seen in 2019 would be equivalent to an 85 km +planetesimal disintegrating into a power law collisional cascade of +objects. This is well below the size of the planetary embryos that +were simulated in this work, however, as mentioned in section 1.2, a +giant impact is likely to produce a vapour condensate population with +dust grains < 1 cm. The total mass of this dust population is heav- +ily dependent on the collision parameters, so a giant impact could +imitate a wide range of collision cascade signals. V488 Persei could +provide an interesting case study to see whether we observe any of +the short-term variation effects highlighted in this work and Watt +et al. (2021). Assuming the 2019 uplift was caused by a collision, we +might see a similar decay trend to ID8 where a rapid increase in flux +was followed by a slower decay in emission over time. A possible +complicating factor when comparing this disk to the results of this +work is the assumed pre-existing debris disk. Secondary collisions +and ongoing instability could help to mask the pure signal of a single +collision that we have simulated in this work. Further observations +of this extremely active disk over the next few years will be useful. +3.8 Methodological Caveats +We employed some simplifications in order to reduce computation +time and maximize the number of simulation runs. These are impor- +tant to keep in mind when evaluating the results presented here. +One of the most important caveats for all the results in this work is +that all simulation runs were based on the Parallel and Perpendicular +SPH configurations outlined in Table 1. In other cases with other +embryo mass ratios, impact velocities, and impact parameters the disk +morphologies and IR emission could change dramatically. However, +these configurations were useful as fixed test cases to determine the +effect of orbital eccentricity. +3.8.1 𝑁-body Simplifications +A number of simplifications were employed in the 𝑁-body code to +ensure the parameter space could be covered in an acceptable amount +of time. +The first simplification, which has already been mentioned in sec- +tion 2, was that we only simulate the dust formed from the vapour +population of the collision - the vapour condensate population. The +primary reason for this is that over such a short simulation time +frame (20 orbits) the vapour condensate population is more likely +to be observationally active. We assume the boulder population will +take much longer to be visible. This is due to the difference in as- +sumed particle formation sizes. Particles from the vapour population +will condense into solids ∼ 1 cm in size whereas boulder population +particles are likely to form at a much larger size than this, typically +kilometres in size. The boulder population will eventually become +observationally active as the particles collide and grind down to +smaller sizes, but this will take 100s to 1000s of orbits (dependent +on disk semi-major axis) and we wanted to focus on the population +that would be immediately observable (Jackson et al. 2014). +The only exception to this simplification is the two most massive +remnants from the collision. These are included in the simulation +because they are likely to have a non-negligible effect on the dy- +namical evolution of the vapour condensate population and can act +as stirring bodies due to their gravitational influence. The primary +driver of any stirring will be the largest remnant, as this body was +much more massive than the second largest remnant. +The second simplification is that the vapour condensate population +is not gravitationally active, so none of the particles interact with +each other. Given the small individual masses which constitute this +population this is a reasonable assumption. We can assume that the +bulk of the dynamical evolution of the system will be governed by the +central star and the two largest remnants of the boulder population. +Finally, we do not simulate any collisions between vapour con- +densate particles. We would expect the vapour condensate disk to +fade away over time because the particles will be ground down to +the blowout size through mutual collisions. In this work we are pri- +marily interested in determining whether short-term variations on +orbital timescales could be a distinct characterising observational +feature of giant impacts. The important result for this investigation +is understanding whether these variations are present in different +configurations rather than their lifetime or observability. +4 CONCLUSIONS +In this work we simulated a number of eccentric debris disks pro- +duced by giant impacts. We examined the resultant morphology and +MNRAS 000, 1–20 (2022) + +Eccentricity and Extreme Debris Disks +19 +infrared emission of these objects to gain an understanding of how +different collision parameters can affect observability. +In total, we ran 84 𝑁-body simulations which covered a broad +parameter space of eccentricities and collision positions along the +eccentric orbit. When examining the morphology of these simulated +disks over time we found the same basic dichotomy between disks +produced by Parallel collisions and disks produced by Perpendicu- +lar collisions. Parallel collision disks were more tightly bound while +Perpendicular collision disks expanded out in spiral ring structures. +We also found that eccentricity and collision positions can alter the +structure of the disk greatly. Increasing eccentricity can either expand +or constrain the disk depending on the collision position. Collisions at +the periapsis of eccentric orbits give the opportunity for large changes +in particle semi-major axis by an effect analogous to the Oberth Ef- +fect. This leads to a more expansive, less tightly constrained disk +which scales with eccentricity. Collisions at apoapsis do not benefit +from this effect, so the disks produced by these collisions are gen- +erally more constrained. Increasing eccentricity in this case reduced +the orbital velocity at apoapsis, further reducing the impact of the +Oberth Effect and leading to an even more tightly bound disk. This +pattern was found in both the Parallel and Perpendicular collision +cases but was much more pronounced in the Perpendicular collision +case, as material is preferentially ejected in directions parallel to the +embryo orbit. In the Perpendicular* case, where the collision oc- +curs perpendicular to both the orbital path and orbital plane of the +centre of mass, we found a very similar morphology pattern to the +Perpendicular case with large expansive spiral arms present across +most of the parameter space. The slight differentiating feature was +the less well-defined spiral rings in the Perpendicular* case which +was attributed to the additional parallel kick component providing a +small offset to the particle orbits. +Beyond the morphology it was also important to examine the ob- +servability of all disks within the parameter space. Using particle po- +sitions from the 𝑁-body simulations, we used RADMC-3D to model +the total infrared emission of the disks in multiple wavelengths over +time. We found periodic, short-term variations in the mid-infrared +flux of all of the disks created by Parallel collisions when observed +from face-on (x-y). These variations were not present in the flux +of the disks created by Perpendicular collisions. This supports the +conclusions from Watt et al. (2021) who found the same result for +circular orbits. The nature of these short-term variations, as with the +disk structure, is highly dependent on centre of mass eccentricity and +collision position. Increasing eccentricity acts to suppress certain flux +dips depending on the collision position. For periapsis collisions, the +dip at each integer orbit was suppressed with increasing eccentricity +until at high eccentricity it is no longer visible. For apoapsis colli- +sions, the dip occurring at each half-integer orbit is suppressed with +increasing eccentricity. In both of these cases the new peaks in flux +coincide with a time when most of the disk material is at periapsis, +implying that the flux variation due to distance from the star is over- +riding any flux variation due to changing optical depth. Increasing +eccentricity also affects the relative magnitudes of flux at different +wavelengths. The 24𝜇m and 10𝜇m intensities begin to converge as +eccentricity is increased until in the e=0.6 and e=0.8 simulations the +10𝜇m flux is greater than the 24𝜇m flux at certain times. This is +likely due to average periapsis of the particles getting closer to the +host star with increased eccentricity. This leads to higher average +dust temperatures and preferential emission at shorter wavelengths. +This work and Watt et al. (2021) both indicate that short-term +variability or ’wiggles’ in infrared emission is a good indicator of the +sudden appearance of a vapour condensate population, most likely +from giant impact. However, this work also points to the conclusion +that the formation of this variability is highly dependent on several +collision variables, including collision orientation, viewing orienta- +tion, eccentricity, and the true anomaly of the collision. Additionally, +lifetime of this variability is expected to be short due to the rapid +evolution of the vapour condensate population. This may help us to +start to understand why we have observed relatively few debris disks +with distinct short-term variations at present. The parameter space +in which we would expect short-term variations is likely to be fairly +narrow, so while there could be a large number of debris disks pro- +duced by giant impacts, the number with distinct variability could be +much smaller. +There is clearly much more work required to make any of the +conclusions above more definitive, but a target for future work could +be understanding the distribution of extreme disk eccentricities. We +found eccentricity plays a key role in the magnitude of short-term +variations, but if the number of EDDs with larger eccentricities is +small then this effect is much less important. Eccentric EDDs take +on the eccentric characteristics of their progenitor embryos, so un- +derstanding the eccentricity distribution of the planetary embryos in +the early Solar System could give strong hints about the distribu- +tion of disk eccentricities. Additionally, simulating both the vapour +condensate and boulder populations concurrently would allow an +understanding of the full evolution of an impact-produced disk. The +collision rate within both of these populations is key to the longevity +of any disk, so quantifying these values would help to refine under- +standing of observability. +5 SOFTWARE +In this work we used the following software: RADMC-3D (Dulle- +mond et al. 2012), numpy (Harris et al. 2020), scipy (Virtanen et al. +2020), and matplotlib (Hunter 2007). +ACKNOWLEDGEMENTS +LW acknowledges financial support from STFC/UKRI (grant +ST/S505274/1). ZL thanks UKRI (grant ST/V000454/1). This +work was carried out using the computational facilities of the +Advanced Computing Research Centre, University of Bristol - +https://www.bristol.ac.uk/acrc/. Thanks to Professor Maughan for +his help and discussion on computing disk scale heights. Thanks to +Dr. Su and the anonymous reviewer for discussion that improved the +quality of this work. +DATA AVAILABILITY STATEMENT +The data underlying this article will be shared on reasonable request +to the corresponding author. +REFERENCES +Agnor C. B., Canup R. M., Levison H. F., 1999, Icarus, 142, 219 +Alexander R., Pascucci I., Andrews S., Armitage P., Cieza L., 2014, in Beuther +H., Klessen R. S., Dullemond C. P., Henning T., eds, , Protostars and +Planets VI. University of Arizona Press, Tucson, pp 475–496 +Aumann H., et al., 1984, The Astrophysical Journal, 278, L23 +Benz W., Slattery W. L., Cameron A. G. W., 1988, Icarus, 74, 516 +Bonsor A., Augereau J.-C., Thébault P., 2012, A&A, 548, A104 +Bonsor A., Raymond S. 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C., Clarke C. J., Booth M., 2011, Celestial Mechanics and Dynam- +ical Astronomy, 111, 1 +Wyatt M. C., Bonsor A., Jackson A. P., Marino S., Shannon A., 2017, Monthly +Notices of the Royal Astronomical Society, 464, 3385 +APPENDICES +Large figures and tables are found in the online supplementary ma- +terial associated with this work. These include: +• Table A1 and A2 which summarise all 𝑁-body simulations. +• Fig. B1 and Fig. B2 which show the full grid of Perpendicular +IR emission viewed in the x-y and y-z planes respectively. +• Fig. B3 and Fig. B4 which show the full grid of Parallel IR +emission viewed in the x-z and y-z planes respectively. +• Fig. B5 and Fig. B6 which show the Perpendicular* IR emission +grid viewed from the x-z and y-z planes respectively. +A set of videos (.mp4) showing how the density of a select number +of disks evolves over time is also available online. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–20 (2022) + +1 +APPENDIX A: SUMMARY OF N-BODY SIMULATIONS +Table A1 and Table A2 summarise all N-body simulations used in this work. +MNRAS 000, 000–000 (0000) +arXiv:2301.03307v1 [astro-ph.EP] 9 Jan 2023 + +2 +Table A1. The various 𝑁 -body simulations which have been analysed in this work. The Sim. column is used throughout this paper to refer to individual +simulations +Sim. +Collision Orientation (radians) +Centre of Mass Orbital Eccentricity +Collision True Anomaly (radians) +PR Drag +0 +0.0𝜋 +0.0 +0.0𝜋 +Off +1 +0.0𝜋 +0.0 +0.25𝜋 +Off +2 +0.0𝜋 +0.0 +0.5𝜋 +Off +3 +0.0𝜋 +0.0 +0.81𝜋 +Off +4 +0.0𝜋 +0.0 +1.0𝜋 +Off +5 +0.0𝜋 +0.05 +0.0𝜋 +Off +6 +0.0𝜋 +0.05 +0.25𝜋 +Off +7 +0.0𝜋 +0.05 +0.5𝜋 +Off +8 +0.0𝜋 +0.05 +0.81𝜋 +Off +9 +0.0𝜋 +0.05 +1.0𝜋 +Off +10 +0.0𝜋 +0.1 +0.0𝜋 +Off +11 +0.0𝜋 +0.1 +0.25𝜋 +Off +12 +0.0𝜋 +0.1 +0.5𝜋 +Off +13 +0.0𝜋 +0.1 +0.81𝜋 +Off +14 +0.0𝜋 +0.1 +1.0𝜋 +Off +15 +0.0𝜋 +0.2 +0.0𝜋 +Off +16 +0.0𝜋 +0.2 +0.25𝜋 +Off +17 +0.0𝜋 +0.2 +0.5𝜋 +Off +18 +0.0𝜋 +0.2 +0.81𝜋 +Off +19 +0.0𝜋 +0.2 +1.0𝜋 +Off +20 +0.0𝜋 +0.4 +0.0𝜋 +Off +21 +0.0𝜋 +0.4 +0.25𝜋 +Off +22 +0.0𝜋 +0.4 +0.5𝜋 +Off +23 +0.0𝜋 +0.4 +0.81𝜋 +Off +24 +0.0𝜋 +0.4 +1.0𝜋 +Off +25 +0.0𝜋 +0.6 +0.0𝜋 +Off +26 +0.0𝜋 +0.6 +0.25𝜋 +Off +27 +0.0𝜋 +0.6 +0.5𝜋 +Off +28 +0.0𝜋 +0.6 +0.81𝜋 +Off +29 +0.0𝜋 +0.6 +1.0𝜋 +Off +30 +0.0𝜋 +0.8 +0.0𝜋 +Off +31 +0.0𝜋 +0.8 +0.25𝜋 +Off +32 +0.0𝜋 +0.8 +0.5𝜋 +Off +33 +0.0𝜋 +0.8 +0.81𝜋 +Off +34 +0.0𝜋 +0.8 +1.0𝜋 +Off +35 +0.5𝜋 +0.0 +0.0𝜋 +Off +36 +0.5𝜋 +0.0 +0.25𝜋 +Off +37 +0.5𝜋 +0.0 +0.5𝜋 +Off +38 +0.5𝜋 +0.0 +0.81𝜋 +Off +39 +0.5𝜋 +0.0 +1.0𝜋 +Off +40 +0.5𝜋 +0.05 +0.0𝜋 +Off +41 +0.5𝜋 +0.05 +0.25𝜋 +Off +42 +0.5𝜋 +0.05 +0.5𝜋 +Off +43 +0.5𝜋 +0.05 +0.81𝜋 +Off +44 +0.5𝜋 +0.05 +1.0𝜋 +Off +45 +0.5𝜋 +0.1 +0.0𝜋 +Off +46 +0.5𝜋 +0.1 +0.25𝜋 +Off +47 +0.5𝜋 +0.1 +0.5𝜋 +Off +48 +0.5𝜋 +0.1 +0.81𝜋 +Off +49 +0.5𝜋 +0.1 +1.0𝜋 +Off +50 +0.5𝜋 +0.2 +0.0𝜋 +Off +51 +0.5𝜋 +0.2 +0.25𝜋 +Off +52 +0.5𝜋 +0.2 +0.5𝜋 +Off +53 +0.5𝜋 +0.2 +0.81𝜋 +Off +54 +0.5𝜋 +0.2 +1.0𝜋 +Off +55 +0.5𝜋 +0.4 +0.0𝜋 +Off +56 +0.5𝜋 +0.4 +0.25𝜋 +Off +57 +0.5𝜋 +0.4 +0.5𝜋 +Off +58 +0.5𝜋 +0.4 +0.81𝜋 +Off +59 +0.5𝜋 +0.4 +1.0𝜋 +Off +60 +0.5𝜋 +0.6 +0.0𝜋 +Off +61 +0.5𝜋 +0.6 +0.25𝜋 +Off +MNRAS 000, 000–000 (0000) + +3 +Table A2. Table A1 continued. +Sim. +Collision Orientation (radians) +Centre of Mass Orbital Eccentricity +Collision True Anomaly (radians) +PR Drag +62 +0.5𝜋 +0.6 +0.5𝜋 +Off +63 +0.5𝜋 +0.6 +0.81𝜋 +Off +64 +0.5𝜋 +0.6 +1.0𝜋 +Off +65 +0.5𝜋 +0.8 +0.0𝜋 +Off +66 +0.5𝜋 +0.8 +0.25𝜋 +Off +67 +0.5𝜋 +0.8 +0.5𝜋 +Off +68 +0.5𝜋 +0.8 +0.81𝜋 +Off +69 +0.5𝜋 +0.8 +1.0𝜋 +Off +70 +0.0𝜋 +0.0 +0.0𝜋 +On +71 +0.0𝜋 +0.0 +1.0𝜋 +On +72 +0.0𝜋 +0.8 +0.0𝜋 +On +73 +0.0𝜋 +0.8 +1.0𝜋 +On +74 +0.5𝜋 +0.0 +0.0𝜋 +On +75 +0.5𝜋 +0.0 +1.0𝜋 +On +76 +0.5𝜋 +0.8 +0.0𝜋 +On +77 +0.5𝜋 +0.8 +1.0𝜋 +On +78 +0.0𝜋 (Out of plane) +0.0 +0.0𝜋 +Off +79 +0.0𝜋 (Out of plane) +0.0 +1.0𝜋 +Off +80 +0.0𝜋 (Out of plane) +0.4 +0.0𝜋 +Off +81 +0.0𝜋 (Out of plane) +0.4 +1.0𝜋 +Off +82 +0.0𝜋 (Out of plane) +0.8 +0.0𝜋 +Off +83 +0.0𝜋 (Out of plane) +0.8 +1.0𝜋 +Off +MNRAS 000, 000–000 (0000) + +4 +Figure B1. Grid of IR emission in the x-y plane for debris disks produced by collisions perpendicular to the centre of mass of the two colliders. The first 0.5 +orbits for all simulations have been cropped for clarity (during this period flux density increases rapidly as the disk expands). The grid shows the effect of varying +centre of mass eccentricity, position of the collision along the orbit, and the observation wavelength. The columns indicate different collision positions along the +orbit, 𝜈 = 0.0𝜋 denotes a collision at periapsis while 𝜈 = 1.0𝜋 denotes one at apoapsis. The rows represent different eccentricities: (a) e=0.0 (Sim. 35, 37, 38, +39 in Table A1, left to right); (b) e=0.2 (Sim. 50, 52, 53, 54 in Table X, left to right); (c) e=0.4 (Sim. 55, 57, 58, 59 in Table A1, left to right); (d) e=0.6 (Sim. +60, 62, 63, 64 in Table A1 and A2, left to right); (e) e=0.8 (Sim. 65, 67, 68, 69 in Table A2, left to right). The different observation wavelengths are denoted by +different line colours - 24𝜇m: blue, 10𝜇m: orange, 4.5𝜇m: green, and 3.6𝜇m: red. +APPENDIX B: ADDITIONAL SIMULATED INFRARED EMISSION GRIDS +Below are the full RADMC-3D infrared emission grids for a variety of orientations and viewing angles. +MNRAS 000, 000–000 (0000) + +. +=0.5m +V=0.811 +1.5 +(KI) n) +1.0 +0.5 +99 +Flux (ly) +1.0 +0.5 +99 +(KI) n) +1.0 +0.5 +99 +d) +() xn +1.0 +0.5 +99 +(K) xn +1.0 +0.5 +0.0 +2 +. +10 +7 +10 +8 +10 +10 +Orbital Periods +Orbital Periods +Orbital Penods +Orbital Penods +24.0μm +10.0μm +4.5μm +3.6μm5 +Figure B2. Same as Fig. B1 but in the y-z plane. +MNRAS 000, 000–000 (0000) + +=0.0π +=0.5π +=0.81π +=1.0π +1.25 +1.00 +(Jy) +0.75 +0.50 +0.25 +0.00 +1.25 +1.00 +(Jy) +0.75 +0.50 +0.25 +1.25 +0.00 +1.00 +(KI) +0.75 +Flux +0.50 +0.25 +0.00 +1.25 +1.00 +D +(KI) +0.75 +0.50 +0.25 +0.00 +1.25 +1.00 +(KI) +0.75 +0.50 +0.25 +0.00 +2 +6 +8 +10 +2 +A +6 +8 +10 +2 +6 +10 +4 +6 +8 +10 +Orbital Periods +Orbital Periods +Orbital Periods +Orbital Periods6 +Figure B3. Same as Fig. B1 but for a collision occurring parallel to the centre of mass path and in the x-z plane. +MNRAS 000, 000–000 (0000) + +V=0.0π +=0.5π +=0.81π +0.8 +e +(KI) +0.6 +Flux +0.4 +0.2 +0.0 +0.8 +b +(KI) +0.6 +Flux +0.4 +M +0.2 +0.0 +0.8 +(KI) +0.6 +0.4 +0.2 +0.0 +0.8 +(Jy) +0.6 +0.4 +0.2 +0.0 +0.8 +(K) +0.6 +Flux +0.4 +0.2 +NN +0.0 +2 +4 +6 +8 +10 +2 +4 +6 +8 +10 +2 +4 +6 +8 +10 +2 +4 +6 +8 +10 +Orbital Periods +Orbital Periods +Orbital Periods +Orbital Periods7 +Figure B4. Same as Fig. B1 but for a collision occurring parallel to the centre of mass path and in the y-z plane. The rows represent different eccentricities: (a) +e=0.0 (Sim. 0, 2, 3, 4 in Table A1, left to right); (b) e=0.2 (Sim. 15, 17, 18, 19 in Table A1, left to right); (c) e=0.4 (Sim. 20, 22, 23, 24 in Table A1, left to right); +(d) e=0.6 (Sim. 25, 27, 28, 29 in Table A1, left to right); (e) e=0.8 (Sim. 30, 32, 33, 34 in Table A1, left to right). +MNRAS 000, 000–000 (0000) + +V : +=0.0π +=0.5π +=0.81π +=1.0π +1.25 +1.00 +(Jy) +0.75 +0.50 +0.25 +1:29 +1.00 +0.75 +Flux +0.50 +0.25 +1:29 +.2 +1.00 +Flux (Jy) +0.75 +0.50 +SASSS +0.25 +1:29 +1.00 +0.75 +0.50 +0.25 +1:29 +1.00 +e. +(K) +0.75 +Flux +0.50 +0.25 +0.00 +2 +4 +6 +8 +10 +2 +4 +6 +8 +10 +2 +6 +8 +10 +2 +4 +6 +8 +10 +Orbital Periods +Orbital Periods +Orbital Periods +Orbital Periods8 +Figure B5. Same as Fig. B1 but for a collision occurring perpendicular to the centre of mass path and orbital plane and in the x-z plane. The rows represent +different eccentricities: (a) e=0.0 (Sim. 78 and 79 in Table A2, left to right); (b) e=0.4 (Sim. 80 and 81 in Table A2, left to right); (c) e=0.8 (Sim. 82 and 83 in +Table A2, left to right). +MNRAS 000, 000–000 (0000) + +V=1.0m +0.5 +24.0μm +4.5μm +(a) +10.0μm +3.6μm +0.4 +0.3 +xnI +0.2 +0.1 +0.0 +0.5 +(b) +0.4 - +(KI) +0.3 +0.2 +0.1 +M +0.0 +0.5 +(c) +0.4 +0.3 +xn +0.2 +0.1 +0.0 +4 +6 +8 +2 +6 +8 +Orbital Periods +Orbital Periods9 +Figure B6. Same as Fig. B1 but for a collision occurring perpendicular to the centre of mass path and orbital plane and in the y-z plane. +MNRAS 000, 000–000 (0000) + +=. +=1.D +0.6 +24.0μm +4.5μm +(a) +0.5 +10.0μm +3.6μm +0.4 +E0 +0.2 +0.1 +0.0 +0.6 +(b) +0.5 +0.4 +(K) +E0 +0.2 +0.1 +0.0 +0.6 +(c) +0.5 +0.4 +() +0.3 +0.2 +0.1 +0.0 +2 +4 +6 +8 +10 +2 +4 +6 +8 +10 +Orbital Periods +Orbital Periods \ No newline at end of file diff --git a/7tE1T4oBgHgl3EQfnQST/content/tmp_files/load_file.txt b/7tE1T4oBgHgl3EQfnQST/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..05dc455f78570c99bfa83656f010eea20f0d70d8 --- /dev/null +++ b/7tE1T4oBgHgl3EQfnQST/content/tmp_files/load_file.txt @@ -0,0 +1,2470 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf,len=2469 +page_content='MNRAS 000, 1–20 (2022) Preprint 10 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 Isolating the Extreme Debris Disk Signature - Explorations of Eccentric Extreme Debris Disks Formed by Giant Impacts Thomas Lewis,★ Lewis Watt, and Zoë M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Leinhardt School of Physics, University of Bristol, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Wills Physics Laboratory, Tyndall Avenue, Bristol, BS8 1TL, UK Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' in original form ZZZ ABSTRACT In this work we used 𝑁-body simulations and a radiative transfer package to model the evolution of eccentric debris disks produced by giant impacts between planetary embryos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This included how the morphology and infrared emission of these disks varied with embryo eccentricity and collision true anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We found that eccentric disks inherit the eccentric properties of the centre of mass orbit of the two colliding embryos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' However, the orientation of the collision with the respect to this orbit plays a key role in determining how closely the disk material resembles the centre of mass orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Additionally, we found that increased eccentricity acted to suppress the formation of certain short-term variations in the disk emission depending on the collision position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These short-term variations have been associated with an observational phenomenon called extreme debris disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Short-term variability has been suggested as a potential signature for giant impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Key words: circumstellar matter – planets and satellites: formation – method: numerical 1 INTRODUCTION Debris disks are one of the most useful observational features of solar systems, as they encode a large amount of information on the evolution of a system and are fairly easy to observe around other stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Additionally, debris disks appear to be quite common with examples in our own Solar System in the form of the Asteroid Belt and the Kuiper Belt, as well as in many other systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Traditional debris disks are belts of material orbiting around stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This material is composed of particles of a range of sizes from larger planetesimals to smaller dust particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Debris disks are most commonly detected through observation of excess infrared emission from a star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The dust grains in debris disks are heated by their host star and re-radiate energy in the infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This dust emission is visible in the spectral energy density (SED) profile of the star as a small bump in the IR wavelengths when compared to a pure stellar blackbody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Debris disks are often characterised using fractional luminosity, 𝑓 = 𝐿𝑑𝑖𝑠𝑘/𝐿∗, which compares the disk luminosity (𝐿𝑑𝑖𝑠𝑘) to that of the host star (𝐿∗) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Wyatt 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Typically, the fractional luminosity of a debris disk is 𝑓 < 10−3 (Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Most debris disks that have been observed are so-called exo-Kuiper belts with inner radii of tens or hundreds of au and similar magnitudes in width, analogous to the Kuiper belt (located between ∼30 au and ∼50 au, Trujillo & Brown 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' A prototypical example of an exo- Kuiper belt is the debris disk around Vega which was first reported in Aumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (1984) and has an inner radius of 86 au and extends out to hundreds of au (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Debris disks are thought to be the remnants of structures called protoplanetary disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These structures are collections of gas and ★ E-mail: tom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='lewis@bristol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='uk dust orbiting around young, newly-formed stars from which planets and planetesimals are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Protoplanetary disks lose mass and dissipate over time through accretion, leaving a dusty debris disk as a remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Debris disks are therefore generally an order of magnitude fainter than protoplanetary disks which have fractional luminosity values of at least 𝑓 ≈ 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Additionally, as a consequence of their origins, debris disks contain little to no gas and exist around more mature stars with ages ≳10 Myrs (Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Finally, debris disks also typically have very low optical depth in optical wavelengths when compared to protoplanetary disks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These four factors – luminosity, system age, gas content, and optical depth, help observationally distinguish debris disks from protoplanetary disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='1 Extreme Debris Disks The standard model for debris disks assumes a steady-state collisional cascade which gradually grinds down large 1-100 km planetesimals into smaller and smaller particles (Wyatt & Dent 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Quillen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Wyatt 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Smaller particles are able to re-emit absorbed radiation in the infrared much more efficiently than larger particles, meaning collisional grinding helps to create a dust population which is observable through its IR emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Sub-micron dust particles are small enough to be affected by radiation pressure from the host star, so most simple models assume these particles are blown out of the debris disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The balance of collisional grinding and radiation pressure blow out leads to a stable dust population in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This implies the observed fractional luminosity should be constant over orbital timescales (Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' However, it is important to note that this steady-state is only maintained while there is sufficient mass in the large planetesimal population to keep a roughly consistent collision rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Once mass runs out at the top of the distribution the © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='03307v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='EP] 9 Jan 2023 2 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The IR excess from the debris disk of ID8 detected by the Spitzer Space Telescope at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 (blue dots) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 (red crosses) 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The error bars (1𝜎) on both sets of data points represent the uncertainty in excess flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Data from Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2019) reproduced here with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' amount of dust produced decreases leading to a drop in fractional luminosity over several hundreds of Myrs (Wyatt 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' However, some debris disks do not seem to be sustained by the traditional steady-state collisional cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This sub-class of debris disks are much brighter than traditional debris disks and often highly variable, so are usually referred to as extreme debris disks (EDDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' EDDs have average fractional luminosities in excess of 10−2 (Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2015), but this value can vary significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Two examples of observed EDDs are ID8 (Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2014) and HD23514 (Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Both of these disks display variability in their infrared output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' ID8 showed a rapid increase in infrared luminosity (in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6𝜇m and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜇m wavebands) of roughly 40% at the start of 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This was followed by a gradual decay in output throughout the rest of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' ID8 also displays short-term, quasi- periodic variability overlaid on the longer-term decay trend Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Observational data of the excess flux from ID8 over 2012 and 2013 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' HD23514 shows a similar decay trend to ID8 in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6𝜇m and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜇m wavebands without any significant short-term variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These two types of variability both occur on timescales of years and decades which is much faster than the Myr evolution timescales associated with dust generated by a collisional cascade Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Additionally, they are both found around fairly young stars with ages of ∼35 Myr and ∼120 Myr respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' At these ages (>10 Myr) protoplanetary disks will likely have been cleared of gas Math- ews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2011), implying the unusual brightness and variability is not related to collisional activity during earliest stages of planet formation (Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The unusual brightness and variability of EDDs, exemplified by ID8 and HD23514, cannot be explained via the traditional, steady- state model alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is because the short-term and medium-term variations in luminosity of disks like ID8 are far too rapid to be attributed to the dust produced by slow, collisional grinding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In- stead, other processes have been suggested to account for these ob- servations, including dynamical instabilities (Bonsor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2013) and comets scattering into the inner regions of the system (Marino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Nesvorný et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Bonsor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In Moór et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) a variety of the possible explanations for EDDs are explored and eval- uated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' One of these explanations that has received significant interest in recent years has been giant impacts (Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Wyatt & Jackson 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Wyatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Giant impacts are a type of collision between large terrestrial bodies such as planets and planetary embryos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Whilst there is no standard definition of the giant in giant impacts, in this work we will assume this refers to rocky planetesimals with a diameter >1500 km Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Giant impacts are highly energetic interactions which can partly melt and vapourise the surface of the colliding embryos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The ejected vapour cools and condenses after the collision to form a cloud of small dust particles in the mm-cm range (Johnson & Melosh 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This cloud of dust would be detectable in the infrared almost immedi- ately after impact due to the small particle size (Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The sudden appearance of this vapour population could explain the rapid increase in luminosity of EDDs like ID8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The decay trend of ID8 and HD23514 could be attributed to the transient nature of the vapour condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Dust particles of this size are small enough to be significantly affected by radiation pressure and Poynting-Robertson drag which dissipates the disk and decreases the total disk luminosity over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The ejected melt material will also cool and solidify into a population of planetesimals which can undergo the traditional colli- sion cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This fresh population could eventually produce visible dust once the collisional cascade has reached steady-state (which could take many thousands of orbits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Giant impacts could therefore produce enough ejected material to form a new transient debris disk which would be observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Numerical simulations have suggested that giant impacts are com- mon in the late stages of terrestrial planet formation (Chambers & Wetherill 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Agnor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Chambers 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Quintana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2016) and could play a key role in planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Evidence of possible giant impacts can be seen across our own Solar System, including the formation of the Moon (Canup & Asphaug 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Hart- mann 2014), the size and location of Mercury (Benz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 1988), the origin of the Pluto-Charon system (McKinnon 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Canup 2010), and collisional family around Haumea (Leinhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This does provide some evidence that giant impacts would be occurring at the right time in stellar evolution to cause the observed EDDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The observation of EDDs has led several authors to a focus on giant impacts as a possible explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 Previous Work on Modelling Extreme Debris Disks Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2014) and Jackson & Wyatt (2012) modelled the dynamical evolution of debris disks produced by planetary collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' One of the key conclusions from both of these projects was that the morphology of the disk is primarily shaped by the collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is the point in space at which the giant impact occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' At the moment of collision all of the source particles which will make up the disk are located at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' During the collision the particles each receive a velocity kick which places the dust on a distribution of defined orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Over time the dust clump will shear out due to differences in their respective orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' However, the collision point remains a fixed point on all of their orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In other words, all particles must pass through the collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In reality, the collision point will not be a single point, but a small volume which depends on the size of the colliding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2014) assumed a single point for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In addition to the collision point, there is also the anti-collision line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is a radial line on the opposite side of the star to the collision point and in the plane of progenitor embryo which all particles will cross at some point in their orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2014) found that this confluence of orbital paths leads to an asymmetry in the disk structure with an over-density of material in these two regions which increases the perceived optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This asymmetry effect should create a quasi-periodic variation in the luminosity of the disk on orbital timescales, although the observability of this variation would MNRAS 000, 1–20 (2022) 20122013 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6μm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5μm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 Excess Flux (mjy) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 56100 56200 56300 56400 56500 Barycentric Modified Julian DateEccentricity and Extreme Debris Disks 3 depend on viewingangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='Disk asymmetry hasbeenused as apossible explanation for the short-term variability observed in ID8, as well as the observable characteristics of EDDs more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2014) showed that this asymmetry eventually smears out as the particle orbits precess over many orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Typically, the asymmetric phase lasts around 1000 orbits, so the observable lifetime largely depends on the semi-major axis of the original planetary embryo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) followed on from this work but took a different approach by simulating the entire process from collision to debris disk evolution, as well as the expected infrared emission of the de- bris post-impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' To simulate collisions between planetary embryos they used a modified version of an SPH (smooth particle hydrody- namics) code called GADGET-2 (Springel 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Carter 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This code was originally developed to model cosmological events, such as galaxy cluster formations, however it has been re-purposed to be used in many other astrophysical contexts, including planet forma- tion and planetary collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The modified version of GADGET-2 allows the use of tabulated equations of state (EOS) to determine the thermodynamic state of the particles (Ćuk & Stewart 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The planetary embryos were initialised with an iron core and forsterite mantle using ANEOS equations of state for these two materials (Mar- cus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Melosh 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The embryos were equilibrated as in previous work (Denman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Performing these simulations required an understanding of the state and composition of the mass ejected from the embryo colli- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As mentioned earlier, numerical simulations have shown that giant impacts with sufficient energy can produce a vapour conden- sate cloud with particles in mm-cm range (Johnson & Melosh 2012), as well as a more standard population of planetesimals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We refer to these two populations of ejecta as the vapour condensate and boulder populations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The boulder population is formed from material that has been melted by the giant impact and then re-solidified into planetesimals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The boulder population would generally contain large km-sized plan- etesimals which grind down through a collision cascade until they reach a steady-state with a fixed size distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This size distribu- tion is usually assumed to resemble a power law based on observa- tions of debris disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In this way the disk formed from the boulder population is similar to a traditional debris disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Conversely, the vapour condensate population forms directly from material vapourised in the collision and is thought to be composed of much smaller particles, generally in the mm/cm range depending on impact velocity and impactor size (Johnson & Melosh 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Dust particles in this size range are able to absorb and re-emit in the in- frared much more efficiently than larger particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This implies that the vapour condensate population would be visible to observers al- most immediately after the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The boulder population would eventually becomevisible in theinfrared, butwould takemuch longer, as it would need time for the large planetesimals to grind down into sufficiently small dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This difference in formation would also likely have an effect on the lifetime on these two populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Assuming we consider the two populations completely separately, the vapour con- densate population has no larger planetesimals to replenish its dust leading to a shorter overall lifetime when compared to the boulder population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Given this dichotomy in the ejected mass, an important aspect of the collision simulation was determining the fraction of mass in liquid and vapour post-impact, as this dictated the relative ratio of the boulder and vapour condensate populations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) assumed that the supercritical ejecta cooled isentropically until the triple point temperature was reached inside a liquid-vapour dome determined from the material equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The vapour fraction of each SPH particle was then calculated using the lever rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This allowed them to determine the mass of the vapour condensate population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) assumed that the vapour condensate population would be visible immediately after the collision and therefore sim- ulated the infrared emission from this dusty debris while ignoring the boulder population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' They found that in certain circumstances the infrared emission of the vapour disk could exhibit short-term vari- ations on orbital timescales, similar to the variations observed in ID8 and P1121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This was assumed to be related the disk asymmetry highlighted in Jackson & Wyatt (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Increased optical depth at the collision point and anti-collision line led to a drop in the ob- served emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' However, the appearance of these variations was highly dependent on the parameters of the collision, in particular the orientation with respect to the orbital path of the centre of mass of the two colliding embryos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Collisions which predominately launched ejecta perpendicular to the centre of mass orbit produced disks with variability while collisions which launched ejecta parallel to the cen- tre of mass orbit did not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Variability is a good indicator that dust has been generated by an impact rather than some other mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Any factor which suppresses variability would make detection and characterisation of giant impacts less likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The question of the link between giant impacts and EDDs is an important one, because there is a fundamental tension between our assumptions and the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Giant impacts are thought to be common during the later stages of planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' They are as- sumed to occur during a separate stage of solar system formation af- ter the conclusion of the oligarchic stage (Kenyon & Bromley 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' If giant impacts can lead to EDDs and if giant impacts are common, why do we not observe more EDDs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Depending on the definition of EDD, the number of observed EDDs is around a few dozen at the time of publication (Moór et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Melis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Kennedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Rieke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This implies there could be something wrong with our assumptions about the regular- ity of giant impacts or perhaps something about the way EDDs are formed which makes them difficult to detect and observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' EDDs give us a vital observational foothold when trying to under- stand planet formation in other solar systems and provide evidence for different planet formation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Despite their rarity, EDDs can play a key role in our understanding of planet formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Through this investigation we hoped to more fully understand the factors which suppress EDD variability and observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 Aims In this project we expanded upon the work first outlined in Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) found that simulated collisions between planetary embryos on circular orbits could produce debris disks with distinct, short-term variability in their infrared output, similar in na- ture to observations of EDDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' They also found that the presence of this variability was highly dependent on the specific parameters of the collision, such as impact parameter, impact speed, and collision orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This result implied a potentially narrow parameter space over which EDDs could be observable, leaving the vast majority of debris disks created by giant impacts observationally indistinguish- able from traditional debris disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' However, not all planetary collisions are likely to occur on per- fectly circular orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Instead, we would expect the population of embryos to exist on orbits with a range of eccentricities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Eccentricity would likely change the morphology of the resultant debris disk and affect its infrared emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In addition, an embryo on an eccentric MNRAS 000, 1–20 (2022) 4 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' orbit will have an instantaneous orbital velocity that varies depend- ing on its position in the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The embryo orbital velocity at the moment before the collision would likely affect the velocity distribu- tion of the ejected material which would change the morphology of the resultant debris disk and again affect the infrared emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The combined impact of these effects is unclear, but it is important to un- derstand whether the parameter space which can generate observable variability is as narrow as Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) concludes when consid- ering more realistic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The main aim of this work was therefore to investigate how embryo eccentricity and collision position affects the observability of these short-term variations in disk flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In section 2 we outline the steps we performed to simulate embryo collisions and subsequent disk evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We then detail the analysis we performed on the simulation data in order to compare the disks produced by different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In section 3, we examine how the morphology and observability of the simulated disks changed with eccentricity and collision position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We then discuss how these results compare to other simulated and observed data and the implications on explanations for the origin of EDDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Finally, in section 4 we summarise the work and suggest areas of future exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2 METHODS Our numerical campaign was broken down into several steps which we will cover briefly in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='1 Modelling the Collisions The first step was modelling the collision between the planetary embryos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) ran a large array of collision scenarios covering a range of impact speeds and impact parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' However, we focussed on a single collision simulation between two 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='1 Earth- mass embryos (containing 4 × 104 SPH particles) with an impact velocity of 10 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Index 8 of Table A1 in Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) gives the full details of this collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The general simulation setup, includ- ing embryo composition and equations of state used, are provided in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) demonstrated that this particular collision generated the most distinct short-term variations in infrared emission and provided a good starting point to study the effect of orbital eccentricity and collision position on variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We used this single SPH simulation to generate three contrasting baseline cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As mentioned earlier in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2, Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) found that infrared variability was highly dependent on whether the collision occurs parallel or perpendicular to the orbital path of the centre of mass of the two embryos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We therefore rotated the initial simulation data by 90◦ to ensure we investigated both the perpendic- ular and parallel cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The impact parameters of both of these cases are summarised in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The Parallel configuration in Table 1 was a set of parameters which Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) showed generated observable short-term variations in the simulated disk emission for a circular orbit, whereas the Perpendicular configuration did not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Collisions that are parallel to the orbital path of the centre of mass of the two embryos are denoted by 𝜃 = 0◦ while collisions that occur perpendicular to that path are denoted by 𝜃 = 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The orientations of these two collisions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We also performed analysis of a third orientation where the col- lision occurs perpendicular to both the preceding cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Both the Parallel and Perpendicular collisions still occurred within the orbital plane of the original centre of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' However, there is another possi- ble orientation where the velocities of the embryos are perpendicular Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The two collision configurations used throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The columns are as follows: 𝑀𝑒𝑚𝑏 is the mass of the projectile and target plan- etary embryos, 𝑣 is the relative impact velocity between the two embryos, 𝑏 is the impact parameter, 𝑎 is the semi-major axis of the target embryo, and 𝜃 is the collision orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The collision orientation tracks how the collision occurs with respect to the orbital path of the centre of mass of the two embryos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In this work we consider parallel and perpendicular collision orientations along with a range of eccentricities and collision positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' All simulations are summarised in Tables A1 and A2 of the online supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Config 𝑀𝑒𝑚𝑏 𝑣 𝑏 𝑎 𝜃 In plane?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Parallel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='1𝑀⊕ 10 km s−1 0 1 au 0◦ In Perpendicular 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='1𝑀⊕ 10 km s−1 0 1 au 0◦ In Perpendicular* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='1𝑀⊕ 10 km s−1 0 1 au 90◦ Out (a) Parallel collision (𝜃=0◦) (b) Perpendicular collision (𝜃=90◦) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' A simple cartoon diagram to demonstrate the two collision cases studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The large black circles represent the planetary embryos involved in the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The dashed black line represents the orbital path of the centre of mass of the two embryos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The thick black arrow indicates the orbital velocity direction of this centre of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The grey clouds and arrows show the direction in which material is preferentially ejected in each collision case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The red arrows indicate the relative velocities of the embryos - i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=', the collision orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' to the orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This case was labelled with Perpendicular* in Table 1 and was not studied by Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' A brief summary of the basic processes of the collision modelling performed by Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) is found in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' For a more comprehensive explanation see the full details in Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 Evolving the Particles through Time In order to evolve the ejecta for several orbits after the collision, the SPH simulation data was handed over to an 𝑁-body integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The output SPH particle data we used had been modified by Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) following the procedure outlined in their work, producing the vapour condensate population and the two largest remnants of the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This simulation had orbital parameters matching our Parallel and Perpendicular configurations from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The largest (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='146𝑀⊕) and second largest (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='001𝑀⊕) remnants account for the majority of the mass in the boulder population, so were also included as they could have a noticeable effect on the evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The largest remnants were identified by examining the kinetic and gravitational potential energies of the SPH particles to determine which particles are bound and which are unbound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This was an itera- tive process which identified the particle with the lowest gravitational potential energy as the seed particle for the largest remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Other particles were then added to this remnant if their kinetic energy was less than their potential energy in the centre of mass frame of the remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The process was repeated for the second largest remnant ignoring the largest remnant particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In addition to the extraction of the two largest remnants and the MNRAS 000, 1–20 (2022) Eccentricity and Extreme Debris Disks 5 vapour mass, the vapour condensate population was upscaled using a process which maintained the velocity distribution and the total mass of the original SPH particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This procedure was originally outlined by Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) and was done to shift the resolution of the simulation to focus on the vapour condensate population, improving the granularity of the simulation when resolving the more complex gravitational interaction of these particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The two largest remnants of the impact were converted directly to 𝑁-body particles, as they were expected to be single gravitationally coherent object rather than a distribution of small dust particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' At the end of this processing we had a set of ∼100,000 particles with individual position and velocity data which matched the distribution of ejecta after the SPH collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We evolved this system of particles using the leapfrog integrator as described in Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We ran 84 𝑁-body simulations using the same SPH data output from the process described above, but in each run we varied the centre of mass orbital eccentricity and the true anomaly of the collision to see how the debris disk morphology and output flux changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' All of these simulation runs are summarised in Table A1 and Table A2 in the online supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The ’Sim.’ value in these tables will be used to refer to individual runs throughout the rest of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We evolved the system in each case for 20 orbits of the pre-collision centre of mass orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' It is important to make clear that particles used in these 𝑁-body simulations are tracers for the mass distribution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In reality, given the average mass of the particles used in the 𝑁-body simulation each particle would be roughly 1 km in radius (assum- ing a particle density of 3 g cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' However, an individual vapour condensate particle is likely to be between a few microns and a few millimetres in radius (Johnson & Melosh 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In these simulations the 𝑁-body particles are being used as "super-particles" to represent a distribution of dust particles and track the spatial distribution of the disk mass rather than the positions of individual particles in a disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Additionally, the 𝑁-body particles only interacted gravitation- ally with the star and the two largest remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 Simulating Disk Emission We also simulated the total flux emitted by the dust particles in the disk over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We used a radiative transfer code package called RADMC-3D (Dullemond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' RADMC-3D takes as input a cubic grid containing particle densities and one or more energy sources, usually a single star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' It then uses this data to generate syn- thetic images/spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In our case we generated synthetic images at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='1 orbit intervals using the following process - we set up a 3 au cubic grid and binned the particles into the grid cells based on their current 𝑁-body posi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In total there were 301 cells along each axis of the grid, giving a total of 3013 cubic cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We input this particle density grid into RADMC-3D alongside a solar-type star as the single energy source for the system - simulated using a 5700K blackbody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This involved specifying stellar mass, stellar radius, position, and stellar spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Additionally, the dust population was assumed to exist in a fixed power law size distribution with particles sizes ranging from 1mm to 1𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The dust opacities were determined from the opacity tool developed to determine the DIANA standard opacities (Toon & Ack- erman 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Dorschner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Woitke 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Finally, we just needed to configure RADMC-3D to produce images at some common observation wavelengths: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6𝜇m, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜇m, 10𝜇m and 24𝜇m, and select the camera angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The camera angle defines from what direction the image is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In this work we limited our observations to the three fundamental planes of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We call these planes x-y, x-z, and y-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In the x-y plane the disk is face-on while in the y-z and x-z planes the disk is edge-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The collision point and collision line for all disks are aligned in the y-z plane at (0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' RADMC-3D produces observation images which gives the flux emitted by the disk as viewed from a specific direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is meant to simulate how the object would look when observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In order to capture the total flux of the disk at a particular timestep, we summed the flux values from each pixel in the observation image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We used RADMC-3D to simulate the disk flux for the entire 20 orbits of the 𝑁-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3 RESULTS AND DISCUSSION In total, we ran 84 𝑁-body simulations using the output of a sin- gle SPH simulation to generate three different collision scenarios (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Using these three configurations as base cases, we varied the centre of mass orbital eccentricity between e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 and e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In addition, we varied the position of the collision along the centre of mass orbit with the true anomaly value, 𝜈, used to track this collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' A value of 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 represented a collision at the periapsis of an eccentric orbit while a value of 𝜈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 represented a collision at the apoapsis of an eccentric orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We varied the collision point between 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 and 𝜈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These parameter limits were chosen because they represented the extremes of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' A maximum eccentricity of 𝑒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 was chosen, because we expect the number of planetary embryos on orbits with eccentricity greater than this value to be quite low based on numerical planet formation simulations of runaway and oligarchic growth, as well as pebble accretion models (Chambers & Wetherill 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Izidoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Levison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Raymond & Izidoro 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Matsumura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Izidoro & Raymond 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' All of the simulation runs are noted in Table A1 and A2 of the online supplementary material with their collision parameters and an associated simulation number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We will use these simulation numbers throughout this section to refer to the different simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Note that Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 0 and Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 35 are the 𝑁-body simulations shown in Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='1 The Morphology of Circular Disks Firstly, we examined how the morphology of the giant impact ejecta evolves through time on a circular orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Simulations from Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2014), Wyatt & Jackson (2016), and Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) show that immediately after the collision the ejecta is clumped together without any discernible structure, however after several orbits of the centre of mass the material shears out to form a clear disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We find that this is true throughout all of our simulations, but the precise shape and structure of the disk varies greatly as we adjust the collision parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The most distinct morphological difference in all of our simu- lations was found between disks created by Parallel collisions and disks created by Perpendicular collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is illustrated for cir- cular orbits in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3 which shows the spatial evolution of a debris disk from a Parallel collision and a Perpendicular collision on a circular orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The snapshots in this figure range from just a few days after the collision to 8 orbits/years after the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Both disks shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3, start with a clump of material at the col- lision point with some anisotropy, but evolve in very different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The disk from the Perpendicular collision (bottom row) produces spiral arm structures which eventually evolve into concentric rings that spread out across a broad range of semi-major axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' On the other MNRAS 000, 1–20 (2022) 6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Evolution of the morphology of two debris disk created from a giant impact of two different configurations over eight orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The top row (a) shows the evolution of a debris disk from Parallel collision (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 0 from Table A1 in the online supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The bottom row (b) shows the evolution of a debris disk from a Perpendicular collision (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The inset figures in the bottom left of the first timestep shows the collision orientation with respect to the centre of mass orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Brighter colours indicate a greater density of material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The anisotropic distribution of the dust at t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='05 in both plots shows the effect of changing the collision orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The two plots at the end of (a) and (b) show the view of each disk in the x-z and y-z planes at the final timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' hand, the Parallel collision produces a largely contiguous disk with few distinguishable rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The clear difference between disks pro- duced by Parallel and Perpendicular collisions is consistent across parameter space explored in this work, although eccentricity and collision position do have an impact on morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The difference in the morphology of disks created by Parallel and Perpendicular collisions can be explained by understanding how the collision affects the distribution of semi-major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' When a collision occurs parallel to the orbital path of the centre of mass of the two embryos (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2a), material is preferentially ejected in a direction perpendicular to the orbital path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The direction of this velocity kick does not greatly change the orbital path of the ejected particles, leading to a tight distribution of semi-major axes and a clumpier disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' On the other hand, when a collision occurs perpendicular to the centre of mass orbital path (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2b), material is preferentially ejected along the orbital path of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This causes a greater change in the velocities of the ejected particles, leading to a broader distribution of semi-major axes and a more extended disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This can be thought of similarly to prograde/retrograde burn by a satellite versus a radial/anti-radial burn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' On the end of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3, we have also included the view looking towards the x-z and y-z planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The y-z plane on both disks has a typical flared-out pattern similar to a bow tie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is because in this view we are looking directly at the collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is the pinch point through which all particles must pass at some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Either side of the collision point the particle orbits flare out slightly as they all follow their slightly different orbit inclinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In the x-z view, we see two denser regions at either end of the disk, marking the collision point (right) and anti-collision line (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In the Parallel collision case (see (a) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3), the anti-collision line region is fairly compact similar to the collision point, however in the Perpendicular case (see (b) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3), this is closer to a series of dense regions in a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This matches what we would expect from the x-y morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Using RADMC-3D to calculate the radiance of these disks over time shows the expected effect on observation of these two different morphologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 4a we see clear periodic variation in the radi- ance of the Parallel collision simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Dips occur every half-orbit which coincides with the collision point and the anti-collision line, since most of the material tracks closely with the largest remnant of the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is what would be expected from the increase in density and optical depth which occurs at these points (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Increased optical depth means the total flux visible to an observer decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is compared to the Perpendicular collision case (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 4b) where we see no distinct periodic variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As highlighted be- fore and in Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021), the difference between these two is likely a result of the different distributions of semi-major axis in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The Parallel case has a tighter distribution of semi-major axis which creates dense regions of material at the collision point and anti-collision line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The Perpendicular case has a wider distribution of semi-major axis which reduces the density of these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 The Morphology of Eccentric Disks As mentioned in section 2, we ran a number of 𝑁-body simulations where the eccentricity of centre of mass orbit and collision position along the centre of mass orbit were varied between 𝑒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 and 𝑒 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 and 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 and 𝜈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 respectively, where 𝜈 is the true anomaly of the collision position with 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 representing a collision at periapsis and 𝜈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 representing a collision at apoapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We wanted to understand how changes in eccentricity of the centre of mass affected the structure of the debris disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 6 show how the morphology of the debris disks change with eccentricity and collision position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' All disks are plotted exactly 10 orbits after the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This was chosen because at this point the structure of the disk had stabilised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Additionally, at this time the largest remnant of the collision and most of the other disk material was passing through the collision point (marked by white arrows in each plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We find the same dichotomy in morphology exists between Paral- lel and Perpendicular collisions in eccentric orbits as with circular orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Debris disks from Parallel collisions are more tightly bound whereas those from Perpendicular collisions are extended over a greater range of semi-major axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Additionally, these debris disks broadly inherit the eccentric characteristics of the centre of mass orbit with the number density of the disk tracing the original embryo orbit (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We also see the same periodic increase in density at the collision point which is responsible for the short-term variations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' MNRAS 000, 1–20 (2022) (a) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='05 t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 t=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 t=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 (b) (n) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='05 t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 t=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 =8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 t=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 0 2 0 2 X (AU) n) AxEccentricity and Extreme Debris Disks 7 (a) Infrared emission of a simulated debris disk produced by a Parallel colli- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is from Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 0 in Table A1 in the online supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (b) Infrared emission of a simulated debris disk produced by a Perpendicular collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is from Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 31 in Table A1 in the online supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The total infrared emission of a simulated extreme debris produced by two different types of collision orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This was simulated using the RADMC-3D package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This region of increased density travels around the disk tracking with the largest remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' A gap is also visible in the denser region which is likely a result of the largest remnant of the collision scatter- ing surrounding material as it approaches the narrow orbital space of the collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The gap travels around the disk in the middle of the dense region, but it becomes most distinct as the largest remnant passes through the collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The eccentricity of the centre of mass also has an effect on the spread of semi-major axis values in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' For the collisions at periapsis in both the Parallel and Perpendicular collision cases, in- creasing eccentricity leads to a greater spread of semi-major axis values, although this effect is much more pronounced in the Per- pendicular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The reverse effect occurs for collisions at apoapsis where the disks are more extended at low eccentricities and become more tightly bound as eccentricity is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This can be explained with reference to Oberth Effect in astronautics (Oberth 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Accel- erating a body while it is at its maximum orbital velocity at periapsis will increase the semi-major of the orbit more efficiently, pushing the apoapsis of the body further from the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Decelerating the body at periapsis will have the opposite effect, circularising the orbit and decreasing the semi-major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The strength of this effect roughly scales with eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In the case of a Perpendicular collision, ma- terial is preferentially kicked parallel or anti-parallel to the velocity of the centre of mass of the two embryos creating a larger range of semi-major axis values amongst the ejected particle population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In the Parallel collision case material is preferentially kicked perpen- dicular to this velocity, meaning the effect is much less pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Accelerating a body while it is travelling more slowly at apoapsis will give a much smaller boost to the semi-major axis, so instead we see a more constrained disk across all eccentricities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Changing the position of the collision along the orbit has a clear effect on the morphology of the disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' For circular orbit and low eccentricity cases shifting the collision position simply rotates the entire density pattern of the disk when comparing to 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 case, but otherwise the morphology remains similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' For example, in row (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 6 the 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 case (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2 and Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 37) is essentially the 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 0 and Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 35) case rotated by 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The higher density region created by the collision point is obviously in a different spatial position and disk expansion direction has also changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As we increase eccentricity in the middle two intermediate cases (𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 and 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='81𝜋), the collision point moves closer to the apparent periapsis of the eccentric disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The higher density region caused by the collision point also shifts accordingly with the collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is to be expected when the true anomaly is fixed and eccentricity of an orbit is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We also see in these two middle cases at high eccentricity that the expansion direction of the disk is no longer on the opposite side of the disk to the collision point, as the periapsis and the collision point no longer align.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This can be seen most clearly in the third column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We do not see a shift in the apoapsis and periapsis collision cases (outer columns of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Both of the higher density regions remain very close to the apoapsis and periapsis respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' An interesting consequence of increasing eccentricity for collisions at apoapsis is that the periapsis of the centre of mass orbit is brought closer to the host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Velocity kicks from the collision can then shift the periapsis of individual ejected particles even closer to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In the most extreme case (bottom right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 6) there is a build-up of material on the host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In reality, this material would be accreted onto the surface of the star, but it does serve to highlight how close material is getting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As will be shown in the next section, this has an effect on the IR output of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Finally, we also looked at how eccentricity and collision position affect the vertical structure of the disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The inset plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 6 show views towards the x-z and y-z planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As with circular orbits, there is a bow tie structure in the y-z plane for collisions at apoapsis and periapsis (first and last columns of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This structure gets flattened as eccentricity is increased in the peri- apsis case, but increases in the height in the apoapsis case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is discussed more quantitatively in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In the x-z plane there is an oval structure with dense regions at each disk ansae for apoapsis and periapsis collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is an effect of the viewing angle as the line of sight through each disk ansae will have a greater column den- sity than the rest of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Similarly to the y-z view, this structure flattens with increasing eccentricity in the periapsis collision case leading to a more homogeneous density distribution along the mid- plane of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Conversely in the apoapsis case, the disk becomes more extended in x-z view, but the disk ansae are still distinctly dense regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In the 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 case, the structures in the x-z and y-z are swapped due to the rotated density structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In the 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='81𝜋 case we see a twisted bow tie shape in both of x-z and y-z planes due to the off-centre location of the collision point from these viewing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 The Morphology Disks from Out-of-plane Collisions We covered a smaller subset of parameters for collisions in the Per- pendicular* orientation from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These results are summarised in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As with all other simulations, the Perpendicular* disks resemble the centre of mass orbit of their progenitor embryos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The morphology pattern of Perpendicular* collisions across the parameter space is broadly similar to the Perpendicular collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We found the dust sheared out quickly, spreading across a large range of semi-major axes in a set of concentric rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Increasing eccentricity also had a similar MNRAS 000, 1–20 (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='40 (A) Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='20 6 8 10 Orbits since collisionhw 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='40 (K[) Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='30 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 5+ 8 10 Orbits since collision8 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Grid of simulated debris disks produced by collisions parallel to the orbital path of the centre of mass of the two colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The grid shows the effect of varying centre of mass eccentricity and position of the collision along the orbital path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Colour in these plots is used to indicate 𝑁 -body particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' All plots are taken from the same timestep which is 10 orbits after the collision and shows the disk in the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The columns in this figure show different positions along the centre of mass orbital path at which a collision has occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The 𝜈 value at the top of the figure tracks the true anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 denotes a collision at periapsis while 𝜈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 denotes a collision at apoapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The rows represent changing eccentricity with (a) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0, (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2, (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4, (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6, and (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The white triangles on each plot point to the spatially fixed collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The two inset figures show the same disk from the x-z and y-z planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' MNRAS 000, 1–20 (2022) V=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0㎡l 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 V=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5l V=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='81π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 V=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 1 1 X (AU)10 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Grid of simulated debris disks produced by collisions perpendicular to the orbital path of the centre of mass of the two colliders and perpendicular to the orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Colour in these plots is used to indicate 𝑁 -body particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' All plots are taken from the same timestep which is 10 orbits after the collision and shows the disk in the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Columns in this figure show different positions along the centre of mass orbital path at which a collision has occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The 𝜈 value at the top of the figure tracks the true anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 denotes a collision at periapsis while 𝜈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 denotes a collision at apoapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The rows represent changing eccentricity with (a) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0, (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4, and (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The white triangles on each plot point to the spatially fixed collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The two inset figures show the same disk from the x-z and y-z planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' effect on the spatial distribution of the rings as in the Perpendicular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' For collisions at periapsis, increasing eccentricity boosted the range of semi-major axes while the opposite occurred for collisions at apoapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The main difference between the Perpendicular and Perpendicular* cases was that the rings appeared much less cleanly defined than in the Perpendicular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This effect is likely a result of more particles receiving both a perpendicular and parallel velocity kick component in the collision compared to the Perpendicular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The additional parallel kick component could help to offset particle orbits slightly, creating rings that are more indistinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Disks in this case are also much thinner in the z-axis than in the other orientations, with average scale heights typically a tenth the size of their corresponding Parallel and Perpendicular disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This can be seen in the x-z and y-z inset figures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Flatter disks were expected given that the material was preferentially ejected into the orbital (x-y) plane, so the velocity kick components in the z-direction are minimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The average scale height of a disk over the first 10 orbits of Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 0 in Table A1 in the online supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The red dotted line is a horizontal linear fit to the data, representing the average scale height across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 Scale Height In order to move beyond simple qualitative descriptions of the simu- lated disks, we quantified the average scale height of each simulated disk over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This allowed us to compare the average height of different debris disks through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Initially, we tried using a simple exponential fit of the particle density against height above orbital plane, however, the density-height profile did not seem to follow an exponential decay in every timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Instead, we went for a more general approach and calculated the 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='7th percentile of the particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' While this may not be exactly equivalent to the scale height, it can provide a rough estimate that can be used for comparative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 8 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 9 summarise this scale height information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 8 shows an example of how scale height varies over the first 10 orbits of Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The most obvious trend in this data is the periodic dips in scale height over the course of a single orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This reinforces our conclusions about the origin of the short-term variations in the simulated emission shown in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The collision point and anti-collision line are spatially fixed regions in the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This can be seen in the bow tie shape of the disks in the y-z plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The x-y plane in all of the simulations is defined by the orbital plane of the centre of mass of the two colliding embryos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As the largest remnant and a large proportion of the disk material passes through the pinch points at the collision point and along the anti-collision line, the average scale height drops because the x-y plane defines the zero point of the height scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This drop in average scale height implies an increase in density at the pinch points which creates the short-term variations seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The second trend seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 8 is the attenuation of the magnitude of the short-term variations over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is likely a result of mate- rial being sheared out over time leading to less pronounced clumping at the collision point and anti-collision line so less variation in the average scale height of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' To gain an understanding of how eccentricity and collision position affects scale height we plotted scale height against these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The results of this are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' One of the primary results from this is that the average scale height decreases with eccentric- ity when the collision occurs at the periapsis of the centre of mass orbit, whereas the average scale height increases with eccentricity when the collision occurs at apoapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is likely related to the variation in orbital velocity at different points in an eccentric orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' MNRAS 000, 1–20 (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='03:0 Height (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0L5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='005 2 t 6 8 Timestep0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 V=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 V=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 N 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content="5 o'T- 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 X (AU)Eccentricity and Extreme Debris Disks 11 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Average scale height variation of simulated debris disks with eccentricity, collision orientation, and collision position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The solid line shows scale height variation for collisions occurring parallel to the orbital path of the centre of mass of the two colliders while the dotted line shows this value for Perpendicular collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The columns show different collision positions around the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The velocity of a body in an eccentric orbit is at its maximum at periapsis and at its minimum at apoapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This effect scales with ec- centricity, meaning increasing eccentricity will increase the velocity at periapsis, but decrease the velocity at apoapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The relative colli- sion velocity between the projectile and the target embryos is fixed at 10 km s−1 across all simulations as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The average scale height of the disk should be dependent on the distribution of particle inclination in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' A wider inclination distribution leads to a greater average disk scale height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The inclination of a particle is much easier to change when its orbital velocity is lower, so this is why we find a higher average scale height when the centre of mass of the two embryos is travelling more slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This pattern is reinforced when looking at collision points between apoapsis and periapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' For example, 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 (a collision occurring halfway between apoapsis and periapsis) results in an average scale height does that not vary significantly with eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Following the reasoning outlined above, this implies that the orbital velocity of the centre of mass at the point of collision does not vary significantly with eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' When 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='81𝜋 we see that average scale height increases with eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This again matches our expectations as the orbital velocity of the centre of mass at the point of collision will decrease as eccentricity is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 9 shows the scale height trends for both Parallel (solid blue line) and Perpendicular collisions (dotted orange line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These lines follow very closely across all 𝜈 and eccentricity values, implying that the overall trend in average scale height is related to orbital speed and collision position rather than collision orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 Infrared Emission from Extreme Disks The figures in this section contain grids of light curves at various wavelengths generated by radiative transfer (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3) which show how the dust emission of the simulated disks varies over the first 10 orbits after the embryo collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Similarly to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 6, these figures show the parameter space we covered during our investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The eccentricity of the centre of mass in the collision increases as you move down the rows while the true anomaly of the collision changes from periapsis to apoapsis as you move from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10, the light curve in the second row of the final column corresponds to a collision occurring at apoapsis (𝜈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋) on an orbit with eccentricity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10 shows the light curves for our simulated collisions occur- ring parallel to the orbital path of the centre of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Short-term variations are found across all collision positions and eccentricities, however the periodicity and magnitude of these variations changes as we traverse the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' At low eccentricity in Parallel collisions the dust emission dips at half-integer orbit intervals, so there are two dips in emission per orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As mentioned in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4, this variation can be related to the average scale height of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 8 and the light curves in the top left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10 are generated from the same simulation and demonstrate dips at similar half-integer intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We also find that as eccentricity is increased one of these emission dips is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' For example, with a Parallel collision at periapsis (𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋, first column in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10), the dip that occurs on each full orbit (vertical dotted lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10) is increasingly suppressed with increasing eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This continues until at e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 there is only a single dip detectable per orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Conversely, when the collision occurs at apoapsis (𝜈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋, final column in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10), the half-integer dip is suppressed as eccentricity is increased, leaving only the dip that occurs on each orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Additionally, increased eccentricity also results in a general increase in the magnitude of dips across all collision positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' It is not entirely clear what is causing these effects, but one of the most likely explanations is that ever-changing distance between the dust and the star in an eccentric orbit creates a periodic variation in the disk emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The amount of flux emitted by dust in a debris disk is dependent on the amount of energy absorbed from the star which in turn is dependent on the distance to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In circular orbits the distance from the star to an orbiting body does not change with time, however in eccentric orbits this distance is continually changing, as a body completes an orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This means the amount of stellar flux received by the dust and therefore the temperature of the dust continually changes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The temperature of the dust should peak at orbital periapsis and reach a minimum at apoapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Hotter dust will emit more total energy and preferentially emit in shorter wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 6 revealed that a denser region of dust is co-located with the largest remnant as it orbits the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is why MNRAS 000, 1–20 (2022) (a) v=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 (b) v=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25π (c) v= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5π (d)v=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='81π (e)v=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='035 (ne) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='010 Parallel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='005 Perpendicular 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 Eecentricity12 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Grid of IR emission in the x-y plane for debris disks produced by collisions parallel to the centre of mass of the two colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The first 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 orbits for all simulations have been cropped for clarity (during this period flux density increases rapidly as the disk expands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The grid shows the effect of varying centre of mass eccentricity, position of the collision along the orbit, and the observation wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The columns indicate different collision positions along the orbit, 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 denotes a collision at periapsis while 𝜈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 denotes one at apoapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The rows represent different eccentricities: (a) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 0, 2, 3, 4 in Table A1, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (b) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 15, 17, 18, 19 in Table A1, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (c) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 20, 22, 23, 24 in Table A1, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (d) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 25, 27, 28, 29 in Table A1, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (e) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 30, 32, 33, 34 in Table A1, left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The different observation wavelengths are denoted by different line colours - 24𝜇m: blue, 10𝜇m: orange, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜇m: green, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6𝜇m: red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' a peak is seen in dust emission as the largest remnant passes through the periapsis of eccentric orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The strength of this dust temperature variation effect would in- crease with eccentricity, as the periapsis gets closer to the star (mov- ing down the columns in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' It is possible as centre of mass eccentricity is increased the impact of this effect overrides the impact of any optical depth variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' To investigate this we plotted the average dust temperature for a set of eccentricities directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 11 shows how the temperature varies across different eccentricities for a Parallel collision orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Average dust temperature was broadly flat in the circular case but had strong peaks in the highly eccentric cases which coincided with the largest remnant passing through periapsis (see bottom right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This suggests tem- perature variation is a major driver of variability in the most eccentric Parallel disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 12 shows the light curves for our simulated collisions occur- ring perpendicular to the orbital path of the target embryo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This grid covers a subset of the total parameter space, because all of the light curves from Perpendicular collisions look broadly similar across most of the parameter space we studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These light curves are sta- ble with time and do not display much variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This reinforces the conclusion from Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) that collisions occurring per- pendicular to the centre of mass orbit do not result in disks with short-term variations in their emission, at least for the collision con- figurations shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As mentioned earlier, the suggested explanation for this result is that collisions perpendicular to the cen- tre of mass orbit preferentially eject material along the orbital path (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This means, on average, that the direction of the velocity kick given to the ejected material from the collision is more likely to be parallel or anti-parallel to the original orbital velocity of the centre of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This leads to a faster shearing out of the dust and prevents a build-up of dust density at the collision point and anti-collision line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' MNRAS 000, 1–20 (2022) =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5斤 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='81 V=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 a 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5μm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='75 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='75 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content="25 DD'0 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content="25 c) DD'T 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content="25 DD'T 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='75 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content="25 e DD'T 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='75 Flux ANNAAAA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 DDO 2 6 10 2 4 6 8 10 2 女 6 8 10 2 6 8 10 Orbital Periods Orbital Periods Orbital Periods Orbital PeriodsEccentricity and Extreme Debris Disks 13 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Average temperature across the entire disk for a set of Parallel collisions occurring at apoapsis (right-most column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10, Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 4, Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 24, and Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 34 in Table A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This increase in density and optical depth causes drops in observed emission, so preventing this build-up removes a source of variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The only notable exceptions to this result are the light curves in the bottom right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These light curves are the result of a collision occurring at the apoapsis of a highly eccentric orbit (e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In these light curves there is some significant variance in emission, but not the consistent, periodic variation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We can attempt to understand this by looking at the morphology of the highly eccentric disk in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 6 (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This disk has the smallest average disk periapsis compared to other disks in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Particles making closer approaches to the host star will be heated to higher dust temperatures and preferentially emit in shorter wavelengths while at the periapsis of their orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This explanation is supported by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 13 which shows how the average dust temperature varies over the timeframe shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Average dust temperature peaks at every half-integer orbit for the first few orbits, aligning with the peaks in emission seen in the bottom right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The collision in this case occurred at apoapsis which implies the emission and temperature peak when the largest remnant and most of the other disk material is passing through the periapsis of their orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This conclusion is further supported by the relative flux levels of the different wavelengths in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The average disk temperature rises over the first 5 orbits before flattening out which broadly mirrors the ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In general, the shorter wavelengths are much stronger compared to the other collisions in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The initial periodic variability tends to peter out after a few orbits - likely due to the rapid Keplerian shearing of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The velocity kicks from the collision set all dust particles on slightly different orbital trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Over several orbits the material becomes increasingly out-of-sync with the original clump of material co-located with the largest remnant until dust is spread more evenly around the disk and there is no periodic variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This shearing effect may also account for the shorter wavelengths peaking slightly earlier in the first few orbits of collision shown in the bottom right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The velocity kicks from the collision will alter the orbit of some amount of material, so that it makes a closer approach to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This material will be slightly out-of-sync with the main bulk of material around the largest remnant, so the shorter wavelength peak from this material will occur at a slightly different time to the main peak tracked by 10𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Another noteworthy observation in both Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 12 is that the 24𝜇m and 10𝜇m flux density lines begin to converge as eccentricity is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In some highly eccentric cases the 10𝜇m line actually exceeds the 24𝜇m line (Row (b) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 12 and rows (d) and (e) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This behaviour is consistent across all collision positions we simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In addition, the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6𝜇m and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜇m lines are both essentially zero across most of the parameter space, except for the most extreme eccentricities (Row (b) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 12 and row (d) and (e) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10) where these flux densities increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This result again supports the idea that the short-term variations at higher eccentric- ities are driven by distance variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As eccentricity increases, the average periapsis of the particles gets closer to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Dust particles approaching closer to the star will be heated to a higher equilibrium temperature and preferentially radiate in shorter wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' An important caveat to these results is that in our 𝑁-body simu- lation particles are only removed when they enter the stellar radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='005 au assuming the host star has the same radius as the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In reality, dust particles are likely to be sublimated at a much larger semi-major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Some of the material that builds up close to the star, as shown in the bottom right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 6, may be re- moved by this effect which could subdue the temperature variability somewhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The persistence of this material may particularly affect the shorter observation wavelengths (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜇m and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6𝜇m) which do not seem to drop as expected when the largest remnant passes through apoapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Material building up close to the star (but outside the stel- lar radius) would help to keep shorter wavelengths more stable than longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The full grid of simulated IR emission for disks produced by Per- pendicular collisions can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B1 in the online supple- mentary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Additionally, a set of animated movies showing the evolution of various disks are included in the online supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='1 Infrared Emission from Out-of-plane Collisions We also simulated the IR emission from our Perpendicular* col- lisions where the velocities of the colliding embryos were perpen- dicular to the centre of mass orbital path and orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These collisions preferentially ejected material into the orbital plane of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 14 shows the light curves for our simulated collisions occurring perpendicular to the orbital path of the target embryo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As might be expected from the morphology pattern discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 14 shows a very similar pattern to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 12, with broadly flat disk emission across most of the parameter space except for the highly eccentric collision at apoapsis (bottom right) where the variability was driven by dust temperature variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The peaks in this case seem to be more distinct and more clearly defined for longer than the Perpendicular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In this orientation some particles are ejected perpendicular to orbital path which may increase the amount of time it takes for the disk to shear out completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The lower shearing rate may help the disk stay more coherent for slightly longer, making the temperature-induced variation appear more distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As with the Perpendicular case, we also see the shorter wavelength flux peaking slightly before the 10𝜇m flux due to Keplerian shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 Observing Emission from the x-z and y-z Planes Up until this point we have focussed on observing the emission of the disk when looking down at the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In other words, we have observed these disks face-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is useful as a starting point for discussions of morphology and observability, but in reality, we are likely to see EDDs from various viewing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' For Perpendicular collisions there is a clear lack of short-term MNRAS 000, 1–20 (2022) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 400 e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 (K) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 temperature 350 300 lisk 250 abei Avel 200 150 6 10 Orbital Periods14 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10 except for simulated debris disks from collisions perpendicular to the orbital path of the centre of mass of the two colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The rows represent different eccentricities: (a) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 35 and 39 in Table A1, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (b) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 65 and 69 in Table A2, left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Average temperature across the entire disk for an e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 Perpen- dicular collision occurring at apoapsis (bottom right corner of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 12, Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 69 in Table A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' variability in the IR emission of the resulting disks when looking at the x-z and y-z planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' For example, the grid in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 15 shows a subset of the simulated IR emission for Perpendicular collisions viewed in the x-z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Similarly, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B2 in the online supplementary material shows the simulated IR emission for Perpendicular collisions viewed in the y-z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The lack of variability in x-z and y-z is consistent with the view from the x-y plane (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 12 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B1 in the online supplementary material) and implies short-term, periodic variability is a good observable indicator of giant impact collision orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The exceptions to this rule are the cases with extreme eccentricity and collision positions closer to apoapsis (bottom right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 15) where there is some variability on orbital timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Similar to the x-y view, this is likely a result of oscillating dust temperature as the largest remnant moves from apoapsis to periapsis and back to apoapsis over the course of a single orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This variation would be reflected in the IR emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As with the x-y results, more complex variability is seen in disks produced by Parallel collisions when viewed in the x-z and y-z planes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B4 in the online supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' At these orientations disk ansae, a product of viewing angle (rather than a mor- phological feature), should create additional variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Disk ansae are the two extreme points at the far end of the disk when viewed edge-on (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The apparent column density should increase at these two ansae points when the largest remnant of the collision passes through them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This should create a drop in total emission sim- ilar in nature to the collision point/anti-collision line effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' However, the disk ansae effect will be obscured at certain viewing angles where the disk ansae and collision point/anti-collision line align from the perspective of the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2019) attribute some of the observed variability in ID8 to the disk ansae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' To understand this further take the example of a circular disk shown in the top left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' A clearer view of the x-z and y-z orientations is shown at the end of the top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In the x-z view the collision point and anti-collision line coincide with disk ansae at either end of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This implies there should only be two dips per orbit corresponding to the collision point and anti-collision line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is exactly what is seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 17 where the IR emission from this disk viewed from x-z and y-z planes are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The magenta line in this figure dips at each integer and half-integer orbit which is what would be expected from the collision point/anti-collision line effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Contrasting this with the y-z view where the morphology resembles a bowtie shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In this case the disk ansae and collision point/anti-collision line should be distinct from the observer’s line of sight with the ansae points found either edge of the disk and the collision point/anti-collision line found at the pinch point of the bowtie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' With this orientation four distinct dips in emission should MNRAS 000, 1–20 (2022) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' V=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 (e) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm (K) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5μm Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 (b) (KI) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='75 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 2 6 8 6 8 10 Orbital Peniods Orbital Peniods(>) : 350 temperature 325 300 275 : disk 250 Average I 225 200 175 4 6 8 10 Orbital PeriodsEccentricity and Extreme Debris Disks 15 Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 12 except for simulated debris disks from collisions perpendicular to both the orbital path and orbital plane of the centre of mass of the two colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The rows represent different eccentricities: (a) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 78 and 79 in Table A2, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (b) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 80 and 81 in Table A2, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (c) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 82 and 83 in Table A2, left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' be seen over the course of an orbit as the largest remnant passes through the collision point, the first disk ansa, the anti-collision line, and finally the second disk ansa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' However, looking at the orange line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 17, although there might be some dips in the first two orbits which could align the disk ansa, broadly there is conspicuous lack of consistent periodic variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The dips corresponding to the two ansae points could be suppressed due to the flaring of the bowtie shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Differences in the orbital inclination of the different particles creates a disk that flares out at the ansae points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This flaring could be enough to reduce the column density and optical depth at the ansae points, eliminating any drop in emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Another possibility, which would explain the apparent lack of dips from the collision point and anti-collision line, is that from this line of sight the disk optical depth is much more consistent over time leading to little variation in emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' There is however some limited evidence that disk ansae could induce or support some variability in disk emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The full light curve grids covering the entire parameter space in the x-z and y-z planes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B4 in the online supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B4 we can see that two dips per orbit is a persistent trend as eccentricity is increased in the X-Z case (where the disk ansae and collision-point/anti-collision line align).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This suggests that line of sight ansae effects reinforcing the collision point/anti-collision line effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Much like the emission in the x-y plane, the magnitude and pe- riodicity of the emission in the x-z and y-z planes is dependent on the centre of mass eccentricity and collision position along the ec- centric orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In general, the x-z case is qualitatively similar across our parameter space to the x-y case, with the magnitude of variations increasing with eccentricity for apoapsis collisions, but decreasing for collisions at periapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Also, some dips appear to be suppressed with increasing eccentricity which appears to align with the expla- nation from the x-y case that the major driver of flux variation in more eccentric disks is the distance to the host star as opposed to changes in optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 18 which shows the disk emission in x-z for a subset of the collision parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In general, we find reduced variability in the y-z case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As with all other cases where we find fairly quiescent disks, the exceptions to this are apoapsis collisions at higher eccentricities which have variability that increases in magnitude with increasing eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' MNRAS 000, 1–20 (2022) =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='D 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5μm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 (a) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 (KI) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 2 8 6 8 10 Orbital Periods Orbital Periods16 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 12 but for a collision occurring perpendicular to the centre of mass orbit and viewed from the x-z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The rows represent different eccentricities: (a) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (b) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' A diagram of a debris disk from a collision at the apoapsis of an eccentric orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Observing the disk edge-on towards the periapsis (as shown in the inset diagram in the top left) creates two ansae, A and B, which would be the most extreme points at either end of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 Comparing with other Published Results Giant impact-produced debris disks have been modelled before, most notably in Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In that work they used analytical models to predict the dynamical evolution of material released by a giant collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Their results for a collision on a circular orbit are qualitatively similar to ours with a clump of material released imme- diately after the collision shearing out into a coiled spiral pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We also recreate their collision point and anti-collision line asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2014) also modelled disks produced from eccentric orbits which included varying the eccentricity and the position of the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' They found broadly similar patterns to our results, including the disk being centred on an elliptical orbit rather than a circular one and additional asymmetry when the collision point is moved away from either apoapsis or periapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' One of the main points they focus on is the interaction between apoapsis of the eccentric disk Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The 24𝜇m flux evolution of the debris disk from Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 0 in Table A1 in the y-z (orange) and x-z (magenta) planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The collision which produces this disk occurs on a circular orbit and is orientated parallel to the centre of mass orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' and collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' They argue that dust particles will spend more time at the apoapsis of their orbit than the periapsis which creates a higher density region at the apoapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Additionally, there is the higher density region around the collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The interaction between these two dense regions can either be constructive or destructive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This description seems to fit the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' For col- lisions occurring at apoapsis and higher eccentricities, the RADMC light curves show a large fall in flux on each integer orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is because the dense regions are aligned, creating an ultra-dense region at the apoapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The opposite is true for a collision at periapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The two dense regions are at opposite ends of the orbit, so we find a drop in flux at half-integer orbits when the bulk of material is passing through the apoapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Material spends much less time at the periap- sis so the effect of the collision point is attenuated as eccentricity MNRAS 000, 1–20 (2022) V=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='D 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 (a) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 Flux 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 :8 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 (AI) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 4 6 8 2 4 6 8 Orbital Periods Orbital PeriodsA B Collision point Diskansaepoints B Viewing direction Anti-collision liney-z X-Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='35 Flux (ly) @ 24μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='15 2 4 6 B 10 Orbits since collisionEccentricity and Extreme Debris Disks 17 Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10 but for a collision occurring parallel to the centre of mass orbit and viewed from the x-z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The rows represent different eccentricities: (a) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (b) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We have also added an understanding of how temperature variation is also a factor in this dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The point where our work significantly differs is that Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2014) assumed an isotropic velocity distribution post-impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As covered in previous sections, we have found that velocity distribution plays a significant role in determining both the disk morphology and flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The presence of short-term variability is strongly tied to the initial dust distribution, so accounting for anisotropy is vital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='7 Comparison with Observed Debris Disks It is also important to consider how our simulated results compare to observational instances of EDDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Direct imaging of debris disks is quite difficult, so instead we will focus on the excess IR emission from the disk as the comparison value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In particular, the average fractional luminosity of the disk and the variability of the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As of the date of publication, there are tens of observed examples of EDDs (Moór et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The most well-studied examples of EDDs currently are ID8 and P1121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='1 ID8 and P1121 ID8 is a young solar-type star in NGC2547 which displays strong infrared excess, implying the presence of a dusty debris disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The average fractional luminosity of ID8 is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 × 10−2 (Olofsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Yearly variation in the 24𝜇𝑚 IR excess of ID8 was first observed in Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Periodicity analysis in Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2014) revealed two significant periods in the IR excess emission of ID8: 𝑃1 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='1 days and 𝑃2 = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These two periods were explained as the combined influence of two orbital effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The collision-point collision-line optical depth asymmetry and the disk ansae viewpoint flux drop for edge-on disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The first effect was described early in this work and results from a confluence of particle orbital paths at the collision point and anti-collision line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The second effect is a result of the viewing angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' If ID8 is edge-on or nearly edge- on the two disk ansae would appear to have greater optical depth than the rest of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Assuming a roughly sinusoidal variation to both of these effects and fitting these to the photometric measurements of the disks gives a rough peak-to-peak amplitude of 6×10−3 fractional luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2014) estimated the semi-major axis of the debris disk in ID8 from periodicity analysis to be roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='33 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is roughly consistent with analysis performed by Olofsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This implies the ID8 disk is slightly smaller than most of the ones simulated in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' P1121 is another solar-type star which has displayed high levels of IR excess with variability (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This star is roughly 120 Myrs old, so as with ID8 it is in the age range where planet formation is ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Observations by Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2019) revealed a long-term flux decay in the IR excess of P1121 with a decay timescale of 𝑡0 = 310 ± 60 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Additionally, they found short-term variability in this emission with a period of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='7 days and an amplitude of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='08 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2019) considers a several explanations for this variability, including a giant-impact produced cloud of debris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As discussed earlier this explanation implies a dip in disk emission as the largest collision remnant passes through the collision point and anti-collision line, increasing dust density in these two regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' As with Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2014) and ID8, Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2019) also considers the effect of the viewing angle on disk emission which can create apparent increases in optical depth at the disk ansae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Combining these effects should lead to emission dips at every half-integer or quarter integer depending on the viewing angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This gives a true orbital period of 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 days for MNRAS 000, 1–20 (2022) V=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0T 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 (a) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 (q) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 2 6 8 10 6 8 10 Orbital Periods Orbital Periods18 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' a face-on disk and 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 days for an edge-on disk and implies that the collision point would be at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 au or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='32 au from P1121 depending on the orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In our results for edge-on disks we do not see the additional dips from the disk ansae as suggested by Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2014) and Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The effect of the disk ansae would only be revealed when looking down at the collision point and anti-collision line, so only certain viewing angles would have the possibility of seeing an ansae effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' However, even when viewed the correct orientation we do not see any additional dips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This could be explained by the flaring out of the disk at the ansae point due to the range of orbital inclinations of the disk particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This may help to reduce the optical depth at the ansae points and avoid a noticeable dip in flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 V488 Persei Beyond ID8 and P1121, there are growing numbers of examples of extremely variable debris disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Recent results by Rieke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) have highlighted a particularly acute example of this type of disk around V488 Persei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' V488 Persei is an 80 Myr-old star (Soderblom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2014) which places it in a similar age range to ID8 and P1121 and, as with those stars, at an age where planet formation is assumed to be ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Observations of the infrared excess of V488 Persei over a number of years revealed a relatively quiescent phase, followed by a major uplift in emission in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' During the quiescent phase the disk is still extremely variable with excess infrared emission varying be- tween 30% to 60% of the peak signal at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6𝜇m and 60% to 75% for 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This variability is possibly periodic on the timescale of a few months, but this is still uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Rieke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) used the Debris Disk Simulator (Wolf & Hillenbrand 2005) to fit a simple three- component, optically thin debris disk model consisting of an inner disk at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='35 AU, an outer disk at 25-45 AU, and a distribution of micron-sized grains extended inward from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='3 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This model estimated the total fractional luminosity for the inner component at ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6% and the outer component between 10-16%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The estimated fractional luminosities ( 𝑓 ≫ 10−3) and variability of this disk marks it clearly as a strong EDD candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Rieke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) suggest that such high fractional luminosity and variability implies a very dense and collisionally active disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' They propose that a massive planet or brown dwarf has perturbed the inner disk, creating an exceptionally collisionally active disk with high levels of dust production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' They estimated the amount of dust mass required to produce the major boost in infrared flux seen in 2019 would be equivalent to an 85 km planetesimal disintegrating into a power law collisional cascade of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is well below the size of the planetary embryos that were simulated in this work, however, as mentioned in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2, a giant impact is likely to produce a vapour condensate population with dust grains < 1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The total mass of this dust population is heav- ily dependent on the collision parameters, so a giant impact could imitate a wide range of collision cascade signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' V488 Persei could provide an interesting case study to see whether we observe any of the short-term variation effects highlighted in this work and Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Assuming the 2019 uplift was caused by a collision, we might see a similar decay trend to ID8 where a rapid increase in flux was followed by a slower decay in emission over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' A possible complicating factor when comparing this disk to the results of this work is the assumed pre-existing debris disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Secondary collisions and ongoing instability could help to mask the pure signal of a single collision that we have simulated in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Further observations of this extremely active disk over the next few years will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 Methodological Caveats We employed some simplifications in order to reduce computation time and maximize the number of simulation runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These are impor- tant to keep in mind when evaluating the results presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' One of the most important caveats for all the results in this work is that all simulation runs were based on the Parallel and Perpendicular SPH configurations outlined in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In other cases with other embryo mass ratios, impact velocities, and impact parameters the disk morphologies and IR emission could change dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' However, these configurations were useful as fixed test cases to determine the effect of orbital eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='1 𝑁-body Simplifications A number of simplifications were employed in the 𝑁-body code to ensure the parameter space could be covered in an acceptable amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The first simplification, which has already been mentioned in sec- tion 2, was that we only simulate the dust formed from the vapour population of the collision - the vapour condensate population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The primary reason for this is that over such a short simulation time frame (20 orbits) the vapour condensate population is more likely to be observationally active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We assume the boulder population will take much longer to be visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is due to the difference in as- sumed particle formation sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Particles from the vapour population will condense into solids ∼ 1 cm in size whereas boulder population particles are likely to form at a much larger size than this, typically kilometres in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The boulder population will eventually become observationally active as the particles collide and grind down to smaller sizes, but this will take 100s to 1000s of orbits (dependent on disk semi-major axis) and we wanted to focus on the population that would be immediately observable (Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The only exception to this simplification is the two most massive remnants from the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These are included in the simulation because they are likely to have a non-negligible effect on the dy- namical evolution of the vapour condensate population and can act as stirring bodies due to their gravitational influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The primary driver of any stirring will be the largest remnant, as this body was much more massive than the second largest remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The second simplification is that the vapour condensate population is not gravitationally active, so none of the particles interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Given the small individual masses which constitute this population this is a reasonable assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We can assume that the bulk of the dynamical evolution of the system will be governed by the central star and the two largest remnants of the boulder population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Finally, we do not simulate any collisions between vapour con- densate particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We would expect the vapour condensate disk to fade away over time because the particles will be ground down to the blowout size through mutual collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In this work we are pri- marily interested in determining whether short-term variations on orbital timescales could be a distinct characterising observational feature of giant impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The important result for this investigation is understanding whether these variations are present in different configurations rather than their lifetime or observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 4 CONCLUSIONS In this work we simulated a number of eccentric debris disks pro- duced by giant impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We examined the resultant morphology and MNRAS 000, 1–20 (2022) Eccentricity and Extreme Debris Disks 19 infrared emission of these objects to gain an understanding of how different collision parameters can affect observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In total, we ran 84 𝑁-body simulations which covered a broad parameter space of eccentricities and collision positions along the eccentric orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' When examining the morphology of these simulated disks over time we found the same basic dichotomy between disks produced by Parallel collisions and disks produced by Perpendicu- lar collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Parallel collision disks were more tightly bound while Perpendicular collision disks expanded out in spiral ring structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We also found that eccentricity and collision positions can alter the structure of the disk greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Increasing eccentricity can either expand or constrain the disk depending on the collision position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Collisions at the periapsis of eccentric orbits give the opportunity for large changes in particle semi-major axis by an effect analogous to the Oberth Ef- fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This leads to a more expansive, less tightly constrained disk which scales with eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Collisions at apoapsis do not benefit from this effect, so the disks produced by these collisions are gen- erally more constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Increasing eccentricity in this case reduced the orbital velocity at apoapsis, further reducing the impact of the Oberth Effect and leading to an even more tightly bound disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This pattern was found in both the Parallel and Perpendicular collision cases but was much more pronounced in the Perpendicular collision case, as material is preferentially ejected in directions parallel to the embryo orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In the Perpendicular* case, where the collision oc- curs perpendicular to both the orbital path and orbital plane of the centre of mass, we found a very similar morphology pattern to the Perpendicular case with large expansive spiral arms present across most of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The slight differentiating feature was the less well-defined spiral rings in the Perpendicular* case which was attributed to the additional parallel kick component providing a small offset to the particle orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Beyond the morphology it was also important to examine the ob- servability of all disks within the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Using particle po- sitions from the 𝑁-body simulations, we used RADMC-3D to model the total infrared emission of the disks in multiple wavelengths over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We found periodic, short-term variations in the mid-infrared flux of all of the disks created by Parallel collisions when observed from face-on (x-y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These variations were not present in the flux of the disks created by Perpendicular collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This supports the conclusions from Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) who found the same result for circular orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The nature of these short-term variations, as with the disk structure, is highly dependent on centre of mass eccentricity and collision position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Increasing eccentricity acts to suppress certain flux dips depending on the collision position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' For periapsis collisions, the dip at each integer orbit was suppressed with increasing eccentricity until at high eccentricity it is no longer visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' For apoapsis colli- sions, the dip occurring at each half-integer orbit is suppressed with increasing eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' In both of these cases the new peaks in flux coincide with a time when most of the disk material is at periapsis, implying that the flux variation due to distance from the star is over- riding any flux variation due to changing optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Increasing eccentricity also affects the relative magnitudes of flux at different wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The 24𝜇m and 10𝜇m intensities begin to converge as eccentricity is increased until in the e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 and e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 simulations the 10𝜇m flux is greater than the 24𝜇m flux at certain times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This is likely due to average periapsis of the particles getting closer to the host star with increased eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This leads to higher average dust temperatures and preferential emission at shorter wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This work and Watt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (2021) both indicate that short-term variability or ’wiggles’ in infrared emission is a good indicator of the sudden appearance of a vapour condensate population, most likely from giant impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' However, this work also points to the conclusion that the formation of this variability is highly dependent on several collision variables, including collision orientation, viewing orienta- tion, eccentricity, and the true anomaly of the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Additionally, lifetime of this variability is expected to be short due to the rapid evolution of the vapour condensate population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This may help us to start to understand why we have observed relatively few debris disks with distinct short-term variations at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The parameter space in which we would expect short-term variations is likely to be fairly narrow, so while there could be a large number of debris disks pro- duced by giant impacts, the number with distinct variability could be much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' There is clearly much more work required to make any of the conclusions above more definitive, but a target for future work could be understanding the distribution of extreme disk eccentricities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' We found eccentricity plays a key role in the magnitude of short-term variations, but if the number of EDDs with larger eccentricities is small then this effect is much less important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Eccentric EDDs take on the eccentric characteristics of their progenitor embryos, so un- derstanding the eccentricity distribution of the planetary embryos in the early Solar System could give strong hints about the distribu- tion of disk eccentricities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Additionally, simulating both the vapour condensate and boulder populations concurrently would allow an understanding of the full evolution of an impact-produced disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The collision rate within both of these populations is key to the longevity of any disk, so quantifying these values would help to refine under- standing of observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 5 SOFTWARE In this work we used the following software: RADMC-3D (Dulle- mond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2012), numpy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2020), scipy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 2020), and matplotlib (Hunter 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' ACKNOWLEDGEMENTS LW acknowledges financial support from STFC/UKRI (grant ST/S505274/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' ZL thanks UKRI (grant ST/V000454/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This work was carried out using the computational facilities of the Advanced Computing Research Centre, University of Bristol - https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='bristol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='uk/acrc/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Thanks to Professor Maughan for his help and discussion on computing disk scale heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Thanks to Dr.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=', Jackson A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=', Marino S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=', Shannon A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=', 2017, Monthly Notices of the Royal Astronomical Society, 464, 3385 APPENDICES Large figures and tables are found in the online supplementary ma- terial associated with this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' These include: Table A1 and A2 which summarise all 𝑁-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B2 which show the full grid of Perpendicular IR emission viewed in the x-y and y-z planes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B4 which show the full grid of Parallel IR emission viewed in the x-z and y-z planes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B6 which show the Perpendicular* IR emission grid viewed from the x-z and y-z planes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' A set of videos (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='mp4) showing how the density of a select number of disks evolves over time is also available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' MNRAS 000, 1–20 (2022) 1 APPENDIX A: SUMMARY OF N-BODY SIMULATIONS Table A1 and Table A2 summarise all N-body simulations used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='03307v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='EP] 9 Jan 2023 2 Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The various 𝑁 -body simulations which have been analysed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' column is used throughout this paper to refer to individual simulations Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Collision Orientation (radians) Centre of Mass Orbital Eccentricity Collision True Anomaly (radians) PR Drag 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='81𝜋 Off 59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 Off 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 Off 61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25𝜋 Off MNRAS 000, 000–000 (0000) 3 Table A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Table A1 continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Collision Orientation (radians) Centre of Mass Orbital Eccentricity Collision True Anomaly (radians) PR Drag 62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 Off 63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='81𝜋 Off 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 Off 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 Off 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25𝜋 Off 67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 Off 68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='81𝜋 Off 69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 Off 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 On 71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 On 72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 On 73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 On 74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 On 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 On 76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 On 77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 On 78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 (Out of plane) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 Off 79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 (Out of plane) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 Off 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 (Out of plane) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 Off 81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 (Out of plane) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 Off 82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 (Out of plane) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 Off 83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 (Out of plane) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 Off MNRAS 000, 000–000 (0000) 4 Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Grid of IR emission in the x-y plane for debris disks produced by collisions perpendicular to the centre of mass of the two colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The first 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 orbits for all simulations have been cropped for clarity (during this period flux density increases rapidly as the disk expands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The grid shows the effect of varying centre of mass eccentricity, position of the collision along the orbit, and the observation wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The columns indicate different collision positions along the orbit, 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 denotes a collision at periapsis while 𝜈 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0𝜋 denotes one at apoapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The rows represent different eccentricities: (a) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 35, 37, 38, 39 in Table A1, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (b) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 50, 52, 53, 54 in Table X, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (c) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 55, 57, 58, 59 in Table A1, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (d) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 60, 62, 63, 64 in Table A1 and A2, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (e) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 65, 67, 68, 69 in Table A2, left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The different observation wavelengths are denoted by different line colours - 24𝜇m: blue, 10𝜇m: orange, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5𝜇m: green, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6𝜇m: red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' APPENDIX B: ADDITIONAL SIMULATED INFRARED EMISSION GRIDS Below are the full RADMC-3D infrared emission grids for a variety of orientations and viewing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5m V=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='811 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 (KI) n) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 99 Flux (ly) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 99 (KI) n) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 99 d) () xn 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 99 (K) xn 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 10 7 10 8 10 10 Orbital Periods Orbital Periods Orbital Penods Orbital Penods 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0μm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5μm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6μm5 Figure B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B1 but in the y-z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0π =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5π =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='81π =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0π 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 (Jy) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 (Jy) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 (KI) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='75 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 D (KI) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 (KI) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 2 6 8 10 2 A 6 8 10 2 6 10 4 6 8 10 Orbital Periods Orbital Periods Orbital Periods Orbital Periods6 Figure B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B1 but for a collision occurring parallel to the centre of mass path and in the x-z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) V=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0π =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5π =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='81π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 e (KI) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 b (KI) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 (KI) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 (Jy) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 NN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 Orbital Periods Orbital Periods Orbital Periods Orbital Periods7 Figure B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B1 but for a collision occurring parallel to the centre of mass path and in the y-z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The rows represent different eccentricities: (a) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 0, 2, 3, 4 in Table A1, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (b) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='2 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 15, 17, 18, 19 in Table A1, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (c) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 20, 22, 23, 24 in Table A1, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (d) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='6 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 25, 27, 28, 29 in Table A1, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (e) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 30, 32, 33, 34 in Table A1, left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) V : =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0π =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='5π =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='81π =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0π 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 (Jy) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='75 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='00 2 4 6 8 10 2 4 6 8 10 2 6 8 10 2 4 6 8 10 Orbital Periods Orbital Periods Orbital Periods Orbital Periods8 Figure B5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' B1 but for a collision occurring perpendicular to the centre of mass path and orbital plane and in the x-z plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' The rows represent different eccentricities: (a) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='0 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 78 and 79 in Table A2, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (b) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='4 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 80 and 81 in Table A2, left to right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' (c) e=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content='8 (Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' 82 and 83 in Table A2, left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE1T4oBgHgl3EQfnQST/content/2301.03307v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) V=1.' metadata={'source': 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Although deep learning methods have +shown remarkable success in the multisource data classification +task, self-supervised learning has rarely been explored. It is +commonly nontrivial to build a robust self-supervised learning +model for multisource data classification, due to the fact that the +semantic similarities of neighborhood regions are not exploited +in existing contrastive learning framework. Furthermore, the +heterogeneous gap induced by the inconsistent distribution of +multisource data impedes the classification performance. To +overcome these disadvantages, we propose a Nearest Neighbor- +based Contrastive Learning Network (NNCNet), which takes +full advantage of large amounts of unlabeled data to learn +discriminative feature representations. Specifically, we propose +a nearest neighbor-based data augmentation scheme to use +enhanced semantic relationships among nearby regions. The +intermodal semantic alignments can be captured more accu- +rately. In addition, we design a bilinear attention module to +exploit the second-order and even high-order feature interactions +between the HSI and LiDAR data. Extensive experiments on +four public datasets demonstrate the superiority of our NNCNet +over state-of-the-art methods. The source codes are available at +https://github.com/summitgao/NNCNet. +Index Terms—hyperspectral image, self-supervised learning, +light detection and ranging, contrastive learning, image classifi- +cation. +I. INTRODUCTION +R +ECENTLY, with the rapid development of satellite sen- +sors, an ever increasing number of multimodal im- +ages (optical, SAR, hyperspectral and LiDAR) are obtained +everyday [1]. Among these multimodal data, hyperspectral +images (HSIs) provide detailed spectral information for the +identification of specified objects on the ground, while LiDAR +data provide elevation information of the area [2] [3] [4]. These +HSI and LiDAR sensors are different in imaging mechanism, +spatial resolution, and even coverage. Therefore, both sensors +capture different properties of the earth, such as spectral +radiance and height information. For example, there are no +significant differences in the spectral domain between the +“trees” on the ground and the “trees” on the hill, but they can +This work was supported in part by the National Key Research and +Development Program of China under Grant 2018AAA0100602, and in part +by the National Natural Science Foundation of China under Grant 42106191. +Meng Wang, Feng Gao, and Junyu Dong are with the School of Information +Science and Engineering, Ocean University of China, Qingdao 266100, China. +(Corresponding author: Feng Gao.) +H. -C. Li is with the Sichuan Provincial Key Laboratory of Information +Coding and Transmission, Southwest Jiaotong University, Chengdu 610031, +China. +Qian Du is with the Department of Electrical and Computer Engineering, +Mississippi State University, Starkville, MS 39762 USA. +Encoder +Momentum +encoder +Similarity +Contrastive loss +Nearest +neighbor +Encoder +Momentum +encoder +Similarity +Contrastive loss +(a) +(b) +Fig. 1. Conceptual comparison of MoCo and the proposed nearest neighbor- +based contrastive learning framework. In the proposed framework, the nearest +neighbors are considered as positive samples. The semantic similarities among +neighborhood regions are exploited. +be distinguished from the LiDAR data [5]. Therefore, the joint +exploitation of HSI and LiDAR data enables us to interpret +ground objects at a more detailed and precise level, which can +hardly be achieved by using single-mode data [6]. Thus, the +classification of cross-modal data has attracted considerable +attention and has been widely applied in multisource image +interpretations [7] [8]. +A great deal of effort has been put into solving the problem +of HSI and LiDAR joint classification. Traditionally, feature- +level fusion models have been proposed, and these models +commonly concatenate the HSI and LiDAR features for clas- +sification [9] [10] [11]. Besides feature-level fusion, decision- +level fusion is another popular solution for HSI and LiDAR +classification. Several classifiers are designed for HSI and +LiDAR data, respectively. The voting strategy is commonly +used to obtain the final classification map [12]. Subsequently, +to further exploit high-level semantic features, convolutional +neural networks (CNNs) are employed for multisource data +classification [13]. Encoder-decoder network [14], coupled +CNNs [15], Gabor CNN [16], cross attention [17], and Trans- +former [18] are used to extract representative multisource +features, and these methods have achieved promising perfor- +mance. +In practice, deep learning models have demonstrated re- +markable success in various multisource data joint classifi- +cation. However, it is non-trivial to build an effective HSI +and LiDAR classification model. One of the critical reasons is +that the deep learning-based model commonly requires a great +number of labeled samples to achieve satisfactory accuracy, +which is expensive and limited in ground object modeling. +Recent research in self-supervised learning encourages the +deep network to learn more representative and interpretable +arXiv:2301.03335v1 [eess.IV] 9 Jan 2023 + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING +2 +features in natural language processing [19] [20] and computer +vision tasks [21] [22]. Self-supervised learning mines the +inherent attributes and semantics of large-scale unlabeled data +to obtain beneficial semantic representations, and it does not +require manually annotated data [23]. After the self-supervised +training finished, the learned features can be transferred to +classification tasks (especially when only small training data +is available) as pretrained models to boost the classification +performance and alleviate overfitting [24] [25] [26] [27]. In +HSI and LiDAR joint classification, self-supervised learning +has rarely been explored, and in this paper, we aim to build +an effective self-supervised model to solve the problem. +It is commonly non-trivial to build a robust self-supervised +learning model for the HSI and LiDAR joint classification task, +due to the following reasons: 1) Data augmentation scheme. +In Momentum Contrast (MoCo) for self-supervised learning +[22], the random color jittering, random horizontal flip, and +random grayscale conversion are used for data augmentation. +However, such data augmentation scheme does not take the +spatial distances between the positive and negative samples +into account, and the semantic similarities of neighborhood +regions are not exploited. Consequently, how to properly +utilize the semantic similarities among nearby regions is a +major challenge. 2) The heterogeneous gap. HSI and LiDAR +joint classification requires a comprehensive understanding +of complex heterogeneous data simultaneously. However, the +heterogeneous gap induced by the inconsistent distributions of +multisource data would greatly impedes its implementation. +Therefore, it is vital to bridge this gap for more robust +multisource data classification. +To address the aforementioned challenges, we propose a +Nearest Neighbor-based Contrast learning Network, NNCNet +for short, which aims to learn an encoder that encodes similar +data of the same kind and makes the encoding results of differ- +ent classes of data as different as possible. To be more specific, +we propose a nearest neighbor-based framework to use the +enhanced semantic relationships among nearby regions. As +illustrated in Fig. 1, nearest neighbors of positive samples +are fed into the encoder for contrastive learning. The feature +representations are learned by encouraging the proximity of +between different views of the same sample and its nearest +neighbors in the spatial domain. Therefore, the contrastive +learning framework is encouraged to generalize to new feature +embeddings that may not be covered by the data augmentation. +In addition, we design a bilinear attention fusion module to +exploit second-order and even higher-order feature interactions +between the HSI and LiDAR data, and the information flow +can be controlled more flexibly. +The contributions of this work are as follows: +• We propose a self-supervision contrastive learning ap- +proach NNCNet, which integrates a nearest neighbor- +based data augmentation scheme. The scheme can exploit +the semantic similarities among neighborhood regions, +and hence capture inter-modal semantic alignments more +accurately. To our best knowledge, we are the first to +apply self-supervised contrastive learning to HSI and Li- +DAR joint classification, which has both great theoretical +and practical significance. +• We propose a bilinear attention fusion module that aims +to enhance the contextual representation of HSI and +LiDAR data. The module captures second-order feature +interactions between multisource data. +• We have conducted extensive experiments on four bench- +mark datasets to validate the effectiveness of our NNC- +Net. Additionally, we have released our codes and pa- +rameters to benefit other researchers. +II. RELATED WORK +A. Morphological Filter-Based Methods for HSI and LiDAR +Classification +The joint use of HSI and LiDAR has already been in- +vestigated for a variety of applications, such as illumination +calibration [28], forest area analysis [29], bushfire monitoring +[30], and urban sprawl modeling [31]. Great efforts have been +devoted to exploiting the complementary information between +multisource data, especially for morphological filter-based +methods. Morphological filters are intensively used to atten- +uate the redundant spatial details and preserve the geometric +structures. Pedergnana et al. [9] used morphological extended +attribute profiles to HSI and LiDAR data for classification. +Features extracted from HSI and LiDAR data are stacked +for classification. Liao et al. [32] computed morphological +attribute profiles from HSI and LiDAR data, and these attribute +profiles are fused using a generalized graph-based method. +Khodadadzadeh et al. [33] pointed out that simple stacking +of morphological attribute profiles from multisource data may +contain redundant features. To solve this issue, they proposed +a multiple feature learning approach based on the multinomial +logistic regression classifier, which can adaptively exploit +the spatially and spectrally derived features. Later, attribute +profiles are considered to be complex and time-consuming +in threshold initialization, and extinction profiles [34] are +proposed to solve the problem. Ghamisi et al. [35] presented a +classification framework based on extinction profiles and deep +learning. +B. CNN-Based Methods for HSI and LiDAR Classification +Recently, deep CNNs have attracted extensive research +attention in the remote sensing data fusion community, and +many CNN-based models have been proposed for multisource +data classification. Xu et al. [36] proposed a two-branch CNN +model, which consists of a 2-D convolutional network and +a 1-D convolutional network. Zhang et al. [37] presented a +patch-to-patch CNN for the joint feature extraction of HSI +and LiDAR data. Chen et al. [38] proposed a CNN and DNN +hybrid model for multisource feature extraction. CNNs are +used to extract informative features from multisource data, +and a DNN is utilized to fuse these heterogeneous features +for robust classification. Li et al. [39] proposed a dual-channel +spatial, spectral and multiscale attention CNN for multisource +data classification. Hang et al. [15] used coupled CNNs for +multisource data classification. The coupled layers reduce the +number of parameters and guide both networks learning from +each other. Zhao et al. [16] proposed a fractional Gabor CNN, + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING +3 +and focused on efficient feature fusion. Fractional Gabor con- +volutional kernels are used for multiscale and multidirectional +feature extraction and yield robust feature representations +against semantic changes. In [40], a multisource graph fusion +network is presented to integrate feature extraction and fusion +into a single network. A multimodal graph is constructed to +guide the multimodal image feature extraction. Gao et al. +[17] proposed a deep-wise feature interaction network for +multisource remote sensing image classification. Consistency +loss, discrimination loss, and classification loss are designed +for parameter optimization. +Although CNNs have been successfully applied to HSI +and LiDAR joint classification, its performance remains un- +satisfactory for practical applications. An important factor +may be the lack of sufficient annotated data. In practical +applications, remote sensing data annotation is costly, making +it difficult to obtain robust deep learning models. To solve this +problem, we aim to build a simple yet effective self-supervised +method for multisource data joint classification. It extracts the +inherent attributes and semantics from unlabeled large-scale +data to capture beneficial feature representations. In addition, +a nearest-neighbor-based data augmentation scheme is used to +exploit the semantic relationships among nearby regions. +III. METHODOLOGY +As shown in Fig. 2, the proposed NNCNet consists of +three parts: nearest neighbor-based data augmentation, bilinear +attention-based encoder, and contrastive loss computation. +Considering that the proposed NNCNet is based on a self- +supervised contrastive learning framework, we first introduce +the nearest neighbor-based contrastive learning framework +and then successively elaborate the bilinear attention-based +encoder. +A. Nearest Neighbor-based Momentum Contrast Learning +Framework +Considering that unlabeled data have no supervised infor- +mation, we aim to extract the supervised information from +large-scale unsupervised HSI and LiDAR data. Our goal is to +train an encoder that keeps the different transformations from +the same sample as close as possible and the different samples +as far away as possible in the feature space. To solve the +problem, He et al. [22] proposed Momentum Contrast (MoCo) +for self-supervised learning. To be specific, a minibatch of +samples is selected from the data. Each sample is handled +by random data augmentation (Gaussian blur, flip, or spectral +distortions) to generate a query sample and a key sample. The +query and key samples are encoded separately to embedding q +and k. The cosine similarity between q and k is computed for +representation learning. The embedding from the same image +is defined as the positive key, and embedding from different +image is defined as the negative key. In MoCo, a dynamic +dictionary is built with a queue and a dynamic encoder. +For remote sensing data classification, we argue that random +augmentations can hardly provide positive pairs for the same +object representation. For the sake of covering more variance +in a given class, we propose nearest neighbor-based contrastive +Algorithm 1 Pseudocode of the Nearest Neighobr-Based +Contrastive Learning in PyTorch Style. +# f_q: encoder network for query +# f_k: encoder network for key +# queue: key dictionary +# r: momentum coefficient +f_k.param = f_q.param # initialize parameters +# load a mini-batch x with N samples +for x in loader: +nn = neighbor(x) # generate neighbors of x +x_q = augment(x) # query randomly augmentation +x_k = augment(x) # key random augmentation +x_n = augment(nn) # neighbors random augmentation +# randomly substitute half samples in x_k by x_n +x_k = substitute(x_k, x_n) +q = f_q.forward(x_q) # queries: NxC +k = f_k.forward(x_k) # keys: NxC +k = k.detach() # no gradient to keys +# positive logits: Nx1 +# bmm: batch matrix multiplication +l_pos = bmm(q.view(N,1,C), k.view(N,C,1)) +# negative logits: NxK +# mm: matrix multiplication +l_neg = mm(q.view(N,C), queue.view(C,K)) +# logits: Nx(1+K) +logits = cat([l_pos, l_neg], dim=1) +# contrastive loss computation +labels = zeros(N) +loss = CrossEntropyLoss(logits/t, labels) +# back propagation, only update the query network +loss.backward() +update(f_q.param) +# dictionary update +f_k.param = r*f_k.param+(1-r)*f_q.param +enqueue(queue, k) # push the current key +dequeue(queue) # pop the earliest key +learning framework. In remote sensing images, the neighbor +labels of one specified position tend to be the same. As +illustrated in Fig. 3, the region within the red box is rooftop, +and the green boxes are its nearest neighbors. Blue boxes +denote regions far from the red box. From the visualized +feature space, it can be observed that the features of the nearest +neighbors are close. In this paper, we use a nearest neighbor- +based contrastive learning framework in which the semantic +similarities among neighborhood regions are exploited. There- +fore, the inter-modal semantic alignments are reinforced. +Nearest Neighbor-Based Contrastive Learning. Algo- +rithm 1 provides the pseudo code of the proposed nearest +neighbor-based contrastive learning. In the proposed frame- +work, a set of sample pairs are selected from HSI and +LiDAR data centered at the same position. During training, +each sample pair is handled by random data augmentation +to generate a query sample xq and a key sample xk. They +are encoded to embeddings q and k, respectively. The em- +beddings from the same image are defined as positive key, +and embeddings from different images are defined as negative +key. A large number of negative key embeddings are stored +in a dictionary {k1, k2, k3, . . .}, while one positive key k+ +is stored separately. Furthermore, we randomly select some +nearest neighbors of q to generate embeddings, which are +denoted as kn+. Next, in a minibatch, half positive keys k+ +are substituted by kn+ to form new positive keys. Hence, +nearest neighbors act small semantic perturbations. In our + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING +4 +Hyperspectral image +LiDAR +Data +augmentation +Nearest neighbor +substitution +Encoder +Momentum +encoder +Push +Pop +Positive key +Dictionary of negative keys +Contrastive loss +Contrastive loss +D +D +D +D +Similarity of positive pairs +Similarity of negative pairs +Slowly update +Fig. 2. Schematic illustration of the nearest-neighbor based contrastive learning. It consists of three components: 1) Nearest neighbor-based data augmentation. +The input samples are handled by random data augmentation to generate query and key samples. In a mini-batch, half positive key samples are substituted by +its nearest neighbors to form new positive key samples. These nearest neighbors act small semantic perturbations. 2) Bilinear attention-based feature encoder. +The query and key samples are fed into the encoder for feature extraction. A bilinear attention fusion module is employed to capture the second-order feature +interactions between multisource data. 3) Contrastive loss computation. Positive and negative keys are stored in a dynamic dictionary, and contrastive loss is +computed to assign high scores for positive keys and low scores for negative keys. +Remote sensing images +Visualized feature space +Fig. 3. +Typical regions in remote sensing images and the corresponding +visualized features. The region within the red box is rooftop, and the green +boxes are its nearest neighbors. Blue boxes denote regions far from the the +red box. In the visualized feature space, it can be observed that features of the +nearest neighbors are close. Therefore, the contextual information is critical +in contrastive learning for remote sensing image classification. +implementations, nearest neighbors denotes a region whose +overlap area with xq is greater than 80%. +We calculate the cosine similarities between q and keys +(both the positive key and negative keys). Then, the results +are stored as {D+, D1, D2, D3, . . . , DK}. Here D+ is the +similarity between q and positive key k+. The rest are the +similarities between q and negative keys. K is the number of +negative keys. +The objective of contrastive learning is to force the query to +match the positive key and far apart from the negative keys. To +be specific, the contrastive loss whose value is low when q is +similar to the positive key k+ and dissimilar to all the negative +keys. Therefore, the contrastive loss function is designed as +follows: +L = − log +exp(D+/τ) +�K +i=1 exp(Di/τ) +(1) +where τ is a temperature hyperparameter. Intuitively, softmax +classifier and cross-entropy loss can be combined into the +above equation. +Dictionary Update and Moving Average Encoder. Similar +to MoCo [22], we maintain the dictionary as a queue which +stores many minibatch of negative samples. The negative sam- +ples in the dictionary are updated progressively. Specifically, +during training, when a new minibatch is pushed into the +dictionary, the oldest minibatch is removed. The length of the +dictionary is flexibly set as a hyperparameter. +Furthermore, the parameters of the encoder for the dic- +tionary are updated slowly. Similar to MoCo, we use a +separate moving average encoder for the key samples. During +training, no backpropagation is done for the key encoder. The +parameters of key encoder are updated as follows: +θk = rθk + (1 − r)θq, +(2) +where θk denotes the parameters of the key encoder, and θq +denotes the parameters of the query encoder. r is a momentum +coefficient that controls the speed of key encoder update. +Only θq is updated by backpropagation during training. In our +implementations, r is set to 0.9, since a slowly evolving key +encoder is critical for robust feature learning. +Shuffling BN. Batch Normalization (BN) is employed in +the encoder to speed up convergence and improve the gener- +alization of the network. Similar to MoCo, we use the shuffling +BN for better feature representation. In particular, we shuffle +the sample order in the current minibatch for the key encoder. +The sample order of the mini-batch for the query encoder is +not changed. +B. Bilinear Attention-Based Multisource Encoder +In this work, the purpose of contrastive learning is to +generate a pretrained model, and the model can be used for the +classification task. To achieve high classification accuracy, the +encoder is an essential part of the contrastive framework. We +design a multisource encoder for hyperspectral and LiDAR +feature modeling, as illustrated in Fig. 4. It contains three +parts: HSI feature extraction, LiDAR feature extraction, and +bilinear attention fusion. +The detailed summary of the encoder in terms of layer +type, kernel size, and output size is illustrated in Table I. + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING +5 +LiDAR +3DConv +3DConv +3DConv +2DConv +2DConv +2DConv +2DConv +HSI +Bilinear +Attention +Output +FC +3DConv +2DConv +3D convolution +2D convolution +Element-wise summation +FC +Fully connected layer +Fig. 4. Bilinear attention-based multisource encoder. +TABLE I +SUMMARY OF THE PROPOSED MULTI-SOURCE ENCODER +HSI feature extraction subnetwork +# +Layer type +Kernel number@size +Output size +Input +— +(11, 11, 30, 1) +1 +3D Conv +8@3×3×9 +(9, 9, 22, 8) +2 +3D Conv +16@3×3×7 +(7, 7, 16, 16) +3 +3D Conv +32@3×3×5 +(5, 5, 12, 32) +4 +Reshape +— +(5, 5, 384) +5 +2D Conv +256@3×3 +(5, 5, 256) +6 +Reshape +— +(25, 256) +LiDAR feature extraction subnetwork +# +Layer type +Kernel numbre@size +Output size +Input +— +(11, 11, 1) +1 +2D Conv +64@3×3 +(9, 9, 64) +2 +2D Conv +128@3×3 +(7, 7, 128) +3 +2D Conv +256@3×3 +(5, 5, 256) +4 +Reshape +— +(25, 256) +Hyperspectral data adopt a network similar to HybridSN [41], +which uses both 3D and 2D convolutions for feature extraction. +Three 3D convolution layers and one 2D convolution layer are +used to derive the HSI feature FH. At the same time, three 2D +convolution layers are used to generate the LiDAR feature +FL. Next, FH and FL are combined to form the fused feature. +A 2D convolution is used for feature embedding. Then, the +fused feature Ffus has the same dimension as FH and FL. +To effectively reduce the inherent redundancy in HSI, and +thereby reduce the amount of data that needs to be processed +in classification, Principal Component Analysis (PCA) is used +to select the best 30 spectral bands for HSI feature extraction. +Finally, FH, FL and Ffus are fed into the bilinear attention +fusion module as Q, K, and V, respectively. The output of the +bilinear attention fusion module is fed into a fully connected +layer to generate the final feature for classification. +C. Bilinear Attention Fusion Module +The attention mechanism has made valuable breakthroughs +in deep neural networks and has been successfully applied to +FC +FC +FC +FC +Bilinear pooling +FC +softmax +C +FC +S +Gate mechanmism +Bilinear pooling +FC +Fully connected layer +Element-wise multiplication +C +Feature concatenation +S +Sigmoid activation +Fig. 5. Bilinear attention fusion module. It can capture second-order interac- +tions between multisource data. +cross-modal tasks (e.g., visual question answering [42], image +captioning [43], and image-text matching [44]). This prompts +recent methods to adopt the attention to trigger the interaction +between multi-modal remote sensing data [45] [46] [47] [48]. +In the conventional attention mechanism, the attention weights +are estimated via linearly fusing the inputs. However, we +argue that conventional attention exploits the first-order feature +interaction and is limited in complex multisource feature +reasoning. +Toward this end, we propose a bilinear attention fusion mod- +ule to exploit the second-order feature interactions between +the hyperspectral and LiDAR data. As illustrated in Fig. 5, +it mainly contains two parts: the multi-head bilinear attention +and the gate mechanism. +Multi-Head Bilinear Attention. Suppose we have query +Q ∈ Rc×d, key K ∈ Rc×d, and value V ∈ Rc×d, where +d denotes the feature dimension, and c is the number of +channels. To enhance the capability of feature representation, +the multi-head scheme is used to model feature interactions +from different subspaces as: +hi = BiAttention(Qi, Ki, Vi), +(3) +where hi is the output of the i-th head, and BiAttention +denotes the bilinear attention. The number of heads is denoted +by H. +The bilinear attention first maps Qi ∈ R +c +H ×d and Ki ∈ +R +c +H ×d into a joint space as: +B1 +i = σ(QiW1 +q) ⊙ σ(KiWk), +(4) +where W1 +q ∈ Rd×d and Wk ∈ Rd×d are weighting matrices, +σ is the ReLU activation and ⊙ denotes the element-wise +multiplication. As such, B1 +i ∈ R +c +H ×d denotes the second-order +representation between query Qi and key Ki. +Similarly, we compute the bilinear representation between +Qi and Vi as: +B2 +i = σ(QiW2 +q) ⊙ σ(ViWv), +(5) +where W2 +q ∈ Rd×d and Wv ∈ Rd×d are weighting matrices. +B2 +i ∈ R +c +H ×d denotes the second-order representation between +query Qi and value Vi. + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING +6 +Next, the bilinear representation B1 +i is projected into atten- +tion weights Watt +i +∈ R +c +H ×d via a linear layer and a softmax +layer as follows: +ˆB1 +i = σ(WBB1 +i ), +(6) +Watt +i = softmax( ˆB1 +i ), +(7) +where WB ∈ R +c +H × c +H is the weight matrix. Next, the attended +feature hi ∈ R +c +H ×d is derived by enhancing the attention +weights as: +hi = Watt +i ⊙ B2 +i +(8) +Gate Mechanism. The aforementioned bilinear attention +exploits the feature interactions among Qi, Ki, and Vi. +However, there may contain noisy information in the query +and key. To adaptively enhance the informative parts and +suppress the useless parts, we design a gate mechanism. To +be specific, for the i-th head, ˆB1 +i is fed into a linear layer and +then handled with a sigmoid function to compute a weight +mask Gi ∈ R +c +H ×1 as: +Gi = sigmoid( ˆB1 +i WB′), +(9) +where WB′ +∈ Rd×1 is the weight matrix. Next, Gi is +expanded to form G′ +i ∈ R +c +H ×d. Then the obtained gating mask +is applied to control the information flow of hi ∈ R +c +H ×d as: +ˆhi = G′ +i ⊙ hi. +(10) +Finally, by concatenating the results of multiple heads, we +obtain the fused representation of multi-source data. In this +work, the size of Q, K and V is 25×256. The number of +heads H is set to 5. +IV. EXPERIMENTAL RESULTS AND ANALYSIS +To validate the effectiveness of the proposed NNCNet, we +conduct extensive experiments on four widely used bench- +mark datasets: Houston 2013 dataset, Trento dataset, MUUFL +dataset and Houston 2018 dataset. We first compare the +proposed NNCNet with state-of-the-art methods. Then we +implemented additional evaluations to investigate the effec- +tiveness of each component of our method. +A. Datasets and Evaluation Metric +Houston 2013 dataset: The dataset was captured by the +National Airborne Center for Laser Mapping, and it was used +as a challenge in the 2013 GRSS Data Fusion Contest. The +HSI was captured by the CASI sensor (144 spectral bands at a +resolution of 2.5 m). Coregistered LiDAR data with the same +resolution are available. A total of 15029 ground truth samples +are distributed in 15 classes. They are divided into train and +test sets containing 2832 and 12197 pixels, respectively. We +used standard training and test sets, and Table II lists the +number of training and test samples. +Trento dataset: The dataset was collected in a rural region +south of Trento, Italy. The HSI image consists of 63 bands with +a wavelength range of 0.42-0.99 µm. The size of the datset +is 166×660 pixels, and the spatial resolution of the datset is +1.0 m. A total of 30214 ground truth samples are distributed +TABLE II +TRAIN-TEST DISTRIBUTION OF SAMPLES FOR THE HOUSTON 2013 +DATASET. +No. +Class Name +Training +Test +1 +Healthy grass +198 +1053 +2 +Stressed grass +190 +1064 +3 +Synthetic grass +192 +505 +4 +Tree +188 +1056 +5 +Soil +186 +1056 +6 +Water +182 +143 +7 +Residential +196 +1072 +8 +Commercial +191 +1053 +9 +Road +193 +1059 +10 +Highway +191 +1036 +11 +Railway +181 +1054 +12 +Parking lot 1 +192 +1041 +13 +Parking lot 2 +184 +285 +14 +Tennis court +181 +247 +15 +Running track +187 +473 +Total +2832 +12197 +TABLE III +TRAIN-TEST DISTRIBUTION OF SAMPLES FOR THE TRENTO DATASET. +No. +Class Name +Training +Test +1 +Apple trees +129 +3905 +2 +Buildings +125 +2778 +3 +Ground +105 +374 +4 +Wood +154 +8969 +5 +Vineyard +184 +10317 +6 +Roads +122 +3052 +Total +819 +29595 +TABLE IV +TRAIN-TEST DISTRIBUTION OF SAMPLES FOR THE MUUFL DATASET. +No. +Class Name +Training +Test +1 +Trees +150 +23096 +2 +Mostly grass +150 +4120 +3 +Mixed ground surface +150 +6732 +4 +Dirt and sand +150 +1676 +5 +Road +150 +6537 +6 +Water +150 +316 +7 +Building shadow +150 +2083 +8 +Building +150 +6090 +9 +Sidewalk +150 +1235 +10 +Yellow curb +150 +33 +11 +Cloth panels +150 +119 +Total +1650 +52037 + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING +7 + (a) Ground truth + (b) FusAtNet + (c) TBCNN + (d) EndNet + (e) MDL + (f) CCNN + (g) S2ENet + (h) w/o Pretraining + (i) Proposed NNCNet +Healthy grass +Stressed grass +Synthetic grass +Tree +Road +Railway +Soil +Water +Residential +Commercial +Highway +Parking lot 1 +Parking lot 2 +Tennis court +Running track +Fig. 6. +Classification maps for the Houston 2013 dataset. (a) Groundtruth. (b) FusAtNet. (c) TBCNN. (d) EndNet. (e) MDL. (f) CCNN. (g) S2ENet. (h) +Proposed NNCNet without pretraining. (i) Proposed NNCNet. +TABLE V +TRAIN-TEST DISTRIBUTION OF SAMPLES FOR THE HOUSTON 2018 +DATASET. +No. +Class Name +Training +Test +1 +Healthy grass +500 +9299 +2 +Stressed grass +500 +32002 +3 +Artificial turf +68 +616 +4 +Evergreen trees +500 +13095 +5 +Deciduous trees +500 +4521 +6 +Bare earth +451 +4065 +7 +Water +26 +240 +8 +Residential buildings +500 +39272 +9 +Non-residential buildings +500 +223252 +10 +Roads +500 +45366 +11 +Sidewalks +500 +33529 +12 +Crosswalks +151 +1367 +13 +Major thoroughfares +500 +45848 +14 +Highways +500 +9365 +15 +Railways +500 +6437 +16 +Paved parking lots +500 +11000 +17 +Unpaved parking lots +14 +132 +18 +Cars +500 +6047 +19 +Trains +500 +4869 +20 +Stadium seats +500 +6324 +Total +8210 +496646 +in 6 classes. Table III lists the distribution of training and test +samples for the Trento dataset. +MUUFL dataset: The MUUFL dataset is captured over +the University of Southern Mississippi Gulf Coast campus +in November 2010. The HSI contains 72 spectral bands, but +the first and last four bands are removed for noise reduction, +leaving 64 bands for classification. The total size of the dataset +is 325×220 pixels. Table IV lists the training and test samples +available for the dataset. In our experiments, we use the entire +data in the pretraining phase, while in the training validation +phase, we use only the portion of the training set for which +labels are given. +Houston 2018 dataset: The dataset was captured by the +Hyperspectral Image Analysis Laboratory and the National +Center for Airborne Laser Mapping (NCALM) at the Uni- +versity of Houston. It was originally released for the 2018 +IEEE GRSS Data Fusion Contest. Hyperspectral data covers +380-1050 nm spectral range with 48 bands at 1.0 m ground +sample distance. The dataset contains a total of 4768×1202 +pixels in which a piece is delineated as the training set with +the size of 2384×601 pixels. Table V lists the distribution of +training and test samples for the Houston 2018 dataset. +The performance of the model is evaluated by Overall Ac- +curacy (OA), Average Accuracy (AA), and Kappa coefficient. +OA denotes the ratio of the model’s correct predictions to the +overall number on all test samples. AA is the ratio between +the number of correct predictions in each category and the +overall number in each category, and finally the average of +the accuracy in each category. Kappa is the percentage of +agreement corrected by the number of agreements that would +be expected purely by chance. +B. Implementation Details +The proposed contrastive learning architecture is used to +generate a pretrained model. In the contrastive learning phase, +we use the Adam optimizer. The mini-batch size is set to 64 +and the learning rate is set to 0.0005. The image patch of + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING +8 +TABLE VI +CLASSIFICATION ACCURACY (%) ON THE HOUSTON 2013 DATASET +Class +FusAtNet [49] +TBCNN [36] +EndNet [14] +MDL [5] +CCNN [15] +S2ENet [50] +NNCNet (ours) +Healthy grass +79.20 +81.01 +78.54 +83.00 +91.55 +82.72 +81.84 +Stressed grass +96.71 +97.93 +96.33 +98.68 +99.72 +100.0 +99.72 +Synthetic grass +97.82 +99.60 +100.0 +99.80 +99.60 +99.60 +99.80 +Tree +97.63 +94.13 +88.26 +93.94 +97.63 +95.74 +99.43 +Soil +100.0 +98.86 +100.0 +99.05 +100.0 +99.81 +100.0 +Water +91.61 +97.90 +100.0 +100.0 +95.80 +97.20 +100.0 +Residential +76.31 +80.50 +83.02 +79.66 +83.12 +91.23 +94.87 +Commercial +74.17 +87.46 +79.96 +80.44 +94.49 +91.55 +94.78 +Road +89.05 +86.50 +93.30 +84.70 +93.20 +95.94 +96.03 +Highway +92.86 +64.86 +92.28 +94.88 +89.96 +84.75 +99.81 +Railway +94.21 +93.74 +85.86 +85.67 +96.39 +94.31 +99.34 +Parking lot 1 +87.32 +74.93 +99.81 +98.75 +99.71 +97.79 +99.81 +Parking lot 2 +84.21 +85.96 +83.16 +82.46 +89.82 +89.47 +90.88 +Tennis court +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +Running track +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +OA +89.70 +87.57 +90.71 +90.80 +94.98 +93.99 +96.77 +AA +90.73 +89.55 +92.03 +92.06 +95.40 +94.67 +97.06 +Kappa +88.81 +86.50 +89.92 +90.01 +94.56 +93.48 +96.49 +TABLE VII +CLASSIFICATION ACCURACY (%) ON THE TRENTO DATASET +Class +FusAtNet [49] +TBCNN [36] +EndNet [14] +MDL [5] +CCNN [15] +S2ENet [50] +NNCNet (ours) +Apple trees +99.95 +99.87 +99.90 +99.90 +99.90 +99.90 +99.13 +Buildings +98.92 +98.81 +99.03 +99.10 +99.10 +98.88 +98.92 +Ground +85.56 +81.02 +85.83 +86.36 +86.90 +86.36 +99.73 +Wood +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +Vineyard +99.68 +98.40 +99.31 +99.61 +99.67 +99.21 +100.0 +Roads +92.07 +89.35 +90.83 +91.12 +91.25 +91.32 +91.88 +OA +98.77 +97.96 +98.52 +98.66 +98.71 +98.53 +98.92 +AA +96.03 +94.57 +95.81 +96.01 +96.13 +95.94 +98.26 +Kappa +98.35 +97.27 +98.01 +98.21 +98.27 +98.03 +98.55 +11×11 pixels is randomly cropped from the dataset as training +samples. +After obtaining the pretrained model, training samples from +the dataset are used for fine-tuning the model. In the fine- +tuning phase, the mini-batch size is set to 128, and the setting +of the optimizer is the same as that in the contrastive learning +phase. +C. Classification Accuracy and Discussion +The proposed NNCNet is implemented on the Houston +2013, Trento, MUUFL and Houston 2018 datasets. To verify +the effectiveness of the proposed NNCNet, we compared it +with six state-of-the-art methods, including FusAtNet [49], +TBCNN [36], EndNet [14], MDL [5], CCNN [15], and +S2ENet [50]. In particular, FusAtNet [49] exploits HSI and +LiDAR features via cross-attention, and attentive spectral and +spatial representations are combined to compute modality- +specific feature embeddings. TBCNN [36] uses a two-branch +CNN for HSI and LiDAR feature extraction. In EndNet [14], +a deep encoder–decoder network is utilized for multimodal +information fusion and classification. MDL [5] presents a +general multimodal deep learning framework. CCNN [15] +presents a coupled network for multimodal information fu- +sion. Feature-level and decision-level fusion are integrated +for heterogeneous feature representation. S2ENet [50] is a +spatial-spectral enhancement network that improves spatial +and spectral feature representations simultaneously. For a +fair comparison, all the compared methods adopt the default +parameters provided in their works. +Table VI shows the classification results on the Houston +2013 dataset. The proposed NNCNet achieves the best per- +formance in terms of OA, AA and Kappa coefficients. Our +NNCNet outperforms the competitor (CCNN and S2ENet) +by 1.78% and 2.78% for OA, respectively. It shows that +the proposed self-supervised framework effectively models +the correlations between multisource samples. Furthermore, +the accuracy for ‘highway’ class (99.81%) is significantly + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING +9 +TABLE VIII +CLASSIFICATION ACCURACY (%) ON THE MUUFL DATASET +Class +FusAtNet [49] +TBCNN [36] +EndNet [14] +MDL [5] +CCNN [15] +S2ENet [50] +NNCNet (ours) +Trees +95.31 +91.18 +90.86 +90.95 +92.40 +93.91 +93.09 +Mostly grass +79.83 +83.98 +83.30 +84.54 +83.52 +88.28 +86.82 +Mixed ground surface +83.69 +83.72 +84.27 +83.01 +84.34 +81.85 +86.29 +Dirt and sand +97.73 +96.12 +96.00 +96.42 +96.72 +97.32 +96.18 +Road +84.86 +91.23 +91.11 +90.44 +91.68 +91.28 +92.35 +Water +99.68 +99.68 +99.68 +99.68 +99.68 +99.68 +99.68 +Building shadow +83.01 +92.85 +92.61 +92.75 +92.61 +88.29 +92.75 +Building +94.70 +96.86 +96.90 +96.70 +96.80 +95.99 +96.21 +Sidewalk +89.80 +87.85 +88.34 +87.85 +89.23 +88.50 +91.34 +Yellow curb +87.88 +90.91 +90.91 +90.91 +90.91 +87.88 +84.85 +Cloth panels +99.16 +99.16 +99.16 +99.16 +99.16 +99.16 +99.16 +OA +90.68 +90.53 +90.39 +90.27 +91.20 +91.61 +92.07 +AA +90.51 +92.14 +92.10 +92.03 +92.45 +92.01 +92.61 +Kappa +87.65 +87.59 +87.42 +87.26 +88.44 +88.93 +89.56 +TABLE IX +CLASSIFICATION ACCURACY (%) ON THE HOUSTON 2018 DATASET +Class +FusAtNet [49] +TBCNN [36] +EndNet [14] +MDL [5] +CCNN [15] +S2ENet [50] +NNCNet (ours) +Healthy grass +89.40 +90.91 +89.55 +93.99 +93.39 +91.29 +93.36 +Stressed grass +90.65 +88.83 +89.51 +88.95 +90.84 +91.97 +91.95 +Artificial turf +98.54 +84.90 +75.97 +98.86 +98.38 +96.59 +98.38 +Evergreen trees +85.30 +71.97 +67.97 +90.60 +94.25 +88.91 +92.00 +Deciduous trees +73.15 +70.87 +66.98 +76.55 +80.62 +79.12 +76.86 +Bare earth +100.0 +99.78 +100.0 +100.0 +99.48 +99.78 +99.98 +Water +99.17 +96.67 +92.92 +98.75 +95.83 +93.75 +96.25 +Residential buildings +97.29 +94.93 +92.54 +86.31 +91.43 +91.31 +87.58 +Non-residential buildings +94.36 +95.78 +96.78 +97.75 +93.93 +95.22 +97.04 +Roads +62.29 +53.26 +42.71 +69.65 +73.14 +70.95 +71.25 +Sidewalks +64.00 +72.67 +71.00 +68.30 +78.85 +76.82 +70.63 +Crosswalks +40.53 +41.84 +03.66 +49.82 +52.38 +56.18 +38.33 +Major thoroughfares +69.77 +78.48 +71.08 +60.56 +76.08 +76.25 +81.58 +Highways +97.16 +98.55 +96.11 +96.18 +98.70 +98.24 +98.54 +Railways +99.43 +99.19 +98.91 +97.84 +99.52 +99.67 +99.94 +Paved parking lots +85.68 +78.73 +75.27 +82.50 +87.32 +91.12 +95.25 +Unpaved parking lots +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +100.0 +Cars +56.13 +72.65 +24.77 +47.87 +89.37 +77.11 +92.16 +Trains +91.29 +65.06 +60.67 +90.37 +76.38 +92.75 +99.01 +Stadium seats +99.62 +99.49 +99.34 +99.59 +99.83 +99.65 +99.78 +OA +85.98 +86.33 +83.84 +86.70 +88.64 +88.87 +89.89 +AA +84.68 +82.72 +75.78 +84.72 +88.48 +88.33 +88.99 +Kappa +81.46 +81.83 +78.06 +82.15 +85.19 +85.38 +86.65 + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING +10 + (a) Ground truth + (b) FusAtNet + (c) TBCNN + (d) EndNet + (e) MDL + (f) CCNN + (g) S2ENet + (h) w/o Pretraining + (i) Proposed NNCNet +Apple trees +Buildings +Ground +Wood +Vineyard +Roads +Fig. 7. Classification maps for the Trento dataset. (a) Groundtruth. (b) FusAtNet. (c) TBCNN. (d) EndNet. (e) MDL. (f) CCNN. (g) S2ENet. (h) Proposed +NNCNet without pretraining. (i) Proposed NNCNet. + (a) Ground truth +(b) FusAtNet + (c) TBCNN + (d) EndNet + (e) MDL +(f) CCNN + (g) S2ENet +(h) w/o Pretraining + (i) Proposed NNCNet +Trees +Mostly grass +Mixed ground +surface +Dirt and sand +Road +Water +Building +Shadow +Building +Sidewalk +Yellow curb +Cloth panels +Fig. 8. Classification maps for the MUUFL dataset. (a) Groundtruth. (b) FusAtNet. (c) TBCNN. (d) EndNet. (e) MDL. (f) CCNN. (g) S2ENet. (h) Proposed +NNCNet without pretraining. (i) Proposed NNCNet. + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING +11 + (a) Group truth + (b) FusAtNet + (c) TBCNN + (d) EndNet + (e) MDL + (f) CCNN + (g) S2ENet + (h) w/o Pretraining + (i) Proposed NNCNet +Healthy grass +Stressed grass +Artificial turf +Evergreen trees +Deciduous trees +Bare earth +Water +Residential +buildings +Non-residential +buildings +Roads +Sidewalks +Crosswalks +Major +thoroughfares +Highways +Railways +Paved parking +lots +Unpaved parking +lots +Cars +Trains +Stadium seats +Fig. 9. +Classification maps for the Houston 2018 dataset. (a) Groundtruth. (b) FusAtNet. (c) TBCNN. (d) EndNet. (e) MDL. (f) CCNN. (g) S2ENet. (h) +Proposed NNCNet without pretraining. (i) Proposed NNCNet. +improved by our NNCNet. There are many unlabeled highway +regions in the Houston 2013 dataset. Therefore, our NNCNet +captured the texture and spectral features of highway via +contrastive learning from unlabeled data. The classification +maps are illustrated in Fig. 6. It can be observed that with- +out pretraining, some highway regions are falsely classified +into road. In contrast, the proposed NNCNet performs better +through contrastive learning. +Table VII illustrates the classification results of different +methods on the Trento dataset. The classification maps are +shown in Fig. 7. It can be seen that without pretraining, +some vineyard regions are falsely classified into apple trees. In +addition, the proposed NNCNet achieves the best performance +in terms of OA, AA, and Kappa. The proposed method +achieves the best OA in ‘ground’. There is only a small amount +of labeled data in this class, but it still accounts for a large +portion of the entire graph. It is evident that our NNCNet is +capable to learning the robust feature representations when +training samples are limited. +Table VIII shows the classification results of different meth- +ods on the MUUFL dataset. The proposed NNCNet obtains +the best performance against the other methods. To be specific, +the proposed method has the best OA (92.07%) and reached +the highest accuracy in five classes (Mixed ground surface, +Road, Water, Sidewalk and Cloth panels). The classification +results of the proposed method for the Mostly building and +Building shadow are quite competitive. Therefore, the com- +parisons demonstrate the superior performance of the proposed +NNCNet on the MUUFL dataset. The classification maps of +the proposed NNCNet with / without pretraining are illustrated +in Fig. 8, it can be observed that the pretraining effectively +improved the classification performance. +Table IX illustrates the classification results of different +methods on the Houston 2018 dataset. Compared to other +methods, the proposed NNCNet achieves the best perfor- +mance. Especially for ‘cars’ and ‘paved parking lots’, our +method achieves 92.16% and 95.25%, which is far ahead of +other methods. The classification maps are shown in Fig. 9. It +can be seen that the results of other methods are not smooth +enough for car classification, while the proposed NNCNet can +depict the clear boundaries of cars and paved parking lots. It +is evident that the proposed NNCNet has strong capabilities +for fine-grained feature representation. +We find that the performance of the proposed NNCNet +on the Houston 2013 dataset and Houston 2018 dataset far +exceeds that on the Trento and MUUFL datasets. We believe +it is due to the higher image resolution of both datasets +(348×1905 and 2384×601 pixels). Therefore, the proposed +NNCNet can exploit better feature representations on large +dataset through contrastive learning. As a result, we believe +that the proposed NNCNet could achieve better classification +results in practical applications, in which more unlabeled data +are available. +D. Ablation Study +To evaluate the effectiveness of different components in +NNCNet, we conducted a series of ablation studies. The effec- +tiveness of each proposed module for improving classification +accuracy is verified through a series of ablation experiments, +and the specific experimental results are listed in Table X. + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING +12 +Healthy grass +Stressed grass +Synthetic grass +Tree +Soil +Water +Residential +Commercial +Road +Highway +Railway +Parking lot 1 +Parking lot 2 +Tennis court +Running track +Features without pretraining +Features with pretraining +(a) Results on the Houston 2013 +Apple trees +Features without pretraining +Features with pretraining +(b) Results on the Trento +Buildings +Ground +Wood +Vineyard +Roads +Trees +Features without pretraining +Features with pretraining +(c) Results on the MUUFL +Mostly grass +Mixed ground surface +Dirt and sand +Road +Water +Building Shadow +Building +Sidewalk +Yellow curb +Cloth panels +Healthy grass +Features without pretraining +Features with pretraining +(d) Results on the Houston 2018 +Stressed grass +Artificial turf +Evergreen trees +Deciduous trees +Bare earth +Water +Residential buildings +Non-residential buildings +Roads +Sidewalks +Crosswalks +Major thoroughfares +Highways +Railways +Paved parking lots +Unpaved parking lots +Cars +Trains +Stadium seats +Features of the final model +Features of the final model +Features of the final model +Features of the final model +Fig. 10. Feature visualizations on different datasets. (a) Results on the Houston 2013 dataset. (b) Results on the Trento dataset. (c) Results on the MUUFL +dataset. (d) Results on the Houston 2018 dataset. The first column denotes features without pretraining, the second column denotes features with pretraining, +the last column represents the features of our final model. The star denotes the cluster center of each class of features. +Effectiveness of the Pretraining and Nearest Neighbor +Learning. We adopt a vanilla convolutional neural network +without pretraining, bilinear attention fusion, and nearest +neighbor contrastive learning as our baseline model. As il- +lustrated in Table X, compared with the baseline model, +pretraining effectively improves classification performance to +some extent on four datasets. It demonstrates that our pretrain- +ing scheme yields parameter initialization that can boost the +classification accuracy. +We further examine our nearest neighbor-based contrastive +learning scheme. As illustrated in Table X, the model with +nearest neighbor learning significantly boosts the classification +performance. The reason is that the semantic similarities of +neighborhood regions are taken into account, and the inter- +modal semantic alignments are enhanced. +To further demonstrate the effectiveness of the pretraining +and nearest neighbor learning, we visualized the features +before and after pretraining in Fig. 10. We visualized the +features without/with pretraining, together with the features in +our final model, respectively. On the Houston 2013, Houston +2018 and Trento datasets, we found that after pretraining, the +features of the same class distributed close to each other and + +★★★IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING +13 +TABLE X +PERFORMANCE COMPARISON OF SEVERAL VARIANTS OF THE PROPOSED MODEL ON DIFFERENT DATASETS +Variant +Pretrain +Bilinear +Attention +Gate +Mechanism +Nearest +Neighbor +Houston 2013 +Trento +MUUFL +Houston 2018 +1 +� +� +� +� +95.20 +98.74 +91.38 +88.21 +2 +� +� +� +� +95.57 +98.80 +91.60 +88.72 +3 +� +� +� +� +96.30 +98.88 +91.83 +89.41 +4 +� +� +� +� +95.64 +98.86 +91.68 +88.79 +5 +� +� +� +� +96.47 +98.90 +92.01 +89.76 +6 +� +� +� +� +95.84 +98.86 +91.68 +88.83 +7 +� +� +� +� +96.77 +98.92 +92.07 +89.89 + +Fig. 11. Classification accuracy for different number of samples. + +Fig. 12. Performance comparison of our model using different data augmen- +tations. +the features of different classes moved far away from each +other. It is evident that our unsupervised framework is effective +on the Houston 2013 and Trento datasets. Furthermore, we +observed that the features after pretraining do not improve +significantly on the MUUFL dataset. The reason may be +that there are more unlabeled data in the Houston 2013, +Houston 2018 and Trento datasets. These unlabeled data play +a critical role in contrastive learning. Therefore, the proposed +contrastive learning framework performs better when more +unlabeled data are available. It is more convenient in practical +applications in which large amounts of unlabeled data are +available. +Number of Training Samples. One of the advantages of +self-supervised learning strategy is its excellent performance in +handling small number of training samples. Therefore, we try +to gradually reduce the number of samples during the training +process, and the results are shown in Fig. 11. On the Houston +2013 dataset, when we use only 375 training samples (25 +samples for each class), the OA value of the proposed method +is 91.86 which is satisfying and encouraging. Furthermore, the +model with pretraining consistently outperforms that without +pretraining on four datasets when small training sets are +used. It is evident that the contrastive learning strategy of the +proposed NNCNet is especially effective for small training +sets. Moreover, we observe that the performance gain of +pretraining on the Houston 2013 and 2018 datasets is better +than that on the Trento and MUUFL datasets. As mentioned +before, there are more unlabeled data on the Houston 2013 +and 2018 datasets. Therefore, the proposed nearest neighbor- +based strategy can exploit rich feature representations on both +datasets. +Effectiveness of Data Augmentation. The purpose of data +augmentation is to enhance the differences between positive +and negative samples as a way to facilitate the training of the +encoder. In the proposed NNCNet, we use four data augmen- +tation schemes, including RandomResizedCrop, RandomHor- +izontalFlip, RandomVerticalFlip and RandomGaussianNoise. +The corresponding results are shown in Fig. 12. We found that +RandomResizedCrop is the key to data augmentation. Since +the image patch is cropped into 11×11 pixels, if the scale +is set too small, the semantic information would easily be +damaged. Therefore, in our implementations, the scale is set +to (0.7, 1). + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING +14 + +Fig. 13. Classification accuracy for different spatial distances, queue sizes, and mini-batch sizes on different datasets. + +Fig. 14. Performance comparison of our model with or without 3D convo- +lution. +E. Parameter Sensitivity +Minimum Spatial Distance between Positive and Neg- +ative Samples. In order to prevent too much similarity be- +tween positive and negative samples, we define a minimum +distance s between them (i.e. the distances between positive +and negative samples need to be greater than s). The results +are shown in Fig. 13(a). In our implementations, the size of +each sample is 11 × 11 pixels. The classification performance +improved slightly when 4 ⩽ s ⩽ 12. It is beneficial to use a +large distance to increase the difference between positive and +negative samples. Therefore, in our implementations, s is set +to 12. +Size of the Negative Key Dictionary. Fig. 13(b) shows +the effect of negative key dictionary size on the classification +performance. The experiments show that a larger dictionary +size will have a positive effect on pretraining, and it is +consistent with our previous assumptions. We believe that the +proposed method works better when more unlabeled data are +available. +Key Encoder Update Speed. We tested different key +encoder update speeds r during pretraining. The experimental +results are shown in Fig. 13(c). We find that the best classifi- +cation performance is achieved when r is set to 0.9. +Effectiveness of 3D Convolution. Inspired by HybridSN +[41], we first use PCA for channel dimensionality reduction. +Then, 3D and 2D convolutions are combined for feature +extraction. To verify the effectiveness of 3D convolution, +we design a network in which the 3D convolutions are +replaced with 2D convolutions (“w/o Conv2d” in Fig. 14). +The experimental results are shown in Fig. 14. We found that +3D convolution can improve the classification performance to +some extent. Although PCA disturbs the spectral continuity of +the hyperspectral data, we argue that 3D convolution can still +generate more discriminative feature maps from the spectral +dimensions than 2D convolution. These discriminative features +generated by 3D convolution can boost the classification +performance. +V. CONCLUSIONS AND FUTURE WORK +In this paper, we propose a self-supervised NNCNet model +to tackle the HSI and LiDAR joint classification problem. +Specifically, we integrate a nearest neighbor-based data aug- +mentation scheme into the contrastive learning framework. Se- +mantic similarities among neighborhood regions are exploited. +The intermodal semantic alignments can be captured more +accurately. In addition, we proposed a bilinear attention fusion +module that can capture second-order feature interactions be- +tween HSI and LiDAR data. Therefore, the module improves +the contextual representation of multisource data effectively. +Extensive experiments on Houston 2013, Trento, MUUFL and +Houston 2018 datasets have demonstrated the superiority of +our model to a wide range of state-of-the-art methods. +In the future, we aim to explicitly explore the semantic and +spatial relations between HSI and LiDAR data. In addition, +we will explore how to further enhance the feature interactions +between HSI and LiDAR data. +Meng Wang received the B.Sc. degree in com- +puter science from Jinan University, Jinan, China, +in 2020. He is currently pursuing the M.Sc. degree +in computer science and applied remote sensing with +the School of Information Science and Technology, +Ocean University of China, Qingdao, China. +His current research interests include computer +vision and remote sensing image processing. + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING +15 +Feng Gao (Member, IEEE) received the B.Sc degree +in software engineering from Chongqing University, +Chongqing, China, in 2008, and the Ph.D. degree +in computer science and technology from Beihang +University, Beijing, China, in 2015. +He is currently an Associate Professor with the +School of Information Science and Engineering, +Ocean University of China. His research interests in- +clude remote sensing image analysis, pattern recog- +nition and machine learning. +Junyu Dong (Member, IEEE) received the B.Sc. +and M.Sc. degrees from the Department of Applied +Mathematics, Ocean University of China, Qingdao, +China, in 1993 and 1999, respectively, and the Ph.D. +degree in image processing from the Department +of Computer Science, Heriot-Watt University, Ed- +inburgh, United Kingdom, in 2003. +He is currently a Professor and Dean with the +School of Computer Science and Technology, Ocean +University of China. His research interests include +visual information analysis and understanding, ma- +chine learning and underwater image processing. +Heng-Chao Li (Senior Member, IEEE) received the +B.Sc. and M.Sc. degrees from Southwest Jiaotong +University, Chengdu, China, in 2001 and 2004, re- +spectively, and the Ph.D. degree from the Graduate +University of Chinese Academy of Sciences, Bei- +jing, China, in 2008. +He is currently a Full Professor with the School +of Information Science and Technology, Southwest +Jiaotong University. His research interests include +statistical analysis of synthetic aperture radar (SAR) +images, remote sensing image processing, and pat- +tern recognition. +Dr. Li is an Editorial Board Member of the Journal of Southwest Jiaotong +University and Journal of Radars. He is an Associate Editor of the IEEE +JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATION AND +REMOTE SENSING. +Qian Du (Fellow, IEEE) received the Ph.D. degree +in electrical engineering from the University of +Maryland at Baltimore, Baltimore, MD, USA, in +2000. +She is currently the Bobby Shackouls Professor +with the Department of Electrical and Computer +Engineering, Mississippi State University, Starkville, +MS, USA. Her research interests include hyperspec- +tral remote sensing image analysis and applications, +and machine learning. +Dr. Du was the recipient of the 2010 Best Re- +viewer Award from the IEEE Geoscience and Remote Sensing Society +(GRSS). She was a Co-Chair for the Data Fusion Technical Committee of +the IEEE GRSS from 2009 to 2013, the Chair for the Remote Sensing +and Mapping Technical Committee of International Association for Pattern +Recognition from 2010 to 2014, and the General Chair for the Fourth IEEE +GRSS Workshop on Hyperspectral Image and Signal Processing: Evolution +in Remote Sensing held at Shanghai, China, in 2012. She was an Associate +Editor for the PATTERN RECOGNITION, and IEEE TRANSACTIONS ON +GEOSCIENCE AND REMOTE SENSING. From 2016 to 2020, she was the +Editor-in-Chief of the IEEE JOURNAL OF SELECTED TOPICS IN APPLIED +EARTH OBSERVATION AND REMOTE SENSING. She is currently a member of +the IEEE Periodicals Review and Advisory Committee and SPIE Publications +Committee. She is a Fellow of SPIE-International Society for Optics and +Photonics (SPIE). +REFERENCES +[1] A. Ma, A. M. Filippi, Z. Wang, Z. Yin, D. Huo, X. Li, and B. 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Li, “S²ENet: Spatial–spectral cross-modal +enhancement network for classification of hyperspectral and LiDAR +data,” IEEE Geoscience and Remote Sensing Letters, vol. 19, pp. 1– +5, 2022. + diff --git a/CtE1T4oBgHgl3EQfpwVv/content/tmp_files/load_file.txt b/CtE1T4oBgHgl3EQfpwVv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a345b77045ec1048c3447cbcd189196fe58059fc --- /dev/null +++ b/CtE1T4oBgHgl3EQfpwVv/content/tmp_files/load_file.txt @@ -0,0 +1,1804 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf,len=1803 +page_content='IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 1 Nearest Neighbor-Based Contrastive Learning for Hyperspectral and LiDAR Data Classification Meng Wang, Feng Gao, Junyu Dong, Heng-Chao Li, Qian Du Abstract—The joint hyperspectral image (HSI) and LiDAR data classification aims to interpret ground objects at more detailed and precise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Although deep learning methods have shown remarkable success in the multisource data classification task, self-supervised learning has rarely been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It is commonly nontrivial to build a robust self-supervised learning model for multisource data classification, due to the fact that the semantic similarities of neighborhood regions are not exploited in existing contrastive learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Furthermore, the heterogeneous gap induced by the inconsistent distribution of multisource data impedes the classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To overcome these disadvantages, we propose a Nearest Neighbor- based Contrastive Learning Network (NNCNet), which takes full advantage of large amounts of unlabeled data to learn discriminative feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Specifically, we propose a nearest neighbor-based data augmentation scheme to use enhanced semantic relationships among nearby regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The intermodal semantic alignments can be captured more accu- rately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In addition, we design a bilinear attention module to exploit the second-order and even high-order feature interactions between the HSI and LiDAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Extensive experiments on four public datasets demonstrate the superiority of our NNCNet over state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The source codes are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='com/summitgao/NNCNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Index Terms—hyperspectral image, self-supervised learning, light detection and ranging, contrastive learning, image classifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' INTRODUCTION R ECENTLY, with the rapid development of satellite sen- sors, an ever increasing number of multimodal im- ages (optical, SAR, hyperspectral and LiDAR) are obtained everyday [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Among these multimodal data, hyperspectral images (HSIs) provide detailed spectral information for the identification of specified objects on the ground, while LiDAR data provide elevation information of the area [2] [3] [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' These HSI and LiDAR sensors are different in imaging mechanism, spatial resolution, and even coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, both sensors capture different properties of the earth, such as spectral radiance and height information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' For example, there are no significant differences in the spectral domain between the “trees” on the ground and the “trees” on the hill, but they can This work was supported in part by the National Key Research and Development Program of China under Grant 2018AAA0100602, and in part by the National Natural Science Foundation of China under Grant 42106191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Meng Wang, Feng Gao, and Junyu Dong are with the School of Information Science and Engineering, Ocean University of China, Qingdao 266100, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (Corresponding author: Feng Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=') H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' -C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Li is with the Sichuan Provincial Key Laboratory of Information Coding and Transmission, Southwest Jiaotong University, Chengdu 610031, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Qian Du is with the Department of Electrical and Computer Engineering, Mississippi State University, Starkville, MS 39762 USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Encoder Momentum encoder Similarity Contrastive loss Nearest neighbor Encoder Momentum encoder Similarity Contrastive loss (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Conceptual comparison of MoCo and the proposed nearest neighbor- based contrastive learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In the proposed framework, the nearest neighbors are considered as positive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The semantic similarities among neighborhood regions are exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' be distinguished from the LiDAR data [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, the joint exploitation of HSI and LiDAR data enables us to interpret ground objects at a more detailed and precise level, which can hardly be achieved by using single-mode data [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Thus, the classification of cross-modal data has attracted considerable attention and has been widely applied in multisource image interpretations [7] [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' A great deal of effort has been put into solving the problem of HSI and LiDAR joint classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Traditionally, feature- level fusion models have been proposed, and these models commonly concatenate the HSI and LiDAR features for clas- sification [9] [10] [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Besides feature-level fusion, decision- level fusion is another popular solution for HSI and LiDAR classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Several classifiers are designed for HSI and LiDAR data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The voting strategy is commonly used to obtain the final classification map [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Subsequently, to further exploit high-level semantic features, convolutional neural networks (CNNs) are employed for multisource data classification [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Encoder-decoder network [14], coupled CNNs [15], Gabor CNN [16], cross attention [17], and Trans- former [18] are used to extract representative multisource features, and these methods have achieved promising perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In practice, deep learning models have demonstrated re- markable success in various multisource data joint classifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' However, it is non-trivial to build an effective HSI and LiDAR classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' One of the critical reasons is that the deep learning-based model commonly requires a great number of labeled samples to achieve satisfactory accuracy, which is expensive and limited in ground object modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Recent research in self-supervised learning encourages the deep network to learn more representative and interpretable arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='03335v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='IV] 9 Jan 2023 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 2 features in natural language processing [19] [20] and computer vision tasks [21] [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Self-supervised learning mines the inherent attributes and semantics of large-scale unlabeled data to obtain beneficial semantic representations, and it does not require manually annotated data [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' After the self-supervised training finished, the learned features can be transferred to classification tasks (especially when only small training data is available) as pretrained models to boost the classification performance and alleviate overfitting [24] [25] [26] [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In HSI and LiDAR joint classification, self-supervised learning has rarely been explored, and in this paper, we aim to build an effective self-supervised model to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It is commonly non-trivial to build a robust self-supervised learning model for the HSI and LiDAR joint classification task, due to the following reasons: 1) Data augmentation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In Momentum Contrast (MoCo) for self-supervised learning [22], the random color jittering, random horizontal flip, and random grayscale conversion are used for data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' However, such data augmentation scheme does not take the spatial distances between the positive and negative samples into account, and the semantic similarities of neighborhood regions are not exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Consequently, how to properly utilize the semantic similarities among nearby regions is a major challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 2) The heterogeneous gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' HSI and LiDAR joint classification requires a comprehensive understanding of complex heterogeneous data simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' However, the heterogeneous gap induced by the inconsistent distributions of multisource data would greatly impedes its implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, it is vital to bridge this gap for more robust multisource data classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To address the aforementioned challenges, we propose a Nearest Neighbor-based Contrast learning Network, NNCNet for short, which aims to learn an encoder that encodes similar data of the same kind and makes the encoding results of differ- ent classes of data as different as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To be more specific, we propose a nearest neighbor-based framework to use the enhanced semantic relationships among nearby regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 1, nearest neighbors of positive samples are fed into the encoder for contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The feature representations are learned by encouraging the proximity of between different views of the same sample and its nearest neighbors in the spatial domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, the contrastive learning framework is encouraged to generalize to new feature embeddings that may not be covered by the data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In addition, we design a bilinear attention fusion module to exploit second-order and even higher-order feature interactions between the HSI and LiDAR data, and the information flow can be controlled more flexibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The contributions of this work are as follows: We propose a self-supervision contrastive learning ap- proach NNCNet, which integrates a nearest neighbor- based data augmentation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The scheme can exploit the semantic similarities among neighborhood regions, and hence capture inter-modal semantic alignments more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To our best knowledge, we are the first to apply self-supervised contrastive learning to HSI and Li- DAR joint classification, which has both great theoretical and practical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We propose a bilinear attention fusion module that aims to enhance the contextual representation of HSI and LiDAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The module captures second-order feature interactions between multisource data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We have conducted extensive experiments on four bench- mark datasets to validate the effectiveness of our NNC- Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Additionally, we have released our codes and pa- rameters to benefit other researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Morphological Filter-Based Methods for HSI and LiDAR Classification The joint use of HSI and LiDAR has already been in- vestigated for a variety of applications, such as illumination calibration [28], forest area analysis [29], bushfire monitoring [30], and urban sprawl modeling [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Great efforts have been devoted to exploiting the complementary information between multisource data, especially for morphological filter-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Morphological filters are intensively used to atten- uate the redundant spatial details and preserve the geometric structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Pedergnana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' [9] used morphological extended attribute profiles to HSI and LiDAR data for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Features extracted from HSI and LiDAR data are stacked for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' [32] computed morphological attribute profiles from HSI and LiDAR data, and these attribute profiles are fused using a generalized graph-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Khodadadzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' [33] pointed out that simple stacking of morphological attribute profiles from multisource data may contain redundant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To solve this issue, they proposed a multiple feature learning approach based on the multinomial logistic regression classifier, which can adaptively exploit the spatially and spectrally derived features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Later, attribute profiles are considered to be complex and time-consuming in threshold initialization, and extinction profiles [34] are proposed to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Ghamisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' [35] presented a classification framework based on extinction profiles and deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' CNN-Based Methods for HSI and LiDAR Classification Recently, deep CNNs have attracted extensive research attention in the remote sensing data fusion community, and many CNN-based models have been proposed for multisource data classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' [36] proposed a two-branch CNN model, which consists of a 2-D convolutional network and a 1-D convolutional network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' [37] presented a patch-to-patch CNN for the joint feature extraction of HSI and LiDAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' [38] proposed a CNN and DNN hybrid model for multisource feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' CNNs are used to extract informative features from multisource data, and a DNN is utilized to fuse these heterogeneous features for robust classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' [39] proposed a dual-channel spatial, spectral and multiscale attention CNN for multisource data classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Hang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' [15] used coupled CNNs for multisource data classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The coupled layers reduce the number of parameters and guide both networks learning from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' [16] proposed a fractional Gabor CNN, IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 3 and focused on efficient feature fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Fractional Gabor con- volutional kernels are used for multiscale and multidirectional feature extraction and yield robust feature representations against semantic changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In [40], a multisource graph fusion network is presented to integrate feature extraction and fusion into a single network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' A multimodal graph is constructed to guide the multimodal image feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' [17] proposed a deep-wise feature interaction network for multisource remote sensing image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Consistency loss, discrimination loss, and classification loss are designed for parameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Although CNNs have been successfully applied to HSI and LiDAR joint classification, its performance remains un- satisfactory for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' An important factor may be the lack of sufficient annotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In practical applications, remote sensing data annotation is costly, making it difficult to obtain robust deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To solve this problem, we aim to build a simple yet effective self-supervised method for multisource data joint classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It extracts the inherent attributes and semantics from unlabeled large-scale data to capture beneficial feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In addition, a nearest-neighbor-based data augmentation scheme is used to exploit the semantic relationships among nearby regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' METHODOLOGY As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 2, the proposed NNCNet consists of three parts: nearest neighbor-based data augmentation, bilinear attention-based encoder, and contrastive loss computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Considering that the proposed NNCNet is based on a self- supervised contrastive learning framework, we first introduce the nearest neighbor-based contrastive learning framework and then successively elaborate the bilinear attention-based encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Nearest Neighbor-based Momentum Contrast Learning Framework Considering that unlabeled data have no supervised infor- mation, we aim to extract the supervised information from large-scale unsupervised HSI and LiDAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Our goal is to train an encoder that keeps the different transformations from the same sample as close as possible and the different samples as far away as possible in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To solve the problem, He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' [22] proposed Momentum Contrast (MoCo) for self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To be specific, a minibatch of samples is selected from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Each sample is handled by random data augmentation (Gaussian blur, flip, or spectral distortions) to generate a query sample and a key sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The query and key samples are encoded separately to embedding q and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The cosine similarity between q and k is computed for representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The embedding from the same image is defined as the positive key, and embedding from different image is defined as the negative key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In MoCo, a dynamic dictionary is built with a queue and a dynamic encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' For remote sensing data classification, we argue that random augmentations can hardly provide positive pairs for the same object representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' For the sake of covering more variance in a given class, we propose nearest neighbor-based contrastive Algorithm 1 Pseudocode of the Nearest Neighobr-Based Contrastive Learning in PyTorch Style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' # f_q: encoder network for query # f_k: encoder network for key # queue: key dictionary # r: momentum coefficient f_k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='param = f_q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='param # initialize parameters # load a mini-batch x with N samples for x in loader: nn = neighbor(x) # generate neighbors of x x_q = augment(x) # query randomly augmentation x_k = augment(x) # key random augmentation x_n = augment(nn) # neighbors random augmentation # randomly substitute half samples in x_k by x_n x_k = substitute(x_k, x_n) q = f_q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='forward(x_q) # queries: NxC k = f_k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='forward(x_k) # keys: NxC k = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='detach() # no gradient to keys # positive logits: Nx1 # bmm: batch matrix multiplication l_pos = bmm(q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='view(N,1,C), k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='view(N,C,1)) # negative logits: NxK # mm: matrix multiplication l_neg = mm(q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='view(N,C), queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='view(C,K)) # logits: Nx(1+K) logits = cat([l_pos, l_neg], dim=1) # contrastive loss computation labels = zeros(N) loss = CrossEntropyLoss(logits/t, labels) # back propagation, only update the query network loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='backward() update(f_q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='param) # dictionary update f_k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='param = r*f_k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='param+(1-r)*f_q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='param enqueue(queue, k) # push the current key dequeue(queue) # pop the earliest key learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In remote sensing images, the neighbor labels of one specified position tend to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 3, the region within the red box is rooftop, and the green boxes are its nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Blue boxes denote regions far from the red box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' From the visualized feature space, it can be observed that the features of the nearest neighbors are close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In this paper, we use a nearest neighbor- based contrastive learning framework in which the semantic similarities among neighborhood regions are exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' There- fore, the inter-modal semantic alignments are reinforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Nearest Neighbor-Based Contrastive Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Algo- rithm 1 provides the pseudo code of the proposed nearest neighbor-based contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In the proposed frame- work, a set of sample pairs are selected from HSI and LiDAR data centered at the same position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' During training, each sample pair is handled by random data augmentation to generate a query sample xq and a key sample xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' They are encoded to embeddings q and k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The em- beddings from the same image are defined as positive key, and embeddings from different images are defined as negative key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' A large number of negative key embeddings are stored in a dictionary {k1, k2, k3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' }, while one positive key k+ is stored separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Furthermore, we randomly select some nearest neighbors of q to generate embeddings, which are denoted as kn+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Next, in a minibatch, half positive keys k+ are substituted by kn+ to form new positive keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Hence, nearest neighbors act small semantic perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In our IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 4 Hyperspectral image LiDAR Data augmentation Nearest neighbor substitution Encoder Momentum encoder Push Pop Positive key Dictionary of negative keys Contrastive loss Contrastive loss D D D D Similarity of positive pairs Similarity of negative pairs Slowly update Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Schematic illustration of the nearest-neighbor based contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It consists of three components: 1) Nearest neighbor-based data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The input samples are handled by random data augmentation to generate query and key samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In a mini-batch, half positive key samples are substituted by its nearest neighbors to form new positive key samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' These nearest neighbors act small semantic perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 2) Bilinear attention-based feature encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The query and key samples are fed into the encoder for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' A bilinear attention fusion module is employed to capture the second-order feature interactions between multisource data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 3) Contrastive loss computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Positive and negative keys are stored in a dynamic dictionary, and contrastive loss is computed to assign high scores for positive keys and low scores for negative keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Remote sensing images Visualized feature space Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Typical regions in remote sensing images and the corresponding visualized features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The region within the red box is rooftop, and the green boxes are its nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Blue boxes denote regions far from the the red box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In the visualized feature space, it can be observed that features of the nearest neighbors are close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, the contextual information is critical in contrastive learning for remote sensing image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' implementations, nearest neighbors denotes a region whose overlap area with xq is greater than 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We calculate the cosine similarities between q and keys (both the positive key and negative keys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Then, the results are stored as {D+, D1, D2, D3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' , DK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Here D+ is the similarity between q and positive key k+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The rest are the similarities between q and negative keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' K is the number of negative keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The objective of contrastive learning is to force the query to match the positive key and far apart from the negative keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To be specific, the contrastive loss whose value is low when q is similar to the positive key k+ and dissimilar to all the negative keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, the contrastive loss function is designed as follows: L = − log exp(D+/τ) �K i=1 exp(Di/τ) (1) where τ is a temperature hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Intuitively, softmax classifier and cross-entropy loss can be combined into the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Dictionary Update and Moving Average Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Similar to MoCo [22], we maintain the dictionary as a queue which stores many minibatch of negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The negative sam- ples in the dictionary are updated progressively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Specifically, during training, when a new minibatch is pushed into the dictionary, the oldest minibatch is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The length of the dictionary is flexibly set as a hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Furthermore, the parameters of the encoder for the dic- tionary are updated slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Similar to MoCo, we use a separate moving average encoder for the key samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' During training, no backpropagation is done for the key encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The parameters of key encoder are updated as follows: θk = rθk + (1 − r)θq, (2) where θk denotes the parameters of the key encoder, and θq denotes the parameters of the query encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' r is a momentum coefficient that controls the speed of key encoder update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Only θq is updated by backpropagation during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In our implementations, r is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='9, since a slowly evolving key encoder is critical for robust feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Shuffling BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Batch Normalization (BN) is employed in the encoder to speed up convergence and improve the gener- alization of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Similar to MoCo, we use the shuffling BN for better feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In particular, we shuffle the sample order in the current minibatch for the key encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The sample order of the mini-batch for the query encoder is not changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Bilinear Attention-Based Multisource Encoder In this work, the purpose of contrastive learning is to generate a pretrained model, and the model can be used for the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To achieve high classification accuracy, the encoder is an essential part of the contrastive framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We design a multisource encoder for hyperspectral and LiDAR feature modeling, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It contains three parts: HSI feature extraction, LiDAR feature extraction, and bilinear attention fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The detailed summary of the encoder in terms of layer type, kernel size, and output size is illustrated in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 5 LiDAR 3DConv 3DConv 3DConv 2DConv 2DConv 2DConv 2DConv HSI Bilinear Attention Output FC 3DConv 2DConv 3D convolution 2D convolution Element-wise summation FC Fully connected layer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Bilinear attention-based multisource encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' TABLE I SUMMARY OF THE PROPOSED MULTI-SOURCE ENCODER HSI feature extraction subnetwork # Layer type Kernel number@size Output size Input — (11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 1) 1 3D Conv 8@3×3×9 (9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 8) 2 3D Conv 16@3×3×7 (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 16) 3 3D Conv 32@3×3×5 (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 32) 4 Reshape — (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 384) 5 2D Conv 256@3×3 (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 256) 6 Reshape — (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 256) LiDAR feature extraction subnetwork # Layer type Kernel numbre@size Output size Input — (11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 1) 1 2D Conv 64@3×3 (9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 64) 2 2D Conv 128@3×3 (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 128) 3 2D Conv 256@3×3 (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 256) 4 Reshape — (25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 256) Hyperspectral data adopt a network similar to HybridSN [41],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' which uses both 3D and 2D convolutions for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Three 3D convolution layers and one 2D convolution layer are used to derive the HSI feature FH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' At the same time, three 2D convolution layers are used to generate the LiDAR feature FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Next, FH and FL are combined to form the fused feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' A 2D convolution is used for feature embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Then, the fused feature Ffus has the same dimension as FH and FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To effectively reduce the inherent redundancy in HSI, and thereby reduce the amount of data that needs to be processed in classification, Principal Component Analysis (PCA) is used to select the best 30 spectral bands for HSI feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Finally, FH, FL and Ffus are fed into the bilinear attention fusion module as Q, K, and V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The output of the bilinear attention fusion module is fed into a fully connected layer to generate the final feature for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Bilinear Attention Fusion Module The attention mechanism has made valuable breakthroughs in deep neural networks and has been successfully applied to FC FC FC FC Bilinear pooling FC softmax C FC S Gate mechanmism Bilinear pooling FC Fully connected layer Element-wise multiplication C Feature concatenation S Sigmoid activation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Bilinear attention fusion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It can capture second-order interac- tions between multisource data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' cross-modal tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=', visual question answering [42], image captioning [43], and image-text matching [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' This prompts recent methods to adopt the attention to trigger the interaction between multi-modal remote sensing data [45] [46] [47] [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In the conventional attention mechanism, the attention weights are estimated via linearly fusing the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' However, we argue that conventional attention exploits the first-order feature interaction and is limited in complex multisource feature reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Toward this end, we propose a bilinear attention fusion mod- ule to exploit the second-order feature interactions between the hyperspectral and LiDAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 5, it mainly contains two parts: the multi-head bilinear attention and the gate mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Multi-Head Bilinear Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Suppose we have query Q ∈ Rc×d, key K ∈ Rc×d, and value V ∈ Rc×d, where d denotes the feature dimension, and c is the number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To enhance the capability of feature representation, the multi-head scheme is used to model feature interactions from different subspaces as: hi = BiAttention(Qi, Ki, Vi), (3) where hi is the output of the i-th head, and BiAttention denotes the bilinear attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The number of heads is denoted by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The bilinear attention first maps Qi ∈ R c H ×d and Ki ∈ R c H ×d into a joint space as: B1 i = σ(QiW1 q) ⊙ σ(KiWk), (4) where W1 q ∈ Rd×d and Wk ∈ Rd×d are weighting matrices, σ is the ReLU activation and ⊙ denotes the element-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' As such, B1 i ∈ R c H ×d denotes the second-order representation between query Qi and key Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Similarly, we compute the bilinear representation between Qi and Vi as: B2 i = σ(QiW2 q) ⊙ σ(ViWv), (5) where W2 q ∈ Rd×d and Wv ∈ Rd×d are weighting matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' B2 i ∈ R c H ×d denotes the second-order representation between query Qi and value Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 6 Next, the bilinear representation B1 i is projected into atten- tion weights Watt i ∈ R c H ×d via a linear layer and a softmax layer as follows: ˆB1 i = σ(WBB1 i ), (6) Watt i = softmax( ˆB1 i ), (7) where WB ∈ R c H × c H is the weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Next, the attended feature hi ∈ R c H ×d is derived by enhancing the attention weights as: hi = Watt i ⊙ B2 i (8) Gate Mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The aforementioned bilinear attention exploits the feature interactions among Qi, Ki, and Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' However, there may contain noisy information in the query and key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To adaptively enhance the informative parts and suppress the useless parts, we design a gate mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To be specific, for the i-th head, ˆB1 i is fed into a linear layer and then handled with a sigmoid function to compute a weight mask Gi ∈ R c H ×1 as: Gi = sigmoid( ˆB1 i WB′), (9) where WB′ ∈ Rd×1 is the weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Next, Gi is expanded to form G′ i ∈ R c H ×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Then the obtained gating mask is applied to control the information flow of hi ∈ R c H ×d as: ˆhi = G′ i ⊙ hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (10) Finally, by concatenating the results of multiple heads, we obtain the fused representation of multi-source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In this work, the size of Q, K and V is 25×256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The number of heads H is set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' EXPERIMENTAL RESULTS AND ANALYSIS To validate the effectiveness of the proposed NNCNet, we conduct extensive experiments on four widely used bench- mark datasets: Houston 2013 dataset, Trento dataset, MUUFL dataset and Houston 2018 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We first compare the proposed NNCNet with state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Then we implemented additional evaluations to investigate the effec- tiveness of each component of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Datasets and Evaluation Metric Houston 2013 dataset: The dataset was captured by the National Airborne Center for Laser Mapping, and it was used as a challenge in the 2013 GRSS Data Fusion Contest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The HSI was captured by the CASI sensor (144 spectral bands at a resolution of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='5 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Coregistered LiDAR data with the same resolution are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' A total of 15029 ground truth samples are distributed in 15 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' They are divided into train and test sets containing 2832 and 12197 pixels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We used standard training and test sets, and Table II lists the number of training and test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Trento dataset: The dataset was collected in a rural region south of Trento, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The HSI image consists of 63 bands with a wavelength range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='42-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='99 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The size of the datset is 166×660 pixels, and the spatial resolution of the datset is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='0 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' A total of 30214 ground truth samples are distributed TABLE II TRAIN-TEST DISTRIBUTION OF SAMPLES FOR THE HOUSTON 2013 DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Class Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Test ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Healthy grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='198 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1053 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Stressed grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='190 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1064 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Synthetic grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='505 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Tree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='188 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1056 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Soil ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='186 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1056 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Water ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='182 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='143 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Residential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='196 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1072 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Commercial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='191 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1053 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Road ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='193 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1059 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Highway ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='191 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1036 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Railway ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='181 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1054 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Parking lot 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1041 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Parking lot 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='184 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='285 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Tennis court ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='181 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='247 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Running track ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='473 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='2832 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='12197 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='TABLE III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='TRAIN-TEST DISTRIBUTION OF SAMPLES FOR THE TRENTO DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Class Name Training Test 1 Apple trees 129 3905 2 Buildings 125 2778 3 Ground 105 374 4 Wood 154 8969 5 Vineyard 184 10317 6 Roads 122 3052 Total 819 29595 TABLE IV TRAIN-TEST DISTRIBUTION OF SAMPLES FOR THE MUUFL DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Class Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Test ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Trees ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='23096 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Mostly grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='4120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Mixed ground surface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='6732 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Dirt and sand ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1676 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Road ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='6537 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Water ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='316 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Building shadow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='2083 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Building ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='6090 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Sidewalk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1235 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Yellow curb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Cloth panels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='119 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1650 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='52037 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='(a) Ground truth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='(b) FusAtNet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='(c) TBCNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='(d) EndNet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='(e) MDL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='(f) CCNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='(g) S2ENet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='(h) w/o Pretraining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='(i) Proposed NNCNet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Healthy grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Stressed grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Synthetic grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Tree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Road ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Railway ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Soil ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Water ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Residential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Commercial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Highway ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Parking lot 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Parking lot 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Tennis court ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Running track ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Classification maps for the Houston 2013 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (a) Groundtruth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (b) FusAtNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (c) TBCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (d) EndNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (e) MDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (f) CCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (g) S2ENet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (h) Proposed NNCNet without pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (i) Proposed NNCNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' TABLE V TRAIN-TEST DISTRIBUTION OF SAMPLES FOR THE HOUSTON 2018 DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Class Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Test ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Healthy grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='9299 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Stressed grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='32002 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Artificial turf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='68 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='616 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Evergreen trees ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='13095 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Deciduous trees ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='4521 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Bare earth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='451 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='4065 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Water ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='240 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Residential buildings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='39272 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Non-residential buildings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='223252 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Roads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='45366 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Sidewalks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='33529 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Crosswalks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='151 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='1367 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Major thoroughfares ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='45848 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Highways ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='9365 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Railways ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='6437 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Paved parking lots ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='11000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Unpaved parking lots ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='132 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Cars ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='6047 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Trains ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='4869 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Stadium seats ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='6324 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='8210 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='496646 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='in 6 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Table III lists the distribution of training and test samples for the Trento dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' MUUFL dataset: The MUUFL dataset is captured over the University of Southern Mississippi Gulf Coast campus in November 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The HSI contains 72 spectral bands, but the first and last four bands are removed for noise reduction, leaving 64 bands for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The total size of the dataset is 325×220 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Table IV lists the training and test samples available for the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In our experiments, we use the entire data in the pretraining phase, while in the training validation phase, we use only the portion of the training set for which labels are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Houston 2018 dataset: The dataset was captured by the Hyperspectral Image Analysis Laboratory and the National Center for Airborne Laser Mapping (NCALM) at the Uni- versity of Houston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It was originally released for the 2018 IEEE GRSS Data Fusion Contest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Hyperspectral data covers 380-1050 nm spectral range with 48 bands at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='0 m ground sample distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The dataset contains a total of 4768×1202 pixels in which a piece is delineated as the training set with the size of 2384×601 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Table V lists the distribution of training and test samples for the Houston 2018 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The performance of the model is evaluated by Overall Ac- curacy (OA), Average Accuracy (AA), and Kappa coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' OA denotes the ratio of the model’s correct predictions to the overall number on all test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' AA is the ratio between the number of correct predictions in each category and the overall number in each category, and finally the average of the accuracy in each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Kappa is the percentage of agreement corrected by the number of agreements that would be expected purely by chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Implementation Details The proposed contrastive learning architecture is used to generate a pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In the contrastive learning phase, we use the Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The mini-batch size is set to 64 and the learning rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The image patch of IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 8 TABLE VI CLASSIFICATION ACCURACY (%) ON THE HOUSTON 2013 DATASET Class FusAtNet [49] TBCNN [36] EndNet [14] MDL [5] CCNN [15] S2ENet [50] NNCNet (ours) Healthy grass 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='20 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='01 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='54 83.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='01 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='21 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='27 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='03 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='55 11×11 pixels is randomly cropped from the dataset as training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' After obtaining the pretrained model, training samples from the dataset are used for fine-tuning the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In the fine- tuning phase, the mini-batch size is set to 128, and the setting of the optimizer is the same as that in the contrastive learning phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Classification Accuracy and Discussion The proposed NNCNet is implemented on the Houston 2013, Trento, MUUFL and Houston 2018 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To verify the effectiveness of the proposed NNCNet, we compared it with six state-of-the-art methods, including FusAtNet [49], TBCNN [36], EndNet [14], MDL [5], CCNN [15], and S2ENet [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In particular, FusAtNet [49] exploits HSI and LiDAR features via cross-attention, and attentive spectral and spatial representations are combined to compute modality- specific feature embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' TBCNN [36] uses a two-branch CNN for HSI and LiDAR feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In EndNet [14], a deep encoder–decoder network is utilized for multimodal information fusion and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' MDL [5] presents a general multimodal deep learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' CCNN [15] presents a coupled network for multimodal information fu- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Feature-level and decision-level fusion are integrated for heterogeneous feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' S2ENet [50] is a spatial-spectral enhancement network that improves spatial and spectral feature representations simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' For a fair comparison, all the compared methods adopt the default parameters provided in their works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Table VI shows the classification results on the Houston 2013 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The proposed NNCNet achieves the best per- formance in terms of OA, AA and Kappa coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Our NNCNet outperforms the competitor (CCNN and S2ENet) by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='78% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='78% for OA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It shows that the proposed self-supervised framework effectively models the correlations between multisource samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Furthermore, the accuracy for ‘highway’ class (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='81%) is significantly IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 9 TABLE VIII CLASSIFICATION ACCURACY (%) ON THE MUUFL DATASET Class FusAtNet [49] TBCNN [36] EndNet [14] MDL [5] CCNN [15] S2ENet [50] NNCNet (ours) Trees 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='31 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='18 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='86 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='95 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='40 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='91 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='09 Mostly grass 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='83 83.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='59 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='42 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='26 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='44 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='93 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='56 TABLE IX CLASSIFICATION ACCURACY (%) ON THE HOUSTON 2018 DATASET Class FusAtNet [49] TBCNN [36] EndNet [14] MDL [5] CCNN [15] S2ENet [50] NNCNet (ours) Healthy grass 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='40 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='91 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='55 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='99 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='39 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='29 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='36 Stressed grass 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='65 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='83 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='51 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='95 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='84 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='97 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='95 Artificial turf 98.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='15 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='87 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='98 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='55 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='62 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='12 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='86 Bare earth 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='78 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='48 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='78 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='98 Water 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='17 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='67 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='92 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='75 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='83 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='75 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='25 Residential buildings 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='29 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='93 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='54 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='31 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='43 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='31 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='58 Non-residential buildings 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='36 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='78 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='78 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='75 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='93 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='22 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='04 Roads 62.' 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100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='0 Cars 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='13 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='65 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='77 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='87 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='37 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='11 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='16 Trains 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='29 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='06 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='67 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='37 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='38 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='75 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='01 Stadium seats 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='62 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='49 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='34 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='59 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='83 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='65 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='78 OA 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='98 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='33 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='84 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='70 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='64 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='87 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='89 AA 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='68 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='72 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='78 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='72 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='48 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='33 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='99 Kappa 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='46 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='83 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='06 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='15 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='19 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='38 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='65 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 10 (a) Ground truth (b) FusAtNet (c) TBCNN (d) EndNet (e) MDL (f) CCNN (g) S2ENet (h) w/o Pretraining (i) Proposed NNCNet Apple trees Buildings Ground Wood Vineyard Roads Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Classification maps for the Trento dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (a) Groundtruth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (b) FusAtNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (c) TBCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (d) EndNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (e) MDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (f) CCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (g) S2ENet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (h) Proposed NNCNet without pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (i) Proposed NNCNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (a) Ground truth (b) FusAtNet (c) TBCNN (d) EndNet (e) MDL (f) CCNN (g) S2ENet (h) w/o Pretraining (i) Proposed NNCNet Trees Mostly grass Mixed ground surface Dirt and sand Road Water Building Shadow Building Sidewalk Yellow curb Cloth panels Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Classification maps for the MUUFL dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (a) Groundtruth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (b) FusAtNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (c) TBCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (d) EndNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (e) MDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (f) CCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (g) S2ENet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (h) Proposed NNCNet without pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (i) Proposed NNCNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 11 (a) Group truth (b) FusAtNet (c) TBCNN (d) EndNet (e) MDL (f) CCNN (g) S2ENet (h) w/o Pretraining (i) Proposed NNCNet Healthy grass Stressed grass Artificial turf Evergreen trees Deciduous trees Bare earth Water Residential buildings Non-residential buildings Roads Sidewalks Crosswalks Major thoroughfares Highways Railways Paved parking lots Unpaved parking lots Cars Trains Stadium seats Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Classification maps for the Houston 2018 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (a) Groundtruth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (b) FusAtNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (c) TBCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (d) EndNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (e) MDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (f) CCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (g) S2ENet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (h) Proposed NNCNet without pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (i) Proposed NNCNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' improved by our NNCNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' There are many unlabeled highway regions in the Houston 2013 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, our NNCNet captured the texture and spectral features of highway via contrastive learning from unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The classification maps are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It can be observed that with- out pretraining, some highway regions are falsely classified into road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In contrast, the proposed NNCNet performs better through contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Table VII illustrates the classification results of different methods on the Trento dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The classification maps are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It can be seen that without pretraining, some vineyard regions are falsely classified into apple trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In addition, the proposed NNCNet achieves the best performance in terms of OA, AA, and Kappa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The proposed method achieves the best OA in ‘ground’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' There is only a small amount of labeled data in this class, but it still accounts for a large portion of the entire graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It is evident that our NNCNet is capable to learning the robust feature representations when training samples are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Table VIII shows the classification results of different meth- ods on the MUUFL dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The proposed NNCNet obtains the best performance against the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To be specific, the proposed method has the best OA (92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='07%) and reached the highest accuracy in five classes (Mixed ground surface, Road, Water, Sidewalk and Cloth panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The classification results of the proposed method for the Mostly building and Building shadow are quite competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, the com- parisons demonstrate the superior performance of the proposed NNCNet on the MUUFL dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The classification maps of the proposed NNCNet with / without pretraining are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 8, it can be observed that the pretraining effectively improved the classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Table IX illustrates the classification results of different methods on the Houston 2018 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Compared to other methods, the proposed NNCNet achieves the best perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Especially for ‘cars’ and ‘paved parking lots’, our method achieves 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='16% and 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='25%, which is far ahead of other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The classification maps are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It can be seen that the results of other methods are not smooth enough for car classification, while the proposed NNCNet can depict the clear boundaries of cars and paved parking lots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It is evident that the proposed NNCNet has strong capabilities for fine-grained feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We find that the performance of the proposed NNCNet on the Houston 2013 dataset and Houston 2018 dataset far exceeds that on the Trento and MUUFL datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We believe it is due to the higher image resolution of both datasets (348×1905 and 2384×601 pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, the proposed NNCNet can exploit better feature representations on large dataset through contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' As a result, we believe that the proposed NNCNet could achieve better classification results in practical applications, in which more unlabeled data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Ablation Study To evaluate the effectiveness of different components in NNCNet, we conducted a series of ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The effec- tiveness of each proposed module for improving classification accuracy is verified through a series of ablation experiments, and the specific experimental results are listed in Table X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Healthy grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Stressed grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Synthetic grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Tree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Soil ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Water ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Residential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Commercial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Road ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Highway ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Railway ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Parking lot 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Parking lot 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Tennis court ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Running track ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Features without pretraining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Features with pretraining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='(a) Results on the Houston 2013 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Apple trees ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Features without pretraining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Features with pretraining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='(b) Results on the Trento ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Buildings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Ground ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Wood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Vineyard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Roads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Trees ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Features without pretraining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Features with pretraining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='(c) Results on the MUUFL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Mostly grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Mixed ground surface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Dirt and sand ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Road ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Water ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Building Shadow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Building ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Sidewalk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Yellow curb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Cloth panels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Healthy grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Features without pretraining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Features with pretraining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='(d) Results on the Houston 2018 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Stressed grass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Artificial turf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Evergreen trees ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Deciduous trees ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Bare earth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Water ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Residential buildings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Non-residential buildings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Roads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Sidewalks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Crosswalks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Major thoroughfares ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Highways ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Railways ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Paved parking lots ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Unpaved parking lots ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Cars ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Trains ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Stadium seats ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Features of the final model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Features of the final model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Features of the final model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Features of the final model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Feature visualizations on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (a) Results on the Houston 2013 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (b) Results on the Trento dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (c) Results on the MUUFL dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' (d) Results on the Houston 2018 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The first column denotes features without pretraining, the second column denotes features with pretraining, the last column represents the features of our final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The star denotes the cluster center of each class of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Effectiveness of the Pretraining and Nearest Neighbor Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We adopt a vanilla convolutional neural network without pretraining, bilinear attention fusion, and nearest neighbor contrastive learning as our baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' As il- lustrated in Table X, compared with the baseline model, pretraining effectively improves classification performance to some extent on four datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It demonstrates that our pretrain- ing scheme yields parameter initialization that can boost the classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We further examine our nearest neighbor-based contrastive learning scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' As illustrated in Table X, the model with nearest neighbor learning significantly boosts the classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The reason is that the semantic similarities of neighborhood regions are taken into account, and the inter- modal semantic alignments are enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To further demonstrate the effectiveness of the pretraining and nearest neighbor learning, we visualized the features before and after pretraining in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We visualized the features without/with pretraining, together with the features in our final model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' On the Houston 2013, Houston 2018 and Trento datasets, we found that after pretraining, the features of the same class distributed close to each other and ★★★IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 13 TABLE X PERFORMANCE COMPARISON OF SEVERAL VARIANTS OF THE PROPOSED MODEL ON DIFFERENT DATASETS Variant Pretrain Bilinear Attention Gate Mechanism Nearest Neighbor Houston 2013 Trento MUUFL Houston 2018 1 � � � � 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='20 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='74 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='38 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='21 2 � � � � 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='57 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='80 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='60 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='72 3 � � � � 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='30 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='88 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='83 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='41 4 � � � � 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='64 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='86 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='68 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='79 5 � � � � 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='47 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='90 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='01 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='76 6 � � � � 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='84 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='86 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='68 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='83 7 � � � � 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='77 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='92 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='07 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='89 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Classification accuracy for different number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Performance comparison of our model using different data augmen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' the features of different classes moved far away from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It is evident that our unsupervised framework is effective on the Houston 2013 and Trento datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Furthermore, we observed that the features after pretraining do not improve significantly on the MUUFL dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The reason may be that there are more unlabeled data in the Houston 2013, Houston 2018 and Trento datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' These unlabeled data play a critical role in contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, the proposed contrastive learning framework performs better when more unlabeled data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It is more convenient in practical applications in which large amounts of unlabeled data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Number of Training Samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' One of the advantages of self-supervised learning strategy is its excellent performance in handling small number of training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, we try to gradually reduce the number of samples during the training process, and the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' On the Houston 2013 dataset, when we use only 375 training samples (25 samples for each class), the OA value of the proposed method is 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='86 which is satisfying and encouraging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Furthermore, the model with pretraining consistently outperforms that without pretraining on four datasets when small training sets are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It is evident that the contrastive learning strategy of the proposed NNCNet is especially effective for small training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Moreover, we observe that the performance gain of pretraining on the Houston 2013 and 2018 datasets is better than that on the Trento and MUUFL datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' As mentioned before, there are more unlabeled data on the Houston 2013 and 2018 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, the proposed nearest neighbor- based strategy can exploit rich feature representations on both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Effectiveness of Data Augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The purpose of data augmentation is to enhance the differences between positive and negative samples as a way to facilitate the training of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In the proposed NNCNet, we use four data augmen- tation schemes, including RandomResizedCrop, RandomHor- izontalFlip, RandomVerticalFlip and RandomGaussianNoise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The corresponding results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We found that RandomResizedCrop is the key to data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Since the image patch is cropped into 11×11 pixels, if the scale is set too small, the semantic information would easily be damaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, in our implementations, the scale is set to (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='7, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Classification accuracy for different spatial distances, queue sizes, and mini-batch sizes on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Performance comparison of our model with or without 3D convo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Parameter Sensitivity Minimum Spatial Distance between Positive and Neg- ative Samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In order to prevent too much similarity be- tween positive and negative samples, we define a minimum distance s between them (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' the distances between positive and negative samples need to be greater than s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 13(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In our implementations, the size of each sample is 11 × 11 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The classification performance improved slightly when 4 ⩽ s ⩽ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' It is beneficial to use a large distance to increase the difference between positive and negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, in our implementations, s is set to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Size of the Negative Key Dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 13(b) shows the effect of negative key dictionary size on the classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The experiments show that a larger dictionary size will have a positive effect on pretraining, and it is consistent with our previous assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We believe that the proposed method works better when more unlabeled data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Key Encoder Update Speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We tested different key encoder update speeds r during pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The experimental results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 13(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We find that the best classifi- cation performance is achieved when r is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Effectiveness of 3D Convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Inspired by HybridSN [41], we first use PCA for channel dimensionality reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Then, 3D and 2D convolutions are combined for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' To verify the effectiveness of 3D convolution, we design a network in which the 3D convolutions are replaced with 2D convolutions (“w/o Conv2d” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The experimental results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' We found that 3D convolution can improve the classification performance to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Although PCA disturbs the spectral continuity of the hyperspectral data, we argue that 3D convolution can still generate more discriminative feature maps from the spectral dimensions than 2D convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' These discriminative features generated by 3D convolution can boost the classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' CONCLUSIONS AND FUTURE WORK In this paper, we propose a self-supervised NNCNet model to tackle the HSI and LiDAR joint classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Specifically, we integrate a nearest neighbor-based data aug- mentation scheme into the contrastive learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Se- mantic similarities among neighborhood regions are exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' The intermodal semantic alignments can be captured more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In addition, we proposed a bilinear attention fusion module that can capture second-order feature interactions be- tween HSI and LiDAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Therefore, the module improves the contextual representation of multisource data effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Extensive experiments on Houston 2013, Trento, MUUFL and Houston 2018 datasets have demonstrated the superiority of our model to a wide range of state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In the future, we aim to explicitly explore the semantic and spatial relations between HSI and LiDAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' In addition, we will explore how to further enhance the feature interactions between HSI and LiDAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Meng Wang received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' degree in com- puter science from Jinan University, Jinan, China, in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' He is currently pursuing the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' degree in computer science and applied remote sensing with the School of Information Science and Technology, Ocean University of China, Qingdao, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' His current research interests include computer vision and remote sensing image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 15 Feng Gao (Member, IEEE) received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Sc degree in software engineering from Chongqing University, Chongqing, China, in 2008, and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' degree in computer science and technology from Beihang University, Beijing, China, in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' He is currently an Associate Professor with the School of Information Science and Engineering, Ocean University of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' His research interests in- clude remote sensing image analysis, pattern recog- nition and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Junyu Dong (Member, IEEE) received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' degrees from the Department of Applied Mathematics, Ocean University of China, Qingdao, China, in 1993 and 1999, respectively, and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' degree in image processing from the Department of Computer Science, Heriot-Watt University, Ed- inburgh, United Kingdom, in 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' He is currently a Professor and Dean with the School of Computer Science and Technology, Ocean University of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' His research interests include visual information analysis and understanding, ma- chine learning and underwater image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Heng-Chao Li (Senior Member, IEEE) received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' degrees from Southwest Jiaotong University, Chengdu, China, in 2001 and 2004, re- spectively, and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' degree from the Graduate University of Chinese Academy of Sciences, Bei- jing, China, in 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' He is currently a Full Professor with the School of Information Science and Technology, Southwest Jiaotong University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' His research interests include statistical analysis of synthetic aperture radar (SAR) images, remote sensing image processing, and pat- tern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Li is an Editorial Board Member of the Journal of Southwest Jiaotong University and Journal of Radars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' He is an Associate Editor of the IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATION AND REMOTE SENSING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Qian Du (Fellow, IEEE) received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' degree in electrical engineering from the University of Maryland at Baltimore, Baltimore, MD, USA, in 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' She is currently the Bobby Shackouls Professor with the Department of Electrical and Computer Engineering, Mississippi State University, Starkville, MS, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Her research interests include hyperspec- tral remote sensing image analysis and applications, and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Du was the recipient of the 2010 Best Re- viewer Award from the IEEE Geoscience and Remote Sensing Society (GRSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' She was a Co-Chair for the Data Fusion Technical Committee of the IEEE GRSS from 2009 to 2013, the Chair for the Remote Sensing and Mapping Technical Committee of International Association for Pattern Recognition from 2010 to 2014, and the General Chair for the Fourth IEEE GRSS Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing held at Shanghai, China, in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' She was an Associate Editor for the PATTERN RECOGNITION, and IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' From 2016 to 2020, she was the Editor-in-Chief of the IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATION AND REMOTE SENSING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' She is currently a member of the IEEE Periodicals Review and Advisory Committee and SPIE Publications Committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' She is a Fellow of SPIE-International Society for Optics and Photonics (SPIE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Ma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Filippi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE1T4oBgHgl3EQfpwVv/content/2301.03335v1.pdf'} +page_content=' Wang, Z.' metadata={'source': 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JupiterTidesFinal +©ESO 2023 +January 9, 2023 +Dynamical tides in Jupiter and the role of interior structure +Yufeng Lin +Department of Earth and Space Sciences, Southern University of Science and Technology, Shenzhen 518055, China +e-mail: linyf@sustech.edu.cn +January 9, 2023 +ABSTRACT +Context. The Juno spacecraft has obtained highly accurate tidal Love numbers, which provide important constraints on the tidal +response and interior structure of Jupiter. +Aims. In order to exploit these observations, it is necessary to develop an approach for accurately calculating the tidal response of +Jupiter for a given interior model and to investigate the role of interior structure. +Methods. We directly solve the linearized tidal equations of a compressible, self-gravitating, rotating and viscous fluid body using a +pseudo-spectral method. The Coriolis force is fully taken into account but the centrifugal effect is neglected. We can simultaneously +obtain the real and imaginary parts of the tidal Love numbers for a given planetary interior model. +Results. We calculate the tidal responses for three simple interior models of Jupiter which may contain a compact rigid core or an +extended dilute core. All of models we consider can explain the fractional correction ∆k22 ≈ −4% due to dynamical tides, but all have +difficulties to reconcile the observed ∆k42 ≈ −11% for the high-degree tidal Love number. We show that the Coriolis force significantly +modifies gravity modes in an extended dilute core at the tidal frequency relevant to the Galilean satellites. We demonstrate that a thin +stable layer in the outer region, if exists, would also influence the tidal responses of Jupiter. +Key words. giant planets – tides – internal structure +1. Introduction +Tidal interactions between Jupiter and the Galilean satellites play +an important role in the orbital evolution of the system and the +internal dynamics of the moons (Lainey et al. 2009). The highly +active volcanic eruptions on Io are believed to be due to strong +tides raised by Jupiter (Peale et al. 1979). Meanwhile, tides are +also raised in Jupiter by its moons, probably dominated by Io +(Gavrilov & Zharkov 1977). The tidal response of a gaseous +body such as Jupiter is conventionally treated as a hydrostatic +deformation, which acquires a small phase lag with respect to +the tidal forcing due to dissipative processes. This is known as +the equilibrium tide. However, the equilibrium tide alone does +not suffice to account for the observed strong tidal dissipation in +Jupiter (Lainey et al. 2009) and the gravitational perturbations +recently measured by the Juno spacecraft (Durante et al. 2020). +In fact, the equilibrium tide does not satisfy the momentum +equation of tidal flows and thus corrections have to be made to +fully account for the tidal response of Jupiter. The corrections to +the equilibrium tide are collectively referred to as the dynamical +tide, which usually involves wave-like motions in the planet and +depends on the tidal frequency and the interior structure (Ogilvie +2014). The dynamical tide may provide extra channels of tidal +dissipation and produce additional gravitational perturbations on +top of the hydrostatic deformation. The Juno spacecraft has ob- +tained highly accurate tidal Love numbers klm (Durante et al. +2020), which quantitatively characterize the tidal response of +Jupiter to a tidal forcing component represented in spherical har- +monics of degree l and order m. The observed tidal Love num- +bers by Juno exhibit non-negligible discrepancies with respect to +the theoretically calculated hydrostatic values (Wahl et al. 2020), +suggesting that the dynamical tide has to be considered to ex- +plain the observed tidal response. Specifically, Juno observations +found ∆k22 ≈ −4% for the dominant tidal component l = 2 and +m = 2 and ∆k42 ≈ −11% for the high-degree tidal component +l = 4 and m = 2, where ∆klm = (klm − k(hs) +lm )/k(hs) +lm represents the +fractional correction to the hydrostatic value k(hs) +lm +(Wahl et al. +2020; Idini & Stevenson 2021, 2022a). +As the dynamical tides is sensitive to the tidal frequency and +the interior structure, the detected gravitational signatures of dy- +namical tides may provide important constraints on the Jupiter’s +interior (Idini & Stevenson 2021; Lai 2021; Idini & Stevenson +2021, 2022b; Dewberry & Lai 2022). Recent studies (Idini & +Stevenson 2021; Lai 2021) have revealed that the discrepancy in +k22 can be mainly attributed to the Coriolis effect on the funda- +mental modes, i.e. f−modes. More recently, Idini & Stevenson +(2022b) proposed that the resonant locking with a gravity mode +in an extended dilute core can explain ∆k42 ≈ −11%. This pro- +vides an independent constraint on the existence of a dilute core +in Jupiter, which has also been suggested by the Juno measure- +ments of gravitational moments of Jupiter (Wahl et al. 2017; Mil- +itzer et al. 2022). However, the tidal constraint on the existence +of a dilute core remains some uncertainties. The calculation of +tidal response in Idini & Stevenson (2022b) inadequately treated +the rotational (Coriolis) effect, which plays an important role in +Jupiter’s tidal responses because the tidal frequencies of Galilean +satellites are comparable to the spin frequency of Jupiter. Includ- +ing the Coriolis force introduces inertial waves in the neutrally +buoyant regions (Ogilvie & Lin 2004; Wu 2005a) and mixed +gravity waves and inertial waves, i.e. gravito-inertial waves, in +the stably stratified region (Dintrans et al. 1999; Xu & Lai 2017). +The mechanism proposed by Idini & Stevenson (2022b) is also +struggling to reconcile both the real part (relevant to the grav- +itational perturbation) and imaginary part (relevant to the tidal +dissipation) of the tidal Love numbers. +Article number, page 1 of 12 +arXiv:2301.02418v1 [astro-ph.EP] 6 Jan 2023 + +A&A proofs: manuscript no. JupiterTidesFinal +In this study, we develop a method to directly calculate the +tidal response of a fully compressible, self-gravitating, rotating +and viscous fluid body. The Coriolis force is fully taken into +account but the centrifugal force is neglected, which allows us +to numerically solve the problem in spherical geometry using a +pseudo-spectral method based on spherical harmonic expansions +(Ogilvie & Lin 2004; Lin & Ogilvie 2017). As we directly solve +the tidally forced problem with explicit viscosity, we can simul- +taneously obtain the real and imaginary parts of the tidal Love +number for a given planetary interior model. Our approach is +different from recent studies on dynamical tides of Jupiter (Lai +2021; Idini & Stevenson 2022b; Dewberry & Lai 2022). They +obtain the eigen modes of the inviscid fluid body first and then +calculate the tidal Love number (only the real part) through pro- +jecting the tidal force onto each eigen modes. We consider three +nominal interior models of Jupiter to investigate the dependence +of the tidal response on the tidal frequency and the interior struc- +ture. We focus on the effect of a compact rigid core, an extended +dilute core and a thin stably stratified layer in the outer region +on tidal responses. All of simplified models can explain the ob- +served ∆k22 ≈ −4% as previous studies have shown. However, +these simplified models are difficult to account for the observed +∆k42 ≈ −11%. Resonances with gravito-inertial modes in an ex- +tended dilute core near the tidal frequency of Io can produce +non-negligible dynamical correction to k42, but it is insufficient +to explain the Juno observation based on our simplified model. +2. Tidal model +We consider linear tidal responses of a rotating gaseous planet +to a tidal potential component of Ψm +l += A(r/R)lYm +l (θ, φ)e−iωt, +where A is the tidal amplitude, R is the radius of the planet, +Ym +l (θ, φ) represents spherical harmonics and ω is the tidal fre- +quency. The resulting tides of the planet produce an external +gravitational potential perturbation Φ′ = B(R/r)l+1Ym +l (θ, φ)e−iωt +(and probably other spherical harmonic components). The ratio +Km +l (ω) = B/A defines the tidal Love number, which depends +on the tidal frequency. The tidal Love number Km +l is a complex +number because there exists a phase lag between the forcing +and the gravitational perturbations due to dissipative processes +(Ogilvie 2014). While the real part klm = Re[Km +l ] measures the +in-phase gravitational perturbations with the tidal forcing, the +imaginary part Im[Km +l ] quantifies the out of phase tidal response +and is related to the dissipation rate. The ratio between the real +and imaginary parts is related to the tidal quality factor +Q = sgn(ω) +klm +Im[Km +l ], +(1) +where sgn(ω) = ±1 is the sign function. Because the phase lag is +generally very small, i.e. Q ≫ 1, the magnitude of the imaginary +part is typically much smaller than the real part. In this study, we +develop an approach to directly and simultaneously calculate the +real and imaginary parts of the tidal Love number for a given +planetary model. +2.1. Linearized equations +For a compressible, self-gravitating and rotating fluid body +which may contain a rigid core of radius Ri, linear perturbations +to a tidal potential Ψ ∝ e−iωt in the rotating frame are described +by the following equations (e.g. Ogilvie & Lin 2004): +−iωu′ = −2Ω × u′ − 1 +ρ0 +∇P′ + ρ′ +ρ2 +0 +∇P0 − ∇Φ′ − ∇Ψ + fν, +(2) +−iωρ′ + ∇ · (ρ0u′) = 0, +(3) +−iω +� P′ +ΓP0 +− ρ′ +ρ0 +� ++ u′ · +�1 +Γ∇ ln P0 − ∇ln ρ0 +� += 0 +(4) +∇2Φ′ = 4πGρ′, +(5) +where u is the velocity, Ω the rotation rate, ρ the density, P the +pressure, Γ the adiabatic index and G the gravitational constant. +In the above equations, the subscript 0 denotes physical quan- +tities in hydrostatic state (without tidal potential) and the nota- +tions with the prime represent Eulerian perturbations induce by +the tidal forcing. In the momentum equation (2), we explicitly +include a viscous force fν defined as +fν = 1 +ρ0 +∇ · (2µS), +(6) +where µ is the dynamic shear viscosity (we neglect the bulk vis- +cosity) and S is the strain-rate tensor: +S = 1 +2 +� +∇u′ + (∇u′)T� +− 1 +3(∇ · u′)I. +(7) +Note that we include the viscous force in the momentum equa- +tion but neglect the viscous heating in the energy equation, i.e. +the density and pressure perturbations are treated as adiabatic. +In this study, we take fully into account the Coriolis force +due to the rotation but neglect the centrifugal distortion for nu- +merical convenience. The centrifugal effect can be measured by +ϵ = Ω/ωdyn, i.e. the ratio between the spin frequency Ω and +the dynamical frequency ωdyn = (GM/R3)1/2, which is not par- +ticularly small for Jupiter (ϵ = 0.288). Indeed, the centrifugal +distortion of Jupiter has non-negligible contributions to the total +Love number klm, especially for the high-degree Love number +k42 because the tidal response at l = m = 2 can produce a gravi- +tational perturbation at l = 4 and m = 2 in an oblate figure (Idini +& Stevenson 2022a). For the hydrostatic k(hs) +42 +of Jupiter due to +Io, 93% of the total value is actually contributed by the centrifu- +gal coupling with k22 and only the remaining 7% is produced by +the tidal forcing at l = 4 and m = 2 (Wahl et al. 2020; Idini +& Stevenson 2022a). In this paper we do not aim to directly +fit the klm observed by Juno, but rather focus on the fractional +corrections ∆klm by the dynamical tides. In terms of the frac- +tional correction ∆klm, the centrifugal contribution to ∆k22 can +be neglected in leading order (Lai 2021). However, the centrifu- +gal contribution to ∆k42 can not be neglected even in leading +order because the k(hs) +42 +is mostly contributed by the centrifugal +coupling with k22. This complicates the comparison between the +calculated ∆k42 in a spherical figure and the observation. Nev- +ertheless, the calculated ∆k42 in a spherical figure can be multi- +plied by the factor 0.07 to account Jupiter’s centrifugal coupling +effect for qualitative comparisons with the observation (Idini & +Stevenson 2022b). Such a comparison would assume that the +tidally-excited internal modes are not significantly modified by +the centrifugal deformation. +By neglecting the centrifugal deformation, the unperturbed +basic state is spherically symmetric, i.e. depends on the radius r +only. Given the density ρ0(r) and pressure P0(r) profiles of the +unperturbed state, the radial gravitational acceleration (inward) +Article number, page 2 of 12 + +Lin: Dynamical tides in Jupiter and the role of interior structure +g(r) and the Brunt-Väisälä frequency N(r) are then determined +by +g = dΦ0 +dr = − 1 +ρ0 +dP0 +dr , +(8) +N2 = g +�1 +Γ +d ln P0 +dr +− d ln ρ0 +dr +� +. +(9) +2.2. Numerical method +In order to obtain the complex Love numbers, we numerically +solve Eqs. (2-5) using a pseudo-spectral method for the pre- +scribed basic states, subject to the relevant boundary conditions. +The numerical scheme is based on the method used in previous +studies (Ogilvie & Lin 2004; Lin & Ogilvie 2017), but we ex- +tend the method to solve the full set of linearized equations (2-5) +without making a low-frequency approximation (Ogilvie 2013). +By introducing h′ = P′/ρ0 and eliminating the density perturba- +tion ρ′, Eqs. (2-5) can be reduced to the following equations: +−iωρ0u′ += +−2ρ0Ω × u′ − ∇(ρ0h′) + g∇2ϕ′/(4πG) +−ρ0∇ϕ′ − ∇Ψ + ∇ · (2µS), +(10) +−iωh′ = −c2 +s(N2u′ +r/g + ∇ · (ρ0u′)/ρ0), +(11) +−iω∇2ϕ′ = −4πG∇ · (ρ0u′), +(12) +where u′ +r is the radial velocity perturbation and c2 +s = ΓP0/ρ0 is +the square of the adiabatic sound speed. +We impose boundary conditions including the regularity of +the gravitational perturbations, zero radial velocity on the rigid +inner boundary and vanishing Lagrange pressure perturbation at +the surface, i.e. δP = P′ + u′ +r/(−iω)∇P0 = 0. In terms of h′ and +u′ +r, the last boundary condition can be written as (Dewberry et al. +2021) +�−iωh′ − gu′ +r +� |r=R = 0. +(13) +As the viscous force is included, additional boundary conditions +are required to complete the boundary value problem. We use the +so-called stress-free conditions, i.e. the tangential stresses van- +ish, at both boundaries. +For a given tidal potential Ψm +l += A(r/R)lYm +l (θ, φ)e−iωt, the +tidal perturbations (including both equilibrium and dynamical +tides) u′, h′ and Φ′ can be expanded as +u′ = +L +� +n=m +um +n (r)Rm +n + +L +� +n=m +vm +n (r)Sm +n + +L +� +n=m +wm +n (r)Tm +n , +(14) +h′ = +L +� +n=m +hm +n (r)Ym +n (θ, φ), +(15) +Φ′ = +L +� +n=m +Φm +n (r)Ym +n (θ, φ), +(16) +where Rm +n , Sm +n , Tm +n are vector spherical harmonics +Rm +n = Ym +n (θ, φ)ˆr, +Sm +n = r∇Ym +n (θ, φ), +Tm +n = r∇ × Rm +n . +(17) +As the basic state is axisymmetric, the perturbations involve +spherical harmonics with the same order m as the tidal potential +Ψm +l , but the Coriolis force would couple all spherical harmon- +ics with degree n ≥ m. For numerical calculations, we have to +make a truncation at certain degree L. Substituting expansions of +Eqs. (14-16) into Eqs. (10-12) and projecting onto spherical har- +monics, we end up with a set of ordinary differential equations +involving um +n (r), vm +n (r), wm +n (r), hm +n (r) and Φm +n (r) . For the radial de- +pendence, we use Chebyshev collocation on Nr Gauss–Lobatto +nodes (Rieutord et al. 2001). The boundary conditions are ap- +plied through replacing the ODEs with the corresponding bound- +ary conditions on the boundary nodes. The regularity of gravita- +tional perturbations requires +rdΦm +n +dr ++ (n + 1)Φm +n = 0 +at r = R, +(18) +rdΦm +n +dr +− nΦm +n = 0 +at r = Ri. +(19) +The vanishing Lagrangian pressure perturbation at the surface +and zero radial velocity at the rigid inner boundary give +−iωhm +n = gum +n +at r = R, +(20) +um +n = 0 +at r = Ri. +(21) +The stress-free boundary condition is given as (Ogilvie 2009) +um +n + rdvm +n +dr − vm +n = 0, +rdwm +n +dr − wm +n = 0, +(22) +at both boundaries. +Using the numerical discretization described above, the +boundary value problem becomes a linear system involving a +large complex block-tridiagonal matrix. The solution of the lin- +ear system is obtained using the standard direct solver. We use +typical truncations of L = 200 and Nr = 100 in this study. +Once the solution of the linear system is obtained numeri- +cally, the complex tidal Love number is readily given by +Km +l = Φm +l (r = R), +(23) +for the tidal potential component Ψm +l = A(r/R)lYm +l (θ, φ)e−iωt (we +simply set A = 1 for the linear tidal response). Note that the so- +lution includes both the equilibrium and dynamical tides. For the +real part of Love numbers, of particular interest is the fractional +correction of dynamical tides +∆klm = (klm − k(hs) +lm )/k(hs) +lm , +(24) +where k(hs) +lm +is the hydrostatic value and is calculated by setting +ω = 0. As our calculations neglect the centrifugal effect which +significantly influences the high-degree Love number k42, the +calculated value of ∆k42 should be multiplied by the factor 0.07 +when compared with the observation as we have discussed in +Sec. 2.1. +We can also calculate the tidal dissipation rate Dν from the +velocity perturbations +Dν = +� +V +2µS2dV, +(25) +Article number, page 3 of 12 + +A&A proofs: manuscript no. JupiterTidesFinal +where the integral is taken over the fluid domain. The dissipation +rate is related to the imaginary part of the tidal Love number +(Ogilvie 2014) +Dν = (2l + 1)RA2 +8πG +ωIm[Km +l ], +(26) +which can be served as an independent validation of the numer- +ical code. The above relation is satisfied to a high degree of ac- +curacy in all of numerical calculations presented in this paper. +2.3. Interior models +In order to solve Eqs. (10-12), we need to prescribe basic state +profiles ρ0(r), g(r) and N2(r) to model Jupiter’s interior. Our un- +derstanding of Jupiter’s interior has been significantly improved +by Juno observations (Stevenson 2020), yet it remains some de- +grees of uncertainties. In this study, we do not aim to build a +realistic model of Jupiter’s interior, but focus on the fractional +contributions of dynamical tides to the tidal Love number for dif- +ferent possible scenarios of Jupiter’s interior. We consider three +nominal interior models (Fig. 1) based on a polytrope of index 1, +which is a good leading order approximation for Jupiter (Steven- +son 2020). +For all of models used in this study, the unperturbed density +and gravity follow a hydrostatic polytrope of index 1, +ρ0 = πM +4R3 +sin kr +kr +, +(27) +g = GM +r2 [sin(kr) − kr cos(kr)] , +(28) +where k = π/R. The first model consists of a small rigid core +of radius 0.25R and an isentropic fluid envelope, i.e. Γ = 2 and +N2 = 0 in the fluid region (Fig. 1(a)). +The second model assumes an extended dilute core of radius +0.7R and an isentropic envelope (Fig. 1(b)).The dilute core is +treated as a stably stratified fluid layer with the Brunt-Väisälä +frequency given by +N2 +ω2 +dyn += ˜N2 sin +�πr +Rc +� +, +(29) +where ˜N2 = 0.25 and Rc = 0.7 for this model. As we fixed the +density and pressure profiles to that of a polytrope, the strati- +fication is effectively realized by adjusting the adiabatic index +(Γ > 2) in the dilute core (Lai 2021). This model is similar to +that used in Idini & Stevenson (2022b), but they adjust the den- +sity profile to model the stable stratification in the dilute core +while fix the adiabatic index Γ = 2. +The third model is based on the model in Fig. 1(a), but we +further add a stably stratified layer between 0.8R and 0.9R (Fig. +1(c)), possibly resulting from H-He immiscibility (Debras & +Chabrier 2019; Stevenson et al. 2022). The Brunt-Väisälä fre- +quency in the top stable layer is prescribed as +N2 +ω2 +dyn += ˜N2 +1 +[1 + e−100(r−0.8)][1 + e100(r−0.9)]. +(30) +The degree of stratification of this layer remains uncertain, but +it is estimated that typical values of N2/ω2 +dyn would be roughly +between 0.1 and 0.8 for Jupiter (Christensen et al. 2020; Gastine +& Wicht 2021). Here we set a moderate value ˜N2 = 0.5. +Note that an interior model with the co-existence of a dilute +core and a top stable layer is also possible (Debras & Chabrier +2019). As this kind of model involves two different stably strat- +ified layers, it would be difficult to characterize the role of the +top stable layer on tides. We consider only the combination of a +compact rigid core and a top stable layer for simplicity. +In all of these models, we set the total mass M, the radius R +and the spin rate Ω such that the ratio ϵ = Ω/ +� +GM/R3 = 0.288, +corresponding the value of Jupiter. Our calculations also require +the fluid viscosity, which is difficult to estimate in detail for giant +planets. We simply assume the dynamic viscosity µ is propor- +tional to the background density ρ0, so the kinematic viscosity +ν = µ/ρ0 is constant. The viscosity can be measured by the di- +mensionless number Ek = ν/(ΩR2), known as the Ekman num- +ber. We set Ek = 10−6 for most of calculations (unless otherwise +specified), roughly corresponding to the effective viscosity based +on mixing-length theory (Guillot et al. 2004). +As we have mentioned that we do not aim to construct a re- +alistic interior model for Jupiter in this study. These simplified +models are designed to investigate the effects of a compact rigid +core, an extended dilute core and a top stable layer on the tidal re- +sponses of Jupiter respectively. Nevertheless, the fractional cor- +rections ∆klm and the tidal quality factor Q for these simplified +models can be used to make some qualitative comparisons with +the observations (Lai 2021; Idini & Stevenson 2021, 2022b). +3. Results +In this paper, we focus on the dominant tidal component Ψ2 +2 and +a high-degree tesseral component Ψ2 +4, for which non-negligible +dynamical corrections have been detected as we have discussed +in Sec. 1. Our calculations are limited to the frequency range of +−2 ≤ ω/Ω ≤ −1, relevant to the tidal frequencies of the Galilean +moons. Note that the negative tidal frequency means that the +tidal forcing is retrograde in the co-rotating frame with the planet +based on our convention. For the real part of Love numbers, we +show the fractional correction ∆klm. In order to make compar- +isons with the Juno observation, the calculated ∆k42 is multi- +plied by 0.07 to compensate the centrifugal effect which is ne- +glected in our calculations. Because of the negative tidal fre- +quency, the imaginary part of Love numbers is also negative +in our calculations and is related to the tidal quality factor by +klm/Ql = −Im[Km +l ] according to Eq. (1). +3.1. Full polytrope model +Before presenting results for the interior models in Fig. 1, we +first show the tidal response of a full isentropic polytrope, i.e. +neutrally buoyant in the whole fluid sphere. This model serves +as a reference for other models and has been used to investigate +the dynamical tides of Jupiter in recent analytical studies (Idini +& Stevenson 2021; Lai 2021). Fig. 2 shows both the real and +imaginary parts of the Love numbers as a function of the tidal +frequency for the full polytrope model. We can see that ∆k22 +is negative in the frequency range we considered and smoothly +varies as the tidal frequency except a burst around ω/Ω = −1.08, +which corresponds to a resonance with an inertial mode. Away +from resonances, our numerical results are consistent with recent +theoretical calculations and produce ∆k22 ≈ −4% at the tidal fre- +quency of Io (Lai 2021; Idini & Stevenson 2021). These studies +also revealed that the dynamical correction ∆k22 can be attributed +to the Coriolis effect on the f-modes. Apart from the f-modes, +the rotating sphere of isentropic fluid also supports smooth iner- +Article number, page 4 of 12 + +Lin: Dynamical tides in Jupiter and the role of interior structure +(a) +(b) +(c) +Fig. 1. Three nominal models of Jupiter’s interior used in this study. The top panel shows the schematic models and the bottom panel shows the +density (normalized by the density at the center) and the Brunt-Väisälä frequency (normalized by the dynamical frequency) as a function of the +radius. The blue shadow in the bottom panel indicates solid regions. (a) A compact rigid core model; (b) an extended dilute core model; (c) a +compact rigid core and an outer stable layer model. +tial modes restored by the Coriolis force in the frequency range +of 0 < |ω/Ω| < 2 (Greenspan 1968; Lockitch & Friedman 1999). +The burst of ∆k22 at ω/Ω = −1.08 indeed is due to the resonant +excitation of the inertial mode as shown in Fig. 3(a), but we no- +tice that the resonance occurs only in a very narrow frequency +range. +However, this inertial mode has more significant contribu- +tions to ∆k42. The angular structure of an inertial mode cannot be +described by a single spherical harmonics in general (Lockitch +& Friedman 1999), but the density perturbations (and thus the +gravitational perturbations) are dominated by the spherical har- +monics Y2 +4(θ, φ) for the resonant inertial mode at ω/Ω = −1.0836 +as we can see from Fig. 3(a). This suggests a likely strong cou- +pling between the tidal potential component Ψ2 +4 and the inertial +mode in Fig. 3(a), i.e. large tidal overlap as described in Wu +(2005b), leading to significant dynamical corrections to k42. The +dynamical correction can reach ∆k42 ≈ −10% (after the centrifu- +gal correction) near the resonance at ω/Ω = −1.0836. However, +the tidal frequencies of the Galilean satellites are too far away +from this resonance. +The curve of ∆k42 also shows a spike around ω/Ω = −1.51, +corresponding to a narrow resonance with a high degree inertial +mode (ρ′ is dominated by Y2 +6(θ, φ) as shown in Fig. 3(b)). Inter- +estingly, the tidal frequency of Io is close to this resonance, but +the dynamical correction caused by this resonant mode is insuffi- +cient to account for the observed ∆k42 ≈ −11%. The frequencies +of inertial modes in Fig. 3 are slightly shifted comparing to that +calculated by Lockitch & Friedman (1999) for a polytrope of +index 1 (see their table 6 and note different conventions for the +sign of frequencies) because they assumed ϵ → 0 whereas we +set ϵ = 0.288. +The imaginary parts of the Love numbers in Fig. 2 show that +resonances with inertial modes significantly enhance the tidal +dissipation. The enhanced dissipation due to resonant inertial +modes in a neutrally buoyant sphere has been demonstrated by +Wu (2005b) but using different density profiles. When the tidal +frequency is away from resonances, the dissipation rate for the +full isentropic polytrope is too small to account for the observed +tidal quality factor Q (Lainey et al. 2009). +3.2. Compact rigid core model +We now consider tidal responses for the interior model with a +compact rigid core. Basically, the inner region (r ≤ 0.25R) of a +whole fluid polytrope becomes solid for this model. Fig. 4 shows +the frequency-dependence of the Love numbers for the compact +rigid core model. We can see that the real parts are largely sim- +ilar to that of a full polytrope, but the imaginary parts are rather +different from that of a full polytrope, showing enhanced tidal +dissipation by introducing the rigid core. The rigid core model +also supports inertial waves in the fluid envelope, but these waves +have some peculiar behaviors due to the singularity in a spher- +Article number, page 5 of 12 + +O1.0 +1.0 +1.0 +p/pc +0.8 +0.8 +0.8 +0.6 +0.6 +0.6 +0.4 +0.4 +0.4 +0.2 +0.2 +0.2 +0.0 +0.0 +0.0 +0.00 +0.25 +0.50 +0.75 +1.00 +0.25 +0.50 +0.75 +1.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.00 +r/R +r/R +r/RA&A proofs: manuscript no. JupiterTidesFinal +Fig. 2. Complex Love number as a function of the tidal frequency for a full isentropic polytrope of index 1. Top panel shows the fractional +correction ∆klm of the real part of the Love numbers. The fractional correction ∆k42 (orange curve in the top panel) is multiplied by 0.07. The +bottom panel shows the minus imaginary part −Im[Km +l ], which is equivalent to klm/Ql. Vertical dashed lines indicate tidal frequencies of four +Galilean Moons of Jupiter (from right to left: Io, Europa, Ganymede, Callisto). The horizontal dashed line in the bottom panel represents the +astrometric observation of the frequency independent k2/Q2 from Lainey et al. (2009). +(a) +(b) +Fig. 3. Density perturbations (left half) and radial velocity perturbations (right half) in the meridional plane to the tidal component Ψ2 +4 at two +resonant frequencies in Fig. 2. Amplitudes are normalized by the maximum absolute values. +ical shell (Stewartson & Rickard 1969). Smooth inertial modes +do not exist generally in a spherical shell even with uniform den- +Article number, page 6 of 12 + +10 +- +1 +K2 +I +- +I +- +5 +K? +- +- +I +I +1 +- +I +Nkim( +0 +1 +1 +- +5 +-10 +-1.4 +-1.0 +-2.0 +-1.8 +-1.6 +-1.2 +101 +- +- +10-1 +- +- +- +10 +3 +- +- +1 +- +- +I +1 +- +I +1 +1 +10-7 +-2.0 +-1.6 +-1.0 +-1.8 +-1.4 +-1.2 +Tidal frequency 0/Q0.5 +0.5 +0 +0 +-0.5 +-0.5 +-1 +1 +/S2 =-1.0836 +Wp +0.5 +0.5 +0 +0 +-0.5 +-0.5 +-1 +1 +3 +/S2 =-1.5117Lin: Dynamical tides in Jupiter and the role of interior structure +Fig. 4. As for Fig. 2 but for the interior model with a compact rigid core. The fractional correction ∆k42 (orange curve in the top panel) is multiplied +by 0.07. +(a) +(b) +Fig. 5. Density perturbations (left half) and gravitational perturbations (right half) in the meridional plane to the tidal component Ψ2 +4 for the +interior model with a compact rigid core at (a) ω/Ω = −1.5092 (resonance) and (b) ω/Ω = 1.53 (non-resonance) with Ek = 10−7. Amplitudes are +normalized by the maximum absolute values. +sity (Rieutord et al. 2001), and localized wave beams spawned +from the critical latitudes propagate in the bulk along the charac- +teristics of the inertial wave equations (e.g. Ogilvie 2009). How- +ever, Lin & Ogilvie (2021) recently revealed that resonant tidal +responses in a spherical shell correspond to eigen modes with +large scale flows hidden beneath localized wave beams using a +Article number, page 7 of 12 + +10 +- +K2 +5 +K? +- +- +0 +- +- +- +-10 +-2.0 +-1.6 +-1.8 +-1.4 +-1.2 +-1.0 +100 +- +- +10-4 +-2.0 +-1.8 +-1.6 +-1.2 +-1.0 +-1.4 +Tidal frequency 0/Qp +0.5 +0.5 +0 +0 +-0.5 +-0.5 +-1 +1 +/S2 =-1.50920.5 +0.5 +0 +0 +-0.5 +-0.5 +-1 +1 +/S2 =-1.5300A&A proofs: manuscript no. JupiterTidesFinal +uniform density model. Furthermore, it was shown that the hid- +den large scale structures basically resemble inertial modes in a +full sphere. This is in line with our results for the non-uniform +density model in this study. The real parts k22 and k42 are relevant +to only large scale density perturbations, which are similar to in- +ertial modes in a full sphere as one can see from Fig. 5. There- +fore, the curves of ∆klm for the rigid core model resemble that of +a full polytrope, but note slight shifts of the resonant frequencies +due to the presence of rigid core. As for the full polytrope, the +compact rigid core model can produce ∆k22 = −4% as observed, +but it cannot produce sufficient dynamical correction in the high- +degree Love number k42 near the tidal frequency of Io to account +for the observed ∆k42 = −11%. +On the other hand, the imaginary parts are largely modified +by the presence of small rigid core. We can see that the tidal dis- +sipation is significantly enhanced by the localized wave beams +spawned from the critical latitudes both in and out of resonances. +The velocity perturbations in Fig. 5(b) indeed exhibit localized +waves propagating in the bulk, which can generate significant +viscous dissipation but do not produce much density and gravi- +tational perturbations. In Fig. 4, we also see that several peaks in +the tidal dissipation (bottom panel) do not lead to obvious fluc- +tuations in ∆klm (top panel), corresponding to resonances with +higher degree modes that have little contributions to the low de- +gree ( i.e. l = 2 and l = 4) gravitational perturbations. +In summary for the compact rigid core model, the tidal dis- +sipation is significantly enhanced with respect to the full poly- +trope case. This is in line with the early work of Ogilvie & Lin +(2004), who have shown the enhanced tidal dissipation due to +inertial waves in the convective envelope of rotating stars and +planets. The averaged dissipation in the tidal frequency range of +Galilean moons gives rise to comparable tidal quality factor as +observed (Lainey et al. 2009). However, the fractional correction +to the real part of Love number ∆k42 is insufficient to explain the +observation. +3.3. Dilute core model +An extended dilute core rather than a compact core in Jupiter has +been suggested recently based on Juno gravitational measure- +ments (Wahl et al. 2017; Militzer et al. 2022). In this subsection, +we consider tidal responses for the interior model with an ex- +tended dilute core as shown in Fig. 1(b). The dilute core is treated +as a stably stratified layer which supports gravity waves restored +by the buoyancy. If the Coriolis force is fully taken into account, +dynamical tides in the dilute core region would be in the the form +of mixed gravity waves and inertial waves, i.e. gravito-inertial +waves (Dintrans et al. 1999; Xu & Lai 2017). Idini & Stevenson +(2022b) recently calculated the tidal response of Jupiter with an +extended dilute core, but they did not fully consider the Coriolis +effect, which turns out to important as we will show. +Fig. 6 shows the frequency-dependence of the Love num- +bers for the dilute core model. For the tidal component Ψ2 +2 (bule +curves), the dynamical correction ∆k22 is generally similar to +that of the full polytrope except the absence of obvious spikes for +the dilute core model. However, the imaginary part exhibits sev- +eral peaks and troughs, suggesting possible resonances with high +degree mixed modes that enhance the tidal dissipation but do not +significantly contribute to the l = 2 gravitational perturbations. +The overall tidal dissipation is also enhanced with respect to the +full polytrope due to the excitation of gravito-inertial waves in +the dilute core and inertial waves in the convective envelope. The +frequency-averaged tidal dissipation tends to be compatible with +the observed tidal quality factor as we can see from Fig. 6. +For the tidal component Ψ2 +4, Fig. 6 also shows results with- +out including the Coriolis force (green curves) for compari- +son. We note that the fractional correction ∆k42 is always pos- +itive when the Coriolis force is neglected, probably because the +pure gravity modes enhance the in-phase gravitational pertur- +bations and thus produce positive dynamical corrections. Nev- +ertheless, we observe distinct resonant responses at certain tidal +frequencies from both real and imaginary parts of the Love num- +ber for the non-Coriolis case. For instance, the resonance at +ω/Ω = −1.5193, which is close to the tidal frequency of Io, +corresponds to the first gravity mode of l = 4 and m = 2 as +shown in Fig. 7 (a). Indeed, Idini & Stevenson (2022b) proposed +the resonant locking between this gravity mode 1 (referred to as +2 +4g1) and the Jupiter-Io orbital evolution to explain the observed +∆k42 for Jupiter. In Idini & Stevenson (2022b), the Coriolis force +is neglected for the calculation of gravity modes, but approxi- +mated rotational corrections are made to obtain the Love num- +ber. However, taking fully into account the the Coriolis force +significantly alter the tidal responses as we can see from Fig. +6. The dynamical correction ∆k42 exhibits several large fluctu- +ations especially in the frequency range of −1.5 < ω/Ω < −1. +This is due to the mixing of gravity modes and inertial modes +in the dilute core, leading to more chances for resonances. The +most significant dynamical corrections are produced near the +tidal frequency ω/Ω = −1.2, which is close to the frequency +of the purely inertial mode as shown in Fig. 3(a). Of course, +the inertial mode is mixed with gravity modes in the dilute core +for this model. The resonance around ω/Ω = −1.2 can pro- +duce more than −10% dynamical corrections in k42 (after the +centrifugal correction), but it is too far away from the tidal fre- +quency of Io. The resonance close to the tidal frequency of Io +(also close to the frequency of pure gravity mode 2 +4g1) occurs at +ω/Ω = −1.4448 when the Coriolis force is considered. Fig. 7 (b) +shows the spatial structure of this resonant response. The Cori- +olis effect not only leads to a non-negligible shift in the mode +frequency, but also largely modifies the mode structure. The per- +turbations are in the from of gravito-inertial waves in the dilute +core and become pure inertial waves in the neutrally buoyant +envelope. Non-negligible dynamical corrections are induced by +this resonance at ω/Ω = −1.4448, but the corrections are in- +sufficient (after the centrifugal correction) to account for the ob- +served ∆k42 = −11%. As the resonance is very narrow, we use +200 equally spaced frequency points in the tidal frequency in- +terval of [-1.45, -1.43]. The peak amplitude of ∆k42 in this fre- +quency interval is comparable to that of using only 20 frequency +points, suggesting that the frequency sampling points are suffi- +cient to capture the resonant peak. +Comparing the orange and green curves in the bottom panel +of Fig. 6, we can see that the tidal dissipation is increased by +about two orders of magnitude when the Coriolis force is in- +cluded. This suggests that the excitation of pure gravity waves +is a less efficient tidal dissipation mechanism (unless resonances +take place) based on our linear calculations, though the nonlinear +interaction or wave breaking of gravity waves may lead to effi- +cient tidal dissipation (e.g. Barker 2011; Weinberg et al. 2012). +3.4. Outer stable layer model +We finally consider the effect of an outer stable layer, which may +exist in Jupiter resulting from H-He immiscibility (Debras & +Chabrier 2019). Fig. 8 shows the Love numbers as a function +1 They used slightly different background density ρ0(r) and Brunt- +Väisälä frequency N(r), so the mode frequency is slightly shifted. +Article number, page 8 of 12 + +Lin: Dynamical tides in Jupiter and the role of interior structure +Fig. 6. As for Fig. 2 but for the interior model with an extended dilute core. Green lines represent results without including the Coriolis force. The +fractional correction ∆k42 (orange and green curves in the top panel) is multiplied by 0.07. +(a) +(b) +Fig. 7. Density perturbations (left half) and radial velocity perturbations (right half) in the meridional plane to the tidal component Ψ2 +4 for the +interior model with an extended dilute core. (a) Without including the Coriolis force at ω/Ω = −1.5193 (resonance); (b) including the Coriolis +force at ω/Ω = −1.4448 (resonance). Amplitudes are normalized by the maximum absolute values. +of the tidal frequency for the interior model (c) in Fig. 1, which +includes a compact rigid core and a top stable layer between 0.8R +Article number, page 9 of 12 + +p +0.5 +0.5 +0 +0 +-0.5 +-0.5 +1 +/S2 =-1.51930.4 +0.4 +0.2 +0.2 +0 +0 +-0.2 +-0.2 +-0.4 +-0.4 +-0.6 +-0.6 +3 +/S2 =-1.4448K +10 +- +- +K(Non-Coriolis) +5 +% +- +Aklm( +0 +- +- +- +1 +-10 +-2.0 +-1.8 +-1.6 +-1.4 +-1.2 +-1.0 +10-1 +Q10-3 +10-5 +- +10-7 +-2.0 +-1.8 +-1.6 +-1.4 +-1.2 +-1.0 +Tidal frequency /QA&A proofs: manuscript no. JupiterTidesFinal +Fig. 8. As for Fig. 2 but for the interior model with a small rigid core and a top stably stratified layer. Green lines represent results at the Ekman +number Ek = 10−7. The fractional correction ∆k42 (orange and green curves in the top panel) is multiplied by 0.07. +(a) +(b) +Fig. 9. Perturbations in the meridional plane to the tidal component Ψ2 +4 at ω/Ω = −1.1650 for the interior model (c) in Fig. 1. (a) Density (left +half) and radial velocity (right half) perturbations; (b) gravitational (left half) and vorticity (right half) perturbations. Amplitudes are normalized +by the maximum absolute values. The dashed lines denote r = 0.8R. +and 0.9R. For the tidal responses to Ψ2 +2, the dynamical correction +∆k22 is similar to the case without the stable layer, but the pres- +ence of the thin stable layer eliminates the spike due to the res- +onant inertial mode at the tidal frequency around ω/Ω = −1.08. +The overall tidal dissipation due to Ψ2 +2 is comparable to the coun- +terpart without the top stable layer (blue curve in the bottom +panel of Fig. 4), but the fluctuation amplitudes, i.e. the differ- +ences between peaks and troughs, are smaller. +Article number, page 10 of 12 + +10 +Ek = 10-6 +5 +Ek = 10-7 +- +% +△klm( +0 +- +1 +-10 +-1.8 +-1.6 +-1.2 +-1.0 +2.0 +-1.4 +100 +10 +Q +klml +10° +-2.0 +-1.8 +-1.6 +-1.4 +-1.2 +-1.0p +0.5 +0.5 +0 +0 +-0.5 +-0.5 +1 +S2 =-1.1650V +0.4 +0.5 +0.3 +0 +0.2 +-0.5 +0.1 +-1 +0 +w/2 =-1.1650Lin: Dynamical tides in Jupiter and the role of interior structure +For the tidal responses to Ψ2 +4, we also show results for Ek = +10−7 (green curves) to illustrate the effect of fluid viscosity in +Fig. 8. One can see that the viscosity has little influence on the +real part of the Love number. The tidal dissipation weakly de- +pends on viscosity at peaks and troughs, but the overall dissipa- +tion tends to be insensitive to viscosity. Indeed, Ogilvie (2013) +has shown the frequency-averaged dissipation is independent of +viscosity. +The dynamical correction ∆k42 is also similar to the case +without the stable layer. We can see large variations of ∆k42 at +the tidal frequency around ω/Ω = −1.165, which corresponds to +a resonant mode as shown in Fig. 9. This mode is complicated +because it involves three different layers for the interior model +considered here. The fluid body is primarily neutrally buoyant +and supports inertial waves. However, the fluid domain is sepa- +rated by the thin stable layer, which suppresses radial fluid mo- +tions and creates a "barrier" for the communication between in- +ertial waves in the inner and outer regions (see the radial velocity +and vorticity perturbations in Fig. 9). In addition, the thin sta- +ble layer supports rotationally modified gravity waves. The den- +sity perturbations are mainly restricted in the stable layer and +the outer envelope, i.e. in the region of r > 0.8R. Despite the +complicated velocity and density perturbations, the gravitational +perturbations are dominated by the l = 4 component with rela- +tively simple radial dependence. In this regard, this complicated +mode is relevant to the l = 4 inertial mode without the stable +layer, leading to large dynamical corrections around the tidal +frequency at ω/Ω ≈ −1.1 as in Fig. 4. However, the dynami- +cal correction ∆k42 is negligible after the centrifugal correction +at the tidal frequency of Io. +4. Conclusions +We have developed a numerical method for calculating the tidal +responses of a compressible, self-gravitating, rotating and vis- +cous fluid body. We take fully into account the Coriolis force but +neglect the centrifugal distortion, which allows us to solve the +problem in the spherical geometry. We use the pseudo-spectral +method based on spherical harmonics in the angular directions +and Chebyshev collocation in the radial direction. Different from +recent studies on Jupiter’s dynamical tides (Lai 2021; Idini & +Stevenson 2022b; Dewberry & Lai 2022), we directly solve the +tidally forced problem and explicitly add the fluid viscosity, +which allows us to simultaneously obtain the real and imaginary +parts of the tidal Love numbers for a given planetary interior +model. +In this study, we considered three simplified interior models +(Fig. 1) of Jupiter based on a polytrope of index 1. We focus +on the tidal components Ψ2 +2 and Ψ2 +4 in the frequency range of +−2 ≤ ω/Ω ≤ −1, which is relevant to the tidal frequencies of +Galilean moons. Our numerical results show that the dynami- +cal correction ∆k22 is generally insensitive to the interior mod- +els. All of models we considered can give rise to the observed +∆k22 ≈ −4% at the tidal frequency of Io, which is also in line +with previous studies (Idini & Stevenson 2021; Lai 2021). The +tidal dissipation is significantly enhanced by the presence of a +compact rigid core model or an extended dilute core with re- +spect to the full polytrope, leading to comparable tidal quality +factor Q as observed (Lainey et al. 2009). +For the tidal responses to the Ψ2 +4 component, all of models +we considered are difficult to give rise to ∆k42 ≈ −11% near +the tidal frequency of Io. For the interior model with a com- +pact rigid core, significant dynamical corrections are generated +at the tidal frequency around ω/Ω ≈ −1.1 due to the resonance +with an inertial mode whose gravitational perturbations are dom- +inated by the spherical harmonics of l = 4 and m = 2. How- +ever, this resonance is too far away from the tidal frequencies +of Galilean moons. For the interior model with an extended di- +lute core, we demonstrate that the gravity modes in the dilute +core can be significantly modified by the Coriolis force, leading +to the mixed gravito-inertial modes. Resonances with gravito- +inertial modes in the dilute core can produce non-negligible dy- +namical corrections, but they are insufficient to explain the ob- +served ∆k42 ≈ −11% near the tidal frequency of Io based on +our simplified model. We also briefly investigated the effect of +a top stable layer on Jupiter’s tides. The thin stable layer acts +as a "barrier" and tends to restrict the density and velocity per- +turbations mainly in the outer envelope. However, our numerical +results show that the top stable layer has little influence on the +real part of tidal Love numbers. +As we have mentioned, we do not aim to construct a realistic +interior model of Jupiter in this study. These simplified models +are designed to characterize the tidal responses of some possi- +ble scenarios of Jupiter’s interior. Because the dynamical tides +highly depend on the tidal frequency, the satellite dependent tidal +Love numbers would provide more constraints on the interior of +Jupiter (Idini & Stevenson 2022b). In addition, seismology is +the most effective approach to determine the interior structure of +planets, though the detection of Jupiter’s oscillations remains a +big challenge (Gaulme et al. 2011). Nevertheless, the numerical +scheme we developed in this study can be also used for theoreti- +cal calculations of oscillation modes of giant planets. +There are some caveats, which should be considered in fu- +ture. First, we do not consider the centrifugal deformation in +order to solve the problem in the spherical geometry. The cen- +trifugal effect plays a significant role in the tidal Love num- +bers of Jupiter, especially for the high-degree tidal components. +Although we have made the centrifugal corrections when the +numerical results are qualitatively compared with the observa- +tions, both the Coriolis and centrifugal effects should be self- +consistently taken into account for quantitative comparisons +with the high precision observations in future. Second, giant +planets exhibit differential rotations, which also influence the os- +cillation modes and thus tidal responses (Dewberry et al. 2021). +Finally, Jupiter has the strongest magnetic field among planets in +the solar system and mainly consists of electrically conducting +fluid (metallic hydrogen), so the magnetic effects (Lin & Ogilvie +2018; Wei 2022) should also play a part in the tides of Jupiter. +Acknowledgements. The author would like to thank an anonymous referee for +constructive comments and Dali Kong for fruitful discussions. This study was +supported by the B-type Strategic Priority Program of the CAS (XDB41000000), +National Natural Science Foundation of China (grant no. 42174215) and the pre- +research project on Civil Aerospace Technologies of CNSA (D020308). Numer- +ical calculations were performed on the Taiyi cluster supported by the Center for +Computational Science and Engineering of Southern University of Science and +Technology. +References +Barker, A. 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D, 96, 083005 +Article number, page 12 of 12 + diff --git a/DdE0T4oBgHgl3EQfggFM/content/tmp_files/load_file.txt b/DdE0T4oBgHgl3EQfggFM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..353dbedbf59bdf5f971af35ae8f5a1d63643ac6b --- /dev/null +++ b/DdE0T4oBgHgl3EQfggFM/content/tmp_files/load_file.txt @@ -0,0 +1,793 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf,len=792 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' JupiterTidesFinal ©ESO 2023 January 9, 2023 Dynamical tides in Jupiter and the role of interior structure Yufeng Lin Department of Earth and Space Sciences, Southern University of Science and Technology, Shenzhen 518055, China e-mail: linyf@sustech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='cn January 9, 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The Juno spacecraft has obtained highly accurate tidal Love numbers, which provide important constraints on the tidal response and interior structure of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In order to exploit these observations, it is necessary to develop an approach for accurately calculating the tidal response of Jupiter for a given interior model and to investigate the role of interior structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We directly solve the linearized tidal equations of a compressible, self-gravitating, rotating and viscous fluid body using a pseudo-spectral method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The Coriolis force is fully taken into account but the centrifugal effect is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We can simultaneously obtain the real and imaginary parts of the tidal Love numbers for a given planetary interior model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We calculate the tidal responses for three simple interior models of Jupiter which may contain a compact rigid core or an extended dilute core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' All of models we consider can explain the fractional correction ∆k22 ≈ −4% due to dynamical tides, but all have difficulties to reconcile the observed ∆k42 ≈ −11% for the high-degree tidal Love number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We show that the Coriolis force significantly modifies gravity modes in an extended dilute core at the tidal frequency relevant to the Galilean satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We demonstrate that a thin stable layer in the outer region, if exists, would also influence the tidal responses of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' giant planets – tides – internal structure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Introduction Tidal interactions between Jupiter and the Galilean satellites play an important role in the orbital evolution of the system and the internal dynamics of the moons (Lainey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The highly active volcanic eruptions on Io are believed to be due to strong tides raised by Jupiter (Peale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Meanwhile, tides are also raised in Jupiter by its moons, probably dominated by Io (Gavrilov & Zharkov 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The tidal response of a gaseous body such as Jupiter is conventionally treated as a hydrostatic deformation, which acquires a small phase lag with respect to the tidal forcing due to dissipative processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' This is known as the equilibrium tide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' However, the equilibrium tide alone does not suffice to account for the observed strong tidal dissipation in Jupiter (Lainey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2009) and the gravitational perturbations recently measured by the Juno spacecraft (Durante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In fact, the equilibrium tide does not satisfy the momentum equation of tidal flows and thus corrections have to be made to fully account for the tidal response of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The corrections to the equilibrium tide are collectively referred to as the dynamical tide, which usually involves wave-like motions in the planet and depends on the tidal frequency and the interior structure (Ogilvie 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The dynamical tide may provide extra channels of tidal dissipation and produce additional gravitational perturbations on top of the hydrostatic deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The Juno spacecraft has ob- tained highly accurate tidal Love numbers klm (Durante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2020), which quantitatively characterize the tidal response of Jupiter to a tidal forcing component represented in spherical har- monics of degree l and order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The observed tidal Love num- bers by Juno exhibit non-negligible discrepancies with respect to the theoretically calculated hydrostatic values (Wahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2020), suggesting that the dynamical tide has to be considered to ex- plain the observed tidal response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Specifically, Juno observations found ∆k22 ≈ −4% for the dominant tidal component l = 2 and m = 2 and ∆k42 ≈ −11% for the high-degree tidal component l = 4 and m = 2, where ∆klm = (klm − k(hs) lm )/k(hs) lm represents the fractional correction to the hydrostatic value k(hs) lm (Wahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Idini & Stevenson 2021, 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' As the dynamical tides is sensitive to the tidal frequency and the interior structure, the detected gravitational signatures of dy- namical tides may provide important constraints on the Jupiter’s interior (Idini & Stevenson 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Lai 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Idini & Stevenson 2021, 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Dewberry & Lai 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Recent studies (Idini & Stevenson 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Lai 2021) have revealed that the discrepancy in k22 can be mainly attributed to the Coriolis effect on the funda- mental modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' f−modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' More recently, Idini & Stevenson (2022b) proposed that the resonant locking with a gravity mode in an extended dilute core can explain ∆k42 ≈ −11%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' This pro- vides an independent constraint on the existence of a dilute core in Jupiter, which has also been suggested by the Juno measure- ments of gravitational moments of Jupiter (Wahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Mil- itzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' However, the tidal constraint on the existence of a dilute core remains some uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The calculation of tidal response in Idini & Stevenson (2022b) inadequately treated the rotational (Coriolis) effect, which plays an important role in Jupiter’s tidal responses because the tidal frequencies of Galilean satellites are comparable to the spin frequency of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Includ- ing the Coriolis force introduces inertial waves in the neutrally buoyant regions (Ogilvie & Lin 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Wu 2005a) and mixed gravity waves and inertial waves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' gravito-inertial waves, in the stably stratified region (Dintrans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Xu & Lai 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The mechanism proposed by Idini & Stevenson (2022b) is also struggling to reconcile both the real part (relevant to the grav- itational perturbation) and imaginary part (relevant to the tidal dissipation) of the tidal Love numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Article number, page 1 of 12 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='02418v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='EP] 6 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' JupiterTidesFinal In this study, we develop a method to directly calculate the tidal response of a fully compressible, self-gravitating, rotating and viscous fluid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The Coriolis force is fully taken into account but the centrifugal force is neglected, which allows us to numerically solve the problem in spherical geometry using a pseudo-spectral method based on spherical harmonic expansions (Ogilvie & Lin 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Lin & Ogilvie 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' As we directly solve the tidally forced problem with explicit viscosity, we can simul- taneously obtain the real and imaginary parts of the tidal Love number for a given planetary interior model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Our approach is different from recent studies on dynamical tides of Jupiter (Lai 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Idini & Stevenson 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Dewberry & Lai 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' They obtain the eigen modes of the inviscid fluid body first and then calculate the tidal Love number (only the real part) through pro- jecting the tidal force onto each eigen modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We consider three nominal interior models of Jupiter to investigate the dependence of the tidal response on the tidal frequency and the interior struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We focus on the effect of a compact rigid core, an extended dilute core and a thin stably stratified layer in the outer region on tidal responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' All of simplified models can explain the ob- served ∆k22 ≈ −4% as previous studies have shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' However, these simplified models are difficult to account for the observed ∆k42 ≈ −11%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Resonances with gravito-inertial modes in an ex- tended dilute core near the tidal frequency of Io can produce non-negligible dynamical correction to k42, but it is insufficient to explain the Juno observation based on our simplified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Tidal model We consider linear tidal responses of a rotating gaseous planet to a tidal potential component of Ψm l = A(r/R)lYm l (θ, φ)e−iωt, where A is the tidal amplitude, R is the radius of the planet, Ym l (θ, φ) represents spherical harmonics and ω is the tidal fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The resulting tides of the planet produce an external gravitational potential perturbation Φ′ = B(R/r)l+1Ym l (θ, φ)e−iωt (and probably other spherical harmonic components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The ratio Km l (ω) = B/A defines the tidal Love number, which depends on the tidal frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The tidal Love number Km l is a complex number because there exists a phase lag between the forcing and the gravitational perturbations due to dissipative processes (Ogilvie 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' While the real part klm = Re[Km l ] measures the in-phase gravitational perturbations with the tidal forcing, the imaginary part Im[Km l ] quantifies the out of phase tidal response and is related to the dissipation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The ratio between the real and imaginary parts is related to the tidal quality factor Q = sgn(ω) klm Im[Km l ], (1) where sgn(ω) = ±1 is the sign function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Because the phase lag is generally very small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Q ≫ 1, the magnitude of the imaginary part is typically much smaller than the real part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In this study, we develop an approach to directly and simultaneously calculate the real and imaginary parts of the tidal Love number for a given planetary model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Linearized equations For a compressible, self-gravitating and rotating fluid body which may contain a rigid core of radius Ri, linear perturbations to a tidal potential Ψ ∝ e−iωt in the rotating frame are described by the following equations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Ogilvie & Lin 2004): −iωu′ = −2Ω × u′ − 1 ρ0 ∇P′ + ρ′ ρ2 0 ∇P0 − ∇Φ′ − ∇Ψ + fν, (2) −iωρ′ + ∇ · (ρ0u′) = 0, (3) −iω � P′ ΓP0 − ρ′ ρ0 � + u′ · �1 Γ∇ ln P0 − ∇ln ρ0 � = 0 (4) ∇2Φ′ = 4πGρ′, (5) where u is the velocity, Ω the rotation rate, ρ the density, P the pressure, Γ the adiabatic index and G the gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In the above equations, the subscript 0 denotes physical quan- tities in hydrostatic state (without tidal potential) and the nota- tions with the prime represent Eulerian perturbations induce by the tidal forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In the momentum equation (2), we explicitly include a viscous force fν defined as fν = 1 ρ0 ∇ · (2µS), (6) where µ is the dynamic shear viscosity (we neglect the bulk vis- cosity) and S is the strain-rate tensor: S = 1 2 � ∇u′ + (∇u′)T� − 1 3(∇ · u′)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (7) Note that we include the viscous force in the momentum equa- tion but neglect the viscous heating in the energy equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' the density and pressure perturbations are treated as adiabatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In this study, we take fully into account the Coriolis force due to the rotation but neglect the centrifugal distortion for nu- merical convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The centrifugal effect can be measured by ϵ = Ω/ωdyn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' the ratio between the spin frequency Ω and the dynamical frequency ωdyn = (GM/R3)1/2, which is not par- ticularly small for Jupiter (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='288).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Indeed, the centrifugal distortion of Jupiter has non-negligible contributions to the total Love number klm, especially for the high-degree Love number k42 because the tidal response at l = m = 2 can produce a gravi- tational perturbation at l = 4 and m = 2 in an oblate figure (Idini & Stevenson 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' For the hydrostatic k(hs) 42 of Jupiter due to Io, 93% of the total value is actually contributed by the centrifu- gal coupling with k22 and only the remaining 7% is produced by the tidal forcing at l = 4 and m = 2 (Wahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Idini & Stevenson 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In this paper we do not aim to directly fit the klm observed by Juno, but rather focus on the fractional corrections ∆klm by the dynamical tides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In terms of the frac- tional correction ∆klm, the centrifugal contribution to ∆k22 can be neglected in leading order (Lai 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' However, the centrifu- gal contribution to ∆k42 can not be neglected even in leading order because the k(hs) 42 is mostly contributed by the centrifugal coupling with k22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' This complicates the comparison between the calculated ∆k42 in a spherical figure and the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Nev- ertheless, the calculated ∆k42 in a spherical figure can be multi- plied by the factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='07 to account Jupiter’s centrifugal coupling effect for qualitative comparisons with the observation (Idini & Stevenson 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Such a comparison would assume that the tidally-excited internal modes are not significantly modified by the centrifugal deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' By neglecting the centrifugal deformation, the unperturbed basic state is spherically symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' depends on the radius r only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Given the density ρ0(r) and pressure P0(r) profiles of the unperturbed state, the radial gravitational acceleration (inward) Article number, page 2 of 12 Lin: Dynamical tides in Jupiter and the role of interior structure g(r) and the Brunt-Väisälä frequency N(r) are then determined by g = dΦ0 dr = − 1 ρ0 dP0 dr , (8) N2 = g �1 Γ d ln P0 dr − d ln ρ0 dr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (9) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Numerical method In order to obtain the complex Love numbers, we numerically solve Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (2-5) using a pseudo-spectral method for the pre- scribed basic states, subject to the relevant boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The numerical scheme is based on the method used in previous studies (Ogilvie & Lin 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Lin & Ogilvie 2017), but we ex- tend the method to solve the full set of linearized equations (2-5) without making a low-frequency approximation (Ogilvie 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' By introducing h′ = P′/ρ0 and eliminating the density perturba- tion ρ′, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (2-5) can be reduced to the following equations: −iωρ0u′ = −2ρ0Ω × u′ − ∇(ρ0h′) + g∇2ϕ′/(4πG) −ρ0∇ϕ′ − ∇Ψ + ∇ · (2µS), (10) −iωh′ = −c2 s(N2u′ r/g + ∇ · (ρ0u′)/ρ0), (11) −iω∇2ϕ′ = −4πG∇ · (ρ0u′), (12) where u′ r is the radial velocity perturbation and c2 s = ΓP0/ρ0 is the square of the adiabatic sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We impose boundary conditions including the regularity of the gravitational perturbations, zero radial velocity on the rigid inner boundary and vanishing Lagrange pressure perturbation at the surface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' δP = P′ + u′ r/(−iω)∇P0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In terms of h′ and u′ r, the last boundary condition can be written as (Dewberry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2021) �−iωh′ − gu′ r � |r=R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (13) As the viscous force is included, additional boundary conditions are required to complete the boundary value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We use the so-called stress-free conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' the tangential stresses van- ish, at both boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' For a given tidal potential Ψm l = A(r/R)lYm l (θ, φ)e−iωt, the tidal perturbations (including both equilibrium and dynamical tides) u′, h′ and Φ′ can be expanded as u′ = L � n=m um n (r)Rm n + L � n=m vm n (r)Sm n + L � n=m wm n (r)Tm n , (14) h′ = L � n=m hm n (r)Ym n (θ, φ), (15) Φ′ = L � n=m Φm n (r)Ym n (θ, φ), (16) where Rm n , Sm n , Tm n are vector spherical harmonics Rm n = Ym n (θ, φ)ˆr, Sm n = r∇Ym n (θ, φ), Tm n = r∇ × Rm n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (17) As the basic state is axisymmetric, the perturbations involve spherical harmonics with the same order m as the tidal potential Ψm l , but the Coriolis force would couple all spherical harmon- ics with degree n ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' For numerical calculations, we have to make a truncation at certain degree L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Substituting expansions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (14-16) into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (10-12) and projecting onto spherical har- monics, we end up with a set of ordinary differential equations involving um n (r), vm n (r), wm n (r), hm n (r) and Φm n (r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' For the radial de- pendence, we use Chebyshev collocation on Nr Gauss–Lobatto nodes (Rieutord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The boundary conditions are ap- plied through replacing the ODEs with the corresponding bound- ary conditions on the boundary nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The regularity of gravita- tional perturbations requires rdΦm n dr + (n + 1)Φm n = 0 at r = R, (18) rdΦm n dr − nΦm n = 0 at r = Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (19) The vanishing Lagrangian pressure perturbation at the surface and zero radial velocity at the rigid inner boundary give −iωhm n = gum n at r = R, (20) um n = 0 at r = Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (21) The stress-free boundary condition is given as (Ogilvie 2009) um n + rdvm n dr − vm n = 0, rdwm n dr − wm n = 0, (22) at both boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Using the numerical discretization described above, the boundary value problem becomes a linear system involving a large complex block-tridiagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The solution of the lin- ear system is obtained using the standard direct solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We use typical truncations of L = 200 and Nr = 100 in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Once the solution of the linear system is obtained numeri- cally, the complex tidal Love number is readily given by Km l = Φm l (r = R), (23) for the tidal potential component Ψm l = A(r/R)lYm l (θ, φ)e−iωt (we simply set A = 1 for the linear tidal response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Note that the so- lution includes both the equilibrium and dynamical tides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' For the real part of Love numbers, of particular interest is the fractional correction of dynamical tides ∆klm = (klm − k(hs) lm )/k(hs) lm , (24) where k(hs) lm is the hydrostatic value and is calculated by setting ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' As our calculations neglect the centrifugal effect which significantly influences the high-degree Love number k42, the calculated value of ∆k42 should be multiplied by the factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='07 when compared with the observation as we have discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We can also calculate the tidal dissipation rate Dν from the velocity perturbations Dν = � V 2µS2dV, (25) Article number, page 3 of 12 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' JupiterTidesFinal where the integral is taken over the fluid domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The dissipation rate is related to the imaginary part of the tidal Love number (Ogilvie 2014) Dν = (2l + 1)RA2 8πG ωIm[Km l ], (26) which can be served as an independent validation of the numer- ical code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The above relation is satisfied to a high degree of ac- curacy in all of numerical calculations presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Interior models In order to solve Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (10-12), we need to prescribe basic state profiles ρ0(r), g(r) and N2(r) to model Jupiter’s interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Our un- derstanding of Jupiter’s interior has been significantly improved by Juno observations (Stevenson 2020), yet it remains some de- grees of uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In this study, we do not aim to build a realistic model of Jupiter’s interior, but focus on the fractional contributions of dynamical tides to the tidal Love number for dif- ferent possible scenarios of Jupiter’s interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We consider three nominal interior models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1) based on a polytrope of index 1, which is a good leading order approximation for Jupiter (Steven- son 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' For all of models used in this study, the unperturbed density and gravity follow a hydrostatic polytrope of index 1, ρ0 = πM 4R3 sin kr kr , (27) g = GM r2 [sin(kr) − kr cos(kr)] , (28) where k = π/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The first model consists of a small rigid core of radius 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='25R and an isentropic fluid envelope, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Γ = 2 and N2 = 0 in the fluid region (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The second model assumes an extended dilute core of radius 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='7R and an isentropic envelope (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='The dilute core is treated as a stably stratified fluid layer with the Brunt-Väisälä frequency given by N2 ω2 dyn = ˜N2 sin �πr Rc � , (29) where ˜N2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='25 and Rc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='7 for this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' As we fixed the density and pressure profiles to that of a polytrope, the strati- fication is effectively realized by adjusting the adiabatic index (Γ > 2) in the dilute core (Lai 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' This model is similar to that used in Idini & Stevenson (2022b), but they adjust the den- sity profile to model the stable stratification in the dilute core while fix the adiabatic index Γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The third model is based on the model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1(a), but we further add a stably stratified layer between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='8R and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='9R (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1(c)), possibly resulting from H-He immiscibility (Debras & Chabrier 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Stevenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The Brunt-Väisälä fre- quency in the top stable layer is prescribed as N2 ω2 dyn = ˜N2 1 [1 + e−100(r−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='8)][1 + e100(r−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='9)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (30) The degree of stratification of this layer remains uncertain, but it is estimated that typical values of N2/ω2 dyn would be roughly between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='8 for Jupiter (Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Gastine & Wicht 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Here we set a moderate value ˜N2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Note that an interior model with the co-existence of a dilute core and a top stable layer is also possible (Debras & Chabrier 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' As this kind of model involves two different stably strat- ified layers, it would be difficult to characterize the role of the top stable layer on tides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We consider only the combination of a compact rigid core and a top stable layer for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In all of these models, we set the total mass M, the radius R and the spin rate Ω such that the ratio ϵ = Ω/ � GM/R3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='288, corresponding the value of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Our calculations also require the fluid viscosity, which is difficult to estimate in detail for giant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We simply assume the dynamic viscosity µ is propor- tional to the background density ρ0, so the kinematic viscosity ν = µ/ρ0 is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The viscosity can be measured by the di- mensionless number Ek = ν/(ΩR2), known as the Ekman num- ber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We set Ek = 10−6 for most of calculations (unless otherwise specified), roughly corresponding to the effective viscosity based on mixing-length theory (Guillot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' As we have mentioned that we do not aim to construct a re- alistic interior model for Jupiter in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' These simplified models are designed to investigate the effects of a compact rigid core, an extended dilute core and a top stable layer on the tidal re- sponses of Jupiter respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Nevertheless, the fractional cor- rections ∆klm and the tidal quality factor Q for these simplified models can be used to make some qualitative comparisons with the observations (Lai 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Idini & Stevenson 2021, 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Results In this paper, we focus on the dominant tidal component Ψ2 2 and a high-degree tesseral component Ψ2 4, for which non-negligible dynamical corrections have been detected as we have discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Our calculations are limited to the frequency range of −2 ≤ ω/Ω ≤ −1, relevant to the tidal frequencies of the Galilean moons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Note that the negative tidal frequency means that the tidal forcing is retrograde in the co-rotating frame with the planet based on our convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' For the real part of Love numbers, we show the fractional correction ∆klm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In order to make compar- isons with the Juno observation, the calculated ∆k42 is multi- plied by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='07 to compensate the centrifugal effect which is ne- glected in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Because of the negative tidal fre- quency, the imaginary part of Love numbers is also negative in our calculations and is related to the tidal quality factor by klm/Ql = −Im[Km l ] according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Full polytrope model Before presenting results for the interior models in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1, we first show the tidal response of a full isentropic polytrope, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' neutrally buoyant in the whole fluid sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' This model serves as a reference for other models and has been used to investigate the dynamical tides of Jupiter in recent analytical studies (Idini & Stevenson 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Lai 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2 shows both the real and imaginary parts of the Love numbers as a function of the tidal frequency for the full polytrope model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We can see that ∆k22 is negative in the frequency range we considered and smoothly varies as the tidal frequency except a burst around ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='08, which corresponds to a resonance with an inertial mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Away from resonances, our numerical results are consistent with recent theoretical calculations and produce ∆k22 ≈ −4% at the tidal fre- quency of Io (Lai 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Idini & Stevenson 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' These studies also revealed that the dynamical correction ∆k22 can be attributed to the Coriolis effect on the f-modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Apart from the f-modes, the rotating sphere of isentropic fluid also supports smooth iner- Article number, page 4 of 12 Lin: Dynamical tides in Jupiter and the role of interior structure (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Three nominal models of Jupiter’s interior used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The top panel shows the schematic models and the bottom panel shows the density (normalized by the density at the center) and the Brunt-Väisälä frequency (normalized by the dynamical frequency) as a function of the radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The blue shadow in the bottom panel indicates solid regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (a) A compact rigid core model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (b) an extended dilute core model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (c) a compact rigid core and an outer stable layer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' tial modes restored by the Coriolis force in the frequency range of 0 < |ω/Ω| < 2 (Greenspan 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Lockitch & Friedman 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The burst of ∆k22 at ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='08 indeed is due to the resonant excitation of the inertial mode as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 3(a), but we no- tice that the resonance occurs only in a very narrow frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' However, this inertial mode has more significant contribu- tions to ∆k42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The angular structure of an inertial mode cannot be described by a single spherical harmonics in general (Lockitch & Friedman 1999), but the density perturbations (and thus the gravitational perturbations) are dominated by the spherical har- monics Y2 4(θ, φ) for the resonant inertial mode at ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0836 as we can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' This suggests a likely strong cou- pling between the tidal potential component Ψ2 4 and the inertial mode in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 3(a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' large tidal overlap as described in Wu (2005b), leading to significant dynamical corrections to k42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The dynamical correction can reach ∆k42 ≈ −10% (after the centrifu- gal correction) near the resonance at ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' However, the tidal frequencies of the Galilean satellites are too far away from this resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The curve of ∆k42 also shows a spike around ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='51, corresponding to a narrow resonance with a high degree inertial mode (ρ′ is dominated by Y2 6(θ, φ) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Inter- estingly, the tidal frequency of Io is close to this resonance, but the dynamical correction caused by this resonant mode is insuffi- cient to account for the observed ∆k42 ≈ −11%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The frequencies of inertial modes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 3 are slightly shifted comparing to that calculated by Lockitch & Friedman (1999) for a polytrope of index 1 (see their table 6 and note different conventions for the sign of frequencies) because they assumed ϵ → 0 whereas we set ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The imaginary parts of the Love numbers in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2 show that resonances with inertial modes significantly enhance the tidal dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The enhanced dissipation due to resonant inertial modes in a neutrally buoyant sphere has been demonstrated by Wu (2005b) but using different density profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' When the tidal frequency is away from resonances, the dissipation rate for the full isentropic polytrope is too small to account for the observed tidal quality factor Q (Lainey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Compact rigid core model We now consider tidal responses for the interior model with a compact rigid core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Basically, the inner region (r ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='25R) of a whole fluid polytrope becomes solid for this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 4 shows the frequency-dependence of the Love numbers for the compact rigid core model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We can see that the real parts are largely sim- ilar to that of a full polytrope, but the imaginary parts are rather different from that of a full polytrope, showing enhanced tidal dissipation by introducing the rigid core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The rigid core model also supports inertial waves in the fluid envelope, but these waves have some peculiar behaviors due to the singularity in a spher- Article number, page 5 of 12 O1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='00 r/R r/R r/RA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' JupiterTidesFinal Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Complex Love number as a function of the tidal frequency for a full isentropic polytrope of index 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Top panel shows the fractional correction ∆klm of the real part of the Love numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The fractional correction ∆k42 (orange curve in the top panel) is multiplied by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The bottom panel shows the minus imaginary part −Im[Km l ], which is equivalent to klm/Ql.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Vertical dashed lines indicate tidal frequencies of four Galilean Moons of Jupiter (from right to left: Io, Europa, Ganymede, Callisto).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The horizontal dashed line in the bottom panel represents the astrometric observation of the frequency independent k2/Q2 from Lainey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Density perturbations (left half) and radial velocity perturbations (right half) in the meridional plane to the tidal component Ψ2 4 at two resonant frequencies in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Amplitudes are normalized by the maximum absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' ical shell (Stewartson & Rickard 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Smooth inertial modes do not exist generally in a spherical shell even with uniform den- Article number, page 6 of 12 10 1 K2 I I 5 K?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' I I 1 I Nkim( 0 1 1 5 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2 101 10-1 10 3 1 I 1 I 1 1 10-7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2 Tidal frequency 0/Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 1 1 /S2 =-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0836 Wp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 1 1 3 /S2 =-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5117Lin: Dynamical tides in Jupiter and the role of interior structure Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' As for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2 but for the interior model with a compact rigid core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The fractional correction ∆k42 (orange curve in the top panel) is multiplied by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Density perturbations (left half) and gravitational perturbations (right half) in the meridional plane to the tidal component Ψ2 4 for the interior model with a compact rigid core at (a) ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5092 (resonance) and (b) ω/Ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='53 (non-resonance) with Ek = 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Amplitudes are normalized by the maximum absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' sity (Rieutord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2001), and localized wave beams spawned from the critical latitudes propagate in the bulk along the charac- teristics of the inertial wave equations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Ogilvie 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' How- ever, Lin & Ogilvie (2021) recently revealed that resonant tidal responses in a spherical shell correspond to eigen modes with large scale flows hidden beneath localized wave beams using a Article number, page 7 of 12 10 K2 5 K?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 0 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 100 10-4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4 Tidal frequency 0/Qp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 1 1 /S2 =-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='50920.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 1 1 /S2 =-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5300A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' JupiterTidesFinal uniform density model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Furthermore, it was shown that the hid- den large scale structures basically resemble inertial modes in a full sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' This is in line with our results for the non-uniform density model in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The real parts k22 and k42 are relevant to only large scale density perturbations, which are similar to in- ertial modes in a full sphere as one can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' There- fore, the curves of ∆klm for the rigid core model resemble that of a full polytrope, but note slight shifts of the resonant frequencies due to the presence of rigid core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' As for the full polytrope, the compact rigid core model can produce ∆k22 = −4% as observed, but it cannot produce sufficient dynamical correction in the high- degree Love number k42 near the tidal frequency of Io to account for the observed ∆k42 = −11%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' On the other hand, the imaginary parts are largely modified by the presence of small rigid core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We can see that the tidal dis- sipation is significantly enhanced by the localized wave beams spawned from the critical latitudes both in and out of resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The velocity perturbations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 5(b) indeed exhibit localized waves propagating in the bulk, which can generate significant viscous dissipation but do not produce much density and gravi- tational perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 4, we also see that several peaks in the tidal dissipation (bottom panel) do not lead to obvious fluc- tuations in ∆klm (top panel), corresponding to resonances with higher degree modes that have little contributions to the low de- gree ( i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' l = 2 and l = 4) gravitational perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In summary for the compact rigid core model, the tidal dis- sipation is significantly enhanced with respect to the full poly- trope case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' This is in line with the early work of Ogilvie & Lin (2004), who have shown the enhanced tidal dissipation due to inertial waves in the convective envelope of rotating stars and planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The averaged dissipation in the tidal frequency range of Galilean moons gives rise to comparable tidal quality factor as observed (Lainey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' However, the fractional correction to the real part of Love number ∆k42 is insufficient to explain the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Dilute core model An extended dilute core rather than a compact core in Jupiter has been suggested recently based on Juno gravitational measure- ments (Wahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Militzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In this subsection, we consider tidal responses for the interior model with an ex- tended dilute core as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The dilute core is treated as a stably stratified layer which supports gravity waves restored by the buoyancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' If the Coriolis force is fully taken into account, dynamical tides in the dilute core region would be in the the form of mixed gravity waves and inertial waves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' gravito-inertial waves (Dintrans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Xu & Lai 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Idini & Stevenson (2022b) recently calculated the tidal response of Jupiter with an extended dilute core, but they did not fully consider the Coriolis effect, which turns out to important as we will show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 6 shows the frequency-dependence of the Love num- bers for the dilute core model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' For the tidal component Ψ2 2 (bule curves), the dynamical correction ∆k22 is generally similar to that of the full polytrope except the absence of obvious spikes for the dilute core model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' However, the imaginary part exhibits sev- eral peaks and troughs, suggesting possible resonances with high degree mixed modes that enhance the tidal dissipation but do not significantly contribute to the l = 2 gravitational perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The overall tidal dissipation is also enhanced with respect to the full polytrope due to the excitation of gravito-inertial waves in the dilute core and inertial waves in the convective envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The frequency-averaged tidal dissipation tends to be compatible with the observed tidal quality factor as we can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' For the tidal component Ψ2 4, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 6 also shows results with- out including the Coriolis force (green curves) for compari- son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We note that the fractional correction ∆k42 is always pos- itive when the Coriolis force is neglected, probably because the pure gravity modes enhance the in-phase gravitational pertur- bations and thus produce positive dynamical corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Nev- ertheless, we observe distinct resonant responses at certain tidal frequencies from both real and imaginary parts of the Love num- ber for the non-Coriolis case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' For instance, the resonance at ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5193, which is close to the tidal frequency of Io, corresponds to the first gravity mode of l = 4 and m = 2 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 7 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Indeed, Idini & Stevenson (2022b) proposed the resonant locking between this gravity mode 1 (referred to as 2 4g1) and the Jupiter-Io orbital evolution to explain the observed ∆k42 for Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In Idini & Stevenson (2022b), the Coriolis force is neglected for the calculation of gravity modes, but approxi- mated rotational corrections are made to obtain the Love num- ber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' However, taking fully into account the the Coriolis force significantly alter the tidal responses as we can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The dynamical correction ∆k42 exhibits several large fluctu- ations especially in the frequency range of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 < ω/Ω < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' This is due to the mixing of gravity modes and inertial modes in the dilute core, leading to more chances for resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The most significant dynamical corrections are produced near the tidal frequency ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2, which is close to the frequency of the purely inertial mode as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Of course, the inertial mode is mixed with gravity modes in the dilute core for this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The resonance around ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2 can pro- duce more than −10% dynamical corrections in k42 (after the centrifugal correction), but it is too far away from the tidal fre- quency of Io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The resonance close to the tidal frequency of Io (also close to the frequency of pure gravity mode 2 4g1) occurs at ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4448 when the Coriolis force is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 7 (b) shows the spatial structure of this resonant response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The Cori- olis effect not only leads to a non-negligible shift in the mode frequency, but also largely modifies the mode structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The per- turbations are in the from of gravito-inertial waves in the dilute core and become pure inertial waves in the neutrally buoyant envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Non-negligible dynamical corrections are induced by this resonance at ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4448, but the corrections are in- sufficient (after the centrifugal correction) to account for the ob- served ∆k42 = −11%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' As the resonance is very narrow, we use 200 equally spaced frequency points in the tidal frequency in- terval of [-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='45, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The peak amplitude of ∆k42 in this fre- quency interval is comparable to that of using only 20 frequency points, suggesting that the frequency sampling points are suffi- cient to capture the resonant peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Comparing the orange and green curves in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 6, we can see that the tidal dissipation is increased by about two orders of magnitude when the Coriolis force is in- cluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' This suggests that the excitation of pure gravity waves is a less efficient tidal dissipation mechanism (unless resonances take place) based on our linear calculations, though the nonlinear interaction or wave breaking of gravity waves may lead to effi- cient tidal dissipation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Barker 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Weinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Outer stable layer model We finally consider the effect of an outer stable layer, which may exist in Jupiter resulting from H-He immiscibility (Debras & Chabrier 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 8 shows the Love numbers as a function 1 They used slightly different background density ρ0(r) and Brunt- Väisälä frequency N(r), so the mode frequency is slightly shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Article number, page 8 of 12 Lin: Dynamical tides in Jupiter and the role of interior structure Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' As for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2 but for the interior model with an extended dilute core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Green lines represent results without including the Coriolis force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The fractional correction ∆k42 (orange and green curves in the top panel) is multiplied by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Density perturbations (left half) and radial velocity perturbations (right half) in the meridional plane to the tidal component Ψ2 4 for the interior model with an extended dilute core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (a) Without including the Coriolis force at ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5193 (resonance);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (b) including the Coriolis force at ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4448 (resonance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Amplitudes are normalized by the maximum absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' of the tidal frequency for the interior model (c) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1, which includes a compact rigid core and a top stable layer between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='8R Article number, page 9 of 12 p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 1 /S2 =-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='51930.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='6 3 /S2 =-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4448K 10 K(Non-Coriolis) 5 % Aklm( 0 1 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 10-1 Q10-3 10-5 10-7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 Tidal frequency /QA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' JupiterTidesFinal Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' As for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2 but for the interior model with a small rigid core and a top stably stratified layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Green lines represent results at the Ekman number Ek = 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The fractional correction ∆k42 (orange and green curves in the top panel) is multiplied by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Perturbations in the meridional plane to the tidal component Ψ2 4 at ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='1650 for the interior model (c) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (a) Density (left half) and radial velocity (right half) perturbations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' (b) gravitational (left half) and vorticity (right half) perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Amplitudes are normalized by the maximum absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The dashed lines denote r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='8R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='9R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' For the tidal responses to Ψ2 2, the dynamical correction ∆k22 is similar to the case without the stable layer, but the pres- ence of the thin stable layer eliminates the spike due to the res- onant inertial mode at the tidal frequency around ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The overall tidal dissipation due to Ψ2 2 is comparable to the coun- terpart without the top stable layer (blue curve in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 4), but the fluctuation amplitudes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' the differ- ences between peaks and troughs, are smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Article number, page 10 of 12 10 Ek = 10-6 5 Ek = 10-7 % △klm( 0 1 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4 100 10 Q klml 10° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='0p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 1 S2 =-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='1650V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='1 1 0 w/2 =-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='1650Lin: Dynamical tides in Jupiter and the role of interior structure For the tidal responses to Ψ2 4, we also show results for Ek = 10−7 (green curves) to illustrate the effect of fluid viscosity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' One can see that the viscosity has little influence on the real part of the Love number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The tidal dissipation weakly de- pends on viscosity at peaks and troughs, but the overall dissipa- tion tends to be insensitive to viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Indeed, Ogilvie (2013) has shown the frequency-averaged dissipation is independent of viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The dynamical correction ∆k42 is also similar to the case without the stable layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We can see large variations of ∆k42 at the tidal frequency around ω/Ω = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='165, which corresponds to a resonant mode as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' This mode is complicated because it involves three different layers for the interior model considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The fluid body is primarily neutrally buoyant and supports inertial waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' However, the fluid domain is sepa- rated by the thin stable layer, which suppresses radial fluid mo- tions and creates a "barrier" for the communication between in- ertial waves in the inner and outer regions (see the radial velocity and vorticity perturbations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In addition, the thin sta- ble layer supports rotationally modified gravity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The den- sity perturbations are mainly restricted in the stable layer and the outer envelope, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' in the region of r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='8R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Despite the complicated velocity and density perturbations, the gravitational perturbations are dominated by the l = 4 component with rela- tively simple radial dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In this regard, this complicated mode is relevant to the l = 4 inertial mode without the stable layer, leading to large dynamical corrections around the tidal frequency at ω/Ω ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='1 as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' However, the dynami- cal correction ∆k42 is negligible after the centrifugal correction at the tidal frequency of Io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Conclusions We have developed a numerical method for calculating the tidal responses of a compressible, self-gravitating, rotating and vis- cous fluid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We take fully into account the Coriolis force but neglect the centrifugal distortion, which allows us to solve the problem in the spherical geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We use the pseudo-spectral method based on spherical harmonics in the angular directions and Chebyshev collocation in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Different from recent studies on Jupiter’s dynamical tides (Lai 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Idini & Stevenson 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Dewberry & Lai 2022), we directly solve the tidally forced problem and explicitly add the fluid viscosity, which allows us to simultaneously obtain the real and imaginary parts of the tidal Love numbers for a given planetary interior model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In this study, we considered three simplified interior models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 1) of Jupiter based on a polytrope of index 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We focus on the tidal components Ψ2 2 and Ψ2 4 in the frequency range of −2 ≤ ω/Ω ≤ −1, which is relevant to the tidal frequencies of Galilean moons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Our numerical results show that the dynami- cal correction ∆k22 is generally insensitive to the interior mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' All of models we considered can give rise to the observed ∆k22 ≈ −4% at the tidal frequency of Io, which is also in line with previous studies (Idini & Stevenson 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Lai 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The tidal dissipation is significantly enhanced by the presence of a compact rigid core model or an extended dilute core with re- spect to the full polytrope, leading to comparable tidal quality factor Q as observed (Lainey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' For the tidal responses to the Ψ2 4 component, all of models we considered are difficult to give rise to ∆k42 ≈ −11% near the tidal frequency of Io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' For the interior model with a com- pact rigid core, significant dynamical corrections are generated at the tidal frequency around ω/Ω ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content='1 due to the resonance with an inertial mode whose gravitational perturbations are dom- inated by the spherical harmonics of l = 4 and m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' How- ever, this resonance is too far away from the tidal frequencies of Galilean moons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' For the interior model with an extended di- lute core, we demonstrate that the gravity modes in the dilute core can be significantly modified by the Coriolis force, leading to the mixed gravito-inertial modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Resonances with gravito- inertial modes in the dilute core can produce non-negligible dy- namical corrections, but they are insufficient to explain the ob- served ∆k42 ≈ −11% near the tidal frequency of Io based on our simplified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' We also briefly investigated the effect of a top stable layer on Jupiter’s tides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The thin stable layer acts as a "barrier" and tends to restrict the density and velocity per- turbations mainly in the outer envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' However, our numerical results show that the top stable layer has little influence on the real part of tidal Love numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' As we have mentioned, we do not aim to construct a realistic interior model of Jupiter in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' These simplified models are designed to characterize the tidal responses of some possi- ble scenarios of Jupiter’s interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Because the dynamical tides highly depend on the tidal frequency, the satellite dependent tidal Love numbers would provide more constraints on the interior of Jupiter (Idini & Stevenson 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' In addition, seismology is the most effective approach to determine the interior structure of planets, though the detection of Jupiter’s oscillations remains a big challenge (Gaulme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Nevertheless, the numerical scheme we developed in this study can be also used for theoreti- cal calculations of oscillation modes of giant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' There are some caveats, which should be considered in fu- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' First, we do not consider the centrifugal deformation in order to solve the problem in the spherical geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The cen- trifugal effect plays a significant role in the tidal Love num- bers of Jupiter, especially for the high-degree tidal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Although we have made the centrifugal corrections when the numerical results are qualitatively compared with the observa- tions, both the Coriolis and centrifugal effects should be self- consistently taken into account for quantitative comparisons with the high precision observations in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Second, giant planets exhibit differential rotations, which also influence the os- cillation modes and thus tidal responses (Dewberry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Finally, Jupiter has the strongest magnetic field among planets in the solar system and mainly consists of electrically conducting fluid (metallic hydrogen), so the magnetic effects (Lin & Ogilvie 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Wei 2022) should also play a part in the tides of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' The author would like to thank an anonymous referee for constructive comments and Dali Kong for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' This study was supported by the B-type Strategic Priority Program of the CAS (XDB41000000), National Natural Science Foundation of China (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' 42174215) and the pre- research project on Civil Aerospace Technologies of CNSA (D020308).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' Numer- ical calculations were performed on the Taiyi cluster supported by the Center for Computational Science and Engineering of Southern University of Science and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE0T4oBgHgl3EQfggFM/content/2301.02418v1.pdf'} +page_content=' References Barker, A.' metadata={'source': 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This is the author’s version of the work. The definitive Version of Record is published in the IEEE International +Symposium on Circuits and Systems (ISCAS), 2023. +TrojanSAINT: Gate-Level Netlist Sampling-Based +Inductive Learning for Hardware Trojan Detection +Hazem Lashen, Lilas Alrahis, Johann Knechtel, and Ozgur Sinanoglu +New York University Abu Dhabi +{hl3372, lma387, jk176, os22}@nyu.edu +Abstract—We propose TrojanSAINT, a graph neural network +(GNN)-based hardware Trojan (HT) detection scheme working at +the gate level. Unlike prior GNN-based art, TrojanSAINT enables +both pre-/post-silicon HT detection. TrojanSAINT leverages a +sampling-based GNN framework to detect and also localize HTs. +For practical validation, TrojanSAINT achieves on average (oa) +78% true positive rate (TPR) and 85% true negative rate (TNR), +respectively, on various TrustHub HT benchmarks. For best-case +validation, TrojanSAINT even achieves 98% TPR and 96% TNR +oa. TrojanSAINT outperforms related prior works and baseline +classifiers. We release our source codes and result artifacts. +Index Terms—Hardware Security, Trojan Detection, GNNs +I. INTRODUCTION +Integrated circuit (IC) design and manufacturing has become +an increasingly outsourced process that involves various third +parties. While this has allowed to increase both productivity +and complexity of ICs, it has also made them more vulnerable +to the introduction of hardware Trojans (HTs), among other +threats. HTs are malicious circuitry, causing system failure, +leaking sensitive information, etc [1], [2]. Thus, methods to +accurately check for HTs become increasingly important. +Conventional methods for HT detection include code re- +view [3] and verification against a “golden reference”, i.e., +a trusted, HT-free version of the design [4]. However, the +former is prone to errors, especially for complex ICs, and +the latter is not always feasible, especially when untrusted +parties are engaged in the design process [4]. Other meth- +ods have been proposed as well, e.g., utilizing side-channel +fingerprinting [5]; however, such are limited to post-silicon +HT detection. Researchers have shown that machine learning +(ML) can successfully adapt to a wide variety of HTs, without +necessitating new techniques for detecting new HT designs [6]. +Using graph neural networks (GNNs) is an emerging and +promising method toward this end [3], [7]–[9]. Thanks to their +ability to work on graph-structured data – such as circuits – +GNNs can leverage both a) the features of each gate and b) +the overall structure of the design for the prediction of HTs. +Still, prior art for GNN-based HT detection suffers from the +following limitations (also summarized in Table I). +HT Localization. State-of-the-art GNN-based detection +schemes, GNN4TJ [3] and HW2VEC [7], predict whether +a design contains a HT or not, but they cannot localize HTs. +However, localizing HTs is essential to identify the part of the +design at fault and name the responsible, malicious party. +Scope. Earlier works [3], [7], [10] are limited to register +transfer level (RTL), unable to handle gate-level netlists (GLNs). +TABLE I +COMPARISON OF GNN-BASED HT DETECTION SCHEMES +Method +HT +HT +Gate-Level +Pre- +Post- +Detection +Localization +Netlist +Silicon +Silicon +GNN4TJ [3] +Yes +No +No +Yes +No +HW2VEC [7] +Yes +No +No +Yes +No +GRFTL [10] +Yes +Yes +No +Yes +No +Our Work +Yes +Yes +Yes +Yes +Yes +Fig. 1. Concept of TrojanSAINT. +Such methods are restricted to pre-silicon assessment; they +cannot detect HTs in the field. Note that only schemes which +can work on GLNs allow for pre- and post-silicon detection. +Associated Research Challenges. Developing a GNN-based +HT detection and localization scheme that can work on GLNs +imposes the following research challenges (RC). +RC1: GLN Complexity. Compared to RTL, GLN designs +are more complex to analyze, as GLNs are flattened (i.e., +hierarchical information is lost) and also considerably larger, +in the range of thousands or even millions of gates and wires. +RC2: Imbalanced Datasets. HTs are stealthy and small in +size; HT gates represent a very small percentage, e.g., 0.14– +11.29% or 1.94% on average for the TrustHub suite considered +in this work. Thus, a highly imbalanced dataset arises (e.g., +the ratio of regular to HT gates reaches up to 719× for the +TrustHub suite), which is difficult to handle for any ML model. +Our Contributions. Here, we propose TrojanSAINT, a GNN- +based method for HT detection and localization that works +well on large-scale GLNs. The concept is outlined in Fig. 1. +As indicated, the graph representations of GLNs are complex +and large, which makes them difficult to handle with traditional +architectures such as graph convolutional networks (GCNs). +This motivates our decision to, without loss of generality +(w/o.l.o.g.), use GraphSAINT [11] for our methodology. Graph- +SAINT is a well-established, sampling-based approach that +extracts smaller sub-graphs for training from the larger original +graph. It has shown good performance for various tasks [11]– +arXiv:2301.11804v1 [cs.CR] 27 Jan 2023 + +Fig. 2. Overview of TrojanSAINT. Black arrows follow the inference process, +orange arrows follow the additional steps needed for training and validation. +In this example, the thresholding value is 0.4. +[13], but it has not been considered for HT detection until now. +We summarize our contributions as follows: +1) A parser for GLN-to-graph conversion (Sec. II-A) which +performs feature extraction tailored for HT detection. +2) A GNN-based method for detection and localization of +HT in GLNs (Sec. II-B), addressing RC1. +3) A procedure for tuning of the classification thresholds to +obtain more accurate predictions, addressing RC2. +4) We demonstrate that our scheme is competitive to tra- +ditional ML baselines and prior art. We also verify the +generalization ability of our scheme – i.e., good prediction +accuracy for unknown HTs on unseen GLNs. +5) We open-source our scheme and related artifacts from +our experimental study [https://github.com/DfX-NYUAD/ +TrojanSAINT]. +II. TROJANSAINT METHODOLOGY +An overview of our methodology is shown in Fig. 2. Next, +we describe all relevant details. +A. GLN Parsing and Feature Vectors +Parser. We develop a parser that converts GLNs (given in +Verilog format) into unweighted and undirected graphs, where +nodes represent gates and edges represent wires. We are dis- +carding directionality for improved representation learning [14]. +Given a set of GLNs, our parser generates one large single +graph, consisting of multiple disjoint graphs, where nodes +are labeled as ‘train,’ ‘validation’ or ‘test,’ depending on the +designation of the GLN they belong to. The graph is encoded +as an adjacency matrix A following a standard procedure. +Feature Vectors. Our parser also generates a matrix X of +feature vectors for all nodes. Vectors cover the following: +• Gate type, represented via one-hot encoding. From exper- +imentation, we are more interested in the functionality of +the gate over the exact implementation. That is, we group +functionally related gates together, e.g., all AND gates +are grouped regardless of the number of inputs and the +driver strengths that the different AND gates support. +• Input, output degrees of gates, i.e., the number of incoming +and outgoing connections. +• Shortest distances to primary inputs/outputs. For gates not +directly connected with a primary input/output, a breadth- +first search is conducted to obtain shortest distances. +For training and validation, we also use a binary label vector +which marks each node as part of some HT or as regular/benign +gate. The related information is derived during parsing. +B. GNN Implementation and Application +Outline. We utilize GraphSAINT [11] for sampling. Further, +we utilize the GNN architecture of GNN-RE [15], along with +GraphSAGE [16]. We tune the classification thresholds for more +accurate predictions. We further utilize a practical validation. +For training and inference, we employ standard procedures. +GNN Architecture. We consider an undirected graph +G (V, A) for representing a GLN, where V is the set of +vertices/nodes/gates, and the adjacency matrix of the graph is +A, where Au,v = 1 and Av,u = 1 if there exists an edge/wire +from vertex/gate u to vertex/gate v. Each vertex u in the initial +graph G has a feature vector xu. This vector represents the +node embedding at layer zero of the GNN. The embedding of +node u is iteratively updated by the GNN, by aggregating the +embedding of the node and its neighbors N(u). The embedding +of a node u after l GNN layers, h(l) +u , is given by: +a(l) +u = AGGREGATE(l) �� +h(l−1) +v +: v ∈ N(u) +�� +(1) +h(l) +u = COMBINE(l) � +h(l−1) +u +, a(l) +u +� +(2) +GNN architectures are defined by their implementation +of AGGREGATE(·) and COMBINE(·). For example, Graph- +SAGE [16], which we also use here, works as follows: +h(l) +u = σ([Wl · AGG({h(l−1) +v +, ∀v ∈ N(u)}), Blh(l−1) +u +]) (3) +AGG = +� +v∈N(u) +h(l−1) +v +|N(u)| +(4) +where σ(.) is an activation function such as ReLU and +Wl and Bl are trainable weight matrices. In GraphSAGE, the +embedding of node h(l) +u is determined by first concatenating the +node’s features from the previous layer h(l−1) +u +with the output +of the AGG function. Then the Wl and Bl transformations +learns the important components of the neighbors’ features +and the node u, respectively. GraphSAGE is compatible with +different AGG functions. Here, we use the mean aggregator +as described in Equation (4). +Thresholding. From experimentation, we observe that the +classification threshold plays a significant role for prediction +performance. This is because of the considerably imbalanced +datasets (Sec. I, RC2), where the GNN model as is can predict +the minority class, i.e., HT nodes, only with low confidence. +The goal of thresholding is to determine a sufficiently small +value so that HT nodes/gates are classified as such the moment +the GNN captures any hint of malicious structures. In other +words, thresholding allows the GNN to focus more on the +minority class, improving the performance of the entire model. +W/o.l.o.g., we tune the threshold between 0–0.5 in steps +of 1,000 and select the threshold that yields the best score +on validation. Here, best score refers to the average of true +positive rate (TPR) and true negative rate (TNR). + +首Algorithm 1 TrojanSAINT training algorithm +Input: Training graph G (V, A); Ground truth Y ; Sampler RWS +Output: Trained GNN +1: Compute normalization coefficients α, λ using RWS +2: for each mini-batch do +3: +Gs (Vs, As) ← Sampled sub-graph of G using RWS +4: +Build GNN on Gs +5: +{yu | u ∈ Vs} ← Propagating α-normalized {xu | u ∈ Vs} +6: +Propagating λ-normalized loss L (yu, yu) to update weights +7: end for +Algorithm 2 TrojanSAINT inference algorithm +Input: Flattened netlist N; Trained GNN; Threshold th +Output: Trojan classification of all nodes/gates +1: Initiate G (V, A) with V ← GLN to graph(N) +2: for each u ∈ V do +3: +zu ← GNN(u) +▷ Compute embedding +4: +cu ← fc(zu, th) +▷ Classify node u based on the threshold +5: end for +Practical Validation. We propose an approach where pre- +dictions are made on unknown HTs residing within circuits +that are neither seen during training nor have golden references. +This represents a real-world scenario, where security engineers +do not know in advance which HT to expect, if any at all, and +further need to test circuits without golden references. Prior +art did not necessarily consider such practical validation. +Training. First, we construct sub-graphs using a standard +random-walk sampler (RWS). TrojanSAINT’s training procedure +is shown in Algorithm 1. Due to the RWS, the network can +become biased towards frequently sampled nodes. To alleviate +this issue, we follow the normalization technique of [11]. We +use stochastic gradient descent as optimizer. Gs is sampled +for each minibatch and a GNN is built on the sub-graph. The +cross-entropy loss is calculated for each node in the sub-graph +and the GNN weights are then updated by backpropagation. +Inference. See Algorithm 2. For all test nodes in the graph, +node embeddings are calculated and passed to a fully-connected +layer with softmax activation, to compute class probabilities. +We then apply our thresholding technique, and finally convert +class probabilities into labels. +III. EXPERIMENTAL STUDY +A. Setup +Software. We use Python for coding and bash scripts +for job/data management. TrojanSAINT extends on GNN- +RE, which is obtained from [14] and is implemented in +PyTorch. Our TrojanSAINT platform is available online [https: +//github.com/DfX-NYUAD/TrojanSAINT]. Baseline models +are implemented using Scikit-Learn, except the fully-connected +neural network (FCNN) in PyTorch. +Computation. Experiments for GNN-RE, TrojanSAINT and +FCNN are conducted on a high-performance cluster with 4x +Nvidia V100 GPUs and 360GB RAM; experiments for others +are conducted on a workstation with Intel i7 CPU and 16GB +RAM. Training of GNN-RE and TrojanSAINT takes ≈15–30 +minutes per model, FCNN ≈10 minutes per model, and all +others ≈3 minutes in total. All inference takes few seconds. +Benchmarks and Model Building. We use 17 exemplary +GLN benchmarks from the TrustHub suite [17]. For each +benchmark, a respective model is trained from scratch. For our +practical validation, each model does not get to see the design +to be tested at all during training.1 +We note that random seeds used in TrojanSAINT’s com- +ponents affect performance significantly. Thus, we conduct +w/o.l.o.g. 6 runs with different seeds and report only results +for each model that performs best on its validation set. +Prior Art, Comparative Study. From Table I, recall that +none of the prior art in GNN-based HT detection works on +GLNs. Thus, a direct comparison is not practical. However, +we consider the following works for comparison. +• GNN-RE [15]: Proposed for reverse engineering of GLNs, +it could also be utilized for HT detection and localization. +This is because GNN-RE seeks to classify gates/nodes +from flattened GLNs into the circuit modules they belong +to; TrojanSAINT’s task of classifying gates/nodes into +begin or HT-infested ones is analogous. +• Related Works [18]–[20]: ML-based, not GNN-based, HT +detection schemes that are working on GLNs. Unlike ours, +these works employ elaborate feature engineering. Also, +these works do not offer native HT localization. +We also implement and run the following well-known +baseline classifiers for a further comparative study. +• XGBoost: A decision tree (DT)-based model that uses +an ensemble of sequentially added DTs. DTs are added +aiming to minimize errors of their predecessor. +• Random Forest: A DT-based model that uses an ensemble +of DTs trained on subsets of the training data. +• Logistic Regression: A classification algorithm that utilizes +the sigmoid function on independent variables. +• Support Vector Machine (SVM): A classification model +that generates a hyperplane to separate different classes. +• FCNN: We implement a three-layer network; each layer +use the SELU activation function [21] and batch normaliza- +tion. The final layer uses sigmoid activation to calculate +classification probabilities. +All these classifiers work on tabular, non-graph data; thus, we +provide them with the feature vectors as inputs. All classifiers, +except SVM, output probabilities; thus, we can study them +considering our proposed thresholding as well. +B. Results +Practical Validation and Impact of Thresholding. In +Table II, we report TPR/TNR results for practical validation +across two scenarios: with thresholding versus without.2 +First, the results show that TrojanSAINT outperforms other +methods for this realistic but challenging scenario of HT +1For example, if rs232t1000 is to be tested, none of the other rs232 designs +are used for training, only for validation. +For s15850t100, the only s15850 design in the suite, we randomly select +three other designs for validation. +2Since thresholding is part of our proposed scheme, we do not consider +TrojanSAINT without. We implement the same thresholding strategy (Sec. II-B) +for all models. SVM directly separates data into classes without computing +probabilities, making thresholding not applicable (N/A). + +TABLE II +TPR/TNR RESULTS FOR PRACTICAL VALIDATION. BEST RESULTS, CONSIDERING AVERAGE OF TPR AND TNR, ARE MARKED IN BOLDFACE. +TrustHub +All With Thresholding +Others Without Thresholding +Benchmark +TrojanSAINT +XGBoost +FCNN +GNN-RE +Logistic +Random +SVM +TrojanSAINT +XGBoost +FCNN +GNN-RE +Logistic +Random +SVM +Regression +Forest +Regression +Forest +rs232t1000 +1.00/0.60 +1.00/0.57 +0.85/0.64 +0.77/0.51 +1.00/0.46 +0.69/0.75 +N/A +1.00/0.60 +0.23/0.91 +0.00/1.00 +0.00/1.00 +0.00/1.00 +0.15/0.92 +0.00/1.00 +rs232t1100 +0.92/0.68 +0.83/0.57 +1.00/0.61 +0.92/0.52 +0.83/0.61 +0.33/0.90 +N/A +0.92/0.68 +0.08/0.91 +0.00/0.94 +0.00/0.94 +0.00/1.00 +0.08/0.92 +0.00/1.00 +rs232t1200 +0.41/0.80 +0.59/0.56 +0.59/0.91 +0.82/0.27 +0.71/0.60 +0.35/0.75 +N/A +0.41/0.80 +0.06/0.90 +0.06/1.00 +0.06/0.93 +0.00/1.00 +0.00/0.91 +0.00/1.00 +rs232t1300 +1.00/0.74 +1.00/0.57 +1.00/0.66 +1.00/0.71 +1.00/0.43 +0.56/0.86 +N/A +1.00/0.74 +0.22/0.91 +0.00/1.00 +0.00/0.98 +0.00/1.00 +0.00/0.92 +0.00/1.00 +rs232t1400 +0.92/0.50 +0.92/0.57 +1.00/0.56 +1.00/0.27 +1.00/0.46 +0.54/0.70 +N/A +0.92/0.50 +0.08/0.91 +0.08/0.71 +0.62/0.94 +0.00/1.00 +0.00/0.92 +0.00/1.00 +rs232t1500 +0.71/0.82 +0.93/0.57 +0.93/0.57 +0.79/0.61 +1.00/0.46 +0.57/0.76 +N/A +0.71/0.82 +0.21/0.91 +0.07/0.77 +0.36/0.94 +0.00/1.00 +0.14/0.91 +0.00/1.00 +rs232t1600 +0.73/0.57 +0.73/0.57 +0.91/0.49 +0.55/0.66 +0.73/0.59 +0.27/0.90 +N/A +0.73/0.57 +0.18/0.91 +0.00/1.00 +0.00/0.83 +0.00/1.00 +0.00/0.91 +0.00/1.00 +s15850t100 +0.35/0.97 +0.77/0.94 +0.92/0.76 +0.88/0.97 +0.96/0.73 +0.85/0.94 +N/A +0.35/0.97 +0.12/1.00 +0.00/1.00 +0.12/1.00 +0.00/1.00 +0.04/1.00 +0.04/1.00 +s35932t100 +1.00/1.00 +0.87/0.98 +0.87/0.64 +0.93/1.00 +0.93/0.44 +1.00/0.98 +N/A +1.00/1.00 +0.20/1.00 +0.00/1.00 +0.00/0.97 +0.00/1.00 +0.13/1.00 +0.07/1.00 +s35932t200 +1.00/1.00 +1.00/0.98 +0.92/0.80 +1.00/1.00 +0.92/0.44 +1.00/0.99 +N/A +1.00/1.00 +0.00/1.00 +0.00/1.00 +0.00/1.00 +0.00/1.00 +0.00/1.00 +0.00/1.00 +s35932t300 +0.97/1.00 +0.94/0.98 +1.00/0.81 +1.00/1.00 +0.40/0.81 +0.97/0.97 +N/A +0.97/1.00 +0.63/1.00 +0.00/1.00 +0.09/1.00 +0.00/1.00 +0.57/1.00 +0.00/1.00 +s38417t100 +0.92/0.92 +1.00/0.82 +0.75/0.77 +1.00/0.92 +1.00/0.35 +0.75/0.90 +N/A +0.92/0.92 +0.33/0.95 +0.00/1.00 +0.00/1.00 +0.00/1.00 +0.42/0.94 +0.00/1.00 +s38417t200 +0.40/0.99 +0.53/0.86 +0.73/0.73 +0.47/0.93 +1.00/0.35 +0.73/0.90 +N/A +0.40/0.99 +0.27/0.95 +0.73/0.90 +0.00/1.00 +0.00/1.00 +0.27/0.94 +0.00/1.00 +s38417t300 +0.98/0.96 +0.98/0.82 +0.18/0.89 +0.98/0.91 +0.16/0.87 +0.95/0.84 +N/A +0.98/0.96 +0.14/0.95 +0.07/1.00 +0.00/0.98 +0.02/1.00 +0.23/0.95 +0.07/1.00 +s38584t100 +1.00/0.95 +1.00/0.87 +1.00/0.87 +1.00/0.92 +1.00/0.52 +1.00/0.93 +N/A +1.00/0.95 +0.22/1.00 +0.00/1.00 +0.00/1.00 +0.00/1.00 +0.22/1.00 +0.00/1.00 +s38584t200 +0.90/0.98 +0.49/0.87 +0.89/0.87 +0.39/0.95 +0.98/0.52 +0.84/0.94 +N/A +0.90/0.98 +0.02/1.00 +0.00/1.00 +0.00/1.00 +0.00/1.00 +0.02/1.00 +0.02/1.00 +s38584t300 +0.13/0.98 +0.08/0.88 +0.47/0.94 +0.23/0.93 +0.94/0.52 +0.45/0.94 +N/A +0.13/0.98 +0.01/1.00 +0.00/1.00 +0.00/1.00 +0.00/1.00 +0.01/1.00 +0.00/1.00 +Average +0.78/0.85 +0.80/0.76 +0.82/0.74 +0.81/0.77 +0.86/0.54 +0.70/0.88 +N/A +0.78/0.85 +0.18/0.95 +0.06/0.96 +0.07/0.97 +0.00/1.00 +0.13/0.95 +0.01/1.00 +detection considering unknown Trojans within unseen circuits. +The GNN framework underlying of TrojanSAINT is superior to +other models. Recall that others take the same feature vectors +as inputs; such direct comparison is fair. Second, thresholding +is crucial for high prediction performance for this task. +Relaxed Validation. We also study a “best case” validation, +using a leave-one-out split where validation and test sets are +the same. Such setting is often considered in the literature, as +it shows the best performance for any model and benchmark. +As indicated, however, it is not as realistic for HT detection. +With thresholding applied, we observe the following average +TPR/TNR values here:3 0.98/0.96 for TrojanSAINT, 0.93/0.93 +for XGBoost, 0.91/0.89 for FCNN, 0.98/0.96 for GNN-RE, +0.89/0.81 for logistic regression, and 0.91/0.994 for random +forest, respectively. Without thresholding applied, we observe +the following average TPR/TNR values: 0.41/0.99 for XGBoost, +0.09/1.00 for FCNN, 0.07/0.97 for GNN-RE, 0.09/1.00 for +logistic regression, 0.40/0.99 for random forest, and 0.11/1.00 +for SVM, respectively. TrojanSAINT is superior to almost all +methods across these two cases; only GNN-RE, and only with +thresholding applied, becomes a close contender. +Related Works. In Table III, we compare to more loosely +related works (Sec. III-A). Results are quoted and rounded. +Numbers of nodes/gates are reported as obtained from our +parser.4 The related works employ leave-one-out or “best case” +validation schemes; thus, we also report TrojanSAINT results +for such “best case” validation here. +TrojanSAINT outperforms these related works for all larger +benchmarks, where the ratio of HT gates/nodes to regular ones +is more challenging—this demonstrates superior scalability for +3Due to limited space, we refrain from reporting a table for this scenario. +4Number of nodes/gates may vary across ours and related works, depending +on parsing approach, technology library etc., but overall ranges remain similar. +TABLE III +BENCHMARK PROPERTIES; TPR/TNR RESULTS FOR RELATED WORKS +TrustHub +Benign +HT +Ratio of Nodes, +R-HTD [18] +[19] +[20] +TrojanSAINT +Benchmark +Nodes +Nodes +HT to Benign +(Orig. Samples) +rs232t1000 +202 +13 +0.064 +1.00/0.94 +1.00/0.99 +1.00/1.00 +1.00/0.94 +rs232t1100 +204 +12 +0.059 +1.00/0.93 +0.50/0.98 +1.00/1.00 +1.00/0.93 +rs232t1200 +199 +17 +0.085 +0.97/0.96 +0.88/1.00 +1.00/1.00 +0.82/0.96 +rs232t1300 +204 +9 +0.044 +1.00/0.95 +1.00/1.00 +0.86/1.00 +1.00/0.98 +rs232t1400 +202 +13 +0.064 +1.00/0.98 +0.98/1.00 +1.00/1.00 +1.00/0.96 +rs232t1500 +202 +14 +0.069 +1.00/0.94 +0.95/1.00 +1.00/1.00 +1.00/0.94 +rs232t1600 +203 +11 +0.054 +0.97/0.92 +0.93/0.99 +0.78/0.99 +1.00/0.88 +s15850t100 +2,156 +26 +0.012 +0.74/0.93 +0.78/1.00 +0.08/1.00 +0.88/0.97 +s35932t100 +5,426 +15 +0.003 +0.80/0.69 +0.73/1.00 +0.08/1.00 +1.00/0.97 +s35932t200 +5,426 +12 +0.002 +0.08/1.00 +0.08/1.00 +0.08/1.00 +1.00/1.00 +s35932t300 +5,427 +35 +0.006 +0.84/1.00 +0.81/1.00 +0.92/1.00 +1.00/1.00 +s38417t100 +5,329 +12 +0.002 +0.67/1.00 +0.33/1.00 +0.09/1.00 +1.00/0.97 +s38417t200 +5,329 +15 +0.003 +0.73/0.99 +0.47/1.00 +0.09/1.00 +1.00/0.97 +s38417t300 +5,329 +44 +0.008 +0.89/1.00 +0.75/1.00 +1.00/1.00 +1.00/0.96 +s38584t100 +6,473 +9 +0.001 +N/A +N/A +0.17/1.00 +1.00/0.99 +s38584t200 +6,473 +83 +0.013 +N/A +N/A +0.18/1.00 +1.00/0.98 +s38584t300 +6,473 +731 +0.113 +N/A +N/A +0.03/1.00 +0.99/0.95 +Average +3,250 +63 +0.035∗ +0.84/0.95 +0.72/1.00 +0.55/1.00 +0.98/0.96 +0.019∗ +∗The first value is averaged across the column; the second value, more representative of +the overall imbalance, is based on re-calculating the ratio using the average node counts. +ours. 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Available: https://trust-hub.org/#/benchmarks/ +chip-level-trojan +[18] K. Hasegawa, S. Hidano, K. Nozawa, S. Kiyomoto, and N. Togawa, +“R-htdetector: Robust hardware-trojan detection based on adversarial +training,” 2022. [Online]. Available: https://arxiv.org/abs/2205.13702 +[19] K. Hasegawa, M. Yanagisawa, and N. Togawa, “Trojan-feature extraction +at gate-level netlists and its application to hardware-trojan detection using +random forest classifier,” in International Symposium on Circuits and +Systems (ISCAS), 2017, pp. 1–4. +[20] T. Kurihara and N. Togawa, “Hardware-trojan classification based on the +structure of trigger circuits utilizing random forests,” in International +Symposium on On-Line Testing and Robust System Design (IOLTS), 2021, +pp. 1–4. +[21] G. Klambauer, T. Unterthiner, A. Mayr, and S. Hochreiter, “Self- +normalizing neural networks,” Advances in neural information processing +systems, vol. 30, 2017. + diff --git a/ENFKT4oBgHgl3EQfZy4X/content/tmp_files/load_file.txt b/ENFKT4oBgHgl3EQfZy4X/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f464beba301ea57d68f74dee6afdef0a0aab0b6c --- /dev/null +++ b/ENFKT4oBgHgl3EQfZy4X/content/tmp_files/load_file.txt @@ -0,0 +1,1056 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf,len=1055 +page_content='© 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' This is the author’s version of the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' The definitive Version of Record is published in the IEEE International Symposium on Circuits and Systems (ISCAS), 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' TrojanSAINT: Gate-Level Netlist Sampling-Based Inductive Learning for Hardware Trojan Detection Hazem Lashen, Lilas Alrahis, Johann Knechtel, and Ozgur Sinanoglu New York University Abu Dhabi {hl3372, lma387, jk176, os22}@nyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='edu Abstract—We propose TrojanSAINT, a graph neural network (GNN)-based hardware Trojan (HT) detection scheme working at the gate level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Unlike prior GNN-based art, TrojanSAINT enables both pre-/post-silicon HT detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' TrojanSAINT leverages a sampling-based GNN framework to detect and also localize HTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' For practical validation, TrojanSAINT achieves on average (oa) 78% true positive rate (TPR) and 85% true negative rate (TNR), respectively, on various TrustHub HT benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' For best-case validation, TrojanSAINT even achieves 98% TPR and 96% TNR oa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' TrojanSAINT outperforms related prior works and baseline classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We release our source codes and result artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Index Terms—Hardware Security, Trojan Detection, GNNs I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' INTRODUCTION Integrated circuit (IC) design and manufacturing has become an increasingly outsourced process that involves various third parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' While this has allowed to increase both productivity and complexity of ICs, it has also made them more vulnerable to the introduction of hardware Trojans (HTs), among other threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' HTs are malicious circuitry, causing system failure, leaking sensitive information, etc [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Thus, methods to accurately check for HTs become increasingly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Conventional methods for HT detection include code re- view [3] and verification against a “golden reference”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=', a trusted, HT-free version of the design [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' However, the former is prone to errors, especially for complex ICs, and the latter is not always feasible, especially when untrusted parties are engaged in the design process [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Other meth- ods have been proposed as well, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=', utilizing side-channel fingerprinting [5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' however, such are limited to post-silicon HT detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Researchers have shown that machine learning (ML) can successfully adapt to a wide variety of HTs, without necessitating new techniques for detecting new HT designs [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Using graph neural networks (GNNs) is an emerging and promising method toward this end [3], [7]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Thanks to their ability to work on graph-structured data – such as circuits – GNNs can leverage both a) the features of each gate and b) the overall structure of the design for the prediction of HTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Still, prior art for GNN-based HT detection suffers from the following limitations (also summarized in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' HT Localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' State-of-the-art GNN-based detection schemes, GNN4TJ [3] and HW2VEC [7], predict whether a design contains a HT or not, but they cannot localize HTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' However, localizing HTs is essential to identify the part of the design at fault and name the responsible, malicious party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Earlier works [3], [7], [10] are limited to register transfer level (RTL), unable to handle gate-level netlists (GLNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' TABLE I COMPARISON OF GNN-BASED HT DETECTION SCHEMES Method HT HT Gate-Level Pre- Post- Detection Localization Netlist Silicon Silicon GNN4TJ [3] Yes No No Yes No HW2VEC [7] Yes No No Yes No GRFTL [10] Yes Yes No Yes No Our Work Yes Yes Yes Yes Yes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Concept of TrojanSAINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Such methods are restricted to pre-silicon assessment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' they cannot detect HTs in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Note that only schemes which can work on GLNs allow for pre- and post-silicon detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Associated Research Challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Developing a GNN-based HT detection and localization scheme that can work on GLNs imposes the following research challenges (RC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' RC1: GLN Complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Compared to RTL, GLN designs are more complex to analyze, as GLNs are flattened (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=', hierarchical information is lost) and also considerably larger, in the range of thousands or even millions of gates and wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' RC2: Imbalanced Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' HTs are stealthy and small in size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' HT gates represent a very small percentage, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='14– 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='29% or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='94% on average for the TrustHub suite considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Thus, a highly imbalanced dataset arises (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=', the ratio of regular to HT gates reaches up to 719× for the TrustHub suite), which is difficult to handle for any ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Our Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Here, we propose TrojanSAINT, a GNN- based method for HT detection and localization that works well on large-scale GLNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' The concept is outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' As indicated, the graph representations of GLNs are complex and large, which makes them difficult to handle with traditional architectures such as graph convolutional networks (GCNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' This motivates our decision to, without loss of generality (w/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' ), use GraphSAINT [11] for our methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Graph- SAINT is a well-established, sampling-based approach that extracts smaller sub-graphs for training from the larger original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' It has shown good performance for various tasks [11]– arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='11804v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='CR] 27 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Overview of TrojanSAINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Black arrows follow the inference process, orange arrows follow the additional steps needed for training and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' In this example, the thresholding value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' [13], but it has not been considered for HT detection until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We summarize our contributions as follows: 1) A parser for GLN-to-graph conversion (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' II-A) which performs feature extraction tailored for HT detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' 2) A GNN-based method for detection and localization of HT in GLNs (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' II-B), addressing RC1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' 3) A procedure for tuning of the classification thresholds to obtain more accurate predictions, addressing RC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' 4) We demonstrate that our scheme is competitive to tra- ditional ML baselines and prior art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We also verify the generalization ability of our scheme – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=', good prediction accuracy for unknown HTs on unseen GLNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' 5) We open-source our scheme and related artifacts from our experimental study [https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='com/DfX-NYUAD/ TrojanSAINT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' TROJANSAINT METHODOLOGY An overview of our methodology is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Next, we describe all relevant details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' GLN Parsing and Feature Vectors Parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We develop a parser that converts GLNs (given in Verilog format) into unweighted and undirected graphs, where nodes represent gates and edges represent wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We are dis- carding directionality for improved representation learning [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Given a set of GLNs, our parser generates one large single graph, consisting of multiple disjoint graphs, where nodes are labeled as ‘train,’ ‘validation’ or ‘test,’ depending on the designation of the GLN they belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' The graph is encoded as an adjacency matrix A following a standard procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Feature Vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Our parser also generates a matrix X of feature vectors for all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Vectors cover the following: Gate type, represented via one-hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' From exper- imentation, we are more interested in the functionality of the gate over the exact implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' That is, we group functionally related gates together, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=', all AND gates are grouped regardless of the number of inputs and the driver strengths that the different AND gates support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Input, output degrees of gates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=', the number of incoming and outgoing connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Shortest distances to primary inputs/outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' For gates not directly connected with a primary input/output, a breadth- first search is conducted to obtain shortest distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' For training and validation, we also use a binary label vector which marks each node as part of some HT or as regular/benign gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' The related information is derived during parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' GNN Implementation and Application Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We utilize GraphSAINT [11] for sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Further, we utilize the GNN architecture of GNN-RE [15], along with GraphSAGE [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We tune the classification thresholds for more accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We further utilize a practical validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' For training and inference, we employ standard procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' GNN Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We consider an undirected graph G (V, A) for representing a GLN, where V is the set of vertices/nodes/gates, and the adjacency matrix of the graph is A, where Au,v = 1 and Av,u = 1 if there exists an edge/wire from vertex/gate u to vertex/gate v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Each vertex u in the initial graph G has a feature vector xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' This vector represents the node embedding at layer zero of the GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' The embedding of node u is iteratively updated by the GNN, by aggregating the embedding of the node and its neighbors N(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' The embedding of a node u after l GNN layers, h(l) u , is given by: a(l) u = AGGREGATE(l) �� h(l−1) v : v ∈ N(u) �� (1) h(l) u = COMBINE(l) � h(l−1) u , a(l) u � (2) GNN architectures are defined by their implementation of AGGREGATE(·) and COMBINE(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' For example, Graph- SAGE [16], which we also use here, works as follows: h(l) u = σ([Wl · AGG({h(l−1) v , ∀v ∈ N(u)}), Blh(l−1) u ]) (3) AGG = � v∈N(u) h(l−1) v |N(u)| (4) where σ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=') is an activation function such as ReLU and Wl and Bl are trainable weight matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' In GraphSAGE, the embedding of node h(l) u is determined by first concatenating the node’s features from the previous layer h(l−1) u with the output of the AGG function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Then the Wl and Bl transformations learns the important components of the neighbors’ features and the node u, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' GraphSAGE is compatible with different AGG functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Here, we use the mean aggregator as described in Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' From experimentation, we observe that the classification threshold plays a significant role for prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' This is because of the considerably imbalanced datasets (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' I, RC2), where the GNN model as is can predict the minority class, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=', HT nodes, only with low confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' The goal of thresholding is to determine a sufficiently small value so that HT nodes/gates are classified as such the moment the GNN captures any hint of malicious structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' In other words, thresholding allows the GNN to focus more on the minority class, improving the performance of the entire model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' W/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=', we tune the threshold between 0–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='5 in steps of 1,000 and select the threshold that yields the best score on validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Here, best score refers to the average of true positive rate (TPR) and true negative rate (TNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' 首Algorithm 1 TrojanSAINT training algorithm Input: Training graph G (V, A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Ground truth Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Sampler RWS Output: Trained GNN 1: Compute normalization coefficients α, λ using RWS 2: for each mini-batch do 3: Gs (Vs, As) ← Sampled sub-graph of G using RWS 4: Build GNN on Gs 5: {yu | u ∈ Vs} ← Propagating α-normalized {xu | u ∈ Vs} 6: Propagating λ-normalized loss L (yu, yu) to update weights 7: end for Algorithm 2 TrojanSAINT inference algorithm Input: Flattened netlist N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Trained GNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Threshold th Output: Trojan classification of all nodes/gates 1: Initiate G (V, A) with V ← GLN to graph(N) 2: for each u ∈ V do 3: zu ← GNN(u) ▷ Compute embedding 4: cu ← fc(zu, th) ▷ Classify node u based on the threshold 5: end for Practical Validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We propose an approach where pre- dictions are made on unknown HTs residing within circuits that are neither seen during training nor have golden references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' This represents a real-world scenario, where security engineers do not know in advance which HT to expect, if any at all, and further need to test circuits without golden references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Prior art did not necessarily consider such practical validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' First, we construct sub-graphs using a standard random-walk sampler (RWS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' TrojanSAINT’s training procedure is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Due to the RWS, the network can become biased towards frequently sampled nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' To alleviate this issue, we follow the normalization technique of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We use stochastic gradient descent as optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Gs is sampled for each minibatch and a GNN is built on the sub-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' The cross-entropy loss is calculated for each node in the sub-graph and the GNN weights are then updated by backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' See Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' For all test nodes in the graph, node embeddings are calculated and passed to a fully-connected layer with softmax activation, to compute class probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We then apply our thresholding technique, and finally convert class probabilities into labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' EXPERIMENTAL STUDY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Setup Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We use Python for coding and bash scripts for job/data management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' TrojanSAINT extends on GNN- RE, which is obtained from [14] and is implemented in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Our TrojanSAINT platform is available online [https: //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='com/DfX-NYUAD/TrojanSAINT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Baseline models are implemented using Scikit-Learn, except the fully-connected neural network (FCNN) in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Experiments for GNN-RE, TrojanSAINT and FCNN are conducted on a high-performance cluster with 4x Nvidia V100 GPUs and 360GB RAM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' experiments for others are conducted on a workstation with Intel i7 CPU and 16GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Training of GNN-RE and TrojanSAINT takes ≈15–30 minutes per model, FCNN ≈10 minutes per model, and all others ≈3 minutes in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' All inference takes few seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Benchmarks and Model Building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We use 17 exemplary GLN benchmarks from the TrustHub suite [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' For each benchmark, a respective model is trained from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' For our practical validation, each model does not get to see the design to be tested at all during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='1 We note that random seeds used in TrojanSAINT’s com- ponents affect performance significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Thus, we conduct w/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' 6 runs with different seeds and report only results for each model that performs best on its validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Prior Art, Comparative Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' From Table I, recall that none of the prior art in GNN-based HT detection works on GLNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Thus, a direct comparison is not practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' However, we consider the following works for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' GNN-RE [15]: Proposed for reverse engineering of GLNs, it could also be utilized for HT detection and localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' This is because GNN-RE seeks to classify gates/nodes from flattened GLNs into the circuit modules they belong to;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' TrojanSAINT’s task of classifying gates/nodes into begin or HT-infested ones is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Related Works [18]–[20]: ML-based, not GNN-based, HT detection schemes that are working on GLNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Unlike ours, these works employ elaborate feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Also, these works do not offer native HT localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We also implement and run the following well-known baseline classifiers for a further comparative study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' XGBoost: A decision tree (DT)-based model that uses an ensemble of sequentially added DTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' DTs are added aiming to minimize errors of their predecessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Random Forest: A DT-based model that uses an ensemble of DTs trained on subsets of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Logistic Regression: A classification algorithm that utilizes the sigmoid function on independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Support Vector Machine (SVM): A classification model that generates a hyperplane to separate different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' FCNN: We implement a three-layer network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' each layer use the SELU activation function [21] and batch normaliza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' The final layer uses sigmoid activation to calculate classification probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' All these classifiers work on tabular, non-graph data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' thus, we provide them with the feature vectors as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' All classifiers, except SVM, output probabilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' thus, we can study them considering our proposed thresholding as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Results Practical Validation and Impact of Thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' In Table II, we report TPR/TNR results for practical validation across two scenarios: with thresholding versus without.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='2 First, the results show that TrojanSAINT outperforms other methods for this realistic but challenging scenario of HT 1For example, if rs232t1000 is to be tested, none of the other rs232 designs are used for training, only for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' For s15850t100, the only s15850 design in the suite, we randomly select three other designs for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' 2Since thresholding is part of our proposed scheme, we do not consider TrojanSAINT without.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We implement the same thresholding strategy (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' II-B) for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' SVM directly separates data into classes without computing probabilities, making thresholding not applicable (N/A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' TABLE II TPR/TNR RESULTS FOR PRACTICAL VALIDATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' BEST RESULTS, CONSIDERING AVERAGE OF TPR AND TNR, ARE MARKED IN BOLDFACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' TrustHub All With Thresholding Others Without Thresholding Benchmark TrojanSAINT XGBoost FCNN GNN-RE Logistic Random SVM TrojanSAINT XGBoost FCNN GNN-RE Logistic Random SVM Regression Forest Regression Forest rs232t1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='00/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='00/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='85/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='64 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='00/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='13/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='01/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='00 detection considering unknown Trojans within unseen circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' The GNN framework underlying of TrojanSAINT is superior to other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Recall that others take the same feature vectors as inputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' such direct comparison is fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Second, thresholding is crucial for high prediction performance for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Relaxed Validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We also study a “best case” validation, using a leave-one-out split where validation and test sets are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Such setting is often considered in the literature, as it shows the best performance for any model and benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' As indicated, however, it is not as realistic for HT detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' With thresholding applied, we observe the following average TPR/TNR values here:3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='98/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='96 for TrojanSAINT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='93/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='93 for XGBoost, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='91/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='89 for FCNN, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='98/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='96 for GNN-RE, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='89/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='81 for logistic regression, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='91/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='994 for random forest, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Without thresholding applied, we observe the following average TPR/TNR values: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='41/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='99 for XGBoost, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='09/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='00 for FCNN, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='07/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='97 for GNN-RE, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='09/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='00 for logistic regression, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='40/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='99 for random forest, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='11/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='00 for SVM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' TrojanSAINT is superior to almost all methods across these two cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' only GNN-RE, and only with thresholding applied, becomes a close contender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Related Works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' In Table III, we compare to more loosely related works (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Results are quoted and rounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Numbers of nodes/gates are reported as obtained from our parser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='4 The related works employ leave-one-out or “best case” validation schemes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' thus, we also report TrojanSAINT results for such “best case” validation here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' TrojanSAINT outperforms these related works for all larger benchmarks, where the ratio of HT gates/nodes to regular ones is more challenging—this demonstrates superior scalability for 3Due to limited space, we refrain from reporting a table for this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' 4Number of nodes/gates may vary across ours and related works, depending on parsing approach, technology library etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=', but overall ranges remain similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' TABLE III BENCHMARK PROPERTIES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' TPR/TNR RESULTS FOR RELATED WORKS TrustHub Benign HT Ratio of Nodes, R-HTD [18] [19] [20] TrojanSAINT Benchmark Nodes Nodes HT to Benign (Orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Samples) rs232t1000 202 13 0.' metadata={'source': 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+page_content='55/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='98/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content='019∗ ∗The first value is averaged across the column;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' the second value, more representative of the overall imbalance, is based on re-calculating the ratio using the average node counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' For the smaller benchmarks, which are not representative of real IC designs, related works achieve better results presum- ably due to feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' In fact, up to 76 features are considered in [19], [20] which reflects on considerable efforts, whereas for ours, some simple feature vectors suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' CONCLUSION We have developed TrojanSAINT, a GNN-based method for detection and localization of HTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' We overcome the HT- inherent issue of class imbalance through threshold tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Through practical validation, ours is capable of generalizing to circuits and HTs it has not seen for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Our method outperforms prior art and a number of strong ML baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' The use of a GNN framework renders TrojanSAINT simple yet competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' For future work, we will study the role of different feature vectors in more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Rajendran, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Zhang, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Sinanoglu, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENFKT4oBgHgl3EQfZy4X/content/2301.11804v1.pdf'} +page_content=' Karri, “High-level synthesis for security and trust,” in International 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Cat´olica de Chile +Casilla 306, Correo 22, Santiago de Chile, Chile +Ra´ul Man´asevich +Departamento de Ingenier´ıa Matem´atica +and Centro de Modelamiento Matem´atico (CNRS IRL2807) +FCFM, Universidad de Chile +Casilla 170 Correo 3, Santiago, Chile +Abstract. We investigate the monotonicity of the minimal period of the periodic +solutions of some quasilinear differential equations and extend results for p = 2 due to +Chow and Wang, and to Chicone, to the case of the p-Laplace operator. Our main +result is the monotonicity of the period for positive solutions of a nonlinear Euler- +Lagrange equation for a minimization problem related with a fundamental interpolation +inequality. In particular we generalize to p greater than 2 recent results of Benguria, +Depassier and Loss. +1. Introduction +In this paper we study monotonicity properties of the minimal period of positive peri- +odic solutions of +� +φp(w′) +�′ + V′(w) = 0 , +(1) +where p ≥ 2, φp(s) = |s|p−2s, and V : R → R is smooth. The potential function V(w) is +assumed to be non-negative for w ≥ 0, V(0) > 0, it has a minimum at w = A > 0 with +V(A) = 0 = V′(A), and satisfies some additional conditions listed in Section 3, which +guarantee that (1) has positive periodic solutions enclosing the critical point (A, 0) in +the phase plane (w, w′). +Date: January 6, 2023. File: DGHM2023.tex. +2020 Mathematics Subject Classification. Primary: 34C25, 35J92. Secondary: 34L30, 34C23. +Key words and phrases. Hamiltonian systems, quasilinear elliptic equations, p-Laplace operator, pe- +riodic solutions, period, energy levels. +∗ Corresponding author: Jean Dolbeault. + +2 +J. DOLBEAULT, M. GARC´IA-HUIDOBRO, AND R. MAN´ASEVICH +The energy E = 1 +p |w′|p + V(w) is conserved if w solves (1) and we are interested in +the positive periodic solutions with energy less than E∗ := V(0) which are enclosed by +the homoclinic orbit attached to (w, w′) = (0, 0). We further assume that V is such that +these solutions are uniquely determined, up to translations, by the energy level E, with +minimal period T(E). +The purpose of this paper is to study under which conditions T is an increasing function +of E in the range 0 ≤ E ≤ E∗ where E∗ is the energy level of the homoclinic orbit. +Furthermore we will consider the asymptotic behaviour of T(E) as E → 0+ and as +E → (E∗)−. Surprisingly enough, the cases p = 2 and p > 2 differ as E → 0+. +Our first result is an extension to p > 2 of a result of Chow and Wang [8, Theorem 2.1]. +Theorem 1. Let p > 2 and assume that V is a C2 function on R+ such that V(A) = +0 = V′(A) and V′′ > 0 on (0, B) with B := min{w > A : V(w) ≥ V(0)}. If w �→ +|V′(w)|2 − p′ V(w) V′′(w) is a positive function, then E �→ T(E) is increasing on (0, E∗). +Notice that w �→ |V′(w)|2 − p′ V(w) V′′(w) is a positive function if and only if w �→ +V(w) |V′(w)|−p′ is a monotone increasing function. +Our second result is also an extension to p > 2 of the monotonicity result in [7, +Theorem A] under Chicone’s condition, which is also a growth condition, but of higher +order in the derivatives. +Theorem 2. Let p > 2 and assume that V is a C3 function on R+ such that V(A) = +0 = V′(A) and let B := min{w > A : V(w) ≥ V(0)}. If V/(V′)2 is a convex function, +then E �→ T(E) is increasing on (0, E∗). +A central motivation for this paper arises from the study of the minimization problem +µ(λ) := +inf +f∈W1,p(S1)\{0} +∥f ′∥2 +Lp(S1) + λ ∥f∥2 +Lp(S1) +∥f∥2 +Lq(S1) +(2) +where q > p is an arbitrary exponent and S1 is the unit circle. The problem can also be +seen as the search for the optimal constant in the interpolation inequality +∥f ′∥2 +Lp(S1) + λ ∥f∥2 +Lp(S1) ≥ µ(λ) ∥f∥2 +Lq(S1) +∀ f ∈ W1,p(S1) . +Testing the inequality with constant functions shows that µ(λ) ≤ ¯µ(λ) := λ |S1| +2 +p − 2 +q . If +p = 2, it is well known from the carr´e du champ method [2, 3] that equality holds if and +only if λ ≤ d/(q−2). If λ > d/(q−2), we have µ(λ) < ¯µ(λ) and optimal functions are non +constant, so that symmetry breaking occurs. The minimization problem problem with +p > 2 was studied in [18]. There is an optimal function for (2) and the corresponding +Euler-Lagrange equation turns out to be the nonlinear differential equation with nonlocal +terms given by +− ∥f ′∥2−p +Lp(S1) +� +φp(f ′) +�′ + λ ∥f∥2−p +Lp(S1) φp(f) = µ(λ) ∥f∥2−q +Lq(S1) φq(f) , +(3) + +MONOTONICITY OF THE PERIOD +3 +where we look for positive solutions on W 1,p(S1)\{0} or equivalently positive 2π-periodic +solutions on R. So far, we do not know the precise value of λ for which there is symmetry +breaking but according to [18] rigidity holds if 0 < λ < λ1 for some explicit λ1 > 0, where +rigidity means that any positive solution of (3) is a constant. In that range, we have +µ(λ) = ¯µ(λ). On the contrary, one can prove that symmetry breaking occurs if λ > λ2 +for some λ2 > λ1, so that µ(λ) < ¯µ(λ) and (3) admits non-constant positive solutions for +any λ > λ2. Using homogeneity, scalings and a suitable change of variables, the study +of (3) is reduced in [18] to the study of positive periodic solutions on R of +� +φp(w′) +�′ + φq(w) − φp(w) = 0 . +(⋆) +In this equation, there are no non-local terms but the minimal period of periodic solutions +is no more given. Equation (⋆) enters in the framework of (1) with A = 1 and potential +V(w) = 1 +q |w|q − 1 +p |w|p − +� 1 +q − 1 +p +� +, +(4) +so that E∗ = 1/p − 1/q. Positive periodic solutions exist only if the energy level satisfies +the condition E < E∗. +Again, let T(E) be the minimal period of such a solution. +Theorems 1 and 2 do not apply easily and we shall prove directly the following result, +which is the main contribution of this paper. +Theorem 3. Let p and q be two exponents such that 2 < p < q and consider the +positive periodic solutions of (⋆). Then the map E �→ T(E) is increasing on (0, E∗) with +limE→0+ T(E) = 0 and limE→(E∗)− T(E) = +∞. +The study of (3) is motivated by rigidity and symmetry breaking results associated +with interpolation inequalities on the unit sphere Sd in one and higher dimensions, that is, +d ≥ 1. If p = 2, a precise description of the threshold value of λ is known in the framework +of Markov processes if q is not too large (see [3] for an overview with historical references +that go back to [2]) and from [5, 11, 14, 15, 16, 17, 13] using entropy methods applied to +nonlinear elliptic and parabolic equations; also see [12] for an overview and extensions to +various related variational problems. +Almost nothing is known beyond [18] if p > 2, even for d = 1. +Our results are +a contribution to a better understanding of the fundamental properties of the solutions +of (1) in the simplest of the cases when p > 2. Without the Assumption that V′(A) = 0 in +Theorems 1 and 2 (which is also satisfied in Theorem 3), it is easy to give similar results +so that E �→ T(E) is decreasing, but in phase plane the solutions are not described +anymore by orbits enclosed by a homoclinic orbit. Some comments on this issue can be +found in Section 2. +In dimension d = 1, the bifurcation problem (3) degenerates in the limit case p = 2, +for which λ1 = λ2 = 1/(q − 2) according to [2]. +We refer to [4, Section 1] for an +introduction to the minimization problem (2) with p = 2, the issue of the branches and +the monotonicity of the period problem. Proving that symmetry breaking occurs if and +only if λ > 1/(q −2) can be reduced to a proof of the monotonicity of the minimal period +using Chicone’s criterion [7, Theorem A]. The study of bifurcation problems using the + +4 +J. DOLBEAULT, M. GARC´IA-HUIDOBRO, AND R. MAN´ASEVICH +period function goes back to [23] in case of equations with cubic non-linearities and was +later extended to various classes of Hamiltonian systems in [22, 21, 10, 9, 19]. +If p′ = p/(p − 1) is the H¨older conjugate of the exponent p and +H(u, v) := V(u) + 1 +p′ |v|p′ , +Equation (1) can be rewritten as the Hamiltonian system of equations +u′ = ∂H +∂v = φp′(v) +and +v′ = − ∂H +∂u = − V′(u) +with w = u and w′ = φp′(v). Although this Hamiltonian structure may superficially +look similar to the conditions of [22, Theorem 1], we have a definitely different set of +assumptions. In [21], a much larger set of Hamiltonian systems is considered but again +our assumptions differ, for instance for the simple reason that the function φp′ is not +of class C2. Further references on the period function can be found in [24]. There are +various other extensions of Chicone’s result [7], see for instance [6]. Also notice that there +is a computation in [6, Section 4] which turns out to be equivalent to an argument used +in the proof of our Theorem 4 (see below in Section 2), although it is stated neither in +that form nor as in Theorem 1. The Hamiltonian version of the method has interesting +applications to Lotka-Volterra systems. +The monotonicity of the minimal period as a function of the energy level is a question +of interest by itself and particularly in the model case of the potential V as in (4), even +in the case p = 2. We quote from [4] that: “It is somewhat surprising that, despite its +ubiquity, the monotonicity of the period function for [this problem] in full generality was +only established recently.” In [20], Miyamoto and Yagasaki proved the monotonicity of +the period function for p = 2 and for q an integer. In [24], Yagasaki generalized the result +to all values of q > 2. Both papers, [20, 24], rely on Chicone’s criterion which is difficult +to apply to non-integer values of q. The purpose of Benguria, Depassier and Loss in [4] +was to give a simplified proof of the monotonicity of the period of the positive solutions +of w′′ + wq−1 − w = 0 (corresponding to p = 2 in our notations). +We point out that in many situations in the paper we will consider the equation +� +φp(w′) +�′ + V ′(w) = 0 +(5) +where V is a potential of class C2 defined on R such that +There are a, b ∈ R with a < 0 < b such that V (a) = V (b) = E∗ > 0, +V ′(a) = V (0) = V ′(0) = 0, V ′′(0) ̸= 0, +and 0 < V (w) < E∗ for all w ∈ (a, 0) ∪ (0, b). +(H1) +The potential V (w) achieves its minimum on (a, b) at x = 0. The relationship of V +with V is given by V (w) = V(w + A), a = − A and b = B − A. The origin w = 0, w′ = 0 +is a stationary point of (5) giving rise to a center surrounded by closed periodic orbits +with minimal period T(E), such that these periodic orbits are enclosed by a homoclinic +orbit attached to (a, 0). + +MONOTONICITY OF THE PERIOD +5 +This paper is organized as follows. Section 2 is devoted to the proof of the p-Laplacian +version of results which are classical when p = 2 and are summarized in Theorems 1 +and 2. +We are not aware of such statements in the existing literature but they are +natural extensions of the case p = 2 and might already be known, so we do not claim +any deep originality. +The result of Theorem 3 is by far more difficult. +In Section 3 +we start with problem (1) by making a change of variables and obtain an expression +for the minimal period following Chicone’s ideas. We also prove some properties of the +minimal period when the energy goes to zero and when it goes to the homoclinic level E∗. +In Section 4 we prove the monotonicity of the minimal period extending, in particular, +the results of [4] for p = 2 to the more general case of the one-dimensional p-Laplacian +operator w �→ +� +φp(w′) +�′, with p > 2. Our main result (Theorem 3) is proved in Section 5, +the proof is highly non-trivial. +2. A p-Laplacian version of some classical results +This section is devoted to the proof of Theorems 1 and 2. We also provide a slightly +more detailed statement of Theorem 1. +We begin by extending [8, Theorem 2.1] by Chow and Wang to the p-Laplacian situa- +tion when p ≥ 2. +We recall that p′ = p/(p − 1) denotes the H¨older conjugate of p. Equation (5) has a +first integral given by +1 +p′ |w′|p + V (w) = E +(6) +for any energy level E ∈ (0, E∗) and the minimal period is given in terms of the energy +by +T(E) = +2 +p′1/p +� w2(E) +w1(E) +dw +� +E − V (w) +�1/p +(7) +where wi(E), i = 1, 2, are two roots of V (w) = E such that +a < w1(E) < 0 < w2(E) < b +and +V (w) < E +∀ w ∈ +� +w1(E), w2(E) +� +. +At this point, let us notice that the map E �→ T(E) is a continuous function if we assume +that w V ′(w) > 0 for any w ∈ (a, 0)∪(0, b), but that it is not the case if V admits another +local minimum than w = 0 in the interval (a, b). Let us define +γ(w, E) := p′ � +E − V (w) +� +, +R(w) := V ′(w)2 − p′ V (w) V ′′(w) +and notice that +∂γ +∂w = − p′ V ′(w) +and +∂γ +∂E = p′ . +The following result is a detailed version of Theorem 1. + +6 +J. DOLBEAULT, M. GARC´IA-HUIDOBRO, AND R. MAN´ASEVICH +Theorem 4. Let p ≥ 2 and consider Equation (5) where we assume that V satisfies (H1). +With the above notations, for any E ∈ (0, E∗), it holds that +dT +dE (E) = +2 +p′ E +� w2(E) +w1(E) +R(w) +γ(w, E)1/p V ′(w)2 dw +(8) +if the integral in the right-hand side is finite. Thus if R is positive on (a, 0) ∪ (0, b), then +the minimal period is increasing. +Notice that from Assumption (H1), we know that V (a) = E∗ > 0 and V ′(a) = 0 so +that limw→a+ V (w) |V ′(w)|−p′ = +∞ and +� +V +|V ′|p′ +�′ += +R V ′ +|V ′|p′+2 +which is incompatible with R being a negative valued function in a neighbourhood of +w = a+. If we remove the assumption that V ′(a) = 0, then it makes sense to assume +that R is a negative function on (a, 0) ∪ (0, b). +In this case, the minimal period is +decreasing. +Proof. The proof relies on the same strategy as for [8, Theorem 2.1]. We skip some details +and emphasize only the changes needed to cover the case p > 2. Let us set +I(E) := +� w2(E) +w1(E) +γ(w, E)1/p′ dw +and +J(E) := +� w2(E) +w1(E) +� +γ(w, E) − p′ E +� +γ(w, E)1/p′ dw . +By differentiating with respect to E, we obtain +dI +dE(E) = +� w2(E) +w1(E) +dw +γ(w, E)1/p = 1 +2 T(E) +and +dJ +dE (E) = +� w2(E) +w1(E) +γ(w, E) − p′E +γ(w, E)1/p +dw , +which implies that +dJ +dE (E) = I(E) − p′ E dI +dE (E) . +(9) +Differentiating once more with respect to E, we get +d2J +dE2(E) = (1 − p′) dI +dE(E) − p′ E d2I +dE2(E) . +(10) +On the other hand, by integrating by parts in +� w2 +w1 +γ +p′+1 +p′ +V ′2 − V V ′′ +V ′2 +dw = +� w2 +w1 +γ +p′+1 +p′ +� V +V ′ +�′ +dw = − p′ + 1 +p′ +� w2 +w1 +γ +1 +p′ V +V ′ +∂γ +∂w dw , +we obtain +J(E) = − +p′ +p′ + 1 +� w2(E) +w1(E) +γ(w, E) +p′+1 +p′ +V ′(w)2 − V (w) V ′′(w) +V ′(w)2 +dw + +MONOTONICITY OF THE PERIOD +7 +by definition of J and γ. See [8] for further details in the case p = 2. By differentiating +twice this expression of J(E) with respect to E, we obtain +d2J +dE2(E) = − p′ +� w2(E) +w1(E) +V ′(w)2 − V (w) V ′′(w) +γ(w, E)1/p V ′(w)2 +dw . +Since T(E) = 2 dI +dE(E), we learn from (10) that +p′ E +2 +dT +dE (E) = p′ E d2I +dE2(E) += (1 − p′) dI +dE (E) − d2J +dE2(E) += (1 − p′) +� w2(E) +w1(E) +dw +γ(w, E)1/p + p′ +� w2(E) +w1(E) +V ′(w)2 − V (w) V ′′(w) +γ(w, E)1/p V ′(w)2 +dw += +� w2(E) +w1(E) +R(w) +γ(w, E)1/p V ′(w)2 dw . +This concludes the proof of (8). +□ +Proof of Theorem 2. Let us consider again Equation (5) with a potential V which satis- +fies (H1). We adapt the proof of [7, Theorem A] to the case p > 2. Let us consider the +function +h(w) := w +|w| +� +V (w) +(11) +for any w ∈ (a, 0) ∪ (0, b) and extend it by h(0) = 0 at w = 0. With the notations of (7), +we have h +� +w1(E) +� += − +√ +E, h +� +w2(E) +� += + +√ +E and we can reparametrize the interval +� +w1(E), w2(E) +� +with some θ ∈ (−π/2, π/2) such that +√ +E sin θ = h(w) . +(12) +With this change of variables, the minimal period can be written as +T(E) = 2 E +1 +2 − 1 +p +p′ +1 +p +� +π +2 +− π +2 +cos θ1− 2 +p +� +h′ ◦ h−1��√ +E sin θ +� dθ . +(13) +Its derivative with respect to E is given by +dT +dE(E) = +�1 +2 − 1 +p +� T(E) +E +− (p′ E)− 1 +p +� +π +2 +− π +2 +h′′(w) +h′(w)3 cos θ1− 2 +p sin θ dθ +where we use the short-hand notation w = h−1�√ +E sin θ +� +. After an integration by parts, +this expression becomes +dT +dE (E) = +�1 +2 − 1 +p +� T(E) +E ++ 1 +2 (p′) +1 +p′ E +1 +2− 1 +p +� +π +2 +− π +2 +3 h′′(w)2 − h′(w) h′′′(w) +h′(w)5 +cos θ3− 2 +p dθ +and one can show that +3 (h′′)2 − h′ h′′′ = |V ′|4 +8 V 2 +� V +|V ′|2 +�′′ + +8 +J. DOLBEAULT, M. GARC´IA-HUIDOBRO, AND R. MAN´ASEVICH +is positive if and only if V/(V ′)2 is a convex function. +This completes the proof of +Theorem 2. +□ +3. Asymptotic results +As in Section 2, let V (w) = V(w + A) and recall that (5) has a first integral given +by (6) where E ≥ 0 is the energy level. In this short section, we shall assume that (H1) +holds with a = − A, define +ω := +� +V ′′(0) = +� +V′′(A) > 0 +(14) +and make the additional hypothesis +lim inf +w→0+ +|V ′(w + a)| +wp−1 += lim inf +w→0+ +|V′(w)| +wp−1 +> 0 . +(H2) +This assumption is satisfied in case of (4) as soon as q > p > 2 and in that case +ω = +� +V′′(1) = √q − p, but the following result holds for a much larger class of potentials. +Lemma 5. Let p > 1. If V is a potential such that (H1) holds, then we have +T(E) ∼ +2 +√ +2 π Γ +� +1 − 1 +p +� +p′1/p ω Γ +�3 +2 − 1 +p +� E +1 +2 − 1 +p +as +E → 0+ +with ω defined by (14). As a consequence, we obtain +lim +E→0+ T(E) = 0 +if +p > 2 , +lim +E→0+ T(E) = 2 π +ω +if +p = 2 , +lim +E→0+ T(E) = + ∞ +if +p ∈ (1, 2) . +Additionally, if (H2) holds, then for any p > 1 we have limE→(E∗)− T(E) = +∞. +Proof. In a neighbourhood of w = 0, we can write V (w) ∼ +1 +2 ω2 w2, use (7) and the +change of variables w = +√ +2 E y/ω to obtain +T(E) ∼ 2 +√ +2 +p′1/p ω E +1 +2− 1 +p +� 1 +−1 +dy +� +1 − y2�1/p +as +E → 0+ . +We obtain the expression of the integral using the formulae [1, 6.2.1 & 6.2.2] for the Euler +Beta function. +Now let us consider the limit as E → (E∗)−. We learn from (H2) that +E∗ − v(w) ≥ ℓ +p (w − a)p +for some ℓ > 0 if w − a > 0 is taken small enough. We deduce from (7) that T(E) +diverges as E → (E∗)−. +□ + +MONOTONICITY OF THE PERIOD +9 +4. The monotonicity of the minimal period +Applying the formulae of Section 2 to study the monotonicity of the minimal period for +periodic solutions of (⋆) leads later to very complicated expressions for our problem with +potential (4). For that reason, it is convenient to introduce a new change of variables as +follows. Let A = − a > 0 and define +h(y) := h(w) = w +|w| +� +V (w) +with +y = (w + A)p +(15) +for any w ∈ (a, 0) ∪ (0, b) and extend it by h(0) = 0 at w = 0. Here h is defined as in +Section 2 (proof of Theorem 1, Eq. (12)) while h is such that +h(y) = h +� +y +1 +p − A +� +∀ y ∈ (0, B) +with B = (A + b)p. We have that +h′(w) = p y +1 +p′ h′(y) . +Let us make the simplifying assumption +w V ′(w) > 0 +∀ w ∈ (a, 0) ∪ (0, b) . +(H3) +Under this assumption, wi(E), i = 1, 2, are the two roots in (a, b) of V (w) = E, as in +Theorem 4, V (w) = E admits no other root in (a, b) for any E ∈ (0, E∗) and the map +E �→ T(E) is continuous. Also notice that +h′(y) > 0 +∀ y ∈ +� +y1(E), Ap� +∪ +� +Ap, y2(E) +� +where y1(E) := +� +A − |w1(E)| +�p and y2(E) := +� +A + w2(E) +�p. +By the above definition of h and (13), the minimal period can now be computed as +T(E) = cp E +1 +2 − 1 +p +� +π +2 +− π +2 +cos θ1− 2 +p +y +1 +p′ h′(y) +dθ +with +cp := +2 +p p′ +1 +p +(16) +using the change of variables y �→ θ ∈ (−π/2, π/2) such that +√ +E sin θ = h(y) = y − A +|y − A| +� +V +� +y1/p − A +� +. +(17) +Let us define +J := +� +π +2 +− π +2 +cos θ1− 2 +p +y +1 +p′ h′(y) +dθ +and notice that J is a function of E as a consequence of the change of variables (17): +y = y(E, θ) is such that +∂y +∂E = +sin θ +2 +√ +E h′(y) +. + +10 +J. DOLBEAULT, M. GARC´IA-HUIDOBRO, AND R. MAN´ASEVICH +By differentiating T(E) in (16) with respect to E, we find that +T ′(E) +T(E) = p − 2 +2 p +1 +E + J′(E) +J(E) +where +J′(E) = − +1 +2 +√ +E +� +π +2 +− π +2 +K(y) cos θ1− 2 +p sin θ dθ , +y is given by (17) and +K(y) := − +1 +h′(y) +d +dy +� +1 +y +1 +p′ h′(y) +� += +y2 h′′(y) + 1 +p′ y h′(y) +y2+ 1 +p′ � +h′(y) +�3 +. +(18) +With p > 2, E �→ T(E) is increasing if J′(E) > 0. Here is a sufficient condition on h, +which is in fact an assumption on V . +Lemma 6. Assume that (H1) and (H3) hold. With the above notations, if the function K +is decreasing on [A, B], then J′ > 0 on (0, E∗) and the minimal period T(E) is a monotone +increasing function of E. +Proof. With y(E, θ) defined by (17), the result is a consequence of +J′(E) = − +1 +2 +√ +E +� +π +2 +0 +� +K +� +y(E, θ) +� +− K +� +y(E, − θ) +�� +cos θ1− 2 +p sin θ dθ +and y(E, − θ) < y(E, θ) if θ ∈ (0, π/2). +□ +We deduce from Lemma 6 a sufficient condition on h to obtain that the minimal period +is monotone increasing. +Corollary 7. Assume that (H1) and (H3) hold. If h and and 1/h′2 are convex functions, +then the minimal period T(E) is a monotone increasing function of E ∈ (0, E∗). +Proof. By convexity of 1/h′2, we have that +0 < 1 +2 +d2 +dy2 +� 1 +h′2 +� += − d +dy +� h′′ +h′3 +� +and h′′ +h′3 is a decreasing function. Next, from (18) written as +K(y) = 1 +y +1 +p′ +h′′(y) +� +h′(y) +�3 + 1 +p′ +1 +y2− 1 +p +1 +� +h′(y) +�2 +if +y > 0 , +(19) +we observe that all the factors on the right hand of this expression are positive decreasing +functions, implying that K is a decreasing function on [A, B]. +□ +5. Proof of the main result +By applying Lemma 6 and Corollary 7, we prove Theorem 3. The main difficulty is to +establish that K is monotone decreasing if 1 < m < 2, which is done in Section 5.3. + +MONOTONICITY OF THE PERIOD +11 +5.1. Notations. Let us consider (1) with V given by (4) and q > p ≥ 2, hence V′(w) = +φq(w) − φp(w), and (1) is reduced to +� +φp(w′) +�′ + φq(w) − φp(w) = 0 . +(⋆) +In particular w = 1 is a trivial solution of this equation. All conditions of Section 1 for V +are satisfied, V (resp. V ) reaches a minimum at w = 1 (resp. w = 0) and +E∗ = q−p +p q = V(B) = V(0) +where +B := +� q +p +� +1 +q−p . +(20) +In the discussion, we shall consider the three cases: m = 2, m > 2 and 1 < m < 2, where +m = q +p . +We have that V (w) = V(w + A) with A = 1, i.e., +V (w) = 1 +q |w + 1|q − 1 +p |w + 1|p − +�1 +q − 1 +p +� +, +With the definitions of (15), we find that +q V (w) = ym − m y − (1 − m) +with +w = y +1 +p − 1 +and the change of variables y �→ θ ∈ (−π/2, π/2) defined by (17) amounts to +√ +E sin θ = h(y) = y − 1 +|y − 1| +� +1 +q +� +ym − m y + m − 1 +� +. +It is convenient to define +γm := m +1 +m−1 = B = +� q +p +� +p +q−p +and +W(y) := ym − m y + m − 1 +∀ y ∈ [0, γm] . +With these notations, we have +0 = W(1) < W(y) < W(0) = W(γm) = m − 1 +∀ y ∈ (0, 1) ∪ (1, γm) . +As a special case, note that W(y) = (y − 1)2 and h(y) = (y − 1)/√q if m = 2. In that +case, the result of Theorem 3 is straightforward. +Lemma 8. If m = 2 and V given by (4), the minimal period T(E) is a monotone +increasing function of E ∈ (0, E∗) with E∗ = +1 +2 p. +Proof. The function K defined by (18) is explicitly given by K(y) = +q2 +p′ y−1/p hence +monotone decreasing and Lemma 6 applies. +□ +5.2. The case m > 2. We obtain the following result. +Lemma 9. If m > 2, h and (h′)−2 are convex. +Proof. Let z = ym−1. With 0 ≤ y ≤ γm, W(y) := ym − m y + m − 1 and the function h +given by h(y) = +y−1 +|y−1| +� +W(y)/q, we find that the expression +4 W 3/2 �√ +W +�′′ = 2 W W ′′ − +� +W ′�2 + +12 +J. DOLBEAULT, M. GARC´IA-HUIDOBRO, AND R. MAN´ASEVICH +has its sign given by +F(y) := − m2 + 2 m (m − 1)2 ym−2 − 2 m2 (m − 2) ym−1 + m (m − 2) y2m−2 . +• If y ≥ 1, then ym−2 ≥ 1, − m2 + 2 m (m − 1)2 ym−2 ≥ m (m − 2) (2 m − 1) and +F(y) ≥ m (m − 2) (z − 1) (z + 1 − 2 m) ≥ 0 . +• If y ≤ 1, then ym−2 ≤ 1, y2m−2 ≤ 1, m (m − 2) y2m−2 − m2 ≤ − 2 m y2m−2 and +− F(y) ≥ 2 m (z − 1) +� +z + (m − 1)2� +≥ 0 . +In both cases, we conclude that h′′ ≥ 0. +The function ((h′)−2)′′ has the sign of +G(y) := 2 (m − 1) (m − 2) − m (2 m − 1) y ++ 2 (m − 1) (2 m − 1) ym−1 − 2 (m − 2) (2 m − 1) ym + (m − 2) y2m−1 . +Since G(1) = G′(1) = 0 and +G′′(y) = 2 (m − 1) (m − 2) (2 m − 1) W(y) ≥ 0 , +we conclude that g ≥ 0 and ((h′)−2)′′ ≥ 0. +□ +This proves Theorem 3 as a consequence of Corollary 7 and Lemma 9 if m > 2. +5.3. The case 1 < m < 2. We cannot apply Corollary 7 and we have to directly rely on +Lemma 6. We recall that m = q/p. Let us start by computing K′. +Lemma 10. The function y �→ − K′(y) has the sign of p2 y2 f(a, m, y, z) where z = ym−1, +the parameters (a, m) are admissible in the sense that +a = 1 +p ∈ +� +0, 1 +2 +� +, +m = q +p ∈ (1, 2) , +and +f(a, m, y, z) = − 3 m y (z − 1)2 (m z − 1) ++ 2 (m − 1 − m y + y z) +� +2 + (1 − 6 m + m2) z + 2 m2 z2� ++ a +� +3 m y (z − 1)3 − 6 (z − 1) (m z − 1) (m − 1 − m y + y z) +� ++ a2 � +2 (z − 1)2 (m − 1 − m y + y z) +� +. +Proof. We set y = xp so that x = y1/p and dx +dy = 1 +p y−1/p′. Let +Φ(x) := W(y) = xmp − m xp + m − 1 +∀ x ∈ +� +0, γ1/p +m +� +, +where W and h are as in Section 5.1, so that +��√q h′(y) +��2 = +����� +Φ′(x) +2 p y1/p′ � +Φ(x) +����� +2 +|x=y1/p +, + +MONOTONICITY OF THE PERIOD +13 +that is, 4 m p3 ��y1/p′ h′(y) +��2 = (Φ′(x))2/Φ(x) and K as in (18) can be rewritten as +K(y) = − 1 +2 y1/p′ d +dy +� +1 +y2/p′ h′(y)2 +� += − 2 m p3 d +dx +� Φ(x) +|Φ′(x)|2 +� +. +Hence − K′ has the sign of +d2 +dx2 (Φ(x) |Φ′(x)|−2), i.e., of 6 Φ |Φ′′|2 − 2 Φ Φ′ Φ′′′ − 3 |Φ′|2 Φ′′ +and the detailed computation shows that +x4 +q2 |Φ′(x)|4 d2 +dx2 +� Φ(x) +|Φ′(x)|2 +� += p2 y2 f(a, m, y, z), +ending the proof of the lemma. +□ +Lemma 11. With V given by (4) and 2 < p < q < 2 p, K defined by (18) is monotone +decreasing. +Proof. Keeping the notations of Lemma 10, our goal is to prove that y �→ f (a, m, y, ym−1) +is nonnegative for any y ∈ (0, γm) whenever the parameters (a, m) are admissible. +Let us start by considering its value at some remarkable points. +• At (y, z) = (0, 0), we have f(a, m, 0, 0) = 2 (1 − a) (2 − a) (m − 1) > 0. +• At (y, z) = (1, 1), we have f(a, m, 1, 1) = 0 but a Taylor expansion shows that +f +� +a, m, y, ym−1� += 1 +12 (m − 1)3 cm,a (y − 1)4 + O +� +(y − 1)5� +as +y → 1 +(21) +for any a ∈ (0, 1/2), where +cm,a = 12 m a (a − m − 1) + m +� +2 m2 + 7 m + 2 +� +≥ cm,1/2 = m (m + 1) (2 m − 1) > 0 . +This proves that y �→ f (a, m, y, ym−1) is positive for any y ∈ (1 − ε, 1) ∪ (1, 1 + ε) for +some ε = ε(a, m) > 0 whenever the parameters (a, m) are admissible. +• At (y, z) = (γm, m), we have +f(a, m, γm, m) = (m−1)3 cm,a , +cm,a = 2 a2 −3 a (2 m+2−m γm)+2 +� +2 m2 + 5 m + 2 +� +. +Using infm∈(1,2) +3 +4 (2 m + 2 − m γm) = limm→1+ +3 +4 (2 m + 2 − m γm) = 3 (1 − e/4) > 1/2, +we have +cm,a > cm,1/2 = (4 − 3 γm) m2 + +� +7 − 3 +2 γm +� +m + 3 +2 +> lim +m→1+ +� +(4 − 3 γm) m2 + +� +7 − 3 +2 γm +� +m + 3 +2 +� += 1 +2 (25 − 9 e) > 0 . +In the limit as m → 2, we have y = z and +f(a, 2, y, z) = 2 (1 − a) (2 − a) (z − 1)4 . +(22) +Hence f (a, 2, y, ym−1) is positive unless y = 1. We are now going to take a given a ∈ +(0, 1/2) and consider m ∈ (1, 2) as a parameter. Let us prove that for some m∗ ∈ (1, 2), +we have f (a, m, y, ym−1) ≥ 0 for any (m, y) such that m∗ < m < 2 and 0 ≤ y ≤ γm. +We assume by contradiction that there are two sequences (mk)k∈N and (yk)k∈N such that +1 < mk < 2 for any k ∈ N, limk→+∞ mk = 2, 0 ≤ yk ≤ γmk and f +� +a, mk, yk, ymk−1 +k +� +< 0 + +14 +J. DOLBEAULT, M. GARC´IA-HUIDOBRO, AND R. MAN´ASEVICH +for any k ∈ N. Up to the extraction of a subsequence, (yk)k∈N converges to some limit +y∞ ∈ [0, 2] and by continuity of f we know that f (a, 2, y∞, y∞) ≤ 0: the only possibility +is y∞ = 1 by (22). Since f +� +a, mk, yk, ymk−1 +k +� +< 0 = f(a, mk, 1, 1), we learn that yk ̸= 1. +Since limk→+∞ yk = 1, this contradicts (21) or, to be precise, |yk − 1| ≥ ε(a, mk), as the +reader is invited to check that lim infk→+∞ ε(a, mk) > 0 because f is a smooth function +of all of its arguments. If we redefine +m∗(a) := inf +� +m ∈ (1, 2) : f +� +a, m, y, ym−1� +≥ 0 ∀ y ∈ [0, γm] +� +, +then we know that for any a ∈ (0, 1/2), we have m∗(a) < 2. +We want to prove that m∗(a) = 1. Again, let us argue by contradiction: if m∗(a) > 1, +and assume that there are two sequences (mk)k∈N and (yk)k∈N such that 1 < mk < m∗(a) +for any k ∈ N, limk→+∞ mk = m∗(a), 0 ≤ yk ≤ γmk and f +� +a, mk, yk, ymk−1 +k +� +< 0 for any +k ∈ N. Up to the extraction of a subsequence, (yk)k∈N converges to some limit y∞ ∈ [0, 2] +and by continuity of f we know that f (a, m∗(a), y∞, ym−1 +∞ +) ≤ 0. For the same reasons as +above, y∞ = 0, y∞ = 1 and y∞ = γm∗(a) are excluded. Altogether, we have proved that +for +m = m∗(a) , +we have f (a, m, y∞, ym−1 +∞ +) = 0 for some y∞ ∈ (0, 1) ∪ (1, γm) and we also have that +f (a, m, y, ym−1) ≥ 0 for any y ∈ (0, 1) ∪ (1, γm), so that y∞ is a local minimizer of +y �→ f (a, m, y, ym−1). As a consequence, we have shown that for m = m∗(a) > 1 and +y = y∞ ̸= 1, we have +f +� +a, m, y, ym−1� += 0 +and +∂ +∂yf +� +a, m, y, ym−1� += 0 . +(23) +As we shall see below, this contradicts Lemma 12. Hence y �→ f (a, m, y, ym−1) takes +nonnegative values for any admissible parameters (a, m) with 1 < m < 2. By Lemma 10, +K′(y) ≤ 0, thus completing the proof. +□ +We still have to prove that (23) has no solution y ∈ (0, 1) ∪ (1, γm). Since +y ∂ +∂yf +� +a, m, y, ym−1� += y ∂f +∂y (a, m, y, z) + (m − 1) z ∂f +∂z (a, m, y, z) , +we can relax the condition z = ym−1 and prove the slightly more general result. +Lemma 12. With the notations of Lemma 10, assume that m > 1, y ∈ (0, γm] and +z ∈ (0, m]. For any admissible parameters (a, m), if +f(a, m, y, z) = 0 , +(24a) +y ∂f +∂y (a, m, y, z) + (m − 1) z ∂f +∂z (a, m, y, z) = 0 . +(24b) +then z = 1. + +MONOTONICITY OF THE PERIOD +15 +Proof. Solving the system (24a)–(24b) is an elimination problem because the function f, +as defined in Lemma 10, is a polynomial in the variables a, y and z. Since (24a) is a first +order equation in y, we can eliminate this variable and find that +y = n(a, m, z) +d(a, m, z) +with +n(a, m, z) := 2 (m − 1) +� +a2 (z − 1)2 − 3 a (z − 1) (m z − 1) ++ 2 m2 z2 + +� +m2 − 6 m + 1 +� +z + 2 +� +, +d(a, m, z) := m +� +2 a2 (z − 1)2 + 3 a (z + 1)2 (z − 1) + 9 z2 + 8 z + 1 +� +− 2 z +� +a2 (z − 1)2 + 3 a (z − 1) + z + 2 +� +− m2 z +� +6 a (z − 1) + z2 + 8 z + 9 +� ++ 2 m3 z (2 z + 1) . +After replacing, solving (24b) under the condition z ̸= 1 is reduced to a second order +equation in z, whose discriminant is +δ(a, m) := − 3 (a − 1)2 (m − 1)2 (a − m)2 � +5 a2 − 10 a (m + 1) − 3 m2 + 14 m − 3 +� +. +Since 5 a2−10 a (m+1)−3 m2 +14 m−3 takes only positive values for admissible (a, m), +there are no other roots than z = 1. This is the desired contradiction, which completes +the proof thanks to Lemma 6. +□ +Proof of Theorem 3. It is a straightforward consequence of Lemmas 6 and 11. +□ +Acknowledgment: J.D. was partially supported by the Project EFI (ANR-17-CE40- +0030) of the French National Research Agency (ANR). M.G-H. was supported by Fonde- +cyt grant 1210241, and R.M. by Centro de Modelamiento Matem´atico (CMM) BASAL +fund FB210005 for center of excellence from ANID-Chile. +© 2023 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. CC-BY 4.0 +References +[1] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, +and mathematical tables. 10th printing, with corrections. National Bureau of Standards. A Wiley- +Interscience Publication. New York etc.: John Wiley & Sons. xiv, 1046 pp. £ 42.70 (1972)., 1972. +[2] D. 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A, 40 (2020), pp. 375– +394. +[19] A. Gasull, A. Guillamon, V. Ma˜nosa, and F. Ma˜nosas, The period function for Hamiltonian +systems with homogeneous nonlinearities, J. Differential Equations, 139 (1997), pp. 237–260. +[20] Y. Miyamoto and K. Yagasaki, Monotonicity of the first eigenvalue and the global bifurcation +diagram for the branch of interior peak solutions, J. Differential Equations, 254 (2013), pp. 342–367. +[21] F. Rothe, Remarks on periods of planar Hamiltonian systems, SIAM J. Math. Anal., 24 (1993), +pp. 129–154. +[22] R. Schaaf, A class of Hamiltonian systems with increasing periods, J. Reine Angew. Math., 363 +(1985), pp. 96–109. +[23] J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equa- +tions, 39 (1981), pp. 269–290. +[24] K. Yagasaki, Monotonicity of the period function for u′′ − u + up = 0 with p ∈ R and p > 1, J. +Differential Equations, 255 (2013), pp. 1988–2001. +Email address: dolbeaul@ceremade.dauphine.fr +Email address: mgarcia@mat.puc.cl +Email address: manasevi@dim.uchile.cl + diff --git a/J9A0T4oBgHgl3EQfCv8A/content/tmp_files/load_file.txt b/J9A0T4oBgHgl3EQfCv8A/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b6ad99e0eef558489ddac99ee58b6aa1e341e4bc --- /dev/null +++ b/J9A0T4oBgHgl3EQfCv8A/content/tmp_files/load_file.txt @@ -0,0 +1,485 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf,len=484 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='01992v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='AP] 5 Jan 2023 MONOTONICITY OF THE PERIOD AND POSITIVE PERIODIC SOLUTIONS OF A QUASILINEAR EQUATION Jean Dolbeault ∗ CEREMADE (CNRS UMR n◦ 7534) PSL University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Universit´e Paris-Dauphine Place de Lattre de Tassigny,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' 75775 Paris 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' France Marta Garc´ıa-Huidobro ∗ Departamento de Matem´aticas Pontificia Universidad Cat´olica de Chile Casilla 306,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Correo 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Santiago de Chile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Chile Ra´ul Man´asevich Departamento de Ingenier´ıa Matem´atica and Centro de Modelamiento Matem´atico (CNRS IRL2807) FCFM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Universidad de Chile Casilla 170 Correo 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Santiago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Chile Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We investigate the monotonicity of the minimal period of the periodic solutions of some quasilinear differential equations and extend results for p = 2 due to Chow and Wang, and to Chicone, to the case of the p-Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Our main result is the monotonicity of the period for positive solutions of a nonlinear Euler- Lagrange equation for a minimization problem related with a fundamental interpolation inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' In particular we generalize to p greater than 2 recent results of Benguria, Depassier and Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Introduction In this paper we study monotonicity properties of the minimal period of positive peri- odic solutions of � φp(w′) �′ + V′(w) = 0 , (1) where p ≥ 2, φp(s) = |s|p−2s, and V : R → R is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The potential function V(w) is assumed to be non-negative for w ≥ 0, V(0) > 0, it has a minimum at w = A > 0 with V(A) = 0 = V′(A), and satisfies some additional conditions listed in Section 3, which guarantee that (1) has positive periodic solutions enclosing the critical point (A, 0) in the phase plane (w, w′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' File: DGHM2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='tex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Primary: 34C25, 35J92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Secondary: 34L30, 34C23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Hamiltonian systems, quasilinear elliptic equations, p-Laplace operator, pe- riodic solutions, period, energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' ∗ Corresponding author: Jean Dolbeault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' DOLBEAULT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' GARC´IA-HUIDOBRO, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' MAN´ASEVICH The energy E = 1 p |w′|p + V(w) is conserved if w solves (1) and we are interested in the positive periodic solutions with energy less than E∗ := V(0) which are enclosed by the homoclinic orbit attached to (w, w′) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We further assume that V is such that these solutions are uniquely determined, up to translations, by the energy level E, with minimal period T(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The purpose of this paper is to study under which conditions T is an increasing function of E in the range 0 ≤ E ≤ E∗ where E∗ is the energy level of the homoclinic orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Furthermore we will consider the asymptotic behaviour of T(E) as E → 0+ and as E → (E∗)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Surprisingly enough, the cases p = 2 and p > 2 differ as E → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Our first result is an extension to p > 2 of a result of Chow and Wang [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let p > 2 and assume that V is a C2 function on R+ such that V(A) = 0 = V′(A) and V′′ > 0 on (0, B) with B := min{w > A : V(w) ≥ V(0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' If w �→ |V′(w)|2 − p′ V(w) V′′(w) is a positive function, then E �→ T(E) is increasing on (0, E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Notice that w �→ |V′(w)|2 − p′ V(w) V′′(w) is a positive function if and only if w �→ V(w) |V′(w)|−p′ is a monotone increasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Our second result is also an extension to p > 2 of the monotonicity result in [7, Theorem A] under Chicone’s condition, which is also a growth condition, but of higher order in the derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let p > 2 and assume that V is a C3 function on R+ such that V(A) = 0 = V′(A) and let B := min{w > A : V(w) ≥ V(0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' If V/(V′)2 is a convex function, then E �→ T(E) is increasing on (0, E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' A central motivation for this paper arises from the study of the minimization problem µ(λ) := inf f∈W1,p(S1)\\{0} ∥f ′∥2 Lp(S1) + λ ∥f∥2 Lp(S1) ∥f∥2 Lq(S1) (2) where q > p is an arbitrary exponent and S1 is the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The problem can also be seen as the search for the optimal constant in the interpolation inequality ∥f ′∥2 Lp(S1) + λ ∥f∥2 Lp(S1) ≥ µ(λ) ∥f∥2 Lq(S1) ∀ f ∈ W1,p(S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Testing the inequality with constant functions shows that µ(λ) ≤ ¯µ(λ) := λ |S1| 2 p − 2 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' If p = 2, it is well known from the carr´e du champ method [2, 3] that equality holds if and only if λ ≤ d/(q−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' If λ > d/(q−2), we have µ(λ) < ¯µ(λ) and optimal functions are non constant, so that symmetry breaking occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The minimization problem problem with p > 2 was studied in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' There is an optimal function for (2) and the corresponding Euler-Lagrange equation turns out to be the nonlinear differential equation with nonlocal terms given by − ∥f ′∥2−p Lp(S1) � φp(f ′) �′ + λ ∥f∥2−p Lp(S1) φp(f) = µ(λ) ∥f∥2−q Lq(S1) φq(f) , (3) MONOTONICITY OF THE PERIOD 3 where we look for positive solutions on W 1,p(S1)\\{0} or equivalently positive 2π-periodic solutions on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' So far, we do not know the precise value of λ for which there is symmetry breaking but according to [18] rigidity holds if 0 < λ < λ1 for some explicit λ1 > 0, where rigidity means that any positive solution of (3) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' In that range, we have µ(λ) = ¯µ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' On the contrary, one can prove that symmetry breaking occurs if λ > λ2 for some λ2 > λ1, so that µ(λ) < ¯µ(λ) and (3) admits non-constant positive solutions for any λ > λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Using homogeneity, scalings and a suitable change of variables, the study of (3) is reduced in [18] to the study of positive periodic solutions on R of � φp(w′) �′ + φq(w) − φp(w) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (⋆) In this equation, there are no non-local terms but the minimal period of periodic solutions is no more given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Equation (⋆) enters in the framework of (1) with A = 1 and potential V(w) = 1 q |w|q − 1 p |w|p − � 1 q − 1 p � , (4) so that E∗ = 1/p − 1/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Positive periodic solutions exist only if the energy level satisfies the condition E < E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Again, let T(E) be the minimal period of such a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Theorems 1 and 2 do not apply easily and we shall prove directly the following result, which is the main contribution of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let p and q be two exponents such that 2 < p < q and consider the positive periodic solutions of (⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Then the map E �→ T(E) is increasing on (0, E∗) with limE→0+ T(E) = 0 and limE→(E∗)− T(E) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The study of (3) is motivated by rigidity and symmetry breaking results associated with interpolation inequalities on the unit sphere Sd in one and higher dimensions, that is, d ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' If p = 2, a precise description of the threshold value of λ is known in the framework of Markov processes if q is not too large (see [3] for an overview with historical references that go back to [2]) and from [5, 11, 14, 15, 16, 17, 13] using entropy methods applied to nonlinear elliptic and parabolic equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' also see [12] for an overview and extensions to various related variational problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Almost nothing is known beyond [18] if p > 2, even for d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Our results are a contribution to a better understanding of the fundamental properties of the solutions of (1) in the simplest of the cases when p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Without the Assumption that V′(A) = 0 in Theorems 1 and 2 (which is also satisfied in Theorem 3), it is easy to give similar results so that E �→ T(E) is decreasing, but in phase plane the solutions are not described anymore by orbits enclosed by a homoclinic orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Some comments on this issue can be found in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' In dimension d = 1, the bifurcation problem (3) degenerates in the limit case p = 2, for which λ1 = λ2 = 1/(q − 2) according to [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We refer to [4, Section 1] for an introduction to the minimization problem (2) with p = 2, the issue of the branches and the monotonicity of the period problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Proving that symmetry breaking occurs if and only if λ > 1/(q −2) can be reduced to a proof of the monotonicity of the minimal period using Chicone’s criterion [7, Theorem A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The study of bifurcation problems using the 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' DOLBEAULT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' GARC´IA-HUIDOBRO, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' MAN´ASEVICH period function goes back to [23] in case of equations with cubic non-linearities and was later extended to various classes of Hamiltonian systems in [22, 21, 10, 9, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' If p′ = p/(p − 1) is the H¨older conjugate of the exponent p and H(u, v) := V(u) + 1 p′ |v|p′ , Equation (1) can be rewritten as the Hamiltonian system of equations u′ = ∂H ∂v = φp′(v) and v′ = − ∂H ∂u = − V′(u) with w = u and w′ = φp′(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Although this Hamiltonian structure may superficially look similar to the conditions of [22, Theorem 1], we have a definitely different set of assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' In [21], a much larger set of Hamiltonian systems is considered but again our assumptions differ, for instance for the simple reason that the function φp′ is not of class C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Further references on the period function can be found in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' There are various other extensions of Chicone’s result [7], see for instance [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Also notice that there is a computation in [6, Section 4] which turns out to be equivalent to an argument used in the proof of our Theorem 4 (see below in Section 2), although it is stated neither in that form nor as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The Hamiltonian version of the method has interesting applications to Lotka-Volterra systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The monotonicity of the minimal period as a function of the energy level is a question of interest by itself and particularly in the model case of the potential V as in (4), even in the case p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We quote from [4] that: “It is somewhat surprising that, despite its ubiquity, the monotonicity of the period function for [this problem] in full generality was only established recently.” In [20], Miyamoto and Yagasaki proved the monotonicity of the period function for p = 2 and for q an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' In [24], Yagasaki generalized the result to all values of q > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Both papers, [20, 24], rely on Chicone’s criterion which is difficult to apply to non-integer values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The purpose of Benguria, Depassier and Loss in [4] was to give a simplified proof of the monotonicity of the period of the positive solutions of w′′ + wq−1 − w = 0 (corresponding to p = 2 in our notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We point out that in many situations in the paper we will consider the equation � φp(w′) �′ + V ′(w) = 0 (5) where V is a potential of class C2 defined on R such that There are a, b ∈ R with a < 0 < b such that V (a) = V (b) = E∗ > 0, V ′(a) = V (0) = V ′(0) = 0, V ′′(0) ̸= 0, and 0 < V (w) < E∗ for all w ∈ (a, 0) ∪ (0, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (H1) The potential V (w) achieves its minimum on (a, b) at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The relationship of V with V is given by V (w) = V(w + A), a = − A and b = B − A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The origin w = 0, w′ = 0 is a stationary point of (5) giving rise to a center surrounded by closed periodic orbits with minimal period T(E), such that these periodic orbits are enclosed by a homoclinic orbit attached to (a, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' MONOTONICITY OF THE PERIOD 5 This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Section 2 is devoted to the proof of the p-Laplacian version of results which are classical when p = 2 and are summarized in Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We are not aware of such statements in the existing literature but they are natural extensions of the case p = 2 and might already be known, so we do not claim any deep originality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The result of Theorem 3 is by far more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' In Section 3 we start with problem (1) by making a change of variables and obtain an expression for the minimal period following Chicone’s ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We also prove some properties of the minimal period when the energy goes to zero and when it goes to the homoclinic level E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' In Section 4 we prove the monotonicity of the minimal period extending, in particular, the results of [4] for p = 2 to the more general case of the one-dimensional p-Laplacian operator w �→ � φp(w′) �′, with p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Our main result (Theorem 3) is proved in Section 5, the proof is highly non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' A p-Laplacian version of some classical results This section is devoted to the proof of Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We also provide a slightly more detailed statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We begin by extending [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='1] by Chow and Wang to the p-Laplacian situa- tion when p ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We recall that p′ = p/(p − 1) denotes the H¨older conjugate of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Equation (5) has a first integral given by 1 p′ |w′|p + V (w) = E (6) for any energy level E ∈ (0, E∗) and the minimal period is given in terms of the energy by T(E) = 2 p′1/p � w2(E) w1(E) dw � E − V (w) �1/p (7) where wi(E), i = 1, 2, are two roots of V (w) = E such that a < w1(E) < 0 < w2(E) < b and V (w) < E ∀ w ∈ � w1(E), w2(E) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' At this point, let us notice that the map E �→ T(E) is a continuous function if we assume that w V ′(w) > 0 for any w ∈ (a, 0)∪(0, b), but that it is not the case if V admits another local minimum than w = 0 in the interval (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let us define γ(w, E) := p′ � E − V (w) � , R(w) := V ′(w)2 − p′ V (w) V ′′(w) and notice that ∂γ ∂w = − p′ V ′(w) and ∂γ ∂E = p′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The following result is a detailed version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' DOLBEAULT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' GARC´IA-HUIDOBRO, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' MAN´ASEVICH Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let p ≥ 2 and consider Equation (5) where we assume that V satisfies (H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' With the above notations, for any E ∈ (0, E∗), it holds that dT dE (E) = 2 p′ E � w2(E) w1(E) R(w) γ(w, E)1/p V ′(w)2 dw (8) if the integral in the right-hand side is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Thus if R is positive on (a, 0) ∪ (0, b), then the minimal period is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Notice that from Assumption (H1), we know that V (a) = E∗ > 0 and V ′(a) = 0 so that limw→a+ V (w) |V ′(w)|−p′ = +∞ and � V |V ′|p′ �′ = R V ′ |V ′|p′+2 which is incompatible with R being a negative valued function in a neighbourhood of w = a+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' If we remove the assumption that V ′(a) = 0, then it makes sense to assume that R is a negative function on (a, 0) ∪ (0, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' In this case, the minimal period is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The proof relies on the same strategy as for [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We skip some details and emphasize only the changes needed to cover the case p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let us set I(E) := � w2(E) w1(E) γ(w, E)1/p′ dw and J(E) := � w2(E) w1(E) � γ(w, E) − p′ E � γ(w, E)1/p′ dw .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' By differentiating with respect to E, we obtain dI dE(E) = � w2(E) w1(E) dw γ(w, E)1/p = 1 2 T(E) and dJ dE (E) = � w2(E) w1(E) γ(w, E) − p′E γ(w, E)1/p dw , which implies that dJ dE (E) = I(E) − p′ E dI dE (E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (9) Differentiating once more with respect to E, we get d2J dE2(E) = (1 − p′) dI dE(E) − p′ E d2I dE2(E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (10) On the other hand, by integrating by parts in � w2 w1 γ p′+1 p′ V ′2 − V V ′′ V ′2 dw = � w2 w1 γ p′+1 p′ � V V ′ �′ dw = − p′ + 1 p′ � w2 w1 γ 1 p′ V V ′ ∂γ ∂w dw , we obtain J(E) = − p′ p′ + 1 � w2(E) w1(E) γ(w, E) p′+1 p′ V ′(w)2 − V (w) V ′′(w) V ′(w)2 dw MONOTONICITY OF THE PERIOD 7 by definition of J and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' See [8] for further details in the case p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' By differentiating twice this expression of J(E) with respect to E, we obtain d2J dE2(E) = − p′ � w2(E) w1(E) V ′(w)2 − V (w) V ′′(w) γ(w, E)1/p V ′(w)2 dw .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Since T(E) = 2 dI dE(E), we learn from (10) that p′ E 2 dT dE (E) = p′ E d2I dE2(E) = (1 − p′) dI dE (E) − d2J dE2(E) = (1 − p′) � w2(E) w1(E) dw γ(w, E)1/p + p′ � w2(E) w1(E) V ′(w)2 − V (w) V ′′(w) γ(w, E)1/p V ′(w)2 dw = � w2(E) w1(E) R(w) γ(w, E)1/p V ′(w)2 dw .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' This concludes the proof of (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' □ Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let us consider again Equation (5) with a potential V which satis- fies (H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We adapt the proof of [7, Theorem A] to the case p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let us consider the function h(w) := w |w| � V (w) (11) for any w ∈ (a, 0) ∪ (0, b) and extend it by h(0) = 0 at w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' With the notations of (7), we have h � w1(E) � = − √ E, h � w2(E) � = + √ E and we can reparametrize the interval � w1(E), w2(E) � with some θ ∈ (−π/2, π/2) such that √ E sin θ = h(w) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (12) With this change of variables, the minimal period can be written as T(E) = 2 E 1 2 − 1 p p′ 1 p � π 2 − π 2 cos θ1− 2 p � h′ ◦ h−1��√ E sin θ � dθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (13) Its derivative with respect to E is given by dT dE(E) = �1 2 − 1 p � T(E) E − (p′ E)− 1 p � π 2 − π 2 h′′(w) h′(w)3 cos θ1− 2 p sin θ dθ where we use the short-hand notation w = h−1�√ E sin θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' After an integration by parts, this expression becomes dT dE (E) = �1 2 − 1 p � T(E) E + 1 2 (p′) 1 p′ E 1 2− 1 p � π 2 − π 2 3 h′′(w)2 − h′(w) h′′′(w) h′(w)5 cos θ3− 2 p dθ and one can show that 3 (h′′)2 − h′ h′′′ = |V ′|4 8 V 2 � V |V ′|2 �′′ 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' DOLBEAULT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' GARC´IA-HUIDOBRO, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' MAN´ASEVICH is positive if and only if V/(V ′)2 is a convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' This completes the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Asymptotic results As in Section 2, let V (w) = V(w + A) and recall that (5) has a first integral given by (6) where E ≥ 0 is the energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' In this short section, we shall assume that (H1) holds with a = − A, define ω := � V ′′(0) = � V′′(A) > 0 (14) and make the additional hypothesis lim inf w→0+ |V ′(w + a)| wp−1 = lim inf w→0+ |V′(w)| wp−1 > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (H2) This assumption is satisfied in case of (4) as soon as q > p > 2 and in that case ω = � V′′(1) = √q − p, but the following result holds for a much larger class of potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' If V is a potential such that (H1) holds, then we have T(E) ∼ 2 √ 2 π Γ � 1 − 1 p � p′1/p ω Γ �3 2 − 1 p � E 1 2 − 1 p as E → 0+ with ω defined by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' As a consequence, we obtain lim E→0+ T(E) = 0 if p > 2 , lim E→0+ T(E) = 2 π ω if p = 2 , lim E→0+ T(E) = + ∞ if p ∈ (1, 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Additionally, if (H2) holds, then for any p > 1 we have limE→(E∗)− T(E) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' In a neighbourhood of w = 0, we can write V (w) ∼ 1 2 ω2 w2, use (7) and the change of variables w = √ 2 E y/ω to obtain T(E) ∼ 2 √ 2 p′1/p ω E 1 2− 1 p � 1 −1 dy � 1 − y2�1/p as E → 0+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We obtain the expression of the integral using the formulae [1, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='1 & 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='2] for the Euler Beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Now let us consider the limit as E → (E∗)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We learn from (H2) that E∗ − v(w) ≥ ℓ p (w − a)p for some ℓ > 0 if w − a > 0 is taken small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We deduce from (7) that T(E) diverges as E → (E∗)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' □ MONOTONICITY OF THE PERIOD 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The monotonicity of the minimal period Applying the formulae of Section 2 to study the monotonicity of the minimal period for periodic solutions of (⋆) leads later to very complicated expressions for our problem with potential (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' For that reason, it is convenient to introduce a new change of variables as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let A = − a > 0 and define h(y) := h(w) = w |w| � V (w) with y = (w + A)p (15) for any w ∈ (a, 0) ∪ (0, b) and extend it by h(0) = 0 at w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Here h is defined as in Section 2 (proof of Theorem 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (12)) while h is such that h(y) = h � y 1 p − A � ∀ y ∈ (0, B) with B = (A + b)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We have that h′(w) = p y 1 p′ h′(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let us make the simplifying assumption w V ′(w) > 0 ∀ w ∈ (a, 0) ∪ (0, b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (H3) Under this assumption, wi(E), i = 1, 2, are the two roots in (a, b) of V (w) = E, as in Theorem 4, V (w) = E admits no other root in (a, b) for any E ∈ (0, E∗) and the map E �→ T(E) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Also notice that h′(y) > 0 ∀ y ∈ � y1(E), Ap� ∪ � Ap, y2(E) � where y1(E) := � A − |w1(E)| �p and y2(E) := � A + w2(E) �p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' By the above definition of h and (13), the minimal period can now be computed as T(E) = cp E 1 2 − 1 p � π 2 − π 2 cos θ1− 2 p y 1 p′ h′(y) dθ with cp := 2 p p′ 1 p (16) using the change of variables y �→ θ ∈ (−π/2, π/2) such that √ E sin θ = h(y) = y − A |y − A| � V � y1/p − A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (17) Let us define J := � π 2 − π 2 cos θ1− 2 p y 1 p′ h′(y) dθ and notice that J is a function of E as a consequence of the change of variables (17): y = y(E, θ) is such that ∂y ∂E = sin θ 2 √ E h′(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' DOLBEAULT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' GARC´IA-HUIDOBRO, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' MAN´ASEVICH By differentiating T(E) in (16) with respect to E, we find that T ′(E) T(E) = p − 2 2 p 1 E + J′(E) J(E) where J′(E) = − 1 2 √ E � π 2 − π 2 K(y) cos θ1− 2 p sin θ dθ , y is given by (17) and K(y) := − 1 h′(y) d dy � 1 y 1 p′ h′(y) � = y2 h′′(y) + 1 p′ y h′(y) y2+ 1 p′ � h′(y) �3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (18) With p > 2, E �→ T(E) is increasing if J′(E) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Here is a sufficient condition on h, which is in fact an assumption on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Assume that (H1) and (H3) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' With the above notations, if the function K is decreasing on [A, B], then J′ > 0 on (0, E∗) and the minimal period T(E) is a monotone increasing function of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' With y(E, θ) defined by (17), the result is a consequence of J′(E) = − 1 2 √ E � π 2 0 � K � y(E, θ) � − K � y(E, − θ) �� cos θ1− 2 p sin θ dθ and y(E, − θ) < y(E, θ) if θ ∈ (0, π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' □ We deduce from Lemma 6 a sufficient condition on h to obtain that the minimal period is monotone increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Assume that (H1) and (H3) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' If h and and 1/h′2 are convex functions, then the minimal period T(E) is a monotone increasing function of E ∈ (0, E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' By convexity of 1/h′2, we have that 0 < 1 2 d2 dy2 � 1 h′2 � = − d dy � h′′ h′3 � and h′′ h′3 is a decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Next, from (18) written as K(y) = 1 y 1 p′ h′′(y) � h′(y) �3 + 1 p′ 1 y2− 1 p 1 � h′(y) �2 if y > 0 , (19) we observe that all the factors on the right hand of this expression are positive decreasing functions, implying that K is a decreasing function on [A, B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Proof of the main result By applying Lemma 6 and Corollary 7, we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The main difficulty is to establish that K is monotone decreasing if 1 < m < 2, which is done in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' MONOTONICITY OF THE PERIOD 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let us consider (1) with V given by (4) and q > p ≥ 2, hence V′(w) = φq(w) − φp(w), and (1) is reduced to � φp(w′) �′ + φq(w) − φp(w) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (⋆) In particular w = 1 is a trivial solution of this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' All conditions of Section 1 for V are satisfied, V (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' V ) reaches a minimum at w = 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' w = 0) and E∗ = q−p p q = V(B) = V(0) where B := � q p � 1 q−p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (20) In the discussion, we shall consider the three cases: m = 2, m > 2 and 1 < m < 2, where m = q p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We have that V (w) = V(w + A) with A = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=', V (w) = 1 q |w + 1|q − 1 p |w + 1|p − �1 q − 1 p � , With the definitions of (15), we find that q V (w) = ym − m y − (1 − m) with w = y 1 p − 1 and the change of variables y �→ θ ∈ (−π/2, π/2) defined by (17) amounts to √ E sin θ = h(y) = y − 1 |y − 1| � 1 q � ym − m y + m − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' It is convenient to define γm := m 1 m−1 = B = � q p � p q−p and W(y) := ym − m y + m − 1 ∀ y ∈ [0, γm] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' With these notations, we have 0 = W(1) < W(y) < W(0) = W(γm) = m − 1 ∀ y ∈ (0, 1) ∪ (1, γm) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' As a special case, note that W(y) = (y − 1)2 and h(y) = (y − 1)/√q if m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' In that case, the result of Theorem 3 is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' If m = 2 and V given by (4), the minimal period T(E) is a monotone increasing function of E ∈ (0, E∗) with E∗ = 1 2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The function K defined by (18) is explicitly given by K(y) = q2 p′ y−1/p hence monotone decreasing and Lemma 6 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The case m > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' If m > 2, h and (h′)−2 are convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let z = ym−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' With 0 ≤ y ≤ γm, W(y) := ym − m y + m − 1 and the function h given by h(y) = y−1 |y−1| � W(y)/q, we find that the expression 4 W 3/2 �√ W �′′ = 2 W W ′′ − � W ′�2 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' DOLBEAULT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' GARC´IA-HUIDOBRO, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' MAN´ASEVICH has its sign given by F(y) := − m2 + 2 m (m − 1)2 ym−2 − 2 m2 (m − 2) ym−1 + m (m − 2) y2m−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' If y ≥ 1, then ym−2 ≥ 1, − m2 + 2 m (m − 1)2 ym−2 ≥ m (m − 2) (2 m − 1) and F(y) ≥ m (m − 2) (z − 1) (z + 1 − 2 m) ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' If y ≤ 1, then ym−2 ≤ 1, y2m−2 ≤ 1, m (m − 2) y2m−2 − m2 ≤ − 2 m y2m−2 and − F(y) ≥ 2 m (z − 1) � z + (m − 1)2� ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' In both cases, we conclude that h′′ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The function ((h′)−2)′′ has the sign of G(y) := 2 (m − 1) (m − 2) − m (2 m − 1) y + 2 (m − 1) (2 m − 1) ym−1 − 2 (m − 2) (2 m − 1) ym + (m − 2) y2m−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Since G(1) = G′(1) = 0 and G′′(y) = 2 (m − 1) (m − 2) (2 m − 1) W(y) ≥ 0 , we conclude that g ≥ 0 and ((h′)−2)′′ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' □ This proves Theorem 3 as a consequence of Corollary 7 and Lemma 9 if m > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The case 1 < m < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We cannot apply Corollary 7 and we have to directly rely on Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We recall that m = q/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let us start by computing K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' The function y �→ − K′(y) has the sign of p2 y2 f(a, m, y, z) where z = ym−1, the parameters (a, m) are admissible in the sense that a = 1 p ∈ � 0, 1 2 � , m = q p ∈ (1, 2) , and f(a, m, y, z) = − 3 m y (z − 1)2 (m z − 1) + 2 (m − 1 − m y + y z) � 2 + (1 − 6 m + m2) z + 2 m2 z2� + a � 3 m y (z − 1)3 − 6 (z − 1) (m z − 1) (m − 1 − m y + y z) � + a2 � 2 (z − 1)2 (m − 1 − m y + y z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We set y = xp so that x = y1/p and dx dy = 1 p y−1/p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let Φ(x) := W(y) = xmp − m xp + m − 1 ∀ x ∈ � 0, γ1/p m � , where W and h are as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='1, so that ��√q h′(y) ��2 = ����� Φ′(x) 2 p y1/p′ � Φ(x) ����� 2 |x=y1/p , MONOTONICITY OF THE PERIOD 13 that is, 4 m p3 ��y1/p′ h′(y) ��2 = (Φ′(x))2/Φ(x) and K as in (18) can be rewritten as K(y) = − 1 2 y1/p′ d dy � 1 y2/p′ h′(y)2 � = − 2 m p3 d dx � Φ(x) |Φ′(x)|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Hence − K′ has the sign of d2 dx2 (Φ(x) |Φ′(x)|−2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=', of 6 Φ |Φ′′|2 − 2 Φ Φ′ Φ′′′ − 3 |Φ′|2 Φ′′ and the detailed computation shows that x4 q2 |Φ′(x)|4 d2 dx2 � Φ(x) |Φ′(x)|2 � = p2 y2 f(a, m, y, z), ending the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' □ Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' With V given by (4) and 2 < p < q < 2 p, K defined by (18) is monotone decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Keeping the notations of Lemma 10, our goal is to prove that y �→ f (a, m, y, ym−1) is nonnegative for any y ∈ (0, γm) whenever the parameters (a, m) are admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let us start by considering its value at some remarkable points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' At (y, z) = (0, 0), we have f(a, m, 0, 0) = 2 (1 − a) (2 − a) (m − 1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' At (y, z) = (1, 1), we have f(a, m, 1, 1) = 0 but a Taylor expansion shows that f � a, m, y, ym−1� = 1 12 (m − 1)3 cm,a (y − 1)4 + O � (y − 1)5� as y → 1 (21) for any a ∈ (0, 1/2), where cm,a = 12 m a (a − m − 1) + m � 2 m2 + 7 m + 2 � ≥ cm,1/2 = m (m + 1) (2 m − 1) > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' This proves that y �→ f (a, m, y, ym−1) is positive for any y ∈ (1 − ε, 1) ∪ (1, 1 + ε) for some ε = ε(a, m) > 0 whenever the parameters (a, m) are admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' At (y, z) = (γm, m), we have f(a, m, γm, m) = (m−1)3 cm,a , cm,a = 2 a2 −3 a (2 m+2−m γm)+2 � 2 m2 + 5 m + 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Using infm∈(1,2) 3 4 (2 m + 2 − m γm) = limm→1+ 3 4 (2 m + 2 − m γm) = 3 (1 − e/4) > 1/2, we have cm,a > cm,1/2 = (4 − 3 γm) m2 + � 7 − 3 2 γm � m + 3 2 > lim m→1+ � (4 − 3 γm) m2 + � 7 − 3 2 γm � m + 3 2 � = 1 2 (25 − 9 e) > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' In the limit as m → 2, we have y = z and f(a, 2, y, z) = 2 (1 − a) (2 − a) (z − 1)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (22) Hence f (a, 2, y, ym−1) is positive unless y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We are now going to take a given a ∈ (0, 1/2) and consider m ∈ (1, 2) as a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Let us prove that for some m∗ ∈ (1, 2), we have f (a, m, y, ym−1) ≥ 0 for any (m, y) such that m∗ < m < 2 and 0 ≤ y ≤ γm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We assume by contradiction that there are two sequences (mk)k∈N and (yk)k∈N such that 1 < mk < 2 for any k ∈ N, limk→+∞ mk = 2, 0 ≤ yk ≤ γmk and f � a, mk, yk, ymk−1 k � < 0 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' DOLBEAULT, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' GARC´IA-HUIDOBRO, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' MAN´ASEVICH for any k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Up to the extraction of a subsequence, (yk)k∈N converges to some limit y∞ ∈ [0, 2] and by continuity of f we know that f (a, 2, y∞, y∞) ≤ 0: the only possibility is y∞ = 1 by (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Since f � a, mk, yk, ymk−1 k � < 0 = f(a, mk, 1, 1), we learn that yk ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Since limk→+∞ yk = 1, this contradicts (21) or, to be precise, |yk − 1| ≥ ε(a, mk), as the reader is invited to check that lim infk→+∞ ε(a, mk) > 0 because f is a smooth function of all of its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' If we redefine m∗(a) := inf � m ∈ (1, 2) : f � a, m, y, ym−1� ≥ 0 ∀ y ∈ [0, γm] � , then we know that for any a ∈ (0, 1/2), we have m∗(a) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' We want to prove that m∗(a) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Again, let us argue by contradiction: if m∗(a) > 1, and assume that there are two sequences (mk)k∈N and (yk)k∈N such that 1 < mk < m∗(a) for any k ∈ N, limk→+∞ mk = m∗(a), 0 ≤ yk ≤ γmk and f � a, mk, yk, ymk−1 k � < 0 for any k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Up to the extraction of a subsequence, (yk)k∈N converges to some limit y∞ ∈ [0, 2] and by continuity of f we know that f (a, m∗(a), y∞, ym−1 ∞ ) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' For the same reasons as above, y∞ = 0, y∞ = 1 and y∞ = γm∗(a) are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Altogether, we have proved that for m = m∗(a) , we have f (a, m, y∞, ym−1 ∞ ) = 0 for some y∞ ∈ (0, 1) ∪ (1, γm) and we also have that f (a, m, y, ym−1) ≥ 0 for any y ∈ (0, 1) ∪ (1, γm), so that y∞ is a local minimizer of y �→ f (a, m, y, ym−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' As a consequence, we have shown that for m = m∗(a) > 1 and y = y∞ ̸= 1, we have f � a, m, y, ym−1� = 0 and ∂ ∂yf � a, m, y, ym−1� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (23) As we shall see below, this contradicts Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Hence y �→ f (a, m, y, ym−1) takes nonnegative values for any admissible parameters (a, m) with 1 < m < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' By Lemma 10, K′(y) ≤ 0, thus completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' □ We still have to prove that (23) has no solution y ∈ (0, 1) ∪ (1, γm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Since y ∂ ∂yf � a, m, y, ym−1� = y ∂f ∂y (a, m, y, z) + (m − 1) z ∂f ∂z (a, m, y, z) , we can relax the condition z = ym−1 and prove the slightly more general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' With the notations of Lemma 10, assume that m > 1, y ∈ (0, γm] and z ∈ (0, m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' For any admissible parameters (a, m), if f(a, m, y, z) = 0 , (24a) y ∂f ∂y (a, m, y, z) + (m − 1) z ∂f ∂z (a, m, y, z) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' (24b) then z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' MONOTONICITY OF THE PERIOD 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Solving the system (24a)–(24b) is an elimination problem because the function f, as defined in Lemma 10, is a polynomial in the variables a, y and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Since (24a) is a first order equation in y, we can eliminate this variable and find that y = n(a, m, z) d(a, m, z) with n(a, m, z) := 2 (m − 1) � a2 (z − 1)2 − 3 a (z − 1) (m z − 1) + 2 m2 z2 + � m2 − 6 m + 1 � z + 2 � , d(a, m, z) := m � 2 a2 (z − 1)2 + 3 a (z + 1)2 (z − 1) + 9 z2 + 8 z + 1 � − 2 z � a2 (z − 1)2 + 3 a (z − 1) + z + 2 � − m2 z � 6 a (z − 1) + z2 + 8 z + 9 � + 2 m3 z (2 z + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' After replacing, solving (24b) under the condition z ̸= 1 is reduced to a second order equation in z, whose discriminant is δ(a, m) := − 3 (a − 1)2 (m − 1)2 (a − m)2 � 5 a2 − 10 a (m + 1) − 3 m2 + 14 m − 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Since 5 a2−10 a (m+1)−3 m2 +14 m−3 takes only positive values for admissible (a, m), there are no other roots than z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' This is the desired contradiction, which completes the proof thanks to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' □ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' It is a straightforward consequence of Lemmas 6 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' □ Acknowledgment: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' was partially supported by the Project EFI (ANR-17-CE40- 0030) of the French National Research Agency (ANR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='G-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' was supported by Fonde- cyt grant 1210241, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' by Centro de Modelamiento Matem´atico (CMM) BASAL fund FB210005 for center of excellence from ANID-Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' © 2023 by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' This paper may be reproduced, in its entirety, for non-commercial purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' CC-BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='0 References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content=' Abramowitz and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='uchile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} +page_content='cl' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9A0T4oBgHgl3EQfCv8A/content/2301.01992v1.pdf'} diff --git a/J9AyT4oBgHgl3EQff_g3/content/tmp_files/2301.00349v1.pdf.txt b/J9AyT4oBgHgl3EQff_g3/content/tmp_files/2301.00349v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f590a6c5b2d574e391c3e1a45f0fe8715f6fa763 --- /dev/null +++ b/J9AyT4oBgHgl3EQff_g3/content/tmp_files/2301.00349v1.pdf.txt @@ -0,0 +1,2369 @@ +EvidenceCap: Towards trustworthy medical image +segmentation via evidential identity cap +Ke Zou1,2,5, Xuedong Yuan1,2,∗, Xiaojing Shen1,3, Yidi Chen4, +Meng Wang5, Rick Siow Mong Goh5, Yong Liu5, Huazhu Fu5,∗ +1 National Key Laboratory of Fundamental Science on Synthetic Vision, +Sichuan University, Chengdu, China +2 College of Computer Science, Sichuan University, Chengdu, China +3 College of Mathematics, Sichuan University, Chengdu, China +4 Department of Radiology, West China Hospital, +Sichuan University, Chengdu, China +5 Institute of High Performance Computing, +Agency for Science, Technology and Research, Singapore +∗ Corresponding authors: X. Yuan (yxd@scu.edu.cn) and +H. Fu (hzfu@ieee.org) +Abstract +Medical image segmentation (MIS) is essential for supporting disease diagnosis and +treatment effect assessment. +Despite considerable advances in artificial intelligence +(AI) for MIS, clinicians remain skeptical of its utility, maintaining low confidence in +such black box systems, with this problem being exacerbated by low generalization +for out-of-distribution (OOD) data. To move towards effective clinical utilization, we +propose a foundation model named EvidenceCap, which makes the box transparent in a +quantifiable way by uncertainty estimation. EvidenceCap not only makes AI visible in +regions of uncertainty and OOD data, but also enhances the reliability, robustness, and +computational efficiency of MIS. Uncertainty is modeled explicitly through subjective +logic theory to gather strong evidence from features. +We show the effectiveness of +EvidenceCap in three segmentation datasets and apply it to the clinic. +Our work +sheds light on clinical safe applications and explainable AI, and can contribute towards +trustworthiness in the medical domain. +1 +arXiv:2301.00349v1 [eess.IV] 1 Jan 2023 + +1 +Introduction +As a result of extensive research into deep learning, medical image segmentation using +Convolutional Neural Networks (CNNs) has greatly facilitated quantitative pathological +assessments [1, 2], diagnostic support systems [3, 4, 5] and tumor analysis [6, 7]. Nonetheless, +clinicians still question the capabilities of artificial intelligence (AI), viewing it as a black +box. This doubt is manifested in clinicians preferring not to use AI-derived results as a +basis for making informed decisions [8, 9]. This situation is exacerbated by AI being prone +to prediction vulnerability that yields unreliable results, especially with out-of-distribution +(OOD) data [10]. These limitations prompted us to introduce EvdenceCap, a new paradigm +for trustworthy medical image segmentation, which acts like an out-of-the-box identity cap +that can quantify what was hitherto a black box (Fig. 1 a). +Researchers have focused on modifying deep network structures for improving the ac- +curacy of segmentation in the last decade. Fully Convolutional Networks (FCN) has been +developed to achieve end-to-end accurate semantic segmentation with notable results [11]. +The U-Net [12] model and its variants [13, 14, 15, 16] were then proposed to obtain better +feature representations and segmentation results. Highly expressive Transformers [17, 18, 19] +have also been used with great success in computer vision and medical image segmentation. +Nonetheless, it is not enough to obtain accurate segmentation results. In particular, +-The above medical image segmentation methods have limited versatility. Due to +OOD data, medical image segmentation performance may drop significantly after deploy- +ment to real systems [10]. Therefore, the awareness of OOD data of the real environment is +very important for the deployed system. After all, it is very time-consuming to re-collect, +label and train data for the current system. +-Current medical image segmentation methods ignore the situations that AI may +make ambiguous decisions. In clinical practice, there are often situations where AI de- +cisions may be not well-informed, and principled mechanisms for quantifying uncertainty +are required for clinically-safe applications [8]. Knowing the unknowns of predictions while +delivering accurate and robust performance will help foster trust in AI technologies among +clinicians. Therefore, uncertainty estimation is an effective way to promote trustworthy +decisions. +Existing uncertainty estimation methods remain poorly utilized in medical +2 + +image segmentation. +Uncertainty quantification methods in medical domain include +Bayesian-[20], ensemble-[21], evidential-[22], and deterministic-based methods[23, 24]. A +simple way to produce uncertainty for medical image segmentation is to use an ensemble of +deep networks [25, 26]. However, deep ensembles require retraining the model from scratch, +which incurs a high computation cost for complex models. Some methods introduce the +dropout in the test phase to estimate lesion-level Bayesian uncertainties [27, 28, 29]. Al- +though this strategy reduces the computational burden, it leads to inconsistent outputs [30]. +Deep deterministic uncertainty [31] is extended for semantic segmentation using feature +space densities. Unfortunately, the above methods inevitably change the network structure +and incur computational costs. A recent study has proposed using a deep feature-extraction +module and an evidential layer to segment lymphomas from positron emission tomography +and computed tomography image [22]. The main aims of these studies remained on guid- +ing uncertainty to improve segmentation performance rather than obtaining more robust +segmentation with calibrated uncertainty, and on generating uncertainty to evaluate the +segmentation results rather than utilizing the calibrated uncertainty to further optimize the +model training. What’s more, there is no sufficient clinically applicable diagnostic studies +using uncertainty estimation to allow AI to filter out low-quality samples and alert OOD +data. +Trustworthy, robust, and computationally efficient uncertainty estimations +provide visible quality assessments for clinical practice. +The main objective of +this study is to introduce trustworthy medical image segmentation and demonstrate its +potential in clinical applications. We develop a trustworthy medical image segmentation +framework named EvidenceCap, which works like an identity cap that provides robustness, +confidence, and high efficiency for medical image segmentation. EvidenceCap renders the +output of the underlying network in an evidence-level manner. This not only estimates a +stable and reasonable pixel-level uncertainty, but also improves the reliability and robust- +ness of segmentation. EvidenceCap derives probabilities and uncertainties for different class +segmentation problems via Subjective Logic (SL) theory, where the Dirichlet distribution +parameterizes the distribution of probabilities for different classes of the segmentation re- +sults. Moreover, EvidenceCap is uncertain for inaccurate segmented regions during initial +training, while remaining confident for accurate regions during subsequent training. To reit- +3 + +erate, EvidenceCap can be flexibly applied to any segmentation backbone without incurring +heavy implementation and excessive computational burden. EvidenceCap can be applied +to detect OOD data and indicate image data quality. We demonstrate here that Evidence- +Cap achieves a superior performance with potential ease of interpretation in medical image +segmentation for diagnostic support and quantitative assessments of diseases. +2 +Results +EvidenceCap pipeline & trustworthy medical image segmentation. +EvidenceCap is a trustworthy medical image segmentation framework based on evidential +deep learning, which provides robust segmentation performance and reliable uncertainty +quantification for diagnostic support. A pipeline of EvidenceCap and its results in under- +taking trustworthy medical image segmentation tasks are shown in Fig. 1 b and c. In the +training phase (Fig. 1 b), EvidenceCap can be applied to any task in numerous medical +domains. +Its trained model visually generates auxiliary diagnostic results, including ro- +bust target segmentation results and reliable uncertainty estimation. In the testing phase, +in order to verify the effectiveness of the method, EvidenceCap was tested for confidence, +robustness, and computational efficiency on different segmentation tasks. +To illustrate, three challenging trustworthy medical image segmentation tasks using +different datasets are undertaken here: (1) dermoscopic images in a 2D setting using the +ISIC2018 dataset; (2) liver CT images in a 3D setting using the LiTS2017 dataset; and (3) +multi-modality MRI images in a multi-modality 3D setting using the BraTS2019 dataset. +In the first task, we hope to use robust segmentation performance and reliable uncertainty +quantification in evaluating skin lesions. In the second task, we hope to obtain credible +3D segmentations for the liver. In the third task, we hope to obtain robust segmentation +results and credible uncertainty estimations for brain tumors under extreme conditions. A +detailed description of the three tasks is presented in App. 4.5. We hope to show through +successful completion of these tasks that segmentation results with uncertainty estimations +of different models on different datasets can contribute to credible disease diagnosis and +treatment effect assessment through medical images. +4 + + i) Traditional Medical image Segmentation + ii) Trustworthy Medical image Segmentation +Confident Robust Efficient +Open the box in a quantitative way: +uncertainty estimation +Medical Image Inputs +Segmentation Results +AI Model: +Invisible Box +Remain Doubts +Knowing the unknows ! +Segmentation results with uncertainty quantifications +AI Model: +Quantitative Box +Alleviate concerns +a +Can I trust them? +Medical Image Inputs +Image Inputs +Segmentation Results +Segmentation results with uncertainty quantifications +Medical Image Inputs +P +U +C +Metric +Score +P +U +C +Metric +Score +Evidence +E=[e1,...,en] +Backbone: +V-Net/AU-Net +Dirichlet +Distribution +α=[α1,...,αn] +Uncertainty U +Predictiton R +GT +Encoder +Decoder +Z +(i,j,k) +(i,j,k) +(i,j,k) +, , +i j k +u +, , +n +i j k + +, , +n +i j k +e +LiTS2017: +3D volume +BraTS2019: +Multi-modal 3D volumes +Backbone: +UNet/V-Net +Encoder +Decoder +Evidence +E=[e1,...,en] +Dirichlet +Distribution +α=[α1,...,αn] +Z +(i,j,k) +(i,j,k) +, , +n +i j k + +, , +n +i j k +e +ISIC2018: +2D slice +Backbone: +UNet/V-Net +Evidence +E=[e1,...,en] +Dirichlet +Distribution +α=[α1,...,αn] +Z Softplus +(i,j,k) +(i,j,k) +, , +n +i j k +e +Backbone: Choose your +own design +Input: 2D/3D/ +Multi-modal 3D +EvidenceCap: +Evidential deep learning +segmentation +Train inputs: 2D slice / 3D volume / Multi-modal 3D volumes +Trustworthy Test: Confidence & Robustness & Efficiency +Input: 2D/3D/ +Multi-modal 3D under +different condition +Single forward pass: +Trained EvidenceCap with +backbone +Output: Robust segmentation +with its uncertainty map +ISIC2018: +dermoscopic images under normal & +abnormal (noise/mask) cases +LiTS2017: CT images +under normal & abnormal (noise/ +Blur/mask) cases +BraTS2019: Multi-modal MRI images +under normal & abnormal (noise/ +Blur/mask) cases +Output: Segmentation with +its uncertainty map +(i,j) +(i,j) +, +n +i j + +, +n +i j +e +Uncertainty U +Predictiton R +GT +Backbone +EvidenceCap +EvidenceCap +Backbone +EvidenceCap +Backbone +R +GT +Original +Noise +Occlusion +GT +Uncertainty U +Predictiton R +R +U +GT +Original +Noise +Blur +Occlusion +(i,j,k) +, , +i j k +u +(i,j) +,i j +u +(i,j) +,i j +u +(i,j) +,i j +u +(i,j) +,i j +u +GT +R +U +U +Original +Noise +Blur +Occlusion +Encoder +Decoder +Softplus +Softplus +CU + +ice +KL + + + +Dice + +CU + +ice +KL + + + +Dice + +CU + +ice +KL + + + +Dice + +b +c +Figure 1: a. The motivation for trustworthy medical image segmentation, i) Traditional +medical image segmentation; ii) Trustworthy medical image segmentation. b. The training +process of our framework. c. The trustworthy test on three datasets. R, U, and GT denote +the prediction, uncertainty map, and ground truth, respectively. +Task 1: EvidenceCap for skin lesion segmentation. +AI makes it possible for dermatologists to quickly diagnose and screen for early stages of +skin diseases using skin lesion boundary segmentation [32, 33]. However, there have been +few studies on skin lesion segmentation with noise interference. +As such, we conducted +studies at different levels of Gaussian noise and random masking based on the ISIC2018 +dataset (2D dermoscopic image) to validate the robustness of our proposed framework. +Comparison with U-Net based methods. We compared the results with Evidence- +5 + +Noised ImageRaw ImageNoisedImagePrediction +Uncertainty +Confidence +SliceScore:0.0206 +SliceScore:0.9747 +VolumeScore:0.0093 +VolumeScore:0.9895 +0.0Predicfion +Uncertainty +Confidence +Slice Score:0.0278 +SliceScore:0.9809 +VolumeScore:0.0105 +VolumeScore:0.9907 +0.0Ground TruthGround TruthPredicionageRawImagePrediction +Uncertainty +Confidence +1.0C&Prediction +Can I trust them? +otsPredicionRawImageR +....... +L10 cm +PR +L +10 cmR +L +10 cmR +L10 cm +PR +........ +L10 cm +PR +I......... +........10 cmR +I......... +........10 cmR +L10 cm +P0RR +LP +10 +cn..R10cm +P10R1010R +L10 cm +PCap with those of other U-Net variants at differing Gaussian noise with variance σ2 = +{0.1, 0.2, 0.3, 0.4, 0.5} and different mask ratios (MR) MR = {0.1, 0.25, 0.4} with eight +patch-size (Fig. 2 a). Performance with U-Net and V-Net degrades slowly, especially at +higher masking ratios and noise, as these confound AI. After applying EvidenceCap, the +results gain partial immunity to interference (Fig. 2 b). The basic network after applying +EvidenceCap contains some anti-interference ability, and the visual uncertainty estimation +graph generated can alert researchers and clinicians to the unreliability of data. +Comparison with uncertainty-based methods. +We compared our framework with +uncertainty-based algorithms and found the PU to be significantly disturbed by noise and +masking, while the underlying network after applying EvidenceCap shows less perturbations +(Fig. 2 a). Comparison of the uncertainty estimation results by ECE and UEO metrics +show that the backbone networks obtained a more robust uncertainty estimation ability +after applying EvidenceCap. Our visualization of the segmentation results and uncertainty +estimates (Fig. 2 b) indicated that the addition of our framework provides higher uncer- +tainty for target edges and the noised or masked pixels, suggesting that our framework can +alert researchers and clinicians to OOD data. +Task 2: EvidenceCap for liver segmentation. +AI can assist clinicians in hepatocellular carcinoma diagnosis and treatment planning for +liver cancers [34]. To verify the reliability and robustness of our method, we conducted +studies with the Liver2017 dataset (3D CT) under differing levels of Gaussian noise, Gaus- +sian blur, and random masking to achieve trustworthy medical image segmentation. +Comparison with U-Net based methods. To verify the robustness of EvidenceCap, +we compared other U-Net-based methods using differing Gaussian noise with variance +σ2 = {0.05, 0.1, 0.2, 0.3, 0.4}, differing Gaussian blur with variance +�� +σ2, k +�� += {(11, 10) , (13, 10) , (15, 20) , (23, 20)}, and differing masking ratios (MR) MR = +{0, 0.1, 0.25, 0.4} with eight patch-size. We found the results of U-Net based methods to +gradually decrease in four metrics with an increase in OOD, but this can be suppressed +when EvidenceCap is applied (Fig. 3 a). The robustness of the base method equipped with +EvidenceCap is higher than those of other methods, as indicated by our method segmenta- +tion results with their uncertainty map under differing conditions (Fig. 3 b). Our framework +6 + +Dice +ASSD +ECE +UEO +i) +ii) +a +Input +U +AU +V +U+Our +V+Our +PU +UE +DU +GT +Certain +Uncertain +1) +2) +3) +4) +5) +6) +Noise +Mask +Original +b +Figure 2: +a. +Quantitative comparisons of U-Net based methods and uncertainty- +based methods using the ISIC2018 dataset under differing Gaussian noise (σ2 += +{0, 0.1, 0.2, 0.3, 0.4, 0.5}) and patch-size mask degradation (MR = {0, 0.1, 0.25, 0.4}): i) +Dice, ASSD, ECE and UEO metrics at differeing Gaussian noise i) the same metrics at dif- +fering masking ratios. b. Visual comparison of skin lesion segmentation results with different +methods: 1) original input and its results; (2-3) input with Gaussian noise (σ2 = 0.2, 0.5) +and its results; (4-5) input with patch-size random mask (σ2 = 0.1, 0.4) and its results; (6) +ground truth. +also provides uncertainty maps that allow further diagnosis and analysis. +Comparison with uncertainty-based methods. We repeated the study with uncertainty- +based algorithms to further compare the calibrated uncertainty for segmentation. Perfor- +mance in uncertainty estimation for all methods degrades with increasing Gaussian noise, +7 + +U +PU +V +AU +UE +DU +U+Our +V+Our90- +1.0 +50 +80 +0.20 +70 +0.8 +40 +60 +0.15 +0.6 +50 . +30 +40 +0.10 +0.4 +20 +20 +0.05 +0.2 +10 - +10 +0.00- +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +Gaussian noise level +Gaussian noise level +Gaussian noise level +Gaussian noise level +90 +35 +1.0 +0.18 +0.9 +0.16 +80 +0.14 - +25 +0.8 +70 20 - +0.7 +09 +0.10 +15 +0.6 +0.08 +50 - +10 +0.5 +0.06 +40 +0.4 +0.04 +0.3 +0.02 +0.1 +0.25 +0.4 +0.1 +0.25 +0.4 +0.1 +0.25 +0.4 +0.1 +0.25 +0.4 +0 +Masking Level +Masking Level +Masking Level +Masking Level10.0 +1.0Certain 0.0 +1.0 UncertainGaussian blur, and random masking (Fig. 3 a). However, the backbones equipped with +EvidenceCap mitigates this degradation. Under normal conditions, our framework provides +higher uncertainty for liver edges compared to other methods. Under OOD conditions, our +framework is more sensitive to unseen regions and provides high uncertainty (Fig. 3 b). +EvidenceCap thus allows clinicians or annotators to better focus on areas of uncertainty in +order to work more efficiently. +Task 3: EvidenceCap for brain tumor segmentation. +AI can allow for the accurate segmentation of brain tumor from different imaging modal- +ities to assess the effectiveness pre- and post-treatment. We conducted studies using the +BraTS2019 dataset (3D multi-modality MRI) with differing levels of Gaussian noise, Gaus- +sian blur, and random masking to achieve trustworthy medical image segmentation. +Comparison with U-Net based methods. To verify the robustness of our model, we +vary Gaussian noise with variance σ2 = {0.5, 1.0, 1.5, 2.0}, Gaussian blur with variance +�� +σ2, k +�� += {(3, 3) , (5, 5) , (7, 7) , (9, 9)}, and masking ratios (MR) MR = {0.1, 0.25, 0.4, 0.6} +with eight patch-size to the voxels of the four modalities (Fig. 4 a). We observe that without +EvidenceCap, V-Net and Attention-UNet show results comparable to those of other meth- +ods, but performance decreases rapidly with additional conditions. V-Net and Attention- +UNet with our framework perform more robustly under increased conditioned levels (Fig. +4 b). +Comparison with uncertainty-based methods. To further quantify the reliability of +the uncertainty estimation, we compared our model to different uncertainty-based methods, +using the elegant uncertainty evaluation metrics of ECE and UEO. The performances of all +uncertainty-based methods decay gradually with increasing levels of Gaussian noise (Figs. 4 +a (2)-(4)), but our method decays more slowly with the benefit of the reliable and robust ev- +idences captured by our trusted segmentation framework. Visually comparing brain tumor +segmentation results from different methods to demonstrate the reliability of the uncertainty +estimations, the V-Net and Attention-UNet with EvidenceCap applied obtained more ac- +curate and robust uncertainty estimations, even under strong noised conditions (Fig. 4 b). +This is due to our not using softmax for output which would lead to over-confidence [23]; +instead, we employed a subjective logical framework to gather more favorable and robust +8 + +i) +ii) +iii) +a +Dice +ASSD +ECE +UEO +Input +U +AU +V +U+Our +V+Our +PU +UE +DU +Certain +Uncertain +1) +2) +3) +4) +5) +6) +7) +8) +Noise +Blur +Original +Mask +GT +b +Figure 3: a. +Quantitative comparisons of U-Net based methods and uncertainty-based +methods with the Liver2017 dataset under differing Gaussian noise, Gaussian blur, and +patch-size random masking degradation conditions: i) Dice, ASSD, ECE, and UEO met- +rics at differing Gaussian noise σ2 = {0, 0.05, 0.1, 0.2, 0.3, 0.4}; ii) four metrics at differ- +ing Gaussian blur +�� +σ2, k +�� += {(11, 10) , (13, 10) , (15, 20) , (23, 20)}; iii) four metrics at +differing random masking MR = {0.1, 0.25, 0.4}. +b. +Visual comparison of liver seg- +mentation results using different methods: +a) original input and its results; (b-c) in- +put with Gaussian noise (σ2 = 0.2, 0.4) and its results; (d-e) input with Gaussian blur +( +�� +σ2, k +�� += {(13, 10) , (15, 20)}) and its results; (f-g) Input with random mask ratios +(σ2 = 0.1, 0.4) and its results; (h) ground truth. +9 + +100 +80 +09 +40 +20 +0.05 +0.1 +0.2 +0.3 +0.4 +Gaussiannoiselevel18 +16 +14 +12 +8 +6 +4 +2 +0 +0 +0.05 +0.1 +0.2 +0.4 +Gaussiannoiselevel0.18 +0.16 +0.14 +0.12 +0.10 +0.08 +0.06 +0.04 +0.02 +0 +0.05 +0.1 +0.2 +0.3 +0.4 +Gaussiannoiselevel1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +0 +0.05 +0.1 +0.2 +0.3 +0.4 +Gaussiannoise level100 +14 +0.12 +12 +90 +0.10 +0.9 +10 +80 +0.08 +0.8 +8 +70 +0.06 +0.7 +09 +0.04 +0.6 +50 +0.02 +0.5 +40 1 +0 +0.00 1 +0.4 +0 +11 +13 +15 +23 +0 +11 +13 +15 +23 +0 +11 +13 +23 +0 +11 +13 +15 +23 +Gaussian blur Level +Gaussian blur Level +Gaussian blur Level +Gaussian blur Level +100 +0.200 +1.0 +50 +0.175 +80 +0.8 +40 +0.150 +0.125 +60 - +30 + +0.6 0.100 +40 +20 +0.075 +0.4 +0.050 +20 - +10 +0.2 +0.025 +0.000 1 +0.0 1 +0.1 +0.25 +0.4 +0.1 +0.25 +0.4 +0 +0.1 +0.25 +0.4 +0 +0.1 +0.25 +0.4 +Masking Level +Masking Level +Masking Level +Masking Level +U +PU +V +AU +UE +DU +U+Our +V+Our。10.0 +1.0evidence from the data. +Inference analysis of uncertainty estimation models for the three +tasks. +DU [35] applied Monte-Carlo dropout (p=0.5) on U-Net before pooling or after upsampling. +UE [21] quantifies the uncertainties by ensembling multiple models. UE shares the same +U-Net structure and trained with different random initialization on the different subsets +(90%) of the training dataset to enhance variability. PU [30] learns a conditional density +model over-segmentation based on a combination of a U-Net with a conditional variational +autoencoder. The methods described above modify the original network structure or re- +duce the efficiency of training. +Our method explicitly quantifies the uncertainty with a +single forward pass through the backbone neural network using subjective logic theory. To +demonstrate the effectiveness of the efficiency (computational cost and accuracy), we pro- +vide more insight into the performances of uncertainty estimation methods on the datasets +of ISIC2018, LiTS2017, and BraTS2019 with Gaussian noise by σ2= {0.3, 0.2, 1.5} (Tab. 1). +The backbones (U/AU/V) applying EvidenceCap considerably outperformed the baseline +uncertainty quantification methods in testing time and FLOPs (Tab. 1). This is because our +method avoids changing the network structure and incurs lower computational costs. More- +over, our method achieves better performance on Dice score, Average Symmetric Surface +Distance (ASSD), Expected Calibration Error (ECE), and Uncertainty-error overlap (UEO), +especially in the case of high noise interference. In contrast, both DU [35] and UE [21] give +unsatisfactory results, especially under high noise conditions. PU [30] improves on these +results, but still struggles to maintain performance under high noise conditions. This is +because PU and DU will sample at the point of testing to obtain uncertainty estimations, +and the UE obtains uncertainty estimations by ensembling multiple models. The success of +EvidenceCap is attributed to employing the subjective logical framework to gather strong +evidence from the data. Additionally, it does not use the softmax layer for output, avoiding +over-confidence in the process. EvidenceCap develops a supervised strategy for uncertainty +estimation to guarantee better performance and calibrated uncertainty. +10 + +i) +ii) +iii) +ASSD +a +Dice +ECE +UEO +Noise +Blur +Original +Mask +Input +U +AU +V +AU+Our +V+Our +PU +UE +DU +U+Our +1) +2) +3) +4) +5) +6) +7) +8) +b +Figure 4: +a. +Quantitative comparisons with U-Net based methods and uncertainty- +based methods with the BraTS2019 dataset under differing Gaussian noise ( σ2 += +{0.5, 1.0, 1.5, 2.0}), Gaussian blur ( +�� +σ2, k +�� += {(3, 3) , (5, 5) , (7, 7) , (9, 9)}) and patch-size +mask degradation conditions (MR = {0.1, 0.25, 0.4, 0.6}: i) Dice, ASSD, ECE, and UEO +metrics with differing Gaussian noise; ii) the same metrics with differing Gaussian blur; +iii) The same metrics of different masking ratios. b.The visual comparison of brain tumor +segmentation results with different methods. 1) Original input (T2 as an example); 2)-3) +Gaussian noise input under (σ2 = {1.0, 2.0}) and its results; 4)-5) Input under Gaussian +blur ( +�� +σ2, k +�� += {(3, 3) , (7, 7)}) and its results; 6)-7) input under random mask ratio +MR = {0.1, 0.4} and its results; 8) ground truth. +11 + +了 +U +PU +V +AU +UE +DU +V+Our +U+Our +AU+Our ++ ++ ++ ++ ++0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0 +0.1 +0.25 +0.4 +0.6 +Masking Level90 +80 +70 +60 +50 +40 +30 +0 +0.5 +1.0 +1.5 +2.0 +Gaussian noise level20.0 +17.5 +15.0 +12.5 +10.0 +7.5 +5.0 +2.5 +0.5 +1.0 +1.5 +2.0 +Gaussian noise +level0.030 +0.025 +0.020 +0.015 +0.010 +0.005 +0 +0.5 +1.0 +1.5 +2.0 +Gaussiannoise +eleve0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0 +0.5 +1.0 +1.5 +2.0 +Gaussian noise level90 +80 +70 +60 +50 +40 +0 +0.5 +1.0 +1.5 +2.0 +Gaussianblurleve +90 +80 +70 +60 +50 +40 +0 +0.1 +0.25 +0.4 +0.6 +Masking Level12 +10 +8 +9 +4 +2 +0 +0.5 +1.0 +1.5 +2.0 +Gaussianblurlevel +10 +9 +8 +7. +6 +5 +4- +3 +2 +1 +0 +0.1 +0.25 +0.4 +0.6 +Masking Level0.030 +0.025 +0.020 +0.015 +0.010 +0.005 +0 +0.5 +1.0 +1.5 +2.0 +Gaussianblurlevel +0.030 +0.025 +0.020 +0.015 +0.010 +0.005 +0 +0.1 +0.25 +0.4 +0.6 +MaskingLevel0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0 +0.5 +1.0 +1.5 +2.0 +GaussianblurleveTable 1: Inference analysis of Uncertainty estimation models on above three datasets under normal condition and Gaussian noise +condition (σ2=0.3, 0.3, 1.5). N and OOD denote the normal condition and out-of-distribution condition, respectively. +Methods +Testing time↓ +Parameter↓ +FLOPs↓ +Dice↑ +ASSD↓ +ECE↓ +UEO↑ +N +OOD +N +OOD +N +OOD +N +OOD +ISIC2018 +UE [21] +57.34 s +7.77 M +137.34 G +0.839 +0.425 +8.28 +30.20 +0.049 +0.142 +0.858 +0.416 +DU [35] +26.16 s +7.77 M +137.34 G +0.849 +0.369 +8.06 +31.78 +0.052 +0.163 +0.865 +0.544 +PU [30] +0.14 s +27.35 M +108.54 G +0.837 +0.580 +9.19 +21.87 +0.055 +0.130 +0.878 +0.665 +U+Ours +0.07 s +7.77 M +13.74 G +0.868 +0.530 +7.41 +21.47 +0.048 +0.119 +0.871 +0.650 +V+Ours +0.08 s +13.07 M +15.45 G +0.874 +0.781 +6.71 +11.35 +0.045 +0.087 +0.905 +0.839 +LiTS2017 +UE [21] +230.22 s +4.75 M +776.04 G +0.856 +0.603 +4.57 +9.58 +0.031 +0.077 +0.858 +0.647 +DU [35] +204.24 s +4.75 M +776.04 G +0.874 +0.607 +3.65 +8.35 +0.034 +0.093 +0.880 +0.670 +PU [30] +1.32 s +5.13 M +157.46 G +0.938 +0.690 +0.92 +8.45 +0.017 +0.086 +0.941 +0.703 +U+Ours +0.73 s +4.75 M +77.60 G +0.933 +0.800 +1.27 +4.29 +0.020 +0.047 +0.935 +0.842 +V+Ours +0.35 s +2.31 M +48.01 G +0.944 +0.797 +0.93 +4.41 +0.017 +0.049 +0.949 +0.825 +BraTS2019 +UE [21] +195.48 s +4.76 M +6941.65 G +0.857 +0.709 +3.42 +4.93 +0.0091 +0.0211 +0.857 +0.727 +DU [35] +105.90 s +4.76 M +6941.65 G +0.825 +0.662 +2.50 +5.79 +0.0062 +0.0104 +0.855 +0.697 +PU [30] +15.4 s +5.13 M +1423.74 G +0.864 +0.470 +1.63 +10.02 +0.0058 +0.0143 +0.886 +0.622 +U+Ours +2.40 s +4.76 M +1263.69 G +0.850 +0.743 +2.89 +4.74 +0.0058 +0.0088 +0.864 +0.838 +AU+Ours +2.72 s +4.77 M +1285.60 G +0.859 +0.803 +1.89 +2.25 +0.0054 +0.0071 +0.880 +0.845 +V+Ours +1.71 s +2.31 M +790.16 G +0.870 +0.721 +1.58 +4.10 +0.0048 +0.0127 +0.895 +0.761 +12 + +Clinically safe applications: out-of-distribution detector & Image +quality indicator +Out-of-distribution detector: EvidenceCap distinguishes OOD data for medical +applications. It is essential that image processing systems identify any OOD samples in +clinical settings. Uncertainty estimation quantifies the uncertainty of the in-distribution and +OOD data to detect inputs that are far outside the training data distribution. EvidenceCap +can thus be used to alert clinicians to areas where lesions may be present in OOD data. We +conducted OOD experiments on the Johns Hopkins OCT dataset and Duke OCT dataset +with Diabetic Macular Edema (DME). The specific experimental details can be found in the +App. 4.5. We found significant differences in the uncertainty results of the in-distribution +and OOD data (Fig. 5 a), with the uncertainty values of some regions with DME lesions +increasing significantly (Fig. 5 a (ii)). There are also marked differences in the uncertainty +of predictions between the in-distribution and OOD (Fig. 5 b). These results combine to +show that EvidenceCap provides a solution for filtering out abnormal areas where lesions +may be present in OOD data. In this way, this ensures that downstream models are only +run on in-distribution data that are likely to perform well, facilitating diagnostic analysis +of clinical data. +Image quality indicator: EvidenceCap indicates the quality of medical images. +As medical data fuels the use of AI in medicine, it is crucial to accurately quantify the value +of data in algorithmic prediction and decision-making. Uncertainty estimation is an intu- +itive and quantitative way to inform clinicians or researchers about the quality of medical +images. EvidenceCap guides image quality quantitatively through the distribution of un- +certainty values and qualitatively through the degree of explicitness of the uncertainty map. +This is due to the significant difference in uncertainty values between high-quality and low- +quality data sources. In what follows, we apply EvidenceCap to indicate the quality of data +for real-world applications. The Digital Retinal Images for Vessel Extraction (DRIVE) and +the Fundus Image Vessel Segmentation (FIVES) datasets are used for quality assessment +experiments. The experimental details can be found in the App. 4.5. As the image quality +decreases, the change in the uncertainty map becomes more evident (Fig. 5 c). We also +found differences in the uncertainty distribution of high and low-quality of images (Fig. 5 +d (1)). The uncertainty sensitivity curve is designed to quantify the quality of data (Fig. 5 +13 + +d (2)). It shows the uncertainty value-ratio at different uncertainty thresholds. The lower +the uncertainty threshold, the higher the uncertainty value-ratio should be. These results +demonstrate that EvidenceCap can serve as an image quality indicator to fairly value per- +sonal data in healthcare and consumer markets. This would help to remove harmful data +while identifying and collecting higher-value data for diagnostic support. +3 +Discussion +Although medical image segmentation methodology is growing considerably, this has not +been matched by a corresponding increase in reliability and robustness [36]. Developing a +reliable method for medical image segmentation is one solution to this issue that provides +sensitivity and explicability for OOD data. In this study, we develop EvidenceCap, the first +reliable medical image segmentation method which works as an identity cap for backbone +networks to generate robust segmentation results and credible uncertainty estimations. We +performed three trustworthy medical image segmentation tasks using three public datasets, +namely ISIC2018 (2D settings), LiTS2017 (3D settings), and BraTS2019 (Multi-modal 3D +settings) respectively, obtaining robust performance and credible uncertainty in each case. +We are confident that our work can potentially benefit researchers in the trustworthy medical +domain. +Robustness. Robustness is a key feature in trustworthy medical image segmentation. +Robustness to adversarial perturbations of the input data is critical to the stability of deep +learning [37]. To verify EvidenceCap’s robustness, we studied different imaging modalities in +2D, 3D and multi-modality 3D under different noised conditions using three datasets. The +robust performance of the basic network framework (U/V/AU) is significantly improved +after applying EvidenceCap (Figure +2, 3 and 4). +This performance improvement can +possibly be due the trustworthy loss function prompting the network to improve accuracy +when beginning training, and then focusing on uncertain regions as learning progresses, +rather than in overly skewing segmentation accuracy. EvidenceCap uses a subjective logic +framework to gather more evidence from the input data, so as to lead the final opinions. +Confidence. Credibility is another key feature in trustworthy medical image segmenta- +tion. Despite recent improvements in the accuracy of medical image segmentation, clinicians +14 + +Input +GT +Prediction +Uncertainty +Confidence +(i) In distribution +Johns Hopkins OCT dataset +(ii) Out-of-distribution +Duke OCT dataset with DME +a +b +Input +GT +Prediction +Uncertainty +Confidence +0.0 +0.7 +High-Quality +Low-Quality +c +i) +ii) +iii) +iv) +d +Input +GT +Prediction +Uncertainty +Confidence +0.1 +0.5 +2) +1) +Figure 5: Clinically safety applications for out-of-distribution detector and quality indica- +tor. a. The qualitative difference between in-distribution (Johns Hopkins dataset without +disease) and out-of-distribution data (Duke dataset with DME). DME: diabetic macular +edema. b. The quantified difference in uncertainty distribution of in-distribution and out- +of-distribution data. c. Qualitative differences between image data of differing qualities. +i-ii) Visualization results of high-quality data on the FIVES dataset. iiii-iv) Low-quality +data visualization results on the FIVES dataset. d. Quantitative difference in uncertainty +distribution and uncertainty sensitivity curve for images of different qualities. 1) Density of +uncertainty of predictions for different quality data. 2) Uncertainty sensitivity curve on the +different quality data. +15 + +Uncertainty Sensitivity Curve +High Quality +0.5 +LowQuality +Uncertainty Value-ratio +0.4 +0.3 +0.2 +0.1 +0.0 +0.1 +0.3 +0.5 +Uncertainty ThresholdingUncertainty of Predictions +20.0 +High Quality +17.5 +Low Quality +15.0 +12.5 +Density +10.0 +7.5 +5.0 +2.5 +0.0 +0.15 +0.3 +0.5 +UncertaintyValueUncertainty of Predictions +10 +Indistribution +Out-of-distribution +8 +Density +6 +4 +0 +0.0 +0.1 +0.3 +0.5 +0.7 +UncertaintyValuestill display little confidence towards this technology. A major reason is for clinicians to +more likely trust intuitive visualizations. To bridge this gap, EvidenceCap provides pixel- +level uncertainty estimations for clinicians while maintaining robustness. We also performed +experiments using three datasets to show calibrated uncertainty under different conditions. +We found EvidenceCap to provide better uncertainty estimates from the more intuitive cases +(Figures 2, 3 and 4). EvidenceCap thus provides clinicians with sufficient visual affirmation +in diagnosing and quantitatively assess diseases. By excluding a softmax layer, we avoid +assigning over-confident scores for incorrect segmentation results; this avoidance is further +aided by the calibrated uncertainty estimation loss function. Of course, the most important +is the subjective logic theory, which provides backbones with more evidence to assist the +final opinion. +Efficient inference. Execution efficiency is a necessary feature of trustworthy medical +image segmentation. To this end, we studied the inference analysis of uncertainty estimation +models. EvidenceCap does not visibly change the underlying network structure compared to +the DU, UE, and PU methods. We added little to the network parameters while maintaining +robustness and calibrated uncertainty (Tab. 1). This is mainly because EvidenceCap does +not require multiple sampling like DU and PU, nor does it require ensemble learning like +UE to estimate uncertainty. EvidenceCap provides reliability and robustness in processing +OOD samples without excessive computational burdens and modifications of the backbone +network. +Differences. We analyzed differences in trusted segmentation networks between the +traditional medical image segmentation methods [38, 39] and the evidential deep learning +method [24]. +Compared with the traditional segmentation methods [33, 34, 38, 39], we +treat the predictions of the backbone neural network as subjective opinions instead of using +a softmax layer to generate predictions with over-confident scores. As a result, our model +provides voxel-wise uncertainty estimations and robust segmentations of the medical image, +which is essential for facilitating interpretation during disease diagnosis. Applying the evi- +dential deep learning method [24], we develop an evidential deep segmentation framework +for use across any backbone network, focusing on trusted medical image segmentation and +providing uncertainty estimations for 2D slices and 3D volumes. We adopt the calibrated +uncertainty supervision strategy in method training to obtain more accurate and certain +16 + +predictions. +Invisible box to quantitative box. The studies described above have verified the +robustness, trust and high computational efficiency for medical image segmentation pro- +vided by EvidenceCap. EvidenceCap converts AI technology from a black box to a system +that is quantifiable. Traditional medical image segmentation methods achieve high numer- +ical performances by training UNet and its variants, and are susceptible to fluctuations in +the OOD data. Because of this, they are regarded as invisible black boxes. EvidenceCap +achieves robust and accurate performance while providing pixel-level confidence. Moreover +uncertainty is a quantifiable indicator that can be used as the loss function to design AI +models, since it is expected to decrease during training. In addition, the change of uncer- +tainty at the pixel level reflects the reliability of the data, which is more sensitive to OOD +data. This helps clinicians in clinical safety applications. +Clinical safety Applications. Finally, we use EvidenceCap in two real-world clinical +safety applications. The first application used EvidenceCap as an identification tool for OOD +data in the medical domain, which assists clinicians in being more sensitive to abnormal +data. We conducted qualitative and quantitative experiments using the Johns Hopkins OCT +dataset and Duke OCT dataset with DME. Experimental results showed that EvidenceCap +can be used to screen out OOD inputs that may appear to be lesions. The second application +used EvidenceCap as a medical image quality diagnostic tool for clinicians to filter unreliable +data. By the way, we designed the uncertainty sensitivity curve to better visualize the quality +of data. Qualitative and quantitative experiments using the DRIVE and FIVES datasets +demonstrate that EvidenceCap can discriminate the quality of data. +Limitations. Although EvidenceCap performs promisingly for segmentation of normal +data, there remains room for improvement. Flaws remain when processing high-level OOD +data for clinical needs. Multi-modality MRI are directly utilized in task 3 as inputs for +segmentation, but we do not progress to estimate uncertainty between the different modali- +ties. Despite this, EvidenceCap provides a reliable shortcut to medical image segmentation +for any backbone network through furnishing robust segmentation results with a visible +uncertainty map for clinicians and researchers. Looking ahead, there is a need to improve +the performance of robust segmentation results and uncertainty estimates under normal +and different levels of OOD data. At the same time, further exploration of multi-modal +17 + +trustworthy medical image segmentation is also needed, as is uncertainty estimation un- +der federated learning. All of these will lead to more trustworthy AI systems for disease +diagnosis and treatment. +In conclusion, our foundation model is analyzed and empirically demonstrated through +EvidenceCap in this study, paving the way for trustworthy medical image segmentation +that generates robust segmentation results and credible uncertainty estimations. Our uni- +fied framework for trusted medical image segmentation reduces excessive both computa- +tional burden and modifications to the backbone network for a model with evidence. We +additionally developed an uncertainty supervised strategy to generate more calibrated un- +certainty and maintain the segmentation performance of the base network. We evaluated +the robustness and ease of interpretation for data generated by EvidenceCap with three +public datasets consisting of different data modalities and different target structures. Our +proposed foundation model can apply for two real-world clinical safety applications. +4 +Methods +For trustworthy medical image segmentation, we adopted the U-Net [12] and its vari- +ants [16, 15] as the backbone networks to obtain the multi-class segmentation results. We +did not use softmax as the output layer, as using the largest softmax output leads to +over-confidence. As such, we considered the Dirichlet distribution to provide more trusted +segmentation results [24]. We further introduced SL [40] to induce probabilities and uncer- +tainties for different classes of segmentation problems. To deal with the unknown pixels in +medical images, we propose a calibrated uncertainty strategy for medical image segmenta- +tion problem. Finally, we designed the overall loss function for trustworthy medical image +segmentation. +4.1 +Constructing EvidenceCap +Backbone. U-Net [12] and its variants [16, 15] have seen recent widespread used across +medical image modalities. We thus employed them as our backbone for capturing contex- +tual information. Furthermore, the backbones only performed down-sampling three times +to reduce information loss and to achieve a balance between GPU memory usage and seg- +18 + +mentation accuracy. EvidenceCap can freely choose different backbones to extract image +features. We only use its decoder output feature vector without the softmax layer. For a +random image X in a medical image domain X, this process can be defined as: +ZX = fω (X) +(1) +Where fω (·) is the different network backbone without the softmax layer. In this study, +we assessed the performance of the three general backbones (U-Net [12], V-Net [16] and +Attention-UNet [15]). +Dirichlet distribution for medical image segmentation. For typical medical image +segmentation tasks [33, 34, 38], the predictions are usually carried by the softmax layer +as the final layer. As mentioned in [23, 41], the softmax layer has a tendency to display +high confidence even for wrong predictions. EvidenceCap alleviates this problem in the +following way: first, the traditional neural network output is followed by an activation +function (Softplus) layer to ensure that the network output is non-negative, which is regarded +as the evidence voxel EX = softplus (ZX). We then obtain a Dirichlet distribution from the +network output, which is considered as the conjugate prior of the multinomial distribution +[40]. This provides a predictive distribution for medical image segmentation and derives +uncertainty from this distribution. For a random image X in a medical image domain X +(∀X ∈ X), the projected probability distribution of multinomial opinions is defined by: +pX = bX + rXU X, +(2) +where bX, rX and U X are the belief mass distribution, base rate distribution and the +uncertainty mass distribution over X, respectively. Then, Dirichlet PDF D(pX | αX) can +be used to represent probability density over pX, which is given by: +D(pX | αX) = +� +� +� +� +� +1 +B(αX) +C� +c=1 +(pc +X)αc +X−1 +for +pX ∈ SC +0 +otherwise +, +(3) +where Dirichlet distribution with parameters αX = +� +α1 +X, . . . , αC +X +� +is considered as belief +mass assignment. B(αX) is the C-dimensional multinomial beta function, and SC is the +C-dimensional unit simplex, given by: +SC = +� +pX +����� +C +� +c=1 +pc +X = 1 +and +0 ≤ p1 +X, . . . , pC +X ≤ 1 +� +. +(4) +19 + +The total strength αX can be denoted as: +αX=EX + rXW. +(5) +To simplify equations (1) and (4), we consider the base rate distribution rX to be 1 and W +to be 1. +4.2 +Uncertainty & deep evidential segmentation +One of the generalizations of Bayesian theory for subjective probability is the Dempster- +Shafer Evidence Theory (DST) [42]. The Dirichlet distribution is formalized as the belief +distribution of DST over the discriminative framework in the Subjective Logic (SL) [40]. +For medical image segmentation, we define a credible segmentation framework through SL +[40], which derives the probability and the uncertainty of the different class segmentation +problem based on the evidence. Using 3D medical image segmentation as an example, SL +provides a belief mass and an uncertainty mass for different classes of segmentation results. +Accordingly, for a 3D image input X and the backbone network output ZX without the +softmax layer, its C + 1 mass values are all non-negative and their sum is one. This can be +defined as follows: +uc +i,j,k + +C +� +c=1 +bc +i,j,k = 1, +(6) +where bc +i,j,k ≥ 0 and uc +i,j,k ≥ 0 denote the probability of the (i, j, k)-th pixel for the c-th class +and the overall uncertainty of the (i, j, k)-th pixel in X, respectively. uc +i,j,k ∈ UX and UX +is the uncertainty for the backbone network output vector ZX (Fig. 1). bc +i,j,k ∈ bX and bX +is the probability for ZX. Then the SL associates the evidence ec +i,j,k having the Dirichlet +distribution with the parameters αc +i,j,k = ec +i,j,k + 1, where ec +i,j,k ≥ 0 and ec +i,j,k ∈ EX. EX is +the evidence for ZX (Fig. 1). Then, the belief mass and the uncertainty of the (i, j, k)-th +pixel can be denoted as: +bc +i,j,k = +ec +i,j,k +S += +αc +i,j,k − 1 +S +and +ui,j,k = C +S , +(7) +where S = +C� +c=1 +αc +i,j,k = +C� +c=1 +� +ec +i,j,k + 1 +� +denotes the Dirichlet strength. This describes such a +phenomenon that the more evidence of the c-th class obtained by the (i, j, k)-th pixel, the +greater its probability. On the contrary, the greater uncertainty for the (i, j, k)-th pixel. +20 + +4.3 +Calibrated uncertainty +EvidenceCap directly learns the uncertainty without sampling. Nevertheless, it may not be +calibrated well enough to handle unknown pixels in the medical image. As pointed out in +the literature [43, 44], a well-calibrated model should be uncertain in its predictions when +being inaccurate, and be confident for the opposite case. To this end, we propose calibrated +uncertainty for medical image segmentation by using the relationship between accuracy +and uncertainty as an anchor. Specifically, we introduce the accuracy versus uncertainty +utility function [44], an optimization method for Calibrated Uncertainty (CU). This enables +the backbone to improve segmentation performance, in addition to learn to provide well- +calibrated uncertainties. It can be defined as: +CU = +NAC + NIU +NAC + NAU + NIC + NIU +. +(8) +where NAC, NAU, NIC and NIU denote the number of the Accurate and Certain (A&C), +Accurate and Uncertain (A&U), the Inaccurate and Certain (I&C) and the Inaccurate +and Uncertain (I&U) samples. +As in the above formula, we hope that the CU will be +larger. In other words, we encourage EvidenceCap to learn more A&C samples in the early +training period and provide more I&U samples later in training. Following the same goal as +[44], we design the uncertainty calibration loss function as Eq. 14. More details about the +training process of calibrated uncertainty and uncertainty calibration loss are presented in +Appendix 4.5. +4.4 +Training to form opinions +Due to the imbalance of organ/tumor, our network is first trained with cross-entropy loss +function, which is defined as: +Lce = +C +� +c=1 +−yc +X log (pc +X), +(9) +where yc +X and pc +X are the label and predicted probability of the m-th sample for class c. +Then, SL associates the Dirichlet distribution with the belief distribution under the frame- +work of evidence theory for obtaining the probability of different classes and uncertainty +of different voxels based on the evidence collected from backbone. Therefore, Eq. 9 can be +21 + +further improved as follows: +Lice = +� � C� +c=1 +−yc +X log(pc +X) +� +1 +B(αX) +C� +c=1 +(pc +X)αc +X−1dpX += +C� +c=1 +yc +X (ψ (SX) − ψ (αc +X)) +, +(10) +where ψ (·) denote the digamma function. pm is the class assignment probabilities on a +simplex. To guarantee that incorrect labels will yield less evidence, even shrinking to 0, the +KL divergence loss function is introduced as below: +LKL = log +� +Γ( +�C +c=1 �αc +X) +Γ(C) �C +c=1 Γ(�αc +X) +� ++ �C +c=1 (�αc +X − 1) +� +ψ (�αc +X) − ψ +��C +c=1 �αc +X +�� +, +(11) +where Γ (·) is the gamma function. +˜αc +X = yc +X + (1 − yc +X) ⊙ αc +X denotes the adjusted +parameters of the Dirichlet distribution, which is used to ensure that ground-truth class +evidence is not mistaken for 0. Furthermore, the Dice score is an important metric for +judging the performance of organ/tumor segmentation. Therefore, we use a soft Dice loss +to optimize the network, which is defined as: +LDice = 1 − 2yc +Xpc +X + e +yc +X + pc +X + e, +(12) +where yc +X and pc +X are the label and probability of the target. So, the segmentation loss +function LS can be define as follows: +LS = Lice + λLKL + (1 − βt) LDice, +(13) +where λ is the balance factor and set to be 0.02. To guide the model optimization at the +early stage of network training, (1 − βt) is noted as the annealing factor, which is defined +by βt=β0e{−(Inβ0/T)t}. T and t are the total epochs and the current epoch, respectively. +Then, according to Sec. 4.3 and Appendix 4.5, the loss function for well-calibrated +uncertainty can be defined as follows: +LCU = −βt +� +i,j,k∈{ˆyi,j,k=yi,j,k} +pi,j,k log (1 − ui,j,k) +− (1 − βt) +� +i,j,k∈{ˆyi,j,k̸=yi,j,k} +(1 − pi,j,k) log (ui,j,k) +(14) +Finally, the overall trustworthy loss function of our proposed network can be defined as +follows: +L = LS + LCU +(15) +22 + +4.5 +Experimental setup & Evaluation. +Experimental Setup: Our proposed network is implemented in PyTorch and trained on +NVIDIA GeForce RTX 2080Ti. We adopt the Adam to optimize the overall parameters. The +initial learning rates for different datasets are set to be 0.0002 (ISIC2018), 0.001 (LiTS2017), +and 0.002 (BraTS2019). +The poly learning strategy is used by decaying each iteration +with a power of 0.9. The maximum of the epoch is set to 200. The batch sizes for the +lesion segmentation, live segmentation, and brain tumor segmentation are set to 16, 4, and +2. All the following experiments adopted a five-fold cross-validation strategy to prevent +performance improvement caused by accidental factors. For the ISIC2018 dataset, we used +the data augmentation by random cropping, flipping, and random rotation as same as [33]. +For the LiTS2017 dataset, we only used the data augmentation by random flipping. For the +BraTS2019 dataset, the data augmentation techniques are similar as [38]. +Evaluation Metrics: The following metrics are employed for quantitative evaluation. +(a) The Dice score (Dice) and (b) Average symmetric surface distance (ASSD) is adopted as +an intuitive evaluation of segmentation accuracy. (c) Expected calibration error (ECE) [45, +46] and (d) Uncertainty-error overlap (UEO) [45, 46] are used as evaluation of uncertainty +estimations. +Dice score. Dice measures the overlap areas between the prediction map R and ground +truth mask G. It can be represented by: +Dice = 2 |R ∩ G| +|R| + |G|, +(16) +ASSD. ASSD calculates the accuracy of segmented boundaries between the point sets of +prediction SR and the point sets of ground truth SG. It can be defined as: +ASSD= +1 +SR+SG +× +� +� � +R∈SR +d (R, SG) + +� +G∈SG +d (G, SR) +� +� , +(17) +where d (r, SG) = ming∈SG (∥r − g∥) represents the minimum Euclidean distance from point +r to all the points in SG. +ECE. ECE approximates the calibration gap between confidence conf (Bm) [45] and accu- +23 + +racy acc (Bm) [45]. It can be expressed as: +ECE= +M +� +m=1 +|Bm| +N +× (acc (Bm) − conf (Bm)) , +(18) +where M is the number of interval bins. Bm denotes the set of indices of samples whose +prediction confidence falls into the interval. N means the number of samples. ECE closer +to zero means better calibration uncertainty. +UEO. UEO measures the overlap between the segmentation error Re and the thresholded +uncertainty Ut, which can be denoted as: +UEO = 2 |Re ∩ u| +|Re| + |u| (u ∈ Ut) , +(19) +where a higher UEO (close to one) indicates a better calibration. +Code Availability +All codes are available at https://github.com/Cocofeat/UMIS +Data Availability +ISIC2018: https://challenge2018.isic-archive.com/. +LiTS2017: https://competitions.codalab.org/competitions/17094. +BraTS2019: https://www.med.upenn.edu/cbica/brats-2019/. +Trustworthy medical image segmentation tasks on above datasets: +https:// +github.com/Cocofeat/UMIS. +Johns Hopkins OCT dataset: https://iacl.ece.jhu.edu/index.php?title=Main_ +Page. +Duke OCT dataset with DME: https://people.duke.edu/~sf59/Chiu_BOE_2014_ +dataset.htm. +DRIVE: https://drive.grand-challenge.org/DRIVE/. +FIVES: https://figshare.com/articles/figure/FIVES_A_Fundus_Image_Dataset_for_ +AI-based_Vessel_Segmentation/19688169/1. +24 + +Acknowledgements +This work was supported in part by AI Singapore Tech Challenge (Open-Theme) Funding +(AISG2-TC-2021-003), the Science and Technology Department of Sichuan Province (Grant +No. 2022YFS0071), and the China Scholarship Council (No. 202206240082). +Author Contributions Statement +Ke Zou: Conceptualization, Methodology, Software, Writing - original draft, Funding ac- +quisition. Xuedong Yuan: Supervision, Project administration, Methodology, Writing - +review & editing. Xiaojing Shen: Supervision, Project administration, Writing - review & +editing. Yidi Chen: Methodology, Writing - review & editing. Meng Wang: Methodol- +ogy, Writing - review & editing. Rick Siow Mong Goh: Supervision, Project administra- +tion, Writing - review & editing. Yong Liu: Supervision, Project administration, Writing - +review & editing. Huazhu Fu: Supervision, Project administration, Methodology, Writing +- review & editing. +Competing Interests Statement +The authors declare no competing interests. +25 + +References +[1] F. Isensee, P. F. Jaeger, S. A. Kohl, J. Petersen, and K. H. Maier-Hein, “nnu-net: a self- +configuring method for deep learning-based biomedical image segmentation,” Nature +methods, vol. 18, no. 2, pp. 203–211, 2021. +[2] S. P. Primakov, A. Ibrahim, J. E. van Timmeren, G. Wu, S. A. Keek, M. Beuque, +R. W. Granzier, E. Lavrova, M. Scrivener, S. 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Trustworthy medical image segmentation tasks. +To evaluate the generalizability of EvidenceCap, we first construct three challenging trust- +worthy medical image segmentation tasks with different imaging modalities in 2D or 3D +on three public datasets, including ISIC2018 (dermoscopic images, 2D settings), LiTS2017 +(liver CT images, 3D setting) and BraTS2019 (multi-modality MRI images, multi-modality +3D setting). Then, EvidenceCap is tested on these tasks to show its reliability, robustness, +and computational efficiency. +Trustworthy 2D medical image segmentation task on ISIC2018 dataset. First, the public +available training set of International Skin Imaging Collaboration (ISIC) 2018 with differ- +ent conditions (such as noise and mask) are constructed for trustworthy 2D medical image +segmentation task. Following the [33], a total of 2594 images and their ground truth are +randomly divided into a training set, validation set, and test set, containing 1814, 260 and +520 images, respectively. To verify the robustness and credibility of different models under +OOD conditions, we add different levels of Gaussian noise and random masks to the test +set, and perform 5-fold cross-validation for final results. First, we added the standard de- +viation of the Gaussian noise ranging from [0.1, 0.2, 0.3, 0.4, 0.5] to the original data with +normalization. Then, the strategy of random mask with 8 pixel-size like [47] ranging from +[0.1, 0.25, 0.4] are deployed for the original data. +Trustworthy 3D medical image segmentation task on LiTS2017 dataset. Second, Liver Tu- +mor Segmentation (LiTS) Challenge 2017 with different conditions (such as noise, blur, and +mask) are constructed for trustworthy 3D medical image segmentation task. It contains +the public 131 and 70 contrast-enhanced 3D abdominal CT scans. Following [34, 48], We +resampled the overall samples to the same resolution 16 × 256 × 256 with the spacing of +0.076 × 0.76 × 1, and randomly divided them into training set and test set containing 105 +(nearly 985 volumes) and 26 cases (nearly 245 volumes), respectively. For the OOD con- +dition, we also add different levels of Gaussian noise, Gaussian blur and random mask to +the test data of 3D volumes. Gaussian noise is added to the normalized data with standard +deviation of the ranging from [0.05, 0.1, 0.2, 0.3, 0.4]. Gaussian blur is added to the test +data with variance varying from 11 to 23 and kernel sizes of 10 to 20, specifically ranging +32 + +from [(11, 10), (13, 10), (15, 20), (23, 20)]. The strategy of random mask with 8 pixel-size +like [47] ranging from [0.1, 0.25, 0.4] are also deployed for the original data. +Trustworthy Multi-modality 3D medical image segmentation task on BraTS2019 dataset. +More importantly, the Brain Tumor Segmentation (BraTS) 2019 challenge [49, 50] with +varying conditions (such as noise, blur and mask) are constructed for trustworthy Multi- +modality 3D medical image segmentation tasks. Four modalities of brain MRI scans with a +volume of 240 × 240 × 155 are used. 335 cases of patients on BraTS2019 with ground-truth +are randomly divided into train dataset, validation dataset, and test dataset with nearly +265, 35, and 35 cases, respectively. The three tumor sub-compartment labels are combined +to segment the whole tumor and all inputs are uniformly adjusted to 128 × 128 × 128 voxels +during the training. The outputs of our network contain 4 classes, which are background +(label 0), necrotic and non-enhancing tumor (label 1), peritumoral edema (label 2), and +GD-enhancing tumor (label 4). Similarly, in order to verify the reliability uncertainty esti- +mation and robust segmentation results of the model under OOD data, five changes were +made to the test set, namely Gaussian noise, Gaussian blur, and random mask. Gaussian +noise is added to the normalized data with standard deviation of the ranging from [0.5, 1.0, +1.5, 2.0]. Gaussian blur is added to the test data with variance varying from 3 to 9 and +kernel sizes of 3 to 9, specifically ranging from [(3, 3), (5, 5), (7, 7), (9, 9)]. The strategy +of random mask with 8 pixel-size like [47] ranging from [0.1, 0.25, 0.4] are also deployed for +the original data. +2. Calibrated uncertainty +We show the four possible toy examples of EvidenceCap output in Fig. 6. The first is a +sloped and sharp Dirichlet simplex specification model that makes accurate and certain +(A&C) predictions (Fig. 6 (a)), as opposed to an unsloped and flat Dirichlet simplex spec- +ification model that makes inaccurate and uncertain (I&U) (Fig. 6 (d)). In addition, the +model may also produce a sloped and flat Dirichlet simplex, that is, accurate and uncer- +tain (A&U) predictions (Fig. 6 (b)), and an unsloped and sharp Dirichlet simplex, that is, +inaccurate and certain (I&C) predictions (Fig. 6 (c)). Following the same goal as [44], we +encourage EvidenceCap to learn a skewed and sharp Dirichlet simplex in the early training +33 + +Figure 6: Examples of Probability Simplex. (a) Accurate and Certain (A&C) (b) Accurate +and Uncertain (A&U) (c) Inaccurate and Certain (I&C) (d) Inaccurate and Uncertain +(I&U). +(Fig. 6 (a)). In addition, we encourage EvidenceCap to provide an unsloped and flat Dirich- +let simplex for incorrect predictions in the late training (Fig. 6 (d)). This stems from the +fact that if a pixel is assigned a high uncertainty, the pixel is more likely to be incorrect, +thereby identifying an unknown pixel. To this end, we design the uncertainty calibration +loss function as Eq. 12, which regularizes EvidenceCap training by maximize the expecta- +tions of A&C and I&U cases (Fig. 6 (a) and Fig. 6 (d)) such that the other cases (A&U in +Fig. 6 (b) and I&C in Fig. 6 (c)) can be discouraged. The first term in Eq. 12 is designed +to give low uncertainty when the model predictions are accurate, while the second term in +Eq. 12 attempts to give high uncertainty when the model predictions are inaccurate. At the +same time, we adopt the annealing weighting factor βt to achieve different penalties. In the +early training stage, inaccurate predictions dominate, so the second term (I&C loss) should +be penalized more, while in late training, accurate predictions dominate, so the first term +(A&U loss) should be penalized more punishment. +3. Experimental details of Clinically safety applications. +Finally, we utilize EvidenceCap on two clinical safe applications as a quality indicator and +OOD detector for clinicians and patients. In clinical, the OOD sample and the value of data +are essential for AI medicine. We employed four real-world clinical datasets for applications +of the quality indicator and OOD detector. In the first application, Johns Hopkins OCT +(JH-OCT) dataset and Duke OCT dataset with Diabetic Macular Edema (Duke-OCT- +DME) are used for the OOD detector. 335 cases of patients on JH-OCT with ground-truth +34 + +(a) A&C +(b) A&U +(c) I&C +(d) I&Uare randomly divided into train dataset, validation dataset and test dataset with nearly +25, 5 and 5 cases, respectively. The 5 cases of test dataset on JH-OCT are used for in- +distribution detection. +In particular, the 10 cases on the Duke-OCT-DME are used as +another test dataset for OOD detection. Every case is uniformly adjusted to 128 × 1024 +voxels during the training and testing. +In the second application, Digital Retinal Images for Vessel Extraction (DRIVE) and +the Fundus Image Vessel Segmentation (FIVES) datasets are used for the quality indicator. +We first train the EvidenceCap on the DRIVE dataset (20 slices) and then test on the +FIVES dataset (600 slices). +In the FIVES dataset, each image was evaluated for three +qualities: lighting and color distortion, blurring, and low-contrast distortion. We tested on +normal images (459 slices with high quality) and images including these three quality ratings +(141 slices with low quality) from the FIVES dataset. Every case is uniformly adjusted to +565 × 584 voxels during the training and testing. +In these applications, the initial learning rate for the dataset are set to be 0.0001. The +poly learning strategy is used by decaying each iteration with a power of 0.9. The maximum +of the epoch is set to 100. The batch sizes for the layer-segmentation and voxel-segmentation +from OCT are set to 8. +4. More visual comparisons +More visual comparisons on ISIC2018, LiTS2017, and BraTS2019 dataset can be seen in +the figures 7, 8 and +9. More visual clinical applications on JH-OCT, Duke-OCT-DME, +DRIVE and FIVES dataset can be seen in the figures 10. +35 + +Input +U +AU +V +U+Our +V+Our +PU +UE +DU +GT +Certain +Uncertain +1) +2) +3) +4) +5) +6) +Noise +Mask +Original +Figure 7: The visual comparison of skin lesion segmentation results with different methods. +1) Original input and its results; (2-3) Input under Gaussian noise (σ2 = 0.2, 0.5) and its +results; (4-5) Input under patch-size mask (σ2 = 0.1, 0.4) and its results; (6) Ground truth. +Input +U +AU +V +U+Our +V+Our +PU +UE +DU +Certain +Uncertain +1) +2) +3) +4) +5) +6) +7) +8) +Noise +Blur +Original +Mask +GT +b +Figure 8: The visual comparison of liver segmentation results with different methods. a) +Original input and its results; (b-c) Input under Gaussian noise (σ2 = 0.2, 0.4) and its +results; (d-e) Input under Gaussian blur ( +�� +σ2, k +�� += {(13, 10) , (15, 20)}) and its results; +(f-g) Input under random mask ratio (σ2 = 0.1, 0.3) and its results; (h) Ground truth. +36 + +10.0 +1.0Certain 0.0 +1.0 +Uncertair10.0 +1.0235 +Certain0.0 +1.0Uncertair1) +2) +3) +4) +5) +6) +7) +8) +9) +10) +11) +12) +Noise +Blur +Original +Mask +Spike +Ghost +Input +U +AU +V +AU+Our +V+Our +PU +UE +DU +GT +Certain +Uncertain +Figure 9: The visual comparison of brain tumor segmentation results with different methods. +1) Original input (T2 as an example); 2)-3) Gaussian noise input under (σ2 = 1, 2) and its +results; 4)-5) Input under Gaussian blur ( +�� +σ2, k +�� += {(3, 3) , (7, 7)}) and its results. 6)-7) +Input under random mask ratio MR = {0.1, 0.4} and its results; 8) Ground Truth. +37 + +0.0 +1.0High-Quality +Low-Quality +i) +ii) +iii) +Input +GT +Prediction +Uncertainty +Confidence +0.1 +0.5 +b +(i) In distribution +Johns Hopkins OCT dataset +(ii) Out-of-distribution +Duke OCT dataset with DME +a +Input +GT +Prediction +Uncertainty +Confidence +0.0 +0.7 +Input +GT +Prediction +Uncertainty +Confidence +Figure 10: Clinically safety applications for out-of-distribution detector and quality indica- +tor. a. The qualitative difference between in-distribution (JH-OCT dataset without disease) +and out-of-distribution data (Duke-OCT-DME). DME: diabetic macular edema. b. Qual- +ity difference between image data of different quality, i) Visualization results of high-quality +data on the FIVES dataset. +ii-iii) Low-quality data visualization results on the FIVES +dataset. +38 + +1 \ No newline at end of file diff --git a/J9AyT4oBgHgl3EQff_g3/content/tmp_files/load_file.txt b/J9AyT4oBgHgl3EQff_g3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d62fa4733015efe8ce9b87b4dd62c3488fa3001 --- /dev/null +++ b/J9AyT4oBgHgl3EQff_g3/content/tmp_files/load_file.txt @@ -0,0 +1,1738 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf,len=1737 +page_content='EvidenceCap: Towards trustworthy medical image segmentation via evidential identity cap Ke Zou1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Meng Wang5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Rick Siow Mong Goh5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Yong Liu5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Huazhu Fu5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='∗ 1 National Key Laboratory of Fundamental Science on Synthetic Vision,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Sichuan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Chengdu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' China 2 College of Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Sichuan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Chengdu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' China 3 College of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Sichuan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Chengdu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' China 4 Department of Radiology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' West China Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Sichuan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Chengdu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' China 5 Institute of High Performance Computing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Agency for Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Technology and Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Singapore ∗ Corresponding authors: X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Yuan (yxd@scu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='cn) and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Fu (hzfu@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='org) Abstract Medical image segmentation (MIS) is essential for supporting disease diagnosis and treatment effect assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Despite considerable advances in artificial intelligence (AI) for MIS, clinicians remain skeptical of its utility, maintaining low confidence in such black box systems, with this problem being exacerbated by low generalization for out-of-distribution (OOD) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To move towards effective clinical utilization, we propose a foundation model named EvidenceCap, which makes the box transparent in a quantifiable way by uncertainty estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap not only makes AI visible in regions of uncertainty and OOD data, but also enhances the reliability, robustness, and computational efficiency of MIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Uncertainty is modeled explicitly through subjective logic theory to gather strong evidence from features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We show the effectiveness of EvidenceCap in three segmentation datasets and apply it to the clinic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Our work sheds light on clinical safe applications and explainable AI, and can contribute towards trustworthiness in the medical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='00349v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='IV] 1 Jan 2023 1 Introduction As a result of extensive research into deep learning, medical image segmentation using Convolutional Neural Networks (CNNs) has greatly facilitated quantitative pathological assessments [1, 2], diagnostic support systems [3, 4, 5] and tumor analysis [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Nonetheless, clinicians still question the capabilities of artificial intelligence (AI), viewing it as a black box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This doubt is manifested in clinicians preferring not to use AI-derived results as a basis for making informed decisions [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This situation is exacerbated by AI being prone to prediction vulnerability that yields unreliable results, especially with out-of-distribution (OOD) data [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' These limitations prompted us to introduce EvdenceCap, a new paradigm for trustworthy medical image segmentation, which acts like an out-of-the-box identity cap that can quantify what was hitherto a black box (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 1 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Researchers have focused on modifying deep network structures for improving the ac- curacy of segmentation in the last decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Fully Convolutional Networks (FCN) has been developed to achieve end-to-end accurate semantic segmentation with notable results [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The U-Net [12] model and its variants [13, 14, 15, 16] were then proposed to obtain better feature representations and segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Highly expressive Transformers [17, 18, 19] have also been used with great success in computer vision and medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Nonetheless, it is not enough to obtain accurate segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In particular, The above medical image segmentation methods have limited versatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Due to OOD data, medical image segmentation performance may drop significantly after deploy- ment to real systems [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Therefore, the awareness of OOD data of the real environment is very important for the deployed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' After all, it is very time-consuming to re-collect, label and train data for the current system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Current medical image segmentation methods ignore the situations that AI may make ambiguous decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In clinical practice, there are often situations where AI de- cisions may be not well-informed, and principled mechanisms for quantifying uncertainty are required for clinically-safe applications [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Knowing the unknowns of predictions while delivering accurate and robust performance will help foster trust in AI technologies among clinicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Therefore, uncertainty estimation is an effective way to promote trustworthy decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Existing uncertainty estimation methods remain poorly utilized in medical 2 image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Uncertainty quantification methods in medical domain include Bayesian-[20], ensemble-[21], evidential-[22], and deterministic-based methods[23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' A simple way to produce uncertainty for medical image segmentation is to use an ensemble of deep networks [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' However, deep ensembles require retraining the model from scratch, which incurs a high computation cost for complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Some methods introduce the dropout in the test phase to estimate lesion-level Bayesian uncertainties [27, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Al- though this strategy reduces the computational burden, it leads to inconsistent outputs [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Deep deterministic uncertainty [31] is extended for semantic segmentation using feature space densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Unfortunately, the above methods inevitably change the network structure and incur computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' A recent study has proposed using a deep feature-extraction module and an evidential layer to segment lymphomas from positron emission tomography and computed tomography image [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The main aims of these studies remained on guid- ing uncertainty to improve segmentation performance rather than obtaining more robust segmentation with calibrated uncertainty, and on generating uncertainty to evaluate the segmentation results rather than utilizing the calibrated uncertainty to further optimize the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' What’s more, there is no sufficient clinically applicable diagnostic studies using uncertainty estimation to allow AI to filter out low-quality samples and alert OOD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Trustworthy, robust, and computationally efficient uncertainty estimations provide visible quality assessments for clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The main objective of this study is to introduce trustworthy medical image segmentation and demonstrate its potential in clinical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We develop a trustworthy medical image segmentation framework named EvidenceCap, which works like an identity cap that provides robustness, confidence, and high efficiency for medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap renders the output of the underlying network in an evidence-level manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This not only estimates a stable and reasonable pixel-level uncertainty, but also improves the reliability and robust- ness of segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap derives probabilities and uncertainties for different class segmentation problems via Subjective Logic (SL) theory, where the Dirichlet distribution parameterizes the distribution of probabilities for different classes of the segmentation re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Moreover, EvidenceCap is uncertain for inaccurate segmented regions during initial training, while remaining confident for accurate regions during subsequent training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To reit- 3 erate, EvidenceCap can be flexibly applied to any segmentation backbone without incurring heavy implementation and excessive computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap can be applied to detect OOD data and indicate image data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We demonstrate here that Evidence- Cap achieves a superior performance with potential ease of interpretation in medical image segmentation for diagnostic support and quantitative assessments of diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 2 Results EvidenceCap pipeline & trustworthy medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap is a trustworthy medical image segmentation framework based on evidential deep learning, which provides robust segmentation performance and reliable uncertainty quantification for diagnostic support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' A pipeline of EvidenceCap and its results in under- taking trustworthy medical image segmentation tasks are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 1 b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In the training phase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 1 b), EvidenceCap can be applied to any task in numerous medical domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Its trained model visually generates auxiliary diagnostic results, including ro- bust target segmentation results and reliable uncertainty estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In the testing phase, in order to verify the effectiveness of the method, EvidenceCap was tested for confidence, robustness, and computational efficiency on different segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To illustrate, three challenging trustworthy medical image segmentation tasks using different datasets are undertaken here: (1) dermoscopic images in a 2D setting using the ISIC2018 dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (2) liver CT images in a 3D setting using the LiTS2017 dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' and (3) multi-modality MRI images in a multi-modality 3D setting using the BraTS2019 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In the first task, we hope to use robust segmentation performance and reliable uncertainty quantification in evaluating skin lesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In the second task, we hope to obtain credible 3D segmentations for the liver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In the third task, we hope to obtain robust segmentation results and credible uncertainty estimations for brain tumors under extreme conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' A detailed description of the three tasks is presented in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We hope to show through successful completion of these tasks that segmentation results with uncertainty estimations of different models on different datasets can contribute to credible disease diagnosis and treatment effect assessment through medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4 i) Traditional Medical image Segmentation ii) Trustworthy Medical image Segmentation Confident Robust Efficient Open the box in a quantitative way: uncertainty estimation Medical Image Inputs Segmentation Results AI Model: Invisible Box Remain Doubts Knowing the unknows !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Segmentation results with uncertainty quantifications AI Model: Quantitative Box Alleviate concerns a Can I trust them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Medical Image Inputs Image Inputs Segmentation Results Segmentation results with uncertainty quantifications Medical Image Inputs P U C Metric Score P U C Metric Score Evidence E=[e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=',en] Backbone: V-Net/AU-Net Dirichlet Distribution α=[α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=',αn] Uncertainty U Predictiton R GT Encoder Decoder Z (i,j,k) (i,j,k) (i,j,k) , , i j k u , , n i j k \uf061 , , n i j k e LiTS2017: 3D volume BraTS2019: Multi-modal 3D volumes Backbone: UNet/V-Net Encoder Decoder Evidence E=[e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=',en] Dirichlet Distribution α=[α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=',αn] Z (i,j,k) (i,j,k) , , n i j k \uf061 , , n i j k e ISIC2018: 2D slice Backbone: UNet/V-Net Evidence E=[e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=',en] Dirichlet Distribution α=[α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='αn] Z Softplus (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='k) (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='k) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='i j k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='Backbone: Choose your ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='own design ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='Input: 2D/3D/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='Multi-modal 3D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='EvidenceCap: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='Evidential deep learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='segmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='Train inputs: 2D slice / 3D volume / Multi-modal 3D volumes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='Trustworthy Test: Confidence & Robustness & Efficiency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='Input: 2D/3D/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='Multi-modal 3D under ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='different condition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='Single forward pass: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='Trained EvidenceCap with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='backbone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='Output: Robust segmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='with its uncertainty map ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='ISIC2018: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='dermoscopic images under normal & ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='abnormal (noise/mask) cases ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='LiTS2017: CT images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='under normal & abnormal (noise/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='Blur/mask) cases ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='BraTS2019: Multi-modal MRI images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='under normal & abnormal (noise/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='Blur/mask) cases ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='Output: Segmentation with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='its uncertainty map ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='j) (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='j) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' n i j \uf061 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' n i j e Uncertainty U Predictiton R GT Backbone EvidenceCap EvidenceCap Backbone EvidenceCap Backbone R GT Original Noise Occlusion GT Uncertainty U Predictiton R R U GT Original Noise Blur Occlusion (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='k) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' i j k u (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='j) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='i j u (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='j) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='i j u (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='j) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='i j u (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='j) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='i j u GT R U U Original Noise Blur Occlusion Encoder Decoder Softplus Softplus CU \uf04c ice KL \uf02b \uf04c \uf04c Dice \uf04c CU \uf04c ice KL \uf02b \uf04c \uf04c Dice \uf04c CU \uf04c ice KL \uf02b \uf04c \uf04c Dice \uf04c b c Figure 1: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The motivation for trustworthy medical image segmentation, i) Traditional medical image segmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' ii) Trustworthy medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The training process of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The trustworthy test on three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' R, U, and GT denote the prediction, uncertainty map, and ground truth, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Task 1: EvidenceCap for skin lesion segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' AI makes it possible for dermatologists to quickly diagnose and screen for early stages of skin diseases using skin lesion boundary segmentation [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' However, there have been few studies on skin lesion segmentation with noise interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' As such, we conducted studies at different levels of Gaussian noise and random masking based on the ISIC2018 dataset (2D dermoscopic image) to validate the robustness of our proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Comparison with U-Net based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We compared the results with Evidence- 5 Noised ImageRaw ImageNoisedImagePrediction Uncertainty Confidence SliceScore:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0206 SliceScore:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='9747 VolumeScore:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0093 VolumeScore:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='9895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0Predicfion Uncertainty Confidence Slice Score:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0278 SliceScore:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='9809 VolumeScore:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0105 VolumeScore:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='9907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0Ground TruthGround TruthPredicionageRawImagePrediction Uncertainty Confidence 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0C&Prediction Can I trust them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' otsPredicionRawImageR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' L10 cm PR L 10 cmR L 10 cmR L10 cm PR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='. L10 cm PR I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='.10 cmR I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='.10 cmR L10 cm P0RR LP 10 cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='.R10cm P10R1010R L10 cm PCap with those of other U-Net variants at differing Gaussian noise with variance σ2 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5} and different mask ratios (MR) MR = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4} with eight patch-size (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 2 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Performance with U-Net and V-Net degrades slowly, especially at higher masking ratios and noise, as these confound AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' After applying EvidenceCap, the results gain partial immunity to interference (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 2 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The basic network after applying EvidenceCap contains some anti-interference ability, and the visual uncertainty estimation graph generated can alert researchers and clinicians to the unreliability of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Comparison with uncertainty-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We compared our framework with uncertainty-based algorithms and found the PU to be significantly disturbed by noise and masking, while the underlying network after applying EvidenceCap shows less perturbations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 2 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Comparison of the uncertainty estimation results by ECE and UEO metrics show that the backbone networks obtained a more robust uncertainty estimation ability after applying EvidenceCap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Our visualization of the segmentation results and uncertainty estimates (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 2 b) indicated that the addition of our framework provides higher uncer- tainty for target edges and the noised or masked pixels, suggesting that our framework can alert researchers and clinicians to OOD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Task 2: EvidenceCap for liver segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' AI can assist clinicians in hepatocellular carcinoma diagnosis and treatment planning for liver cancers [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To verify the reliability and robustness of our method, we conducted studies with the Liver2017 dataset (3D CT) under differing levels of Gaussian noise, Gaus- sian blur, and random masking to achieve trustworthy medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Comparison with U-Net based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To verify the robustness of EvidenceCap, we compared other U-Net-based methods using differing Gaussian noise with variance σ2 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4}, differing Gaussian blur with variance �� σ2, k �� = {(11, 10) , (13, 10) , (15, 20) , (23, 20)}, and differing masking ratios (MR) MR = {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4} with eight patch-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We found the results of U-Net based methods to gradually decrease in four metrics with an increase in OOD, but this can be suppressed when EvidenceCap is applied (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 3 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The robustness of the base method equipped with EvidenceCap is higher than those of other methods, as indicated by our method segmenta- tion results with their uncertainty map under differing conditions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 3 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Our framework 6 Dice ASSD ECE UEO i) ii) a Input U AU V U+Our V+Our PU UE DU GT Certain Uncertain 1) 2) 3) 4) 5) 6) Noise Mask Original b Figure 2: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Quantitative comparisons of U-Net based methods and uncertainty- based methods using the ISIC2018 dataset under differing Gaussian noise (σ2 = {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5}) and patch-size mask degradation (MR = {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4}): i) Dice, ASSD, ECE and UEO metrics at differeing Gaussian noise i) the same metrics at dif- fering masking ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Visual comparison of skin lesion segmentation results with different methods: 1) original input and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (2-3) input with Gaussian noise (σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5) and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (4-5) input with patch-size random mask (σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4) and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (6) ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' also provides uncertainty maps that allow further diagnosis and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Comparison with uncertainty-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We repeated the study with uncertainty- based algorithms to further compare the calibrated uncertainty for segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Perfor- mance in uncertainty estimation for all methods degrades with increasing Gaussian noise, 7 U PU V AU UE DU U+Our V+Our90- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 50 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='20 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='8 40 60 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5 Gaussian noise level Gaussian noise level Gaussian noise level Gaussian noise level 90 35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='16 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='14 - 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='8 70 20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='7 09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='08 50 - 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='06 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4 0 Masking Level Masking Level Masking Level Masking Level10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0Certain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 UncertainGaussian blur, and random masking (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 3 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' However, the backbones equipped with EvidenceCap mitigates this degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Under normal conditions, our framework provides higher uncertainty for liver edges compared to other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Under OOD conditions, our framework is more sensitive to unseen regions and provides high uncertainty (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 3 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap thus allows clinicians or annotators to better focus on areas of uncertainty in order to work more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Task 3: EvidenceCap for brain tumor segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' AI can allow for the accurate segmentation of brain tumor from different imaging modal- ities to assess the effectiveness pre- and post-treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We conducted studies using the BraTS2019 dataset (3D multi-modality MRI) with differing levels of Gaussian noise, Gaus- sian blur, and random masking to achieve trustworthy medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Comparison with U-Net based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To verify the robustness of our model, we vary Gaussian noise with variance σ2 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0}, Gaussian blur with variance �� σ2, k �� = {(3, 3) , (5, 5) , (7, 7) , (9, 9)}, and masking ratios (MR) MR = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='6} with eight patch-size to the voxels of the four modalities (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4 a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We observe that without EvidenceCap, V-Net and Attention-UNet show results comparable to those of other meth- ods, but performance decreases rapidly with additional conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' V-Net and Attention- UNet with our framework perform more robustly under increased conditioned levels (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Comparison with uncertainty-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To further quantify the reliability of the uncertainty estimation, we compared our model to different uncertainty-based methods, using the elegant uncertainty evaluation metrics of ECE and UEO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The performances of all uncertainty-based methods decay gradually with increasing levels of Gaussian noise (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4 a (2)-(4)), but our method decays more slowly with the benefit of the reliable and robust ev- idences captured by our trusted segmentation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Visually comparing brain tumor segmentation results from different methods to demonstrate the reliability of the uncertainty estimations, the V-Net and Attention-UNet with EvidenceCap applied obtained more ac- curate and robust uncertainty estimations, even under strong noised conditions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This is due to our not using softmax for output which would lead to over-confidence [23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' instead, we employed a subjective logical framework to gather more favorable and robust 8 i) ii) iii) a Dice ASSD ECE UEO Input U AU V U+Our V+Our PU UE DU Certain Uncertain 1) 2) 3) 4) 5) 6) 7) 8) Noise Blur Original Mask GT b Figure 3: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Quantitative comparisons of U-Net based methods and uncertainty-based methods with the Liver2017 dataset under differing Gaussian noise, Gaussian blur, and patch-size random masking degradation conditions: i) Dice, ASSD, ECE, and UEO met- rics at differing Gaussian noise σ2 = {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' ii) four metrics at differ- ing Gaussian blur �� σ2, k �� = {(11, 10) , (13, 10) , (15, 20) , (23, 20)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' iii) four metrics at differing random masking MR = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Visual comparison of liver seg- mentation results using different methods: a) original input and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (b-c) in- put with Gaussian noise (σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4) and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (d-e) input with Gaussian blur ( �� σ2, k �� = {(13, 10) , (15, 20)}) and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (f-g) Input with random mask ratios (σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4) and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (h) ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 9 100 80 09 40 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4 Gaussiannoiselevel18 16 14 12 8 6 4 2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4 Gaussiannoiselevel0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4 Masking Level Masking Level Masking Level Masking Level U PU V AU UE DU U+Our V+Our。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0evidence from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Inference analysis of uncertainty estimation models for the three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' DU [35] applied Monte-Carlo dropout (p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5) on U-Net before pooling or after upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' UE [21] quantifies the uncertainties by ensembling multiple models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' UE shares the same U-Net structure and trained with different random initialization on the different subsets (90%) of the training dataset to enhance variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' PU [30] learns a conditional density model over-segmentation based on a combination of a U-Net with a conditional variational autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The methods described above modify the original network structure or re- duce the efficiency of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Our method explicitly quantifies the uncertainty with a single forward pass through the backbone neural network using subjective logic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To demonstrate the effectiveness of the efficiency (computational cost and accuracy), we pro- vide more insight into the performances of uncertainty estimation methods on the datasets of ISIC2018, LiTS2017, and BraTS2019 with Gaussian noise by σ2= {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5} (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The backbones (U/AU/V) applying EvidenceCap considerably outperformed the baseline uncertainty quantification methods in testing time and FLOPs (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This is because our method avoids changing the network structure and incurs lower computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' More- over, our method achieves better performance on Dice score, Average Symmetric Surface Distance (ASSD), Expected Calibration Error (ECE), and Uncertainty-error overlap (UEO), especially in the case of high noise interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In contrast, both DU [35] and UE [21] give unsatisfactory results, especially under high noise conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' PU [30] improves on these results, but still struggles to maintain performance under high noise conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This is because PU and DU will sample at the point of testing to obtain uncertainty estimations, and the UE obtains uncertainty estimations by ensembling multiple models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The success of EvidenceCap is attributed to employing the subjective logical framework to gather strong evidence from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Additionally, it does not use the softmax layer for output, avoiding over-confidence in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap develops a supervised strategy for uncertainty estimation to guarantee better performance and calibrated uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 10 i) ii) iii) ASSD a Dice ECE UEO Noise Blur Original Mask Input U AU V AU+Our V+Our PU UE DU U+Our 1) 2) 3) 4) 5) 6) 7) 8) b Figure 4: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Quantitative comparisons with U-Net based methods and uncertainty- based methods with the BraTS2019 dataset under differing Gaussian noise ( σ2 = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0}), Gaussian blur ( �� σ2, k �� = {(3, 3) , (5, 5) , (7, 7) , (9, 9)}) and patch-size mask degradation conditions (MR = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='6}: i) Dice, ASSD, ECE, and UEO metrics with differing Gaussian noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' ii) the same metrics with differing Gaussian blur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' iii) The same metrics of different masking ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='The visual comparison of brain tumor segmentation results with different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 1) Original input (T2 as an example);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 2)-3) Gaussian noise input under (σ2 = {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0}) and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4)-5) Input under Gaussian blur ( �� σ2, k �� = {(3, 3) , (7, 7)}) and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 6)-7) input under random mask ratio MR = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4} and its results;' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='721 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='58 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='761 12 Clinically safe applications: out-of-distribution detector & Image quality indicator Out-of-distribution detector: EvidenceCap distinguishes OOD data for medical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' It is essential that image processing systems identify any OOD samples in clinical settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Uncertainty estimation quantifies the uncertainty of the in-distribution and OOD data to detect inputs that are far outside the training data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap can thus be used to alert clinicians to areas where lesions may be present in OOD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We conducted OOD experiments on the Johns Hopkins OCT dataset and Duke OCT dataset with Diabetic Macular Edema (DME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The specific experimental details can be found in the App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We found significant differences in the uncertainty results of the in-distribution and OOD data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 5 a), with the uncertainty values of some regions with DME lesions increasing significantly (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 5 a (ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' There are also marked differences in the uncertainty of predictions between the in-distribution and OOD (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 5 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' These results combine to show that EvidenceCap provides a solution for filtering out abnormal areas where lesions may be present in OOD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In this way, this ensures that downstream models are only run on in-distribution data that are likely to perform well, facilitating diagnostic analysis of clinical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Image quality indicator: EvidenceCap indicates the quality of medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' As medical data fuels the use of AI in medicine, it is crucial to accurately quantify the value of data in algorithmic prediction and decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Uncertainty estimation is an intu- itive and quantitative way to inform clinicians or researchers about the quality of medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap guides image quality quantitatively through the distribution of un- certainty values and qualitatively through the degree of explicitness of the uncertainty map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This is due to the significant difference in uncertainty values between high-quality and low- quality data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In what follows, we apply EvidenceCap to indicate the quality of data for real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The Digital Retinal Images for Vessel Extraction (DRIVE) and the Fundus Image Vessel Segmentation (FIVES) datasets are used for quality assessment experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The experimental details can be found in the App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' As the image quality decreases, the change in the uncertainty map becomes more evident (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 5 c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We also found differences in the uncertainty distribution of high and low-quality of images (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 5 d (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The uncertainty sensitivity curve is designed to quantify the quality of data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 5 13 d (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' It shows the uncertainty value-ratio at different uncertainty thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The lower the uncertainty threshold, the higher the uncertainty value-ratio should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' These results demonstrate that EvidenceCap can serve as an image quality indicator to fairly value per- sonal data in healthcare and consumer markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This would help to remove harmful data while identifying and collecting higher-value data for diagnostic support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 3 Discussion Although medical image segmentation methodology is growing considerably, this has not been matched by a corresponding increase in reliability and robustness [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Developing a reliable method for medical image segmentation is one solution to this issue that provides sensitivity and explicability for OOD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In this study, we develop EvidenceCap, the first reliable medical image segmentation method which works as an identity cap for backbone networks to generate robust segmentation results and credible uncertainty estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We performed three trustworthy medical image segmentation tasks using three public datasets, namely ISIC2018 (2D settings), LiTS2017 (3D settings), and BraTS2019 (Multi-modal 3D settings) respectively, obtaining robust performance and credible uncertainty in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We are confident that our work can potentially benefit researchers in the trustworthy medical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Robustness is a key feature in trustworthy medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Robustness to adversarial perturbations of the input data is critical to the stability of deep learning [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To verify EvidenceCap’s robustness, we studied different imaging modalities in 2D, 3D and multi-modality 3D under different noised conditions using three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The robust performance of the basic network framework (U/V/AU) is significantly improved after applying EvidenceCap (Figure 2, 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This performance improvement can possibly be due the trustworthy loss function prompting the network to improve accuracy when beginning training, and then focusing on uncertain regions as learning progresses, rather than in overly skewing segmentation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap uses a subjective logic framework to gather more evidence from the input data, so as to lead the final opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Credibility is another key feature in trustworthy medical image segmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Despite recent improvements in the accuracy of medical image segmentation, clinicians 14 Input GT Prediction Uncertainty Confidence (i) In distribution Johns Hopkins OCT dataset (ii) Out-of-distribution Duke OCT dataset with DME a b Input GT Prediction Uncertainty Confidence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='7 High-Quality Low-Quality c i) ii) iii) iv) d Input GT Prediction Uncertainty Confidence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5 2) 1) Figure 5: Clinically safety applications for out-of-distribution detector and quality indica- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The qualitative difference between in-distribution (Johns Hopkins dataset without disease) and out-of-distribution data (Duke dataset with DME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' DME: diabetic macular edema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The quantified difference in uncertainty distribution of in-distribution and out- of-distribution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Qualitative differences between image data of differing qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' i-ii) Visualization results of high-quality data on the FIVES dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' iiii-iv) Low-quality data visualization results on the FIVES dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Quantitative difference in uncertainty distribution and uncertainty sensitivity curve for images of different qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 1) Density of uncertainty of predictions for different quality data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 2) Uncertainty sensitivity curve on the different quality data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 15 Uncertainty Sensitivity Curve High Quality 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5 LowQuality Uncertainty Value-ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5 Uncertainty ThresholdingUncertainty of Predictions 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 High Quality 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5 Low Quality 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5 Density 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5 UncertaintyValueUncertainty of Predictions 10 Indistribution Out-of-distribution 8 Density 6 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='7 UncertaintyValuestill display little confidence towards this technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' A major reason is for clinicians to more likely trust intuitive visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To bridge this gap, EvidenceCap provides pixel- level uncertainty estimations for clinicians while maintaining robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We also performed experiments using three datasets to show calibrated uncertainty under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We found EvidenceCap to provide better uncertainty estimates from the more intuitive cases (Figures 2, 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap thus provides clinicians with sufficient visual affirmation in diagnosing and quantitatively assess diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' By excluding a softmax layer, we avoid assigning over-confident scores for incorrect segmentation results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' this avoidance is further aided by the calibrated uncertainty estimation loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Of course, the most important is the subjective logic theory, which provides backbones with more evidence to assist the final opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Efficient inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Execution efficiency is a necessary feature of trustworthy medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To this end, we studied the inference analysis of uncertainty estimation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap does not visibly change the underlying network structure compared to the DU, UE, and PU methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We added little to the network parameters while maintaining robustness and calibrated uncertainty (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This is mainly because EvidenceCap does not require multiple sampling like DU and PU, nor does it require ensemble learning like UE to estimate uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap provides reliability and robustness in processing OOD samples without excessive computational burdens and modifications of the backbone network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We analyzed differences in trusted segmentation networks between the traditional medical image segmentation methods [38, 39] and the evidential deep learning method [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Compared with the traditional segmentation methods [33, 34, 38, 39], we treat the predictions of the backbone neural network as subjective opinions instead of using a softmax layer to generate predictions with over-confident scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' As a result, our model provides voxel-wise uncertainty estimations and robust segmentations of the medical image, which is essential for facilitating interpretation during disease diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Applying the evi- dential deep learning method [24], we develop an evidential deep segmentation framework for use across any backbone network, focusing on trusted medical image segmentation and providing uncertainty estimations for 2D slices and 3D volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We adopt the calibrated uncertainty supervision strategy in method training to obtain more accurate and certain 16 predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Invisible box to quantitative box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The studies described above have verified the robustness, trust and high computational efficiency for medical image segmentation pro- vided by EvidenceCap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap converts AI technology from a black box to a system that is quantifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Traditional medical image segmentation methods achieve high numer- ical performances by training UNet and its variants, and are susceptible to fluctuations in the OOD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Because of this, they are regarded as invisible black boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap achieves robust and accurate performance while providing pixel-level confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Moreover uncertainty is a quantifiable indicator that can be used as the loss function to design AI models, since it is expected to decrease during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In addition, the change of uncer- tainty at the pixel level reflects the reliability of the data, which is more sensitive to OOD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This helps clinicians in clinical safety applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Clinical safety Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Finally, we use EvidenceCap in two real-world clinical safety applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The first application used EvidenceCap as an identification tool for OOD data in the medical domain, which assists clinicians in being more sensitive to abnormal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We conducted qualitative and quantitative experiments using the Johns Hopkins OCT dataset and Duke OCT dataset with DME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Experimental results showed that EvidenceCap can be used to screen out OOD inputs that may appear to be lesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The second application used EvidenceCap as a medical image quality diagnostic tool for clinicians to filter unreliable data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' By the way, we designed the uncertainty sensitivity curve to better visualize the quality of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Qualitative and quantitative experiments using the DRIVE and FIVES datasets demonstrate that EvidenceCap can discriminate the quality of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Although EvidenceCap performs promisingly for segmentation of normal data, there remains room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Flaws remain when processing high-level OOD data for clinical needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Multi-modality MRI are directly utilized in task 3 as inputs for segmentation, but we do not progress to estimate uncertainty between the different modali- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Despite this, EvidenceCap provides a reliable shortcut to medical image segmentation for any backbone network through furnishing robust segmentation results with a visible uncertainty map for clinicians and researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Looking ahead, there is a need to improve the performance of robust segmentation results and uncertainty estimates under normal and different levels of OOD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' At the same time, further exploration of multi-modal 17 trustworthy medical image segmentation is also needed, as is uncertainty estimation un- der federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' All of these will lead to more trustworthy AI systems for disease diagnosis and treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In conclusion, our foundation model is analyzed and empirically demonstrated through EvidenceCap in this study, paving the way for trustworthy medical image segmentation that generates robust segmentation results and credible uncertainty estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Our uni- fied framework for trusted medical image segmentation reduces excessive both computa- tional burden and modifications to the backbone network for a model with evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We additionally developed an uncertainty supervised strategy to generate more calibrated un- certainty and maintain the segmentation performance of the base network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We evaluated the robustness and ease of interpretation for data generated by EvidenceCap with three public datasets consisting of different data modalities and different target structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Our proposed foundation model can apply for two real-world clinical safety applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4 Methods For trustworthy medical image segmentation, we adopted the U-Net [12] and its vari- ants [16, 15] as the backbone networks to obtain the multi-class segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We did not use softmax as the output layer, as using the largest softmax output leads to over-confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' As such, we considered the Dirichlet distribution to provide more trusted segmentation results [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We further introduced SL [40] to induce probabilities and uncer- tainties for different classes of segmentation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To deal with the unknown pixels in medical images, we propose a calibrated uncertainty strategy for medical image segmenta- tion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Finally, we designed the overall loss function for trustworthy medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 Constructing EvidenceCap Backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' U-Net [12] and its variants [16, 15] have seen recent widespread used across medical image modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We thus employed them as our backbone for capturing contex- tual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Furthermore, the backbones only performed down-sampling three times to reduce information loss and to achieve a balance between GPU memory usage and seg- 18 mentation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap can freely choose different backbones to extract image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We only use its decoder output feature vector without the softmax layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' For a random image X in a medical image domain X, this process can be defined as: ZX = fω (X) (1) Where fω (·) is the different network backbone without the softmax layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In this study, we assessed the performance of the three general backbones (U-Net [12], V-Net [16] and Attention-UNet [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Dirichlet distribution for medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' For typical medical image segmentation tasks [33, 34, 38], the predictions are usually carried by the softmax layer as the final layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' As mentioned in [23, 41], the softmax layer has a tendency to display high confidence even for wrong predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EvidenceCap alleviates this problem in the following way: first, the traditional neural network output is followed by an activation function (Softplus) layer to ensure that the network output is non-negative, which is regarded as the evidence voxel EX = softplus (ZX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We then obtain a Dirichlet distribution from the network output, which is considered as the conjugate prior of the multinomial distribution [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This provides a predictive distribution for medical image segmentation and derives uncertainty from this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' For a random image X in a medical image domain X (∀X ∈ X), the projected probability distribution of multinomial opinions is defined by: pX = bX + rXU X, (2) where bX, rX and U X are the belief mass distribution, base rate distribution and the uncertainty mass distribution over X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Then, Dirichlet PDF D(pX | αX) can be used to represent probability density over pX, which is given by: D(pX | αX) = � � � � � 1 B(αX) C� c=1 (pc X)αc X−1 for pX ∈ SC 0 otherwise , (3) where Dirichlet distribution with parameters αX = � α1 X, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' , αC X � is considered as belief mass assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' B(αX) is the C-dimensional multinomial beta function, and SC is the C-dimensional unit simplex, given by: SC = � pX ����� C � c=1 pc X = 1 and 0 ≤ p1 X, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' , pC X ≤ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (4) 19 The total strength αX can be denoted as: αX=EX + rXW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (5) To simplify equations (1) and (4), we consider the base rate distribution rX to be 1 and W to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2 Uncertainty & deep evidential segmentation One of the generalizations of Bayesian theory for subjective probability is the Dempster- Shafer Evidence Theory (DST) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The Dirichlet distribution is formalized as the belief distribution of DST over the discriminative framework in the Subjective Logic (SL) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' For medical image segmentation, we define a credible segmentation framework through SL [40], which derives the probability and the uncertainty of the different class segmentation problem based on the evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Using 3D medical image segmentation as an example, SL provides a belief mass and an uncertainty mass for different classes of segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Accordingly, for a 3D image input X and the backbone network output ZX without the softmax layer, its C + 1 mass values are all non-negative and their sum is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This can be defined as follows: uc i,j,k + C � c=1 bc i,j,k = 1, (6) where bc i,j,k ≥ 0 and uc i,j,k ≥ 0 denote the probability of the (i, j, k)-th pixel for the c-th class and the overall uncertainty of the (i, j, k)-th pixel in X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' uc i,j,k ∈ UX and UX is the uncertainty for the backbone network output vector ZX (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' bc i,j,k ∈ bX and bX is the probability for ZX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Then the SL associates the evidence ec i,j,k having the Dirichlet distribution with the parameters αc i,j,k = ec i,j,k + 1, where ec i,j,k ≥ 0 and ec i,j,k ∈ EX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' EX is the evidence for ZX (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Then, the belief mass and the uncertainty of the (i, j, k)-th pixel can be denoted as: bc i,j,k = ec i,j,k S = αc i,j,k − 1 S and ui,j,k = C S , (7) where S = C� c=1 αc i,j,k = C� c=1 � ec i,j,k + 1 � denotes the Dirichlet strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This describes such a phenomenon that the more evidence of the c-th class obtained by the (i, j, k)-th pixel, the greater its probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' On the contrary, the greater uncertainty for the (i, j, k)-th pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3 Calibrated uncertainty EvidenceCap directly learns the uncertainty without sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Nevertheless, it may not be calibrated well enough to handle unknown pixels in the medical image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' As pointed out in the literature [43, 44], a well-calibrated model should be uncertain in its predictions when being inaccurate, and be confident for the opposite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To this end, we propose calibrated uncertainty for medical image segmentation by using the relationship between accuracy and uncertainty as an anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Specifically, we introduce the accuracy versus uncertainty utility function [44], an optimization method for Calibrated Uncertainty (CU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This enables the backbone to improve segmentation performance, in addition to learn to provide well- calibrated uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' It can be defined as: CU = NAC + NIU NAC + NAU + NIC + NIU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (8) where NAC, NAU, NIC and NIU denote the number of the Accurate and Certain (A&C), Accurate and Uncertain (A&U), the Inaccurate and Certain (I&C) and the Inaccurate and Uncertain (I&U) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' As in the above formula, we hope that the CU will be larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In other words, we encourage EvidenceCap to learn more A&C samples in the early training period and provide more I&U samples later in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Following the same goal as [44], we design the uncertainty calibration loss function as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' More details about the training process of calibrated uncertainty and uncertainty calibration loss are presented in Appendix 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4 Training to form opinions Due to the imbalance of organ/tumor, our network is first trained with cross-entropy loss function, which is defined as: Lce = C � c=1 −yc X log (pc X), (9) where yc X and pc X are the label and predicted probability of the m-th sample for class c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Then, SL associates the Dirichlet distribution with the belief distribution under the frame- work of evidence theory for obtaining the probability of different classes and uncertainty of different voxels based on the evidence collected from backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 9 can be 21 further improved as follows: Lice = � � C� c=1 −yc X log(pc X) � 1 B(αX) C� c=1 (pc X)αc X−1dpX = C� c=1 yc X (ψ (SX) − ψ (αc X)) , (10) where ψ (·) denote the digamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' pm is the class assignment probabilities on a simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To guarantee that incorrect labels will yield less evidence, even shrinking to 0, the KL divergence loss function is introduced as below: LKL = log � Γ( �C c=1 �αc X) Γ(C) �C c=1 Γ(�αc X) � + �C c=1 (�αc X − 1) � ψ (�αc X) − ψ ��C c=1 �αc X �� , (11) where Γ (·) is the gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' ˜αc X = yc X + (1 − yc X) ⊙ αc X denotes the adjusted parameters of the Dirichlet distribution, which is used to ensure that ground-truth class evidence is not mistaken for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Furthermore, the Dice score is an important metric for judging the performance of organ/tumor segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Therefore, we use a soft Dice loss to optimize the network, which is defined as: LDice = 1 − 2yc Xpc X + e yc X + pc X + e, (12) where yc X and pc X are the label and probability of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' So, the segmentation loss function LS can be define as follows: LS = Lice + λLKL + (1 − βt) LDice, (13) where λ is the balance factor and set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To guide the model optimization at the early stage of network training, (1 − βt) is noted as the annealing factor, which is defined by βt=β0e{−(Inβ0/T)t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' T and t are the total epochs and the current epoch, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Then, according to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3 and Appendix 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5, the loss function for well-calibrated uncertainty can be defined as follows: LCU = −βt � i,j,k∈{ˆyi,j,k=yi,j,k} pi,j,k log (1 − ui,j,k) − (1 − βt) � i,j,k∈{ˆyi,j,k̸=yi,j,k} (1 − pi,j,k) log (ui,j,k) (14) Finally, the overall trustworthy loss function of our proposed network can be defined as follows: L = LS + LCU (15) 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5 Experimental setup & Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Experimental Setup: Our proposed network is implemented in PyTorch and trained on NVIDIA GeForce RTX 2080Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We adopt the Adam to optimize the overall parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The initial learning rates for different datasets are set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0002 (ISIC2018), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='001 (LiTS2017), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='002 (BraTS2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The poly learning strategy is used by decaying each iteration with a power of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The maximum of the epoch is set to 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The batch sizes for the lesion segmentation, live segmentation, and brain tumor segmentation are set to 16, 4, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' All the following experiments adopted a five-fold cross-validation strategy to prevent performance improvement caused by accidental factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' For the ISIC2018 dataset, we used the data augmentation by random cropping, flipping, and random rotation as same as [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' For the LiTS2017 dataset, we only used the data augmentation by random flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' For the BraTS2019 dataset, the data augmentation techniques are similar as [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Evaluation Metrics: The following metrics are employed for quantitative evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (a) The Dice score (Dice) and (b) Average symmetric surface distance (ASSD) is adopted as an intuitive evaluation of segmentation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (c) Expected calibration error (ECE) [45, 46] and (d) Uncertainty-error overlap (UEO) [45, 46] are used as evaluation of uncertainty estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Dice score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Dice measures the overlap areas between the prediction map R and ground truth mask G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' It can be represented by: Dice = 2 |R ∩ G| |R| + |G|, (16) ASSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' ASSD calculates the accuracy of segmented boundaries between the point sets of prediction SR and the point sets of ground truth SG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' It can be defined as: ASSD= 1 SR+SG × � � � R∈SR d (R, SG) + � G∈SG d (G, SR) � � , (17) where d (r, SG) = ming∈SG (∥r − g∥) represents the minimum Euclidean distance from point r to all the points in SG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' ECE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' ECE approximates the calibration gap between confidence conf (Bm) [45] and accu- 23 racy acc (Bm) [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' It can be expressed as: ECE= M � m=1 |Bm| N × (acc (Bm) − conf (Bm)) , (18) where M is the number of interval bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Bm denotes the set of indices of samples whose prediction confidence falls into the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' N means the number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' ECE closer to zero means better calibration uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' UEO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' UEO measures the overlap between the segmentation error Re and the thresholded uncertainty Ut, which can be denoted as: UEO = 2 |Re ∩ u| |Re| + |u| (u ∈ Ut) , (19) where a higher UEO (close to one) indicates a better calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Code Availability All codes are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='com/Cocofeat/UMIS Data Availability ISIC2018: https://challenge2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='isic-archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='com/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' LiTS2017: https://competitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='codalab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='org/competitions/17094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' BraTS2019: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='edu/cbica/brats-2019/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Trustworthy medical image segmentation tasks on above datasets: https:// github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='com/Cocofeat/UMIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Johns Hopkins OCT dataset: https://iacl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='ece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='jhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='edu/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='title=Main_ Page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Duke OCT dataset with DME: https://people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='duke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='edu/~sf59/Chiu_BOE_2014_ dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='htm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' DRIVE: https://drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='grand-challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='org/DRIVE/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' FIVES: https://figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='com/articles/figure/FIVES_A_Fundus_Image_Dataset_for_ AI-based_Vessel_Segmentation/19688169/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 24 Acknowledgements This work was supported in part by AI Singapore Tech Challenge (Open-Theme) Funding (AISG2-TC-2021-003), the Science and Technology Department of Sichuan Province (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 2022YFS0071), and the China Scholarship Council (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 202206240082).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Author Contributions Statement Ke Zou: Conceptualization, Methodology, Software, Writing - original draft, Funding ac- quisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Xuedong Yuan: Supervision, Project administration, Methodology, Writing - review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Xiaojing Shen: Supervision, Project administration, Writing - review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Yidi Chen: Methodology, Writing - review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Meng Wang: Methodol- ogy, Writing - review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Rick Siow Mong Goh: Supervision, Project administra- tion, Writing - review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Yong Liu: Supervision, Project administration, Writing - review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Huazhu Fu: Supervision, Project administration, Methodology, Writing review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Competing Interests Statement The authors 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Rempfler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Crimi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Shinohara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Berger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Ha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Rozycki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=', “Identifying the best machine learning algorithms for brain tumor segmentation, progression assessment, and overall survival prediction in the brats challenge,” arXiv preprint arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='02629, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 31 Appendix 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Trustworthy medical image segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To evaluate the generalizability of EvidenceCap, we first construct three challenging trust- worthy medical image segmentation tasks with different imaging modalities in 2D or 3D on three public datasets, including ISIC2018 (dermoscopic images, 2D settings), LiTS2017 (liver CT images, 3D setting) and BraTS2019 (multi-modality MRI images, multi-modality 3D setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Then, EvidenceCap is tested on these tasks to show its reliability, robustness, and computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Trustworthy 2D medical image segmentation task on ISIC2018 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' First, the public available training set of International Skin Imaging Collaboration (ISIC) 2018 with differ- ent conditions (such as noise and mask) are constructed for trustworthy 2D medical image segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Following the [33], a total of 2594 images and their ground truth are randomly divided into a training set, validation set, and test set, containing 1814, 260 and 520 images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To verify the robustness and credibility of different models under OOD conditions, we add different levels of Gaussian noise and random masks to the test set, and perform 5-fold cross-validation for final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' First, we added the standard de- viation of the Gaussian noise ranging from [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5] to the original data with normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Then, the strategy of random mask with 8 pixel-size like [47] ranging from [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4] are deployed for the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Trustworthy 3D medical image segmentation task on LiTS2017 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Second, Liver Tu- mor Segmentation (LiTS) Challenge 2017 with different conditions (such as noise, blur, and mask) are constructed for trustworthy 3D medical image segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' It contains the public 131 and 70 contrast-enhanced 3D abdominal CT scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Following [34, 48], We resampled the overall samples to the same resolution 16 × 256 × 256 with the spacing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='076 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='76 × 1, and randomly divided them into training set and test set containing 105 (nearly 985 volumes) and 26 cases (nearly 245 volumes), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' For the OOD con- dition, we also add different levels of Gaussian noise, Gaussian blur and random mask to the test data of 3D volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Gaussian noise is added to the normalized data with standard deviation of the ranging from [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Gaussian blur is added to the test data with variance varying from 11 to 23 and kernel sizes of 10 to 20, specifically ranging 32 from [(11, 10), (13, 10), (15, 20), (23, 20)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The strategy of random mask with 8 pixel-size like [47] ranging from [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4] are also deployed for the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Trustworthy Multi-modality 3D medical image segmentation task on BraTS2019 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' More importantly, the Brain Tumor Segmentation (BraTS) 2019 challenge [49, 50] with varying conditions (such as noise, blur and mask) are constructed for trustworthy Multi- modality 3D medical image segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Four modalities of brain MRI scans with a volume of 240 × 240 × 155 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 335 cases of patients on BraTS2019 with ground-truth are randomly divided into train dataset, validation dataset, and test dataset with nearly 265, 35, and 35 cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The three tumor sub-compartment labels are combined to segment the whole tumor and all inputs are uniformly adjusted to 128 × 128 × 128 voxels during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The outputs of our network contain 4 classes, which are background (label 0), necrotic and non-enhancing tumor (label 1), peritumoral edema (label 2), and GD-enhancing tumor (label 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Similarly, in order to verify the reliability uncertainty esti- mation and robust segmentation results of the model under OOD data, five changes were made to the test set, namely Gaussian noise, Gaussian blur, and random mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Gaussian noise is added to the normalized data with standard deviation of the ranging from [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Gaussian blur is added to the test data with variance varying from 3 to 9 and kernel sizes of 3 to 9, specifically ranging from [(3, 3), (5, 5), (7, 7), (9, 9)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The strategy of random mask with 8 pixel-size like [47] ranging from [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4] are also deployed for the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Calibrated uncertainty We show the four possible toy examples of EvidenceCap output in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The first is a sloped and sharp Dirichlet simplex specification model that makes accurate and certain (A&C) predictions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 6 (a)), as opposed to an unsloped and flat Dirichlet simplex spec- ification model that makes inaccurate and uncertain (I&U) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 6 (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In addition, the model may also produce a sloped and flat Dirichlet simplex, that is, accurate and uncer- tain (A&U) predictions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 6 (b)), and an unsloped and sharp Dirichlet simplex, that is, inaccurate and certain (I&C) predictions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 6 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Following the same goal as [44], we encourage EvidenceCap to learn a skewed and sharp Dirichlet simplex in the early training 33 Figure 6: Examples of Probability Simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (a) Accurate and Certain (A&C) (b) Accurate and Uncertain (A&U) (c) Inaccurate and Certain (I&C) (d) Inaccurate and Uncertain (I&U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 6 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In addition, we encourage EvidenceCap to provide an unsloped and flat Dirich- let simplex for incorrect predictions in the late training (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 6 (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' This stems from the fact that if a pixel is assigned a high uncertainty, the pixel is more likely to be incorrect, thereby identifying an unknown pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' To this end, we design the uncertainty calibration loss function as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 12, which regularizes EvidenceCap training by maximize the expecta- tions of A&C and I&U cases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 6 (a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 6 (d)) such that the other cases (A&U in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 6 (b) and I&C in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 6 (c)) can be discouraged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 12 is designed to give low uncertainty when the model predictions are accurate, while the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 12 attempts to give high uncertainty when the model predictions are inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' At the same time, we adopt the annealing weighting factor βt to achieve different penalties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In the early training stage, inaccurate predictions dominate, so the second term (I&C loss) should be penalized more, while in late training, accurate predictions dominate, so the first term (A&U loss) should be penalized more punishment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Experimental details of Clinically safety applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Finally, we utilize EvidenceCap on two clinical safe applications as a quality indicator and OOD detector for clinicians and patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In clinical, the OOD sample and the value of data are essential for AI medicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We employed four real-world clinical datasets for applications of the quality indicator and OOD detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In the first application, Johns Hopkins OCT (JH-OCT) dataset and Duke OCT dataset with Diabetic Macular Edema (Duke-OCT- DME) are used for the OOD detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 335 cases of patients on JH-OCT with ground-truth 34 (a) A&C (b) A&U (c) I&C (d) I&Uare randomly divided into train dataset, validation dataset and test dataset with nearly 25, 5 and 5 cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The 5 cases of test dataset on JH-OCT are used for in- distribution detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In particular, the 10 cases on the Duke-OCT-DME are used as another test dataset for OOD detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Every case is uniformly adjusted to 128 × 1024 voxels during the training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In the second application, Digital Retinal Images for Vessel Extraction (DRIVE) and the Fundus Image Vessel Segmentation (FIVES) datasets are used for the quality indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We first train the EvidenceCap on the DRIVE dataset (20 slices) and then test on the FIVES dataset (600 slices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In the FIVES dataset, each image was evaluated for three qualities: lighting and color distortion, blurring, and low-contrast distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' We tested on normal images (459 slices with high quality) and images including these three quality ratings (141 slices with low quality) from the FIVES dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Every case is uniformly adjusted to 565 × 584 voxels during the training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' In these applications, the initial learning rate for the dataset are set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The poly learning strategy is used by decaying each iteration with a power of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The maximum of the epoch is set to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The batch sizes for the layer-segmentation and voxel-segmentation from OCT are set to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' More visual comparisons More visual comparisons on ISIC2018, LiTS2017, and BraTS2019 dataset can be seen in the figures 7, 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' More visual clinical applications on JH-OCT, Duke-OCT-DME, DRIVE and FIVES dataset can be seen in the figures 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 35 Input U AU V U+Our V+Our PU UE DU GT Certain Uncertain 1) 2) 3) 4) 5) 6) Noise Mask Original Figure 7: The visual comparison of skin lesion segmentation results with different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 1) Original input and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (2-3) Input under Gaussian noise (σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5) and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (4-5) Input under patch-size mask (σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4) and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (6) Ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Input U AU V U+Our V+Our PU UE DU Certain Uncertain 1) 2) 3) 4) 5) 6) 7) 8) Noise Blur Original Mask GT b Figure 8: The visual comparison of liver segmentation results with different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' a) Original input and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (b-c) Input under Gaussian noise (σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4) and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (d-e) Input under Gaussian blur ( �� σ2, k �� = {(13, 10) , (15, 20)}) and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (f-g) Input under random mask ratio (σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='3) and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' (h) Ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 36 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0Certain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 Uncertair10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0235 Certain0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0Uncertair1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) Noise Blur Original Mask Spike Ghost Input U AU V AU+Our V+Our PU UE DU GT Certain Uncertain Figure 9: The visual comparison of brain tumor segmentation results with different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 1) Original input (T2 as an example);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 2)-3) Gaussian noise input under (σ2 = 1, 2) and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 4)-5) Input under Gaussian blur ( �� σ2, k �� = {(3, 3) , (7, 7)}) and its results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 6)-7) Input under random mask ratio MR = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='4} and its results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 8) Ground Truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0High-Quality Low-Quality i) ii) iii) Input GT Prediction Uncertainty Confidence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='5 b (i) In distribution Johns Hopkins OCT dataset (ii) Out-of-distribution Duke OCT dataset with DME a Input GT Prediction Uncertainty Confidence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content='7 Input GT Prediction Uncertainty Confidence Figure 10: Clinically safety applications for out-of-distribution detector and quality indica- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' The qualitative difference between in-distribution (JH-OCT dataset without disease) and out-of-distribution data (Duke-OCT-DME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' DME: diabetic macular edema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' Qual- ity difference between image data of different quality, i) Visualization results of high-quality data on the FIVES dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' ii-iii) Low-quality data visualization results on the FIVES dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} +page_content=' 38 1' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9AyT4oBgHgl3EQff_g3/content/2301.00349v1.pdf'} diff --git a/JdE1T4oBgHgl3EQfGAPd/content/tmp_files/2301.02910v1.pdf.txt b/JdE1T4oBgHgl3EQfGAPd/content/tmp_files/2301.02910v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..230935c4ab298006850ed473bc753a97033666e1 --- /dev/null +++ b/JdE1T4oBgHgl3EQfGAPd/content/tmp_files/2301.02910v1.pdf.txt @@ -0,0 +1,626 @@ +Universality in odd-even harmonic generation +and application in terahertz waveform sampling +Doan-An Trieu,1 Ngoc-Loan Phan,1, ∗ Quan-Hao Truong,1 Hien T. +Nguyen,2 Cam-Tu Le,3, 4 DinhDuy Vu,1 and Van-Hoang Le1, † +1Computational Physics Key Laboratory K002, Department of Physics, +Ho Chi Minh City University of Education, Ho Chi Minh City 72711, Vietnam +2Tay Nguyen University, Buon Ma Thuot City 63161, Vietnam +3Atomic Molecular and Optical Physics Research Group, Advanced Institute of Materials Science, +Ton Duc Thang University, Ho Chi Minh City 72912, Vietnam +4Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City 72912, Vietnam +(Dated: January 10, 2023) +Odd-even harmonics emitted from a laser-target system imprint rich, subtle information character- +izing the system’s dynamical asymmetry, which is desirable to decipher. In this Letter, we discover +a simple universal relation between the odd-even harmonics and the asymmetry of the THz-assisted +laser-atomic system – atoms in a fundamental mid-IR laser pulse combined with a THz laser. First, +we demonstrate numerically and then analytically formulize the harmonic even-to-odd ratio as a +function of the THz electric field, the source of the system’s asymmetry. Notably, we suggest a scal- +ing that makes the obtained rule universal, independent of the parameters of both the fundamental +pulse and atomic target. This universality facilitates us to propose a general pump-probe scheme +for THz waveform sampling from the even-to-odd ratio, measurable within a conventional compact +setup. +Introduction — Recently, high-order harmonic gener- +ation (HHG) from an asymmetric laser-target system +has been paid much attention since it triggers more +subtle structures in HHG spectra [1–6]. A prominent +feature is the appearance of even harmonics, which is +remarkably distinguishable from the pure-odd harmonic +generation of the symmetric laser-target system [7]. De- +ciphering the odd-even HHG is substantial for charac- +terizing the dynamical asymmetry of the laser-target +system [1–6]. For this purpose, specifying a universal +rule reflecting the system’s asymmetry from odd-even +HHG is desirable. +One simple way for symmetry breaking is to add a +weaker external static electric field to the fundamental +multi-cycle laser pulse [8–18]. However, it requires an +extremely strong static field (∼MV/cm) which is prac- +tically hard to achieve in HHG experiments. +Never- +theless, the recent development of powerful terahertz +(THz) sources allows for replacing the static electric +field in generating high-order harmonics [19–25]. This +THz-assisted HHG has been studied mostly in the har- +monic conversion efficiency or the plateau structure [8– +22] but less in other notable aspects such as odd-even +harmonic spectra. Some initial estimations have been +made but are still in their infancy [10, 25]. It still lacks +a direct characterization of odd-even responses to the +asymmetric THz-assisted laser-atomic system. There- +fore, unraveling the connection between the odd-even +HHG and the asymmetry of the system by an analytical +rule may be critically meaningful in extracting quantum +dynamics inside atoms, manipulating electron trajecto- +ries on the attosecond time scale, or in THz waveform +sampling, whose issue has been intensively fascinating +recently [26, 27]. +The goal of this Letter is twofold. +First, we thor- +oughly explore the response of the odd-even HHG to +the changing of the THz electric field in a THz-assisted +laser-atom system. The goal is to achieve a connection +between the measurable harmonic even-to-odd ratio and +the THz electric field, the reason for the system’s asym- +metry. Interestingly, we discover a universal and sim- +ple rule for the even-to-odd ratio as a function of the +scaled THz electric field. This finding stimulates us to +go ahead with the second goal of proposing a general +method for sampling the waveform of the THz pulse. +Our method can retrieve the waveform with high accu- +racy from the harmonic even-to-odd ratio by a simple +design shown in Fig. 1. We emphasize that waveform +detection is another important area of THz science that +is intensively investigated besides its generation [26, 27]. +The validation of the available THz detection techniques +is strongly controlled by choosing active matter; thus, +finding a new temporal detection scheme free from ex- +ternal factors is experimentally meaningful. +Universal response of even-to-odd ratio to THz elec- +tric field — We first look thoroughly into the compre- +hensive response of resolved odd-even harmonics emit- +ted from a hydrogen atom in the combined linearly po- +larized fundamental mid-IR laser pulse and THz field, +as shown in Fig. 2(a). Here, HHG data are calculated +using the wave functions obtained by the TDSE method +(numerically solving the time-dependent Schr¨odinger +equation) [28]. +Since electron wave packets mostly +spread along the laser polarization direction, the one- +dimensional model is sufficient for HHG study with low +computational cost and a qualitative picture compared +with the three-dimensional model [29, 30]. We consider +an example of a used mid-IR pulse with a 10-cycle trape- +arXiv:2301.02910v1 [quant-ph] 7 Jan 2023 + +2 +FIG. 1. Pump-probe procedure for THz waveform sampling with sequential steps: (a) experimental setup; (b) measurements +of intensities of even and two adjacent odd harmonics at the cutoff with pump-probe time delays; (c) extracting step with the +(c1) even-to-odd ratio and (c2) extracted THz electric field using the universal rule. (d) Comparison between the extracted +(dotted curve) and benchmark (solid curve) THz pulses. The lower panel presents their relative deviation. The five-cycle +trapezoidal fundamental pulses with intensity of 2.5 × 1014 W/cm2 and wavelength of 2000 nm are used as probe lasers. +zoidal envelope, an intensity of 2.5 × 1014 W/cm2, and +a wavelength of 2000 nm. +It is clear that when the +added THz field ET is extremely small (< 10−5 a.u. +(51 kV/cm)) to break the symmetry of the laser-atomic +system, the spectrum is pure-odd harmonics [7]. Then, +the strengthening of the THz field gradually breaks the +symmetry (half-period time translation combined with +spatial inversion), leading to the emergence of even har- +monics. +For a more quantitative illustration, Fig. 2(b) shows +the response of harmonic intensity with the encoded +symmetry-breaking factor (THz field) for the two se- +lected even (500th) and odd (501st) harmonics at the +cutoff, denoted respectively as H500 and H501. Their +behavior can be visually partitioned into two regions. +(i) For the first region, the intensity of the even har- +monics gradually takes off, while that of the odd har- +monics is almost unchanged. (ii) After the first meeting +at ET ≈ 4×10−5 a.u. (∼ 0.2 MV/cm) that indicates the +comparability of the even and odd harmonics, the sec- +ond region begins with the intensity fluctuation of the +odd and even harmonics around their average values. +Notably, the odd and even harmonics at some specific +THz field values alternatively undergo deep minimums +corresponding to pure-even or pure-odd spectra, as il- +lustrated in Fig. 2(a). +Since the overall harmonic efficiency is strongly gov- +erned by the laser field, with the keep-in-mind goal of +uncovering a universal odd-even rule, we try to cancel +out the effect of the fundamental laser by introducing +a dimensionless quantity by taking the intensity ratio +between the even harmonic and the average of the two +adjacent odd ones. It is briefly referred to as the even- +to-odd ratio, and its response to the THz field is exhib- +ited in Fig 2(c). However, the fundamental laser still +affects the THz-dependent even-to-odd ratio, causing +it to respond diversely to different laser intensities and +wavelengths, as shown in Figs. 3(a) and 3(b). Specif- +ically, the wavelength modification [Fig 3(b)] disturbs +the even-to-odd ratio much more than changing the in- +tensity [Fig 3(c)]. For this reason, we make further effort +to eradicate the fundamental-field effect by scaling the +THz electric field ET by the factor E0/ω3 +0 as +γ = E0 +ω3 +0 +ET , +(1) +where E0 and ω0 are the fundamental laser’s peak am- +plitude and carrier frequency. Surprisingly, the results +presented in the right panels of Fig. 3 show an almost +identical even-to-odd ratio versus dimensionless scaled +THz electric fields for γ ≳ 0.1 regardless of different +fundamental laser intensities and wavelengths. +Here, +the region γ below 0.1 can be disregarded because of +the numerical signal noise. +We also numerically identify that the scaled-THz- +dependent even-to-odd ratio is indistinguishable when +changing the duration (5, 10, 15, 20 cycles) of the funda- +mental laser pulse and, more interestingly, for different +active atomic targets (H, He, Ne, Ar) [28]. Besides, we +realize that the found universality is observed not only +for harmonics at the cutoff but also those below cutoff +when washing out long electron trajectories under good +phase-matching conditions in HHG experiments. +See +Supplementary [28] for the details. + +2.558- +ET +Time delay (104 a.u.) +-2 +0 +1 +-1 +2 +H2n +(c2) +H2n+1)/ +5 +250 +input +a.u.) +(cl) +extracted +4 +200 +(10-5 +H +13 +150 +2 +field ( +100 +Electric +50 +11 +Electric +10 +0 +-50 +103 +(c) Extracting +Error +101 +Time del +10- +0.4 +0.2 +-0.2 +0.0 +0.6 +0.6 +Time delay (ps) +(b) HHG measurement +(d) Results +(a) Experimental setup3 +FIG. 2. Response to changing of THz electric field for (a) +Intensity of resolved odd-even harmonic spectra, (b) Selected +odd (H501) and even (H500) harmonics at the cutoff, and +(c) Harmonic even-to-odd ratio. The fundamental mid-IR +laser pulse with the same parameters as in Fig. 1 is used for +the HHG process; the THz field with a frequency of 1.3 THz +(231 µm) is used; the color bar in Panel (a) decodes the +HHG intensity in arbitrary units; the dotted horizontal line +in Panel (c) shows the unity. +Analytical formula — The universal relation between +the harmonic even-to-odd ratio and scaled THz elec- +tric field observed numerically above motivates us to +uncover its underlying physics. We approach the even +and odd harmonics by the quantum-orbit theory [31], +where the generation of harmonic order N is indeed the +coherent interference of attosecond bursts emitted on +each half of an optical cycle: H(Nω0) ≈ |D1e−iΦ1 + +D2e−iΦ2|2, where D1(2) and Φ1(2) are their amplitude +and phase. Therefore, the even-to-odd ratio is strongly +governed by the relative intensity |D2|2/|D1|2 and phase +difference ∆Φ between attosecond bursts which in turn +is associated with the difference of quasi-classical ac- +tion ∆S of free electrons in the laser electric field as +∆Φ = Nπ − Re(∆S). Adding a THz field considerably +weak compared to the fundamental intense laser almost +does not change the intensity of attosecond bursts, i.e., +|D2|2/|D1|2 ≈ 1. However, it still distorts the quasi- +classical action by ST = Cγ, leading to the change in +FIG. 3. Dependence of harmonic even-to-odd ratio on pure +(left panels) and scaled (right panels) THz electric field +for a hydrogen atom in various THz-assisted fundamental +laser pulses with (a) different intensities (fixed wavelength +of 2000 nm) and (b) different wavelengths (fixed intensity of +2.5I0), where I0 = 1×1014 W/cm2. The grey-shaded areas in +panels (c) and (d) highlight the region of stable even-to-odd +ratio. +the phase of attosecond bursts, which in turn renovates +the even-to-odd ratio as +η = tan2 (Cγ) . +(2) +Here, C = 2 sin θ (∆θ cos ∆θ − sin ∆θ) with θ = ω0(tr + +ti)/2 and ∆θ = ω0(tr − ti)/2 is a real dimensionless +coefficient depending on harmonic ionization ti and re- +combination tr instants. Specifically for harmonics at +the cutoff, C = 2.558. See more detailed calculations in +Supplementary [28]. +The analytical formula (2) demonstrates a direct con- +nection between the even-to-odd ratio η (a normalized +quantity characterizing the asymmetry of measurable +output) and the scaled THz electric field γ (the normal- +ized factor breaking the symmetry of the laser-atomic +system) where coefficient C is a constant for each har- +monic energy. The independence of the obtained for- +mula (2) from the fundamental laser pulse parameters +and the used symmetric targets implies the universal- +ity of the response of harmonic even-to-odd ratio to the +scaled THz field. Besides, this analytical expression also +shows a periodic modulation of the harmonic even-to- +odd ratio with the period of π/C. It alternatively under- +goes maxima and minima, characterizing the pure-even +and pure-odd harmonic spectra. Based on the relation +(2), we further refer to the quantity γ as a parameter +describing the asymmetric degree of the laser-target sys- +tem and call it the asymmetry parameter. +Figure 4 shows an overall visualization of the even-to- + +496 +498 +500 +502 +504 +506 +0.06 +0.04 +Er (a.u.) +10-4 +0.02 +10-5 +0.00 +495 +497 +499 +501 +503 +505 +507 +HHG order +(b) +10 +10 +H501 +H500 +103 +(c) +Even-to-odd ratio +101 +10- +10-3 +10-5 +10-4 +ET (a.u.)3(a) 2000nm +(c) +103 +.- 1.51o +2.5o +101 +Even-to-odd ratio +10-1 +(b) 2.5Io +(d) +103 +1400nm +-2000nm +.-.-.- 2800nm +101 +10-1 +10-5 +10-4 +10-1 +100 +ET (a.u.) +人4 +FIG. 4. Response of harmonic even-to-odd ratio to a wide +range of scaled THz fields calculated by the analytical for- +mula η = tan2(2.558 γ) (black solid curve) and numerical +TDSE method (dashed and dotted color curves) for harmon- +ics at the cutoff using different fundamental lasers enclosed +in the legend. The grey-, cream-, and mauve- shaded areas +cover three regions with different underlying physics mecha- +nisms. +odd ratio versus the asymmetry parameter γ for the har- +monics at the cutoff. This visualization is based on the +data obtained in two different ways, by the analytical +relation and by the direct numerical calculation (TDSE +method). +Comparing the results of the two methods +reveals three major regions implying completely differ- +ent physical mechanisms. +(i) For a small asymmetry +parameter 0.1 ≲ γ ≲ 0.6 (a gray-shaded area), the an- +alytical formula exactly predicts the even-to-odd ratio +numerically calculated by the TDSE method with var- +ious fundamental laser parameters and atomic targets. +It is reasonable since the THz field is considerably weak, +so it perturbs only the electron quasi-classical motion in +the continuum energy region but does not involve in the +ionization and recombination steps. (ii) For more in- +tense asymmetric parameter 0.6 ≲ γ ≲ 3.5π/C ≡ 4.3 (a +cream-shaded area), the relation (2) fails to characterize +the magnitude but is still right to predict the oscilla- +tion period of the harmonic even-to-odd ratio. Indeed, +the numerical even-to-odd ratio in this area does not +pure-harmonically oscillate and even reverses its phase. +Moreover, the relation is no longer independent of the +fundamental laser’s parameters. However, the oscilla- +tion period remains as same as predicted by the ana- +lytical formula. The main reason is that the consider- +ably strong THz field participates in the ionization step +besides distorting the electron quasi-classical motion. +It causes the imbalance between adjacent attosecond +bursts, thus, jumbling their interference contrast and +manifesting in the irregular modulation of the even-to- +odd ratio. (iii) With γ ≳ 4.3 (a mauve-shaded area), the +high-order disorders are observed in numerical even-to- +odd ratio, and the classical description (2) fails to pre- +dict both the oscillation magnitude and period. Here, +the intense THz field modifies ionized electrons’ travel +time compared to the THz field absence case. Addition- +ally, intense THz fields adjust the electron trajectories +in the continuum energy region, altering the plateau +structure of HHG spectra. +In short, the universality reveals two different aspects +- the even-to-odd magnitude itself in the first region and +its oscillation period in the second region. It is worth +noting that the first universal rule is satisfied if the +fundamental laser’s parameters vary in an appropriate +working range (intensity within [1.0−4.0]×1014 W/cm2 +and wavelength longer than 1200 nm (mid-IR laser)) +[28]. Indeed, the intensity needs to be high enough to +ensure a low controlled ratio ET /E0 to avoid the de- +forming HHG plateau structure and not exceed the in- +tensity saturation. Also, the wavelength should be long +so that the asymmetry parameter γ and, consequently, +the even-to-odd ratio η is above the threshold that can +be measured experimentally. +Application in THz waveform sampling — First, we +focus on exploiting the universal rule in the first re- +gion for THz waveform sampling. We propose a gen- +eral pump-probe scheme illustrated in Fig. 1(a), where a +THz pulse plays as a pump pulse, and the probe mid-IR +lasers (delayed in time) then irradiate to atomic gas to +generate harmonics. The odd-even HHG [Fig. 1(b)], and +consequently, even-to-odd ratio [Fig. 1(c1)] is recorded +at each time delay. +With the analytical formula (2), +the waveform of the THz pulse can be easily extracted +[Fig. 1(c2)]. +Figure 1 shows an exemplification of the wave- +form +detection +for +the +THz +pulse +ET (t) += +ET 0 exp +� +−ω2 +T t2/36π2� +sin ωT t +with +the +intensity +of 8.8 × 107 W/cm2 and frequency ωT = 1.3 THz, +mimicking the one generated in practice recently [32]. +The five-cycle trapezoidal fundamental pulses with +2.5 × 1014 W/cm2 intensity and 2000 nm wavelength +are used as probe lasers. +The time resolution of the +THz waveform detection is related to the duration and +is about 33 fs for the five-cycle laser pulse. +Figure 1(d) demonstrates the validity of our proposed +procedure. It indicates a good consistency between the +extracted THz waveform and the benchmark input data. +Their relative deviation in the lower panel is much less +than 10 % in the entire time domain except the two +pulse tails where THz fields go to naught. We note that +the carrier-envelope phase of the detected THz pulse +might be flipped by π since the proposed method can +extract the magnitude only but not its sign. +Exploiting the first universal rule can detect the THz +electric field within a wide THz-field range of about +[30, 2000] kV/cm since the working-range limitations of +the fundamental pulses [28]. +We emphasize that the +detectable range of the THz field can be expanded if +employing the second region of universal rule, i.e., its +stable oscillation period. We sketch the route utilizing +this rule to detect the THz field as an outlook. By look- + +-- 2000nm. 1.5I +--- 2000nm, 2.5Io +.- 2800nm, 2.5Io +analytical +104 +Even-to-odd ratio +102 +100 +10-2 +10-4 +0 +2 +3 +5 +65 +ing at the analytical formula (2), a new dimensionless +quantity ζ = (1−η)/(1+η) can be defined as a harmonic +oscillation as +ζ = cos +�2CE0 +ω3 +0 +ET +� +. +It offers to measure continuously ζ by changing the value +E0/ω3 +0, either by scanning the mid-IR intensity or wave- +length, which is practically feasible [33]. Afterward, the +Fourier transform of this function ζ(2CE0/ω3 +0) gives a +peak at ET , leading to the detection of the THz field. +In conclusion, we have both numerically and analyt- +ically demonstrated the universality of the even-to-odd +ratio as a function of the scaled THz electric field (asym- +metry parameter of the system). The approach to un- +cover the universal rule in this Letter is so general that +it can be applied to other asymmetric laser-target sys- +tems that may be substantial in probing atomic quan- +tum dynamics, controlling electron dynamics within at- +tosecond time scale, or extracting asymmetric factors of +laser-target systems. +Based on this universal rule, we have proposed a re- +liable pump-probe method for THz waveform sampling +exploiting the even-to-odd ratio, which is measurable +with current compact laser setups. +Unlike the previ- +ous methods for THz detection, where the THz pulse +triggers electronic or optical properties of targets, our +proposed method retrieves the THz field, which gov- +erns only the dynamics of quasi-free electrons in the +continuum energy region leading to modulation of the +even-to-odd ratio but does not directly interact with the +targets. 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A 126, 8588 (2022). + diff --git a/JdE1T4oBgHgl3EQfGAPd/content/tmp_files/load_file.txt b/JdE1T4oBgHgl3EQfGAPd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c30218146cc1f676f8a6d57b907066f908dbe9b5 --- /dev/null +++ b/JdE1T4oBgHgl3EQfGAPd/content/tmp_files/load_file.txt @@ -0,0 +1,560 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf,len=559 +page_content='Universality in odd-even harmonic generation and application in terahertz waveform sampling Doan-An Trieu,1 Ngoc-Loan Phan,1, ∗ Quan-Hao Truong,1 Hien T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Nguyen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='2 Cam-Tu Le,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Ho Chi Minh City 72711,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Vietnam 2Tay Nguyen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Buon Ma Thuot City 63161,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Vietnam 3Atomic Molecular and Optical Physics Research Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Advanced Institute of Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Ton Duc Thang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Ho Chi Minh City 72912,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Vietnam 4Faculty of Applied Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Ton Duc Thang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Ho Chi Minh City 72912,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Vietnam (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 2023) Odd-even harmonics emitted from a laser-target system imprint rich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' subtle information character- izing the system’s dynamical asymmetry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' which is desirable to decipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' In this Letter, we discover a simple universal relation between the odd-even harmonics and the asymmetry of the THz-assisted laser-atomic system – atoms in a fundamental mid-IR laser pulse combined with a THz laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' First, we demonstrate numerically and then analytically formulize the harmonic even-to-odd ratio as a function of the THz electric field, the source of the system’s asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Notably, we suggest a scal- ing that makes the obtained rule universal, independent of the parameters of both the fundamental pulse and atomic target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' This universality facilitates us to propose a general pump-probe scheme for THz waveform sampling from the even-to-odd ratio, measurable within a conventional compact setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Introduction — Recently, high-order harmonic gener- ation (HHG) from an asymmetric laser-target system has been paid much attention since it triggers more subtle structures in HHG spectra [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' A prominent feature is the appearance of even harmonics, which is remarkably distinguishable from the pure-odd harmonic generation of the symmetric laser-target system [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' De- ciphering the odd-even HHG is substantial for charac- terizing the dynamical asymmetry of the laser-target system [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' For this purpose, specifying a universal rule reflecting the system’s asymmetry from odd-even HHG is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' One simple way for symmetry breaking is to add a weaker external static electric field to the fundamental multi-cycle laser pulse [8–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' However, it requires an extremely strong static field (∼MV/cm) which is prac- tically hard to achieve in HHG experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Never- theless, the recent development of powerful terahertz (THz) sources allows for replacing the static electric field in generating high-order harmonics [19–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' This THz-assisted HHG has been studied mostly in the har- monic conversion efficiency or the plateau structure [8– 22] but less in other notable aspects such as odd-even harmonic spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Some initial estimations have been made but are still in their infancy [10, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' It still lacks a direct characterization of odd-even responses to the asymmetric THz-assisted laser-atomic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' There- fore, unraveling the connection between the odd-even HHG and the asymmetry of the system by an analytical rule may be critically meaningful in extracting quantum dynamics inside atoms, manipulating electron trajecto- ries on the attosecond time scale, or in THz waveform sampling, whose issue has been intensively fascinating recently [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The goal of this Letter is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' First, we thor- oughly explore the response of the odd-even HHG to the changing of the THz electric field in a THz-assisted laser-atom system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The goal is to achieve a connection between the measurable harmonic even-to-odd ratio and the THz electric field, the reason for the system’s asym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Interestingly, we discover a universal and sim- ple rule for the even-to-odd ratio as a function of the scaled THz electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' This finding stimulates us to go ahead with the second goal of proposing a general method for sampling the waveform of the THz pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Our method can retrieve the waveform with high accu- racy from the harmonic even-to-odd ratio by a simple design shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' We emphasize that waveform detection is another important area of THz science that is intensively investigated besides its generation [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The validation of the available THz detection techniques is strongly controlled by choosing active matter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' thus, finding a new temporal detection scheme free from ex- ternal factors is experimentally meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Universal response of even-to-odd ratio to THz elec- tric field — We first look thoroughly into the compre- hensive response of resolved odd-even harmonics emit- ted from a hydrogen atom in the combined linearly po- larized fundamental mid-IR laser pulse and THz field, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Here, HHG data are calculated using the wave functions obtained by the TDSE method (numerically solving the time-dependent Schr¨odinger equation) [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Since electron wave packets mostly spread along the laser polarization direction, the one- dimensional model is sufficient for HHG study with low computational cost and a qualitative picture compared with the three-dimensional model [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' We consider an example of a used mid-IR pulse with a 10-cycle trape- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='02910v1 [quant-ph] 7 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Pump-probe procedure for THz waveform sampling with sequential steps: (a) experimental setup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' (b) measurements of intensities of even and two adjacent odd harmonics at the cutoff with pump-probe time delays;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' (c) extracting step with the (c1) even-to-odd ratio and (c2) extracted THz electric field using the universal rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' (d) Comparison between the extracted (dotted curve) and benchmark (solid curve) THz pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The lower panel presents their relative deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The five-cycle trapezoidal fundamental pulses with intensity of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='5 × 1014 W/cm2 and wavelength of 2000 nm are used as probe lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' zoidal envelope, an intensity of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='5 × 1014 W/cm2, and a wavelength of 2000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' It is clear that when the added THz field ET is extremely small (< 10−5 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' (51 kV/cm)) to break the symmetry of the laser-atomic system, the spectrum is pure-odd harmonics [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Then, the strengthening of the THz field gradually breaks the symmetry (half-period time translation combined with spatial inversion), leading to the emergence of even har- monics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' For a more quantitative illustration, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 2(b) shows the response of harmonic intensity with the encoded symmetry-breaking factor (THz field) for the two se- lected even (500th) and odd (501st) harmonics at the cutoff, denoted respectively as H500 and H501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Their behavior can be visually partitioned into two regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' (i) For the first region, the intensity of the even har- monics gradually takes off, while that of the odd har- monics is almost unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' (ii) After the first meeting at ET ≈ 4×10−5 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='2 MV/cm) that indicates the comparability of the even and odd harmonics, the sec- ond region begins with the intensity fluctuation of the odd and even harmonics around their average values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Notably, the odd and even harmonics at some specific THz field values alternatively undergo deep minimums corresponding to pure-even or pure-odd spectra, as il- lustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Since the overall harmonic efficiency is strongly gov- erned by the laser field, with the keep-in-mind goal of uncovering a universal odd-even rule, we try to cancel out the effect of the fundamental laser by introducing a dimensionless quantity by taking the intensity ratio between the even harmonic and the average of the two adjacent odd ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' It is briefly referred to as the even- to-odd ratio, and its response to the THz field is exhib- ited in Fig 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' However, the fundamental laser still affects the THz-dependent even-to-odd ratio, causing it to respond diversely to different laser intensities and wavelengths, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 3(a) and 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Specif- ically, the wavelength modification [Fig 3(b)] disturbs the even-to-odd ratio much more than changing the in- tensity [Fig 3(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' For this reason, we make further effort to eradicate the fundamental-field effect by scaling the THz electric field ET by the factor E0/ω3 0 as γ = E0 ω3 0 ET , (1) where E0 and ω0 are the fundamental laser’s peak am- plitude and carrier frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Surprisingly, the results presented in the right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 3 show an almost identical even-to-odd ratio versus dimensionless scaled THz electric fields for γ ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='1 regardless of different fundamental laser intensities and wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Here, the region γ below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='1 can be disregarded because of the numerical signal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' We also numerically identify that the scaled-THz- dependent even-to-odd ratio is indistinguishable when changing the duration (5, 10, 15, 20 cycles) of the funda- mental laser pulse and, more interestingly, for different active atomic targets (H, He, Ne, Ar) [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Besides, we realize that the found universality is observed not only for harmonics at the cutoff but also those below cutoff when washing out long electron trajectories under good phase-matching conditions in HHG experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' See Supplementary [28] for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='558- ET Time delay (104 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=') 2 0 1 1 2 H2n (c2) H2n+1)/ 5 250 input a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=') (cl) extracted 4 200 (10-5 H 13 150 2 field ( 100 Electric 50 11 Electric 10 0 50 103 (c) Extracting Error 101 Time del 10- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='6 Time delay (ps) (b) HHG measurement (d) Results (a) Experimental setup3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Response to changing of THz electric field for (a) Intensity of resolved odd-even harmonic spectra, (b) Selected odd (H501) and even (H500) harmonics at the cutoff, and (c) Harmonic even-to-odd ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The fundamental mid-IR laser pulse with the same parameters as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 1 is used for the HHG process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' the THz field with a frequency of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='3 THz (231 µm) is used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' the color bar in Panel (a) decodes the HHG intensity in arbitrary units;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' the dotted horizontal line in Panel (c) shows the unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Analytical formula — The universal relation between the harmonic even-to-odd ratio and scaled THz elec- tric field observed numerically above motivates us to uncover its underlying physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' We approach the even and odd harmonics by the quantum-orbit theory [31], where the generation of harmonic order N is indeed the coherent interference of attosecond bursts emitted on each half of an optical cycle: H(Nω0) ≈ |D1e−iΦ1 + D2e−iΦ2|2, where D1(2) and Φ1(2) are their amplitude and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Therefore, the even-to-odd ratio is strongly governed by the relative intensity |D2|2/|D1|2 and phase difference ∆Φ between attosecond bursts which in turn is associated with the difference of quasi-classical ac- tion ∆S of free electrons in the laser electric field as ∆Φ = Nπ − Re(∆S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Adding a THz field considerably weak compared to the fundamental intense laser almost does not change the intensity of attosecond bursts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=', |D2|2/|D1|2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' However, it still distorts the quasi- classical action by ST = Cγ, leading to the change in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Dependence of harmonic even-to-odd ratio on pure (left panels) and scaled (right panels) THz electric field for a hydrogen atom in various THz-assisted fundamental laser pulses with (a) different intensities (fixed wavelength of 2000 nm) and (b) different wavelengths (fixed intensity of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='5I0), where I0 = 1×1014 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The grey-shaded areas in panels (c) and (d) highlight the region of stable even-to-odd ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' the phase of attosecond bursts, which in turn renovates the even-to-odd ratio as η = tan2 (Cγ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' (2) Here, C = 2 sin θ (∆θ cos ∆θ − sin ∆θ) with θ = ω0(tr + ti)/2 and ∆θ = ω0(tr − ti)/2 is a real dimensionless coefficient depending on harmonic ionization ti and re- combination tr instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Specifically for harmonics at the cutoff, C = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' See more detailed calculations in Supplementary [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The analytical formula (2) demonstrates a direct con- nection between the even-to-odd ratio η (a normalized quantity characterizing the asymmetry of measurable output) and the scaled THz electric field γ (the normal- ized factor breaking the symmetry of the laser-atomic system) where coefficient C is a constant for each har- monic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The independence of the obtained for- mula (2) from the fundamental laser pulse parameters and the used symmetric targets implies the universal- ity of the response of harmonic even-to-odd ratio to the scaled THz field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Besides, this analytical expression also shows a periodic modulation of the harmonic even-to- odd ratio with the period of π/C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' It alternatively under- goes maxima and minima, characterizing the pure-even and pure-odd harmonic spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Based on the relation (2), we further refer to the quantity γ as a parameter describing the asymmetric degree of the laser-target sys- tem and call it the asymmetry parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Figure 4 shows an overall visualization of the even-to- 496 498 500 502 504 506 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='04 Er (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=') 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='02 10-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='00 495 497 499 501 503 505 507 HHG order (b) 10 10 H501 H500 103 (c) Even-to-odd ratio 101 10- 10-3 10-5 10-4 ET (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' )3(a) 2000nm (c) 103 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='51o 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='5o 101 Even-to-odd ratio 10-1 (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='5Io (d) 103 1400nm 2000nm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='- 2800nm 101 10-1 10-5 10-4 10-1 100 ET (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=') 人4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Response of harmonic even-to-odd ratio to a wide range of scaled THz fields calculated by the analytical for- mula η = tan2(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='558 γ) (black solid curve) and numerical TDSE method (dashed and dotted color curves) for harmon- ics at the cutoff using different fundamental lasers enclosed in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The grey-, cream-, and mauve- shaded areas cover three regions with different underlying physics mecha- nisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' odd ratio versus the asymmetry parameter γ for the har- monics at the cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' This visualization is based on the data obtained in two different ways, by the analytical relation and by the direct numerical calculation (TDSE method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Comparing the results of the two methods reveals three major regions implying completely differ- ent physical mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' (i) For a small asymmetry parameter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='1 ≲ γ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='6 (a gray-shaded area), the an- alytical formula exactly predicts the even-to-odd ratio numerically calculated by the TDSE method with var- ious fundamental laser parameters and atomic targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' It is reasonable since the THz field is considerably weak, so it perturbs only the electron quasi-classical motion in the continuum energy region but does not involve in the ionization and recombination steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' (ii) For more in- tense asymmetric parameter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='6 ≲ γ ≲ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='5π/C ≡ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='3 (a cream-shaded area), the relation (2) fails to characterize the magnitude but is still right to predict the oscilla- tion period of the harmonic even-to-odd ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Indeed, the numerical even-to-odd ratio in this area does not pure-harmonically oscillate and even reverses its phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Moreover, the relation is no longer independent of the fundamental laser’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' However, the oscilla- tion period remains as same as predicted by the ana- lytical formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The main reason is that the consider- ably strong THz field participates in the ionization step besides distorting the electron quasi-classical motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' It causes the imbalance between adjacent attosecond bursts, thus, jumbling their interference contrast and manifesting in the irregular modulation of the even-to- odd ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' (iii) With γ ≳ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='3 (a mauve-shaded area), the high-order disorders are observed in numerical even-to- odd ratio, and the classical description (2) fails to pre- dict both the oscillation magnitude and period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Here, the intense THz field modifies ionized electrons’ travel time compared to the THz field absence case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Addition- ally, intense THz fields adjust the electron trajectories in the continuum energy region, altering the plateau structure of HHG spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' In short, the universality reveals two different aspects the even-to-odd magnitude itself in the first region and its oscillation period in the second region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' It is worth noting that the first universal rule is satisfied if the fundamental laser’s parameters vary in an appropriate working range (intensity within [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='0−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='0]×1014 W/cm2 and wavelength longer than 1200 nm (mid-IR laser)) [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Indeed, the intensity needs to be high enough to ensure a low controlled ratio ET /E0 to avoid the de- forming HHG plateau structure and not exceed the in- tensity saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Also, the wavelength should be long so that the asymmetry parameter γ and, consequently, the even-to-odd ratio η is above the threshold that can be measured experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Application in THz waveform sampling — First, we focus on exploiting the universal rule in the first re- gion for THz waveform sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' We propose a gen- eral pump-probe scheme illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 1(a), where a THz pulse plays as a pump pulse, and the probe mid-IR lasers (delayed in time) then irradiate to atomic gas to generate harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The odd-even HHG [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 1(b)], and consequently, even-to-odd ratio [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 1(c1)] is recorded at each time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' With the analytical formula (2), the waveform of the THz pulse can be easily extracted [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 1(c2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Figure 1 shows an exemplification of the wave- form detection for the THz pulse ET (t) = ET 0 exp � −ω2 T t2/36π2� sin ωT t with the intensity of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='8 × 107 W/cm2 and frequency ωT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='3 THz, mimicking the one generated in practice recently [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The five-cycle trapezoidal fundamental pulses with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='5 × 1014 W/cm2 intensity and 2000 nm wavelength are used as probe lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The time resolution of the THz waveform detection is related to the duration and is about 33 fs for the five-cycle laser pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Figure 1(d) demonstrates the validity of our proposed procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' It indicates a good consistency between the extracted THz waveform and the benchmark input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Their relative deviation in the lower panel is much less than 10 % in the entire time domain except the two pulse tails where THz fields go to naught.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' We note that the carrier-envelope phase of the detected THz pulse might be flipped by π since the proposed method can extract the magnitude only but not its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Exploiting the first universal rule can detect the THz electric field within a wide THz-field range of about [30, 2000] kV/cm since the working-range limitations of the fundamental pulses [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' We emphasize that the detectable range of the THz field can be expanded if employing the second region of universal rule, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=', its stable oscillation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' We sketch the route utilizing this rule to detect the THz field as an outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' By look- -- 2000nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='5I --- 2000nm, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='5Io .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='- 2800nm, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='5Io analytical 104 Even-to-odd ratio 102 100 10-2 10-4 0 2 3 5 65 ing at the analytical formula (2), a new dimensionless quantity ζ = (1−η)/(1+η) can be defined as a harmonic oscillation as ζ = cos �2CE0 ω3 0 ET � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' It offers to measure continuously ζ by changing the value E0/ω3 0, either by scanning the mid-IR intensity or wave- length, which is practically feasible [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Afterward, the Fourier transform of this function ζ(2CE0/ω3 0) gives a peak at ET , leading to the detection of the THz field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' In conclusion, we have both numerically and analyt- ically demonstrated the universality of the even-to-odd ratio as a function of the scaled THz electric field (asym- metry parameter of the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The approach to un- cover the universal rule in this Letter is so general that it can be applied to other asymmetric laser-target sys- tems that may be substantial in probing atomic quan- tum dynamics, controlling electron dynamics within at- tosecond time scale, or extracting asymmetric factors of laser-target systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Based on this universal rule, we have proposed a re- liable pump-probe method for THz waveform sampling exploiting the even-to-odd ratio, which is measurable with current compact laser setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Unlike the previ- ous methods for THz detection, where the THz pulse triggers electronic or optical properties of targets, our proposed method retrieves the THz field, which gov- erns only the dynamics of quasi-free electrons in the continuum energy region leading to modulation of the even-to-odd ratio but does not directly interact with the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' This independence of targets and probe laser’s parameters makes the method feasible to detect a wide range of THz electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Acknowledgments - This work was funded by Vin- group and supported by Vingroup Innovation Founda- tion (VINIF) under project code VINIF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='DA00031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' The calculations were executed by the high-performance computing cluster at Ho Chi Minh City University of Education, Vietnam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' ∗ loanptn@hcmue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='vn † hoanglv@hcmue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='vn [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Frumker, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Kajumba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Bertrand, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Phan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Le, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Vu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Tran, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Le, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' A 105, 023106 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' [7] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Ben-Tal, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Moiseyev, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Beswick, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' B: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 26, 3017 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Lohr, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Becker, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Kleber, Laser Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 7, 615 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' [10] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Li, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Fu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' B: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 31, 1961 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' [11] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Li, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Fu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' A 59, 2894 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' [12] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Borca, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Flegel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Frolov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Manakov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Miloˇsevi´c, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Starace, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 85, 732 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' B 17, 1509 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Odˇzak and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Miloˇsevi´c, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' A 72, 033407 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' [15] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Hong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Lu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Cao, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Lan, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' B: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 40, 2321 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' [16] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Hong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Lu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Lan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Li, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Liao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' A 77, 033410 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' [17] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Zhao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Guo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Shao, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Xue, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 13, 093035 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' [18] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Yuan and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Bandrauk, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' A 83, 063422 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' [19] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Hong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Lu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Lan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Zhang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Zeng, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Yang, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Express 23, 32222 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Express 23, 8837 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' [23] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Miloˇsevi´c, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} +page_content=' 47, 1669 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE1T4oBgHgl3EQfGAPd/content/2301.02910v1.pdf'} 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Glassera, Hong Qina,b +aPrinceton Plasma Physics Laboratory Princeton University Princeton New Jersey 08543 +bDepartment of Astrophysical Sciences Princeton University Princeton New Jersey 08544 +Abstract +The Yee algorithm for electromagnetic simulations is widely known to have many advantages, including +the following crucial two: (i) Its calculations are local and therefore efficiently parallelizable—enabling +simulations that capitalize on the speed and scalability of high-performance computing architecture. +(ii) Yee’s method faithfully preserves the symplectic geometry of Maxwell’s equations, improving its accu- +racy in long-time numerical simulations. Whereas previous geometric generalizations of Yee’s method have +sacrificed its scalability, in this article the Yee algorithm is generalized to higher order and unstructured +meshes in a manner that fully preserves both its scalability and geometric naturalness. This generalization is +achieved by prioritizing the locality of the algorithm, reflecting the physical locality of Maxwell’s equations. +Specifically, we demonstrate that Yee’s method is but a special case of a larger family of symplectic, finite +element exterior calculus (FEEC) methods that use scalable, local approximations of mass matrices. We +discuss the numerical advantages of this family of methods, which we call scalable FEEC (SFEEC) methods. +1. Introduction +The Yee algorithm [1, 2]—alternatively, the finite +difference time domain (FDTD) method—defines +electromagnetic fields on a cubic mesh. +It asso- +ciates to each edge a component of the electric +field E, and to each face a component of the mag- +netic field B. +This discretization reflects a natu- +ral geometric description of Maxwell’s equations, in +which one defines E ∈ Λ1(R3) as a differential 1-form +on R3 and B ∈ Λ2(R3) as a differential 2-form on +R3. In this way, Yee’s method employs a technique +adopted in many structure-preserving algorithms [3], +wherein differential k-forms are discretized by asso- +ciating them with k-dimensional features of a mesh +[4, 5, 6, 7]. +Structure-preserving algorithms have been widely +adopted in many subfields of computational physics, +including gravitational simulations [8, 9, 10, 11, 12], +geophysics [13, 14] and plasma physics [15, 16, 17, +18, 19, 20, 21, 22, 23]. Such algorithms generally de- +rive from variational principles or Hamiltonian sys- +tems. As a result, they preserve essential mathemat- +ical features of their underlying physical systems, in- +cluding symplectic structure, topology, symmetries, +and conservation laws. These properties contribute +to the numerical fidelity of structure-preserving al- +gorithms, especially in long-time numerical simu- +lations. +Despite Yee’s omission of any overt La- +grangian or Hamiltonian formulations in his original +work [1], Yee’s method (apparently serendipitously) +is one of the most historically successful examples of +a structure-preserving algorithm [24]. +More recently, +the methodology of structure- +preserving algorithms has led to the development +of advanced algorithms for the simulation of plas- +mas, whose electromagnetic fields are constructed +using the formalism of finite element exterior cal- +culus (FEEC) [6, 7, 20, 23]. Although such meth- +Preprint submitted to Elsevier +January 5, 2023 +arXiv:2301.01753v1 [math.NA] 4 Jan 2023 + +ods are in principle readily generalizable to unstruc- +tured meshes and high order finite elements, they lack +the computational efficiency and scalability of Yee’s +method. In particular, the time evolution of electro- +magnetic fields in these FEEC methods requires com- +munication between all nodes of a simulation, thereby +destroying their parallelism. As we shall discuss and +address, this problem arises because sparse finite el- +ement mass matrices generally have dense inverses. +On the other hand, there have also been numer- +ous efforts (e.g. [25, 26, 27, 28]) to generalize Yee’s +method using scalable finite element methods, called +finite element time domain (FETD) methods. Such +efforts leverage the crucial technique of sparse ap- +proximate inverse (SPAI) mass matrices, and pre- +serve the scalability of Yee’s method. However, these +methods’ preservation of symplectic structure has not +been established. +Motivated by the desire to overcome the scalabil- +ity limitation of structure-preserving FEEC plasma +methods, and to guarantee the structure preservation +of FETD methods, in this article we develop scal- +able FEEC (SFEEC) methods, a family of symplec- +tic finite element methods for electromagnetic fields +that includes Yee’s method as a special case. SFEEC +methods enable the higher order simulation of elec- +tromagnetic fields on structured and unstructured +meshes in a manner that preserves the two aforemen- +tioned crucial advantages of Yee’s method, namely: +(i) its scalability on modern architectures; and (ii) its +symplectic geometry (and the resulting conservation +of electric charge and Gauss’ law). +From a finite element point of view, the scalability +of Yee’s method will be reframed as a result of its sim- +plified (or ‘pruned’) approximation of finite element +mass matrices. +To retain their scalability, SFEEC +methods employ a comparable, if more flexible treat- +ment of mass matrices. Relative to Yee’s method, SF- +FEC methods afford a greater flexibility to improve +algorithmic accuracy without sacrificing scalability. +Their use of finite elements and the FEEC formal- +ism further enables their viability on more general +meshes. +While Yee’s algorithm has been generalized nu- +merous times in the literature, including via higher +order and finite element schemes (e.g. [24, 25, 27, +28, 29, 30, 31, 32]), no such generalization is known +to us that simultaneously affords higher order ac- +curacy and scalability while ensuring the geometric +structure-preservation of Yee’s method. +In partic- +ular, the Yee method’s preservation of symplectic +structure is an important aspect of the method’s sta- +bility and long-term accuracy [33]. In this work, we +will review this symplectic structure of Yee’s method +and demonstrate its exact preservation in the SFEEC +family of algorithms we define. +In so doing, we +also offer a means to overcome the scalability limita- +tions of structure-preserving FEEC algorithms (e.g. +[20, 23]) in a massively parallel, high-performance +computing architecture. +The remainder of this article is organized as fol- +lows. In Section 2, the formalism of finite element ex- +terior calculus (FEEC) [6, 7] and its discretization of +electromagnetic fields will be reviewed. In Section 3, +it will be demonstrated that Yee’s algorithm can be +interpreted as an FEEC method with simplified mass +matrices. In Section 4 we will define SFEEC meth- +ods, which extend Yee’s method to higher order finite +element schemes on a general mesh. +In Section 5, +numerical results will be presented that demonstrate +the improved higher order accuracy of the resulting +SFEEC methods relative to Yee’s method, without +sacrificing its scalability. Section 6 will then summa- +rize and conclude. +2. Finite Element Exterior Calculus (FEEC) +for Electromagnetic Simulations +In this section, we briefly review aspects of FEEC +[6, 7] and its application in electromagnetic simula- +tions. We refer the reader to [20, 23] for additional +background. We let Λp(Th) denote a vector space of +finite element differential p-forms on a simplicial or +cubical complex Th ⊂ Rn (where the subscript h de- +notes the maximal diameter, or edge length, of any +simplex in Th). For all 0 ≤ p ≤ n, Λp(Th) may be de- +fined as the span of a finite (Np-dimensional) basis +Λp, whose ith basis element Λp +i ∈ Λp(Th) is a piece- +wise polynomial p-form on Th. Such a basis element +typically has support localized to one or more adja- +cent cells in Th. An arbitrary p-form S ∈ Λp(Th) can +2 + +be expressed in the Λp basis as +S(x) = s · Λp(x) = siΛp +i (x) +(1) +∀ s ∈ RNp and x ∈ |Th|, the convex hull of Th. (Ein- +stein summation convention is used for the repeated +index in Eq. (1) and hereafter.) Individual compo- +nents of S ∈ Λp(Th) will be denoted +S(x)µ1···µp = s · Λp(x)µ1···µp = siΛp +i (x)µ1···µp +(2) +where Greek letters denote coordinate indices. For +example, given |Th| ⊂ R3, the µth component of the +1-form basis element Λ1 +i (x) may be written as Λ1 +i (x)µ +∀ µ ∈ {1, 2, 3}, such that Λ1 +i (x) = Λ1 +i (x)µdxµ. +Because each basis Λp is finite ∀ 0 ≤ p ≤ n, the +exterior derivative d : Λp(Th) → Λp+1(Th) of a fi- +nite element p-form on Th can be computed in the +Λ0, . . . , Λn bases by straightforward matrix multipli- +cation. +A choice of basis for each Λp(Th) in three +dimensions (Th ⊂ R3) determines, for example, ma- +trices that represent the gradient (G), curl (C), and +divergence (D)—as defined in Table 1. +To apply FEEC in a physical setting, it will also +be essential to compute mass matrices on Th for each +basis Λp of finite element p-forms. Specifically, the +mass matrix Mp ∈ RNp×Np for Λp is defined by +(Mp)ij = +� +|Th| +dx +� +Λp +i , Λp +j +� +p +(3) +where (·, ·)p denotes the pointwise inner product on +p-forms induced by the metric gµν—specifically +(α, β)p = 1 +p!αµ1···µpβµ1···µp += 1 +p!αµ1···µpβν1···νpgµ1ν1 · · · gµpνp. +(4) +p-Form +Abstract d +Matrix d +Defined by +S = s · Λ0 +- +- +- +A = a · Λ1 +A = dS +a = Gs +GT Λ1 = dΛ0 +B = b · Λ2 +B = dA +b = Ca +CT Λ2 = dΛ1 +C = c · Λ3 +C = dB +c = Db +DT Λ3 = dΛ2 +Table 1: FEEC matrix implementation of d on Th ⊂ R3. +The property d ◦ d = 0 implies that CG = 0 and DC = 0. +For p = 1, 2 on R3, for example, Eq. (4) defines the +inner product (α, β)p as the standard inner product +α · β. After all, 1- and 2-forms each have three inde- +pendent components in R3, and the metric gµν (and +its inverse gµν) is given by the Kronecker delta— +gµν = δµν. We note that (α, β)p is symmetric, such +that Eq. (3) is symmetric and MT +p = Mp. Eq. (3) also +implies that Mp is typically sparse (so long as each ba- +sis element Λp +i has only local support). Importantly, +however, its inverse matrix M−1 +p +is typically dense. +To apply the FEEC formalism to electromagnetic +simulations, we will first discretize the magnetic vec- +tor potential as an FEEC 1-form, that is, +A = a · Λ1 +(5) +with degrees of freedom given by the vector of coef- +ficients a ∈ RN1. (We work in the temporal gauge, +such that the electric potential φ vanishes, φ = 0.) +The magnetic field is then given by a 2-form, +B = b · Λ2 = Ca · Λ2 = a · CT Λ2 = a · dΛ1 = dA +(6) +with b ∈ RN2, in keeping with the geometry of +Maxwell’s equations and the FEEC implementation +of the curl, as in Table 1. Lastly, the electric field +will be defined as a discrete 1-form +E = e · M−1 +1 +· Λ1 +(7) +with e ∈ RN1. +Following [23], we use a convention +wherein the coefficients e determine the electric field +E with an additional factor of M−1 +1 . We shall regard +the pair (a, e) ∈ R2N1 as defining the degrees of free- +dom of the electromagnetic finite element dynamical +system. +To describe the dynamics of these discrete fields +via Maxwell’s equations, we first recall the canonical +symplectic structure of electromagnetic fields in the +continuum, defined with Poisson bracket and Hamil- +tonian in SI units as [34] +{F, G}EM = 1 +ϵ0 +� +dx +�δF +δE · δG +δA − δG +δE · δF +δA +� +HEM = 1 +2 +� +dx +� +ϵ0 |E|2 + 1 +µ0 +|∇ × A|2 +� +. +(8) +3 + +We now substitute the FEEC fields A = a · Λ1 and +E = e · M−1 +1 +· Λ1 into Eq. (8) to derive (see [23]) the +following discrete Poisson bracket {·, ·}∆ and discrete +Hamiltonian H∆ approximating this system, +{F, G}∆ = 1 +ϵ0 +�∂F +∂e · ∂G +∂a − ∂G +∂e · ∂F +∂a +� +H∆ = 1 +2 +� +ϵ0eT M−1 +1 e + 1 +µ0 +aT CT M2Ca +� +, +(9) +where we have applied dΛ1 = CT Λ2 as in Table 1. +Equations of motion are now readily calculated as +usual with a Poisson bracket. +We plug the FEEC +coefficient vectors a and e into {·, ·}∆ with H∆, to +find +˙a = {a, H∆}∆ = −M−1 +1 e +˙e = {e, H∆}∆ = c2CT M2Ca. +(10) +These equations are discrete forms of Maxwell’s equa- +tions, ˙A = −E and ˙E = c2∇ × ∇ × A, respectively. +In Section 5, the discrete approximation of the curl- +of-curl operator +∇ × ∇× ≈ M−1 +1 CT M2C, +(11) +whose calculation (until our later simplification) in- +corporates the dense matrix M−1 +1 , will be a central +focus of our study. +Note, it will also be convenient to rewrite Eq. (10) +in terms of the magnetic field by substituting b = Ca +(the discrete realization of B = ∇ × A) to find +˙b = −CM−1 +1 e +˙e = c2CT M2b. +(12) +Because of the gauge symmetry of Maxwell’s equa- +tions, +Eq. (12) is an equivalent formulation of +Eq. (10) and exactly preserves the physical degrees +of freedom of interest [22, 23]. +To algorithmically evolve Eq. (10) in discrete time, +we follow [17] (see also [22]) and define a splitting +algorithm that decomposes H∆ of Eq. (9) into two +pieces: +H∆ = Ha + He +where +Ha = +1 +2µ0 +aT CT M2Ca +He = ϵ0 +2 eT M−1 +1 e. +(13) +These ‘sub-Hamiltonians’ define two different subsys- +tems which may be evolved separately, that is, +Ha +� +˙a += {a, Ha}∆ = 0 +˙e += {e, Ha}∆ = c2CT M2Ca. +He +� +˙a += {a, He}∆ = −M−1 +1 e +˙e += {e, He}∆ = 0. +(14) +We see that a is constant in the subsystem Ha, while +e is constant in the subsystem He. As a result, each +subsystem is exactly solvable over a finite time inter- +val ∆t, in particular, +Ha : +e(t0 + ∆t) = e(t0) + ∆t · c2CT M2Ca(t0) +He : +a(t0 + ∆t) = a(t0) − ∆t · M−1 +1 e(t0). +(15) +By choosing an appropriate sequence of these sub- +systems’ discrete time evolutions, a splitting method +may be implemented that approximates a timestep +of the entire electromagnetic system H∆. +A typical example of such a splitting scheme is +called Strang splitting [35], which evolves a and e +according to the approximation +�a +e +� +t0+∆t += exp (∆tH∆) · +�a +e +� +t0 +≈ exp +�∆t +2 He +� +exp (∆tHa) exp +�∆t +2 He +� +· +�a +e +� +t0 +. +(16) +Here, we use the notation exp(∆tHi) to denote a dis- +crete time mapping—as appears in Eq. (15)—that +evolves according to the Hamiltonian Hi for a time +interval ∆t. Since exp(∆tHa) and exp(∆tHe) do not +commute, the second line of Eq. (16) is only an ap- +proximation of the first. Nevertheless, such a split- +ting method is an effective algorithmic implementa- +tion of our dynamical system. It is exactly solvable, +accurate to second order, and preserves the symplec- +tic structure of the system as defined by the Poisson +bracket of Eq. (9). Splitting methods of arbitrarily +high accuracy in ∆t can also be constructed (see, e.g. +[3]). +4 + +3. Reinterpreting Yee’s Method via FEEC +We now rederive Yee’s method using the finite el- +ement formalism of the previous section. +We pro- +ceed in three steps: (i) we first choose generalized +Whitney forms as an FEEC basis on a cubical mesh; +(ii) we then maximally simplify the mass matrices of +the corresponding electromagnetic algorithm; and fi- +nally (iii) we choose Strang splitting as our method +of time evolution. These three steps will exactly re- +cover Yee’s method, demonstrating its interpretation +as an implementation of the FEEC electromagnetic +algorithm defined in Eq. (15)—with simplified mass +matrices. +We begin by considering a cubical lattice Th ⊂ R3 +with lattice spacings {∆x, ∆y, ∆z}. +We make the +simplest possible choice for an FEEC basis on +Th, namely, the generalized Whitney forms, a fam- +ily of piecewise polynomial finite elements denoted +Q− +1 Λp(Th) [36]). This finite element basis is also use- +fully defined in [37] Example 5.2. +A +generalized +Whitney +p-form +can +be +read- +ily understood by its 1-to-1 correspondence with +a p-dimensional feature of a mesh. +In partic- +ular, +we define the generalized Whitney p-form +Wσp ∈ Q− +1 Λp(Th) +associated +to +a +given +p-face +σp ⊂ Th ⊂ Rn by requiring that +1 +|τ p| +� +τ p Wσp = +� +1 +τ p = σp +0 +τ p ̸= σp. +(17) +In this way, generalized Whitney p-forms are dual +to p-faces of Th via integration. Here, |τ p| denotes +the p-volume of τ p (defined as 1, length, area, and +volume respectively for p = 0, 1, 2, 3). This normal- +ization is chosen to mimic fields as they are typically +represented in Yee’s method. +A concrete example of this is represented on +Th ⊂ R3 in Fig. 1. +The 1-form Wx1x2 ∈ Q− +1 Λ1(Th) +and the 2-form Wx1x2x4x3 ∈ Q− +1 Λ2(Th) are depicted, +defined respectively by +Wx1x2 = +� +1 − +y +∆y +� � +1 − +z +∆z +� +dx +Wx1x2x4x3 = +� +1 − +z +∆z +� +dx ∧ dy. +(18) +As in Eq. (17), Wx1x2 is the least-order piecewise +polynomial 1-form satisfying +� +x1x2 Wx1x2 = |x1x2| +and +� +σ1 Wx1x2 = 0 ∀ σ1 ̸= x1x2. +Wx1x2x4x3 +is +analogously defined so that its integral satisfies +� +x1x2x4x3 Wx1x2x4x3 = |x1x2x4x3| and vanishes on +any other face. +In later calculations, it will also +be useful to explicitly define the following Whitney +2-form associated to the face x1x5x6x2 in Fig. 1, +Wx1x5x6x2 = +� +1 − +y +∆y +� +dz ∧ dx. +(19) +Now consider the exterior derivative d of these +forms. +The curl matrix C : RN1 → RN2 is a linear +operator that precisely computes d for 1-forms, map- +ping the coefficients of a discrete 1-form—expanded +in the Q− +1 Λ1(Th) basis—to coefficients of a discrete +2-form—expanded in the Q− +1 Λ2(Th) basis. The 1-to-1 +correspondence between these generalized Whitney +forms and elements of the mesh leads to a convenient +interpretation of C in that basis, as follows. +We find by directly computing the exterior deriva- +tive of Wx1x2 using Eqs. (18-19), for example, that +dWx1x2 = +1 +∆y +Wx1x2x4x3 − 1 +∆z +Wx1x5x6x2. +(20) +We observe that a Whitney 2-form Wσ2 appears on +the right hand side above if and only if its associated +Figure 1: The generalized Whitney 1-forms Wx1x2 and +Wx1x2x4x3 are schematically depicted, respectively, on the +left and right. Wx1x2 evaluates to the length |x1x2| when +integrated along the edge x1x2 (in blue) and vanishes on +all other edges (in orange). +Wx1x2x4x3 likewise yields +the area |x1x2x4x3| when integrated over the blue face +x1x2x4x3, and vanishes on all other faces. +5 + +07 +5 +6 +C3 +C 4 +C28 +05 +M6 +y +C3 +C4 +C2face σ2 contains the edge x1x2 on its boundary. Its +coefficient ±1/∆xµ is determined by the µth dimen- +sion in which the associated face extends x1x2, and +its sign is set by the relative orientation between the +edge and face. +Note, this interpretation of d—as a mapping with +coefficients ±1/∆xµ from 1-forms to the 2-forms of +faces which contain them on the boundary—is ap- +plicable only to the particular (cubical) FEEC mesh +and (Whitney form) FEEC basis that we have chosen. +In this context, however, our interpretation leads us +to note that the curl operator C, (which implements +d as defined in Table 1), acts on on the generalized +Whitney forms of a cubical lattice as nothing more +than a finite difference operator. To see this more +explicitly, let us denote by aσ1 the entry of a ∈ RN1 +corresponding to the Whitney 1-form basis element +Wσ1. Likewise, we let bσ2 denote the coefficient in +b = Ca ∈ RN2 corresponding to Wσ2. With reference +again to Fig. 1, we find that +bx1x2x4x3 = (Ca)x1x2x4x3 += +1 +∆x +� +ax2x4 − ax1x3 +� +− 1 +∆y +� +ax3x4 − ax1x2 +� +. +(21) +Under the map C, the finite differences of the edge +coefficients are therefore summed to derive the coef- +ficient on a face. +The transpose of the curl operator CT , yields an +analogous result, with (CT b)σ1 computed via finite +differences between the faces adjoining edge σ1. Us- +ing Fig. 2 as a reference, for example, we find +(CT b)x1x5 = +1 +∆x +� +bx1x5x6x2 − bx10x12x5x1 +� +− 1 +∆y +� +bx1x3x7x5 − bx9x1x5x11 +� +. +(22) +The observation that C acts as a finite difference +operator on cubical generalized Whitney forms marks +a first step toward recovering Yee’s method from +Eq. (15). +As a second step, let us first consider what M1 and +M2 look like for our chosen Whitney form FEEC ba- +sis on a cubical mesh. Ignoring boundary cells for +Figure 2: +A depiction of the degrees of freedom in- +volved in CT —the transposed curl operator—for gener- +alized Whitney forms on a cubic mesh. +simplicity, we use Eq. (3) to integrate inner products +of forms such as those appearing in Eq. (18) to find +that +(M1)σ1,τ 1 = ∆V · +� +� +� +� +� +4/9 +σ1 = τ 1 +1/9 +σ1 ∥ τ 1 and ∃ τ 2 ⊃ {σ1, τ 1} +1/36 +σ1 ∥ τ 1 and ∃ τ 3 ⊃ {σ1, τ 1} +(M2)σ2,τ 2 = ∆V · +� +2/3 +σ2 = τ 2 +1/6 +σ2 ∥ τ 2 and ∃ τ 3 ⊃ {σ2, τ 2} +(23) +where ∆V = ∆x∆y∆z denotes a cell volume. Given +these matrix entries, each row (and column) of M1 +and M2 can be readily shown to sum to ∆V . In the +interest of recovering Yee’s method, therefore, we de- +fine the simplified mass matrices +MY +1 = ∆V · 1N1×N1 +MY +2 = ∆V · 1N2×N2. +(24) +Here, MY +p signifies the ‘Yee-modified’ mass matrix for +p-forms, and 1 denotes the identity matrix. This pro- +cess is often referred to as ‘lumping’ the mass matrix +(of Eq. (23), for example) into diagonal form. +Importantly, using such simplified mass matrices +leaves the symplectic structure defined by {·, ·}∆ in +6 + +C12 +5 +6 +y +11 +C3 +10 +2Eq. (9) entirely undisturbed. Rather, a substitution +of the simplified mass matrix Mp → MY +p is properly +viewed as taking place within the discrete Hamilto- +nian H∆ in Eq. (9). While this coarser approximation +of HEM reduces the accuracy of the resulting (Yee’s) +algorithm, it has no deleterious effect on its charac- +terization as a structure-preserving algorithm. +We now complete our program to recover Yee’s +method. Let us consider again the Strang splitting +defined in Eq. (16). We can rewrite this system more +explicitly as +�a +e +� +t0+∆t += +H∆t/2 +e +H∆t +a H∆t/2 +e +�a +e +� +t0 +where +H∆t +e += +� +1 +−∆tM−1 +1 +0 +1 +� +, H∆t +a += +� +1 +0 +∆tc2CT M2C +1 +� +(25) +or equivalently, in terms of b = Ca: +� +b +e +� +t0+∆t += +H∆t/2 +e +H∆t +b H∆t/2 +e +�b +e +� +t0 +where +H∆t +e += +� +1 +−∆tCM−1 +1 +0 +1 +� +, +H∆t +b = +� +1 +0 +∆tc2CT M2 +1 +� +. +(26) +Eq. +(26) +is +iterated +in +a +simulation, +so +that +after +n + 1 +timesteps +its +evolution +operator +is +H∆t/2 +e +� +H∆t +e H∆t +b +�n H∆t/2 +e +. +Therefore, applying the +‘finite-difference operators’ C and CT from Eqs. (21- +22) and the mass matrix approximations MY +p from +Eq. (24), let us examine our splitting method at ‘mid- +step’—after [H∆t +e H∆t +b ]H∆t/2 +e +—and compare it with +Yee’s method. +Since H∆t/2 +e +only evolves b, we regard this as an +‘initialization step’ that gives us +H∆t/2 +e +: +� +b[t0] +e[t0] +� +�→ +� +b +� +t1/2 +� +e [t0] +� +. +(27) +In particular, this half-step establishes initial field +data as it appears in Yee’s method. Next, H∆t +b +only +evolves e, such that +H∆t +b : +� +b +� +t1/2 +� +e [t0] +� +�→ +� +b +� +t1/2 +� +e [t1] +� +where e [t1] = e [t0] + ∆t∆V c2CT b[t1/2]. +(28) +This +timestep +(e [t1] − e [t0])/∆tc2 = ∆V CT b[t1/2] +replicates the Yee method’s evolution of the electric +field, e.g. +1 +c2∆t +� +En+1 +x +� +i + 1 +2, j, k +� +− En +x +� +i + 1 +2, j, k +�� += +1 +∆y +� +B +n+ 1 +2 +z +� +i + 1 +2, j + 1 +2, k +� +− B +n+ 1 +2 +z +� +i + 1 +2, j − 1 +2, k +�� +− 1 +∆z +� +B +n+ 1 +2 +y +� +i + 1 +2, j, k + 1 +2 +� +− B +n+ 1 +2 +y +� +i + 1 +2, j, k − 1 +2 +�� +. +(29) +As depicted in Fig. 2 and described in Eq. (22), the +operator CT implements precisely the finite differ- +ences of Eq. (29). The additional constant factor ∆V +in Eq. (28) has no effect other than changing the ef- +fective unit in which b is expressed. +Finally, H∆t +e +only evolves b, such that +H∆t +e +: +� +b +� +t1/2 +� +e [t1] +� +�→ +� +b +� +t3/2 +� +e [t1] +� +where b +� +t3/2 +� += b +� +t1/2 +� +− (∆t/∆V )Ce[t1]. +(30) +This +timestep +(b +� +t3/2 +� +− b +� +t1/2 +� +)∆V /∆t = Ce[t1] +replicates Yee’s evolution of the magnetic field, e.g. +1 +∆t +� +B +n+ 1 +2 +x +� +i, j + 1 +2, k + 1 +2 +� +− B +n− 1 +2 +x +� +i, j + 1 +2, k + 1 +2 +�� += +1 +∆z +� +En +y +� +i, j + 1 +2, k + 1 +� +− En +y +� +i, j + 1 +2, k +�� +− 1 +∆y +� +En +z +� +i, j + 1, k + 1 +2 +� +− En +z +� +i, j, k + 1 +2 +�� +. +(31) +As described in Eq. (21), C implements precisely the +finite differences of Eq. (31). The factor of ∆V again +only impacts the unit of b. +Consequently, we have demonstrated that Yee’s +algorithm is equivalent to the structure-preserving +FEEC method of Eq. (15), using Whitney forms on a +cubical mesh, maximally simplified (diagonal) mass +matrices, and Strang splitting time evolution. +4. Generalizing Yee’s Method +The previous section demonstrates that Yee’s +method is a special case of the FEEC structure- +7 + +Yee’s Method +SFEEC methods +Splitting method +Strang +any (e.g. Lie-Trotter [38], Strang) +Mesh +cubical +any (e.g. simplicial, cubical) +Finite elements +Whitney forms +any FEEC basis (e.g. P− +r Λk, PrΛk) +M−1 +1 +and M2 +diagonal approximation +any sparse approximation +Table 2: The SFEEC generalization of Yee’s method affords considerable flexibility in the choice of splitting scheme, +mesh, and finite element basis in its implementation of Eq. (15). SFEEC methods are symplectic and are also required +to be scalable. Therefore, they require sparse mass matrices and their sparse approximate inversions. +preserving algorithm defined by Eq. (15). +Specifi- +cally, Yee’s method is seen to make several selections +for the FEEC algorithm, including +1. timesteps given by a Strang splitting scheme; +2. a cubical grid; +3. the least-order FEEC finite element basis— +cubical Whitney forms; and +4. mass matrices simplified into diagonal form. +The last of these—the mass matrix approximation— +can be seen not so much as a ‘selection’ but as a ‘de- +parture’ from the algorithm defined by Eq. (15). As +previously described, however, a simplified treatment +of mass matrices occurs at the level of the Hamilto- +nian H∆ in Eq. (9), and does not affect the Poisson +bracket {·, ·}∆. As a result, Yee’s method remains +symplectic and—though it loses accuracy—retains +all of the advantages of a structure-preserving algo- +rithm. Yee’s method thereby preserves the geomet- +ric structure of Maxwell’s equations while employing +a parsimonious description of electromagnetic fields. +Moreover, its sacrifice of accuracy in mass matrices +is arguably compensated by the scalability the Yee +method achieves. +Here, we propose to relax all four of the above se- +lections made for the Yee scheme, and in so doing +we define a new class of algorithms, scalable FEEC +(SFEEC) methods. In particular, Table 2 enumer- +ates the generalizations of Yee’s method afforded by +SFEEC methods. +As defined, SFEEC methods offer considerable +flexibility in generalizing the Yee scheme. They have +the full flexibility of FEEC behind them, and can +therefore handle quite general meshes and high order +finite elements. Moreover, the splitting scheme used +for SFEEC methods can be chosen to be of arbitrarily +high order accuracy in time. +As we shall see, however, this improvement in accu- +racy is constrained by the requirement that SFEEC +methods remain scalable. In particular, while mass +matrices will no longer be maximally simplified into +diagonal form (as in Yee’s approach), M−1 +1 +will re- +quire sparse approximation. Even with this caveat, +we shall demonstrate that SFEEC methods offer a +significant improvement upon the accuracy of Yee’s +method, at limited additional cost in computational +effort. +Let us consider in greater detail the need for sim- +plified mass matrices in large scale implementations +of Eq. (15). We note that essentially any choice of +FEEC finite element has only local support, so that +the mass matrices M1 and M2 are generally sparse +by definition. However, Eq. (15) requires the use of +M−1 +1 , and the inverse of a sparse matrix is generally +dense. +Given M−1 +1 +dense, however, a computation +that evolves subsystem He of Eq. (15) would require +every node of a simulation to pass its local e data to +every other node, so that a could be evolved. This +communication would clearly spoil the efficacy of par- +allelization. +On the other hand, we may leverage the spirit of +Yee’s algorithm as seen through an FEEC lens, and +consider more parsimonious approximations of the +matrix M−1 +1 . We emphasize from our discussion above +that we can ‘prune’ M−1 +1 +as minimally or as maximally +as desired: Any approximation of M−1 +1 +will produce, +as Yee’s method produces, a symplectic algorithm. +The choice of approximation should be guided, there- +8 + +fore, strictly by the trade-off between its scalability +and its accuracy in approximating the electromag- +netic Hamiltonian, H∆ ≈ HEM. +Before proceeding to describe our numerical results +in the next section, we first describe our method to +find sparse approximations of M−1 +1 . Our general strat- +egy (following [26],[39]) is to identify a desirable spar- +sity pattern a priori, and find a matrix Q ≈ M−1 +1 +of the +desired sparsity pattern that best approximates M−1 +1 . +We shall denote the best fit for M−1 +1 +with sparsity pat- +tern S(A) as QS(A), where S(A) denotes the sparsity +pattern of an appropriate matrix A. The goodness of +fit may be determined by the Froebenius norm ∥·∥F , +in which case Q can be solved for column-by-column, +since +∥M1Q − 1∥2 +F = +N1 +� +ℓ=1 +∥M1qℓ − 1ℓ∥2 . +(32) +Here, qℓ is the ℓth column of Q and 1ℓ is the ℓth col- +umn of the identity matrix, (a standard basis vector). +Thus, qℓ can be found ∀ ℓ independently—in parallel, +if desired—as the solution to a least squares problem, +in which the only entries of qℓ permitted to vary are +those that fit the desired sparsity pattern for the ℓth +column of Q. +More explicitly, consider the problem of solving for +the entries of one such column qℓ. We may define +an index Iℓ ⊂ {1, . . . , N1} corresponding to the row +numbers of nonzero entries permitted in qℓ according +to the desired sparsity pattern S(Q). Denote the car- +dinality of such an index |Iℓ| ≤ N1. Then let M1(:, Iℓ) +denote the N1 × |Iℓ| submatrix of M1 comprised of +the subset of its columns indicated by Iℓ. For each +1 ≤ ℓ ≤ N1 we minimize ∥M1(:, Iℓ)qℓ(Iℓ) − 1ℓ∥, that +is, +arg min +qℓ(Iℓ) +� +1≤i≤N1 +j∈Iℓ +� +(M1)ij(qℓ)j − δiℓ +�2 +. +(33) +In the numerical examples we describe below, we +solve Eq. (33) directly via +qℓ(Iℓ) = [M1(:, Iℓ)T M1(:, Iℓ)]−1M1(:, Iℓ)T 1ℓ. +(34) +Most implementations of Eq. (33) will be sparse +enough to be solved this way. Denser choices of the +sparsity pattern (|Iℓ| ≫ 1), however, for which the in- +version in Eq. (34) is more difficult, can also be read- +ily solved using, for example, the conjugate gradient +method (see [40]) or, if desired, constrained versions +thereof (see [41]). +5. Numerical Results +To measure the accuracy of an approximation +Q ≈ M−1 +1 +in a manner relevant to our electromag- +netic problem, we examine the curl-of-curl oper- +ator ∇ × ∇×, +discretized by M−1 +1 CT M2C in the +FEEC +setting, +as +in +Eq. +(11). +For +simplic- +ity, +we work with simplicial finite elements in +2-D, and generate Delaunay triangulations Th of +a 2-torus domain |Th| = [0, Lx] × [0, Ly] ⊂ R2 with +periodic boundary conditions. +On |Th|, we con- +sider a sinusoidal vector potential A = sin(kny)dx +for kn = 2πn/Ly and n ∈ N, and canonically project +it onto some choice of discrete FEEC basis [6], +a · Λ1 ∈ Λ1(Th). +We measure the L2Λ1 error be- +tween ∇ × ∇ × A = k2 +n sin(kny)dx (exactly repre- +senting ˙E/c2 in the continuum) and its FEEC dis- +cretization, M−1 +1 CT M2Ca (representing M−1 +1 ˙e/c2). Fi- +nally, we investigate how the error of approximating +M−1 +1 +with Q—i.e., of approximating ∇ × ∇ × A with +QCT M2Ca—varies with the sparsity pattern S(Q). +We repeat this procedure over a range of frequen- +cies kn = 2πn/Ly and mesh diameters h, (the latter +achieved by varying the number of vertices in T2 and +generating a Delaunay triangulation Th for each). We +examine two different FEEC bases on this 2-D sim- +plicial mesh: P− +1 Λp(Th), the Whitney 1-forms (de- +scribed above for a cubical mesh), and their counter- +parts at the next order of accuracy, P− +2 Λp(Th)—the +second-order trimmed polynomial family of finite el- +ements. +For each basis, we compare four different +sparsity patterns for our approximation of M−1 +1 , in- +cluding: diagonal (Yee’s implicit pattern), M1 spar- +sity, (M1)2 = M1 · M1 sparsity,1 and dense (the exact +1(M1)ij is nonzero whenever basis elements Λ1 +i and Λ1 +j of +Λ1 have nontrivial ‘overlap,’ in the sense of Eq. (3). (M1)2 +ij +is generally nonzero whenever Λ1 +i and Λ1 +j each has nontrivial +overlap with a third basis finite element Λ1 +k (where k ∈ {i, j} +9 + +Figure 3: The figures above depict the L2Λ1 log relative error in an FEEC approximation of ˙E = c2∇ × ∇ × A vs. +log cell size h. The x-axis measures the number of cells (of size h) per wavelength λA = 2π/kn of the vector potential +A = sin(kny)dx. The left plot uses the first order (Whitney form) P− +1 Λp(Th) FEEC basis, while the right plot uses +a more accurate second order basis, P− +2 Λp(Th). For each basis, we plot the relative errors of four approximations of +the FEEC curl-of-curl operator, ∇ × ∇× ≈ QiCT M2C, i ∈ {1, . . . , 4}, and fit their power scalings against h. Here, Qi +is an approximation of the inverse mass matrix M−1 +1 +with i indicating one of four sparsity patterns: diagonal, M1, +(M1)2, and dense (which recovers the exact FEEC operator). The left plot demonstrates that the accuracy of Yee’s +method, (which effectively uses Whitney forms and diagonal mass matrices), is meaningfully improved upon by an +approximation of M−1 +1 +that has the sparsity pattern of M1. The right plot demonstrates that approximating M−1 +1 +by +a matrix with the sparsity pattern of (M1)2 achieves much of the improved accuracy possible with second order finite +elements. It is seen that the power scalings extend to a finite resolution, at which point relative error saturates. +inverse). The results of this investigation are depicted +in Fig. 3. +With the setup described above, the two subplots +of Fig. 3 display the log relative L2Λ1 errors in our +approximations of ˙E = c2∇ × ∇ × A. (For brevity, +we use ˙E merely as a shorthand notation.) In partic- +is also permitted). In our context, therefore, (M1)2 is strictly +denser than M1. By extension, the Neumann representation of +M−1 +1 +demonstrates that S((M1)n) → S(M−1 +1 ) as n approaches +N1 [39]. +ular, given the L2Λ1 norm (see Eq. (4)) +|| ˙E||L2Λ1 = +� +|Th| +( ˙E, ˙E)1dx, +(35) +we compute +||ˆ˙E − ˙E||L2Λ1/|| ˙E||L2Λ1 +(36) +where ˆ˙E = c2QCT M2C denotes the relevant finite +element approximation of the continuum 1-form +˙E = c2k2 +n sin(kny)dx. +10 + +Pi-A→ (Whitney 1-forms) +0.2 +0r +0 +-0.5 +-0.2 +IL2△1 +-1 +-0.4 +-1.5 +二 +-0.6 +log10 +-2 +-0.8 +-Diagonal (“"Yee")△α h0.56 +-Diagonal （“"Yee"）△ α h0.56 +-M1 +-2.5 +△ α h0.73 +-M1 +△ α h1.19 +-1 +(M1)2 +△ α h0.81 +(Mi)2 +△ α h1.7 +Dense (FEEC) +△ α h0.82 +Dense (FEEC) +△ α h1.98 +-1.2 +-3 +20 +9.46 +4.47 +2.11 +1 +20 +9.46 +4.47 +2.11 +入A/h +入A/hThe left plot demonstrates that the FEEC curl- +of-curl operator M−1 +1 CT M2C is well approximated for +a Whitney form basis using QS(M1)CT M2C (where +QS(M1) denotes the approximation of M−1 +1 +with a spar- +sity pattern of M1). In particular, the relative error of +QS(M1)CT M2C scales with cell size as h0.73, while the +exact FEEC operator M−1 +1 CT M2C error scales com- +parably as h0.82. +Remarkably, the first plot demonstrates that a +substantial improvement in accuracy over a diago- +nal sparsity pattern analogous to Yee’s method is +achieved by using only a slightly less parsimonious +estimate of M−1 +1 +in the curl-of-curl FEEC operator. +In particular, while QS(M1)CT M2C scales as h0.73 (as +noted just above), the accuracy of QS(1)CT M2C (with +QS(1) diagonal-patterned) scales as h0.56. +More important than this modest improvement in +power scaling, however, is its applicability to finer +meshes. +The first plot of Fig. 3 shows that the +power scaling of the diagonal mass matrix curl-of- +curl approximation can only be assumed for meshes +of relatively low resolution. The relative error in the +QS(1) discrete differential operator is seen to saturate +when the number of cells per signal wavelength climbs +above 5—that is λA/h ≳ 5. This is in contrast to the +QS(M1) power scaling, which continues to hold even +at the higher resolution of λA/h ∼ 20. This could be +a particular advantage in simulations of small-scale +electromagnetic structure, such as in turbulent plas- +mas or nanomaterials. +It must be further emphasized that the additional +computational cost introduced by this less parsimo- +nious QS(M1) approximation of M−1 +1 +will generally be +fairly modest. The pairs of nodes that communicate +in a simulation with an M1-sparsity approximation +are typically the same as those that communicate +with a diagonal-sparsity approximation, such that +no ‘new channels’ of communication are introduced. +Each communication between nodes will pass more +data, however—in our 2-D example, roughly 5 times +as much. Nevertheless, the amount of data passed is +small to begin with, as it scales only with the number +of boundary cells of a node. +The right plot of Fig. 3 demonstrates that, to +achieve the accuracy of higher order finite elements, +denser approximations of M−1 +1 +must be used in the +curl-of-curl operator. Nevertheless, greater accuracy +is readily attained. In our 2-D simplicial example, +we see that much of the improved accuracy possi- +ble with second order finite elements is achieved by +using a curl-of-curl operator QS(M2 +1)CT M2C, which ap- +proximates M−1 +1 +with a sparsity pattern of (M1)2. In- +deed, this approximate FEEC operator achieves an +error scaling with cell size of h1.70, while the exact +FEEC curl-of-curl operator error scales in our results +as h1.98. It is worth noting, however, that the approx- +imation’s scaling holds only until a mesh resolution +of roughly λA/h ≳ 5. Nevertheless, due to the higher +order finite element, this saturation occurs at a sig- +nificantly higher accuracy overall. +6. Discussion +We have demonstrated that Yee’s method can +be interpreted as a structure-preserving splitting +method using an FEEC formalism on a cubical mesh +with simplified mass matrices. In so doing, we have +identified Yee’s algorithm to be a special case of +a larger family of methods we call SFEEC (scal- +able finite element exterior calculus) methods, sum- +marized by Eq. (15) and Table 2. +These methods +are structure-preserving, as Yee’s method is, and +are therefore accurate in long-time numerical simu- +lations. They also respect the topological and geo- +metric properties of Maxwell’s equations. Crucially, +SFEEC methods are also scalable; they adopt the +strategy implicit in Yee’s method of using sparse +approximations of finite element mass matrices to +achieve computational efficiency. +The classification of SFEEC methods enabled us +to identify higher order extensions of Yee’s method +that preserve its scalability nevertheless (see Fig. 3). +Depending on the computing architecture employed +and the desired accuracy of the problem of interest, +these alternative methods may enable greater long- +time accuracy than Yee’s method provides, with lit- +tle additional computational cost. The applicability +of SFEEC methods on general meshes may also be of +particular benefit in problems with irregular geome- +tries. +11 + +The saturation of the relative errors plotted in +Fig. 3 suggest that it would be fruitful to investigate +in future work whether improved sparse approxima- +tions of the discrete codifferential operator, M−1 +1 CT M2 +can be found. Such efforts might be geared to pre- +serve higher order finite element error scaling at ar- +bitrarily high mesh resolution. +7. Acknowledgments +Thank you to Josh Burby and Tyrus Berry for +helpful discussions, and to Phil Morrison for his sup- +port. +This research was further supported by the +U.S. Department of Energy (DE-AC02-09CH11466), +as well as the U.S. Department of Energy Fusion En- +ergy Sciences Postdoctoral Research Program admin- +istered by the Oak Ridge Institute for Science and +Education (ORISE) for the DOE. ORISE is managed +by Oak Ridge Associated Universities (ORAU) un- +der DOE contract number DE-SC0014664. All opin- +ions expressed in this paper are the authors’ and do +not necessarily reflect the policies and views of DOE, +ORAU, or ORISE. +References +[1] K. 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Shariff, “A constrained conjugate gradient +method and the solution of linear equations,” +Computers & Mathematics with Applications, +vol. 30, no. 11, pp. 25–37, Dec. 1995. +14 + diff --git a/KdAzT4oBgHgl3EQfyf4o/content/tmp_files/load_file.txt b/KdAzT4oBgHgl3EQfyf4o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..051957afef6b417d742e8d96477b611ad7002142 --- /dev/null +++ b/KdAzT4oBgHgl3EQfyf4o/content/tmp_files/load_file.txt @@ -0,0 +1,673 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf,len=672 +page_content='Generalizing Yee’s method: Scalable geometric higher-order FEEC algorithms for Maxwell’s equations on an unstructured mesh Alexander S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Glassera,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Hong Qina,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='b aPrinceton Plasma Physics Laboratory Princeton University Princeton New Jersey 08543 bDepartment of Astrophysical Sciences Princeton University Princeton New Jersey 08544 Abstract The Yee algorithm for electromagnetic simulations is widely known to have many advantages,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' including the following crucial two: (i) Its calculations are local and therefore efficiently parallelizable—enabling simulations that capitalize on the speed and scalability of high-performance computing architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (ii) Yee’s method faithfully preserves the symplectic geometry of Maxwell’s equations, improving its accu- racy in long-time numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Whereas previous geometric generalizations of Yee’s method have sacrificed its scalability, in this article the Yee algorithm is generalized to higher order and unstructured meshes in a manner that fully preserves both its scalability and geometric naturalness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' This generalization is achieved by prioritizing the locality of the algorithm, reflecting the physical locality of Maxwell’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Specifically, we demonstrate that Yee’s method is but a special case of a larger family of symplectic, finite element exterior calculus (FEEC) methods that use scalable, local approximations of mass matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We discuss the numerical advantages of this family of methods, which we call scalable FEEC (SFEEC) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Introduction The Yee algorithm [1, 2]—alternatively, the finite difference time domain (FDTD) method—defines electromagnetic fields on a cubic mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' It asso- ciates to each edge a component of the electric field E, and to each face a component of the mag- netic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' This discretization reflects a natu- ral geometric description of Maxwell’s equations, in which one defines E ∈ Λ1(R3) as a differential 1-form on R3 and B ∈ Λ2(R3) as a differential 2-form on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In this way, Yee’s method employs a technique adopted in many structure-preserving algorithms [3], wherein differential k-forms are discretized by asso- ciating them with k-dimensional features of a mesh [4, 5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Structure-preserving algorithms have been widely adopted in many subfields of computational physics, including gravitational simulations [8, 9, 10, 11, 12], geophysics [13, 14] and plasma physics [15, 16, 17, 18, 19, 20, 21, 22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Such algorithms generally de- rive from variational principles or Hamiltonian sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' As a result, they preserve essential mathemat- ical features of their underlying physical systems, in- cluding symplectic structure, topology, symmetries, and conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' These properties contribute to the numerical fidelity of structure-preserving al- gorithms, especially in long-time numerical simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Despite Yee’s omission of any overt La- grangian or Hamiltonian formulations in his original work [1], Yee’s method (apparently serendipitously) is one of the most historically successful examples of a structure-preserving algorithm [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' More recently, the methodology of structure- preserving algorithms has led to the development of advanced algorithms for the simulation of plas- mas, whose electromagnetic fields are constructed using the formalism of finite element exterior cal- culus (FEEC) [6, 7, 20, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Although such meth- Preprint submitted to Elsevier January 5, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='01753v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='NA] 4 Jan 2023 ods are in principle readily generalizable to unstruc- tured meshes and high order finite elements, they lack the computational efficiency and scalability of Yee’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In particular, the time evolution of electro- magnetic fields in these FEEC methods requires com- munication between all nodes of a simulation, thereby destroying their parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' As we shall discuss and address, this problem arises because sparse finite el- ement mass matrices generally have dense inverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' On the other hand, there have also been numer- ous efforts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' [25, 26, 27, 28]) to generalize Yee’s method using scalable finite element methods, called finite element time domain (FETD) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Such efforts leverage the crucial technique of sparse ap- proximate inverse (SPAI) mass matrices, and pre- serve the scalability of Yee’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' However, these methods’ preservation of symplectic structure has not been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Motivated by the desire to overcome the scalabil- ity limitation of structure-preserving FEEC plasma methods, and to guarantee the structure preservation of FETD methods, in this article we develop scal- able FEEC (SFEEC) methods, a family of symplec- tic finite element methods for electromagnetic fields that includes Yee’s method as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' SFEEC methods enable the higher order simulation of elec- tromagnetic fields on structured and unstructured meshes in a manner that preserves the two aforemen- tioned crucial advantages of Yee’s method, namely: (i) its scalability on modern architectures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' and (ii) its symplectic geometry (and the resulting conservation of electric charge and Gauss’ law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' From a finite element point of view, the scalability of Yee’s method will be reframed as a result of its sim- plified (or ‘pruned’) approximation of finite element mass matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' To retain their scalability, SFEEC methods employ a comparable, if more flexible treat- ment of mass matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Relative to Yee’s method, SF- FEC methods afford a greater flexibility to improve algorithmic accuracy without sacrificing scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Their use of finite elements and the FEEC formal- ism further enables their viability on more general meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' While Yee’s algorithm has been generalized nu- merous times in the literature, including via higher order and finite element schemes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' [24, 25, 27, 28, 29, 30, 31, 32]), no such generalization is known to us that simultaneously affords higher order ac- curacy and scalability while ensuring the geometric structure-preservation of Yee’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In partic- ular, the Yee method’s preservation of symplectic structure is an important aspect of the method’s sta- bility and long-term accuracy [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In this work, we will review this symplectic structure of Yee’s method and demonstrate its exact preservation in the SFEEC family of algorithms we define.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In so doing, we also offer a means to overcome the scalability limita- tions of structure-preserving FEEC algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' [20, 23]) in a massively parallel, high-performance computing architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The remainder of this article is organized as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In Section 2, the formalism of finite element ex- terior calculus (FEEC) [6, 7] and its discretization of electromagnetic fields will be reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In Section 3, it will be demonstrated that Yee’s algorithm can be interpreted as an FEEC method with simplified mass matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In Section 4 we will define SFEEC meth- ods, which extend Yee’s method to higher order finite element schemes on a general mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In Section 5, numerical results will be presented that demonstrate the improved higher order accuracy of the resulting SFEEC methods relative to Yee’s method, without sacrificing its scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Section 6 will then summa- rize and conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Finite Element Exterior Calculus (FEEC) for Electromagnetic Simulations In this section, we briefly review aspects of FEEC [6, 7] and its application in electromagnetic simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We refer the reader to [20, 23] for additional background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We let Λp(Th) denote a vector space of finite element differential p-forms on a simplicial or cubical complex Th ⊂ Rn (where the subscript h de- notes the maximal diameter, or edge length, of any simplex in Th).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' For all 0 ≤ p ≤ n, Λp(Th) may be de- fined as the span of a finite (Np-dimensional) basis Λp, whose ith basis element Λp i ∈ Λp(Th) is a piece- wise polynomial p-form on Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Such a basis element typically has support localized to one or more adja- cent cells in Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' An arbitrary p-form S ∈ Λp(Th) can 2 be expressed in the Λp basis as S(x) = s · Λp(x) = siΛp i (x) (1) ∀ s ∈ RNp and x ∈ |Th|, the convex hull of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (Ein- stein summation convention is used for the repeated index in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (1) and hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=') Individual compo- nents of S ∈ Λp(Th) will be denoted S(x)µ1···µp = s · Λp(x)µ1···µp = siΛp i (x)µ1···µp (2) where Greek letters denote coordinate indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' For example, given |Th| ⊂ R3, the µth component of the 1-form basis element Λ1 i (x) may be written as Λ1 i (x)µ ∀ µ ∈ {1, 2, 3}, such that Λ1 i (x) = Λ1 i (x)µdxµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Because each basis Λp is finite ∀ 0 ≤ p ≤ n, the exterior derivative d : Λp(Th) → Λp+1(Th) of a fi- nite element p-form on Th can be computed in the Λ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' , Λn bases by straightforward matrix multipli- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' A choice of basis for each Λp(Th) in three dimensions (Th ⊂ R3) determines, for example, ma- trices that represent the gradient (G), curl (C), and divergence (D)—as defined in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' To apply FEEC in a physical setting, it will also be essential to compute mass matrices on Th for each basis Λp of finite element p-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Specifically, the mass matrix Mp ∈ RNp×Np for Λp is defined by (Mp)ij = � |Th| dx � Λp i , Λp j � p (3) where (·, ·)p denotes the pointwise inner product on p-forms induced by the metric gµν—specifically (α, β)p = 1 p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='αµ1···µpβµ1···µp = 1 p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='αµ1···µpβν1···νpgµ1ν1 · · · gµpνp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (4) p-Form Abstract d Matrix d Defined by S = s · Λ0 A = a · Λ1 A = dS a = Gs GT Λ1 = dΛ0 B = b · Λ2 B = dA b = Ca CT Λ2 = dΛ1 C = c · Λ3 C = dB c = Db DT Λ3 = dΛ2 Table 1: FEEC matrix implementation of d on Th ⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The property d ◦ d = 0 implies that CG = 0 and DC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' For p = 1, 2 on R3, for example, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (4) defines the inner product (α, β)p as the standard inner product α · β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' After all, 1- and 2-forms each have three inde- pendent components in R3, and the metric gµν (and its inverse gµν) is given by the Kronecker delta— gµν = δµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We note that (α, β)p is symmetric, such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (3) is symmetric and MT p = Mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (3) also implies that Mp is typically sparse (so long as each ba- sis element Λp i has only local support).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Importantly, however, its inverse matrix M−1 p is typically dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' To apply the FEEC formalism to electromagnetic simulations, we will first discretize the magnetic vec- tor potential as an FEEC 1-form, that is, A = a · Λ1 (5) with degrees of freedom given by the vector of coef- ficients a ∈ RN1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (We work in the temporal gauge, such that the electric potential φ vanishes, φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=') The magnetic field is then given by a 2-form, B = b · Λ2 = Ca · Λ2 = a · CT Λ2 = a · dΛ1 = dA (6) with b ∈ RN2, in keeping with the geometry of Maxwell’s equations and the FEEC implementation of the curl, as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Lastly, the electric field will be defined as a discrete 1-form E = e · M−1 1 Λ1 (7) with e ∈ RN1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Following [23], we use a convention wherein the coefficients e determine the electric field E with an additional factor of M−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We shall regard the pair (a, e) ∈ R2N1 as defining the degrees of free- dom of the electromagnetic finite element dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' To describe the dynamics of these discrete fields via Maxwell’s equations, we first recall the canonical symplectic structure of electromagnetic fields in the continuum, defined with Poisson bracket and Hamil- tonian in SI units as [34] {F, G}EM = 1 ϵ0 � dx �δF δE · δG δA − δG δE · δF δA � HEM = 1 2 � dx � ϵ0 |E|2 + 1 µ0 |∇ × A|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (8) 3 We now substitute the FEEC fields A = a · Λ1 and E = e · M−1 1 Λ1 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (8) to derive (see [23]) the following discrete Poisson bracket {·, ·}∆ and discrete Hamiltonian H∆ approximating this system, {F, G}∆ = 1 ϵ0 �∂F ∂e · ∂G ∂a − ∂G ∂e · ∂F ∂a � H∆ = 1 2 � ϵ0eT M−1 1 e + 1 µ0 aT CT M2Ca � , (9) where we have applied dΛ1 = CT Λ2 as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Equations of motion are now readily calculated as usual with a Poisson bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We plug the FEEC coefficient vectors a and e into {·, ·}∆ with H∆, to find ˙a = {a, H∆}∆ = −M−1 1 e ˙e = {e, H∆}∆ = c2CT M2Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (10) These equations are discrete forms of Maxwell’s equa- tions, ˙A = −E and ˙E = c2∇ × ∇ × A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In Section 5, the discrete approximation of the curl- of-curl operator ∇ × ∇× ≈ M−1 1 CT M2C, (11) whose calculation (until our later simplification) in- corporates the dense matrix M−1 1 , will be a central focus of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Note, it will also be convenient to rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (10) in terms of the magnetic field by substituting b = Ca (the discrete realization of B = ∇ × A) to find ˙b = −CM−1 1 e ˙e = c2CT M2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (12) Because of the gauge symmetry of Maxwell’s equa- tions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (12) is an equivalent formulation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (10) and exactly preserves the physical degrees of freedom of interest [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' To algorithmically evolve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (10) in discrete time, we follow [17] (see also [22]) and define a splitting algorithm that decomposes H∆ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (9) into two pieces: H∆ = Ha + He where Ha = 1 2µ0 aT CT M2Ca He = ϵ0 2 eT M−1 1 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (13) These ‘sub-Hamiltonians’ define two different subsys- tems which may be evolved separately, that is, Ha � ˙a = {a, Ha}∆ = 0 ˙e = {e, Ha}∆ = c2CT M2Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' He � ˙a = {a, He}∆ = −M−1 1 e ˙e = {e, He}∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (14) We see that a is constant in the subsystem Ha, while e is constant in the subsystem He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' As a result, each subsystem is exactly solvable over a finite time inter- val ∆t, in particular, Ha : e(t0 + ∆t) = e(t0) + ∆t · c2CT M2Ca(t0) He : a(t0 + ∆t) = a(t0) − ∆t · M−1 1 e(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (15) By choosing an appropriate sequence of these sub- systems’ discrete time evolutions, a splitting method may be implemented that approximates a timestep of the entire electromagnetic system H∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' A typical example of such a splitting scheme is called Strang splitting [35], which evolves a and e according to the approximation �a e � t0+∆t = exp (∆tH∆) · �a e � t0 ≈ exp �∆t 2 He � exp (∆tHa) exp �∆t 2 He � �a e � t0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (16) Here, we use the notation exp(∆tHi) to denote a dis- crete time mapping—as appears in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (15)—that evolves according to the Hamiltonian Hi for a time interval ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Since exp(∆tHa) and exp(∆tHe) do not commute, the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (16) is only an ap- proximation of the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Nevertheless, such a split- ting method is an effective algorithmic implementa- tion of our dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' It is exactly solvable, accurate to second order, and preserves the symplec- tic structure of the system as defined by the Poisson bracket of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Splitting methods of arbitrarily high accuracy in ∆t can also be constructed (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Reinterpreting Yee’s Method via FEEC We now rederive Yee’s method using the finite el- ement formalism of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We pro- ceed in three steps: (i) we first choose generalized Whitney forms as an FEEC basis on a cubical mesh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (ii) we then maximally simplify the mass matrices of the corresponding electromagnetic algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' and fi- nally (iii) we choose Strang splitting as our method of time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' These three steps will exactly re- cover Yee’s method, demonstrating its interpretation as an implementation of the FEEC electromagnetic algorithm defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (15)—with simplified mass matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We begin by considering a cubical lattice Th ⊂ R3 with lattice spacings {∆x, ∆y, ∆z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We make the simplest possible choice for an FEEC basis on Th, namely, the generalized Whitney forms, a fam- ily of piecewise polynomial finite elements denoted Q− 1 Λp(Th) [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' This finite element basis is also use- fully defined in [37] Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' A generalized Whitney p-form can be read- ily understood by its 1-to-1 correspondence with a p-dimensional feature of a mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In partic- ular, we define the generalized Whitney p-form Wσp ∈ Q− 1 Λp(Th) associated to a given p-face σp ⊂ Th ⊂ Rn by requiring that 1 |τ p| � τ p Wσp = � 1 τ p = σp 0 τ p ̸= σp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (17) In this way, generalized Whitney p-forms are dual to p-faces of Th via integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Here, |τ p| denotes the p-volume of τ p (defined as 1, length, area, and volume respectively for p = 0, 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' This normal- ization is chosen to mimic fields as they are typically represented in Yee’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' A concrete example of this is represented on Th ⊂ R3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The 1-form Wx1x2 ∈ Q− 1 Λ1(Th) and the 2-form Wx1x2x4x3 ∈ Q− 1 Λ2(Th) are depicted, defined respectively by Wx1x2 = � 1 − y ∆y � � 1 − z ∆z � dx Wx1x2x4x3 = � 1 − z ∆z � dx ∧ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (18) As in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (17), Wx1x2 is the least-order piecewise polynomial 1-form satisfying � x1x2 Wx1x2 = |x1x2| and � σ1 Wx1x2 = 0 ∀ σ1 ̸= x1x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Wx1x2x4x3 is analogously defined so that its integral satisfies � x1x2x4x3 Wx1x2x4x3 = |x1x2x4x3| and vanishes on any other face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In later calculations, it will also be useful to explicitly define the following Whitney 2-form associated to the face x1x5x6x2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 1, Wx1x5x6x2 = � 1 − y ∆y � dz ∧ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (19) Now consider the exterior derivative d of these forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The curl matrix C : RN1 → RN2 is a linear operator that precisely computes d for 1-forms, map- ping the coefficients of a discrete 1-form—expanded in the Q− 1 Λ1(Th) basis—to coefficients of a discrete 2-form—expanded in the Q− 1 Λ2(Th) basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The 1-to-1 correspondence between these generalized Whitney forms and elements of the mesh leads to a convenient interpretation of C in that basis, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We find by directly computing the exterior deriva- tive of Wx1x2 using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (18-19), for example, that dWx1x2 = 1 ∆y Wx1x2x4x3 − 1 ∆z Wx1x5x6x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (20) We observe that a Whitney 2-form Wσ2 appears on the right hand side above if and only if its associated Figure 1: The generalized Whitney 1-forms Wx1x2 and Wx1x2x4x3 are schematically depicted, respectively, on the left and right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Wx1x2 evaluates to the length |x1x2| when integrated along the edge x1x2 (in blue) and vanishes on all other edges (in orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Wx1x2x4x3 likewise yields the area |x1x2x4x3| when integrated over the blue face x1x2x4x3, and vanishes on all other faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 5 07 5 6 C3 C 4 C28 05 M6 y C3 C4 C2face σ2 contains the edge x1x2 on its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Its coefficient ±1/∆xµ is determined by the µth dimen- sion in which the associated face extends x1x2, and its sign is set by the relative orientation between the edge and face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Note, this interpretation of d—as a mapping with coefficients ±1/∆xµ from 1-forms to the 2-forms of faces which contain them on the boundary—is ap- plicable only to the particular (cubical) FEEC mesh and (Whitney form) FEEC basis that we have chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In this context, however, our interpretation leads us to note that the curl operator C, (which implements d as defined in Table 1), acts on on the generalized Whitney forms of a cubical lattice as nothing more than a finite difference operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' To see this more explicitly, let us denote by aσ1 the entry of a ∈ RN1 corresponding to the Whitney 1-form basis element Wσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Likewise, we let bσ2 denote the coefficient in b = Ca ∈ RN2 corresponding to Wσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' With reference again to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 1, we find that bx1x2x4x3 = (Ca)x1x2x4x3 = 1 ∆x � ax2x4 − ax1x3 � − 1 ∆y � ax3x4 − ax1x2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (21) Under the map C, the finite differences of the edge coefficients are therefore summed to derive the coef- ficient on a face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The transpose of the curl operator CT , yields an analogous result, with (CT b)σ1 computed via finite differences between the faces adjoining edge σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Us- ing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 2 as a reference, for example, we find (CT b)x1x5 = 1 ∆x � bx1x5x6x2 − bx10x12x5x1 � − 1 ∆y � bx1x3x7x5 − bx9x1x5x11 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (22) The observation that C acts as a finite difference operator on cubical generalized Whitney forms marks a first step toward recovering Yee’s method from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' As a second step, let us first consider what M1 and M2 look like for our chosen Whitney form FEEC ba- sis on a cubical mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Ignoring boundary cells for Figure 2: A depiction of the degrees of freedom in- volved in CT —the transposed curl operator—for gener- alized Whitney forms on a cubic mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' simplicity, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (3) to integrate inner products of forms such as those appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (18) to find that (M1)σ1,τ 1 = ∆V · � � � � � 4/9 σ1 = τ 1 1/9 σ1 ∥ τ 1 and ∃ τ 2 ⊃ {σ1, τ 1} 1/36 σ1 ∥ τ 1 and ∃ τ 3 ⊃ {σ1, τ 1} (M2)σ2,τ 2 = ∆V · � 2/3 σ2 = τ 2 1/6 σ2 ∥ τ 2 and ∃ τ 3 ⊃ {σ2, τ 2} (23) where ∆V = ∆x∆y∆z denotes a cell volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Given these matrix entries, each row (and column) of M1 and M2 can be readily shown to sum to ∆V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In the interest of recovering Yee’s method, therefore, we de- fine the simplified mass matrices MY 1 = ∆V · 1N1×N1 MY 2 = ∆V · 1N2×N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (24) Here, MY p signifies the ‘Yee-modified’ mass matrix for p-forms, and 1 denotes the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' This pro- cess is often referred to as ‘lumping’ the mass matrix (of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (23), for example) into diagonal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Importantly, using such simplified mass matrices leaves the symplectic structure defined by {·, ·}∆ in 6 C12 5 6 y 11 C3 10 2Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (9) entirely undisturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Rather, a substitution of the simplified mass matrix Mp → MY p is properly viewed as taking place within the discrete Hamilto- nian H∆ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' While this coarser approximation of HEM reduces the accuracy of the resulting (Yee’s) algorithm, it has no deleterious effect on its charac- terization as a structure-preserving algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We now complete our program to recover Yee’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Let us consider again the Strang splitting defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We can rewrite this system more explicitly as �a e � t0+∆t = H∆t/2 e H∆t a H∆t/2 e �a e � t0 where H∆t e = � 1 −∆tM−1 1 0 1 � , H∆t a = � 1 0 ∆tc2CT M2C 1 � (25) or equivalently, in terms of b = Ca: � b e � t0+∆t = H∆t/2 e H∆t b H∆t/2 e �b e � t0 where H∆t e = � 1 −∆tCM−1 1 0 1 � , H∆t b = � 1 0 ∆tc2CT M2 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (26) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (26) is iterated in a simulation, so that after n + 1 timesteps its evolution operator is H∆t/2 e � H∆t e H∆t b �n H∆t/2 e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Therefore, applying the ‘finite-difference operators’ C and CT from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (21- 22) and the mass matrix approximations MY p from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (24), let us examine our splitting method at ‘mid- step’—after [H∆t e H∆t b ]H∆t/2 e —and compare it with Yee’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Since H∆t/2 e only evolves b, we regard this as an ‘initialization step’ that gives us H∆t/2 e : � b[t0] e[t0] � �→ � b � t1/2 � e [t0] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (27) In particular, this half-step establishes initial field data as it appears in Yee’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Next, H∆t b only evolves e, such that H∆t b : � b � t1/2 � e [t0] � �→ � b � t1/2 � e [t1] � where e [t1] = e [t0] + ∆t∆V c2CT b[t1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (28) This timestep (e [t1] − e [t0])/∆tc2 = ∆V CT b[t1/2] replicates the Yee method’s evolution of the electric field, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 1 c2∆t � En+1 x � i + 1 2, j, k � − En x � i + 1 2, j, k �� = 1 ∆y � B n+ 1 2 z � i + 1 2, j + 1 2, k � − B n+ 1 2 z � i + 1 2, j − 1 2, k �� − 1 ∆z � B n+ 1 2 y � i + 1 2, j, k + 1 2 � − B n+ 1 2 y � i + 1 2, j, k − 1 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (29) As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 2 and described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (22), the operator CT implements precisely the finite differ- ences of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The additional constant factor ∆V in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (28) has no effect other than changing the ef- fective unit in which b is expressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Finally, H∆t e only evolves b, such that H∆t e : � b � t1/2 � e [t1] � �→ � b � t3/2 � e [t1] � where b � t3/2 � = b � t1/2 � − (∆t/∆V )Ce[t1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (30) This timestep (b � t3/2 � − b � t1/2 � )∆V /∆t = Ce[t1] replicates Yee’s evolution of the magnetic field, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 1 ∆t � B n+ 1 2 x � i, j + 1 2, k + 1 2 � − B n− 1 2 x � i, j + 1 2, k + 1 2 �� = 1 ∆z � En y � i, j + 1 2, k + 1 � − En y � i, j + 1 2, k �� − 1 ∆y � En z � i, j + 1, k + 1 2 � − En z � i, j, k + 1 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (31) As described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (21), C implements precisely the finite differences of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The factor of ∆V again only impacts the unit of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Consequently, we have demonstrated that Yee’s algorithm is equivalent to the structure-preserving FEEC method of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (15), using Whitney forms on a cubical mesh, maximally simplified (diagonal) mass matrices, and Strang splitting time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Generalizing Yee’s Method The previous section demonstrates that Yee’s method is a special case of the FEEC structure- 7 Yee’s Method SFEEC methods Splitting method Strang any (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Lie-Trotter [38], Strang) Mesh cubical any (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' simplicial, cubical) Finite elements Whitney forms any FEEC basis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' P− r Λk, PrΛk) M−1 1 and M2 diagonal approximation any sparse approximation Table 2: The SFEEC generalization of Yee’s method affords considerable flexibility in the choice of splitting scheme, mesh, and finite element basis in its implementation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' SFEEC methods are symplectic and are also required to be scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Therefore, they require sparse mass matrices and their sparse approximate inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' preserving algorithm defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Specifi- cally, Yee’s method is seen to make several selections for the FEEC algorithm, including 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' timesteps given by a Strang splitting scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' a cubical grid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' the least-order FEEC finite element basis— cubical Whitney forms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' mass matrices simplified into diagonal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The last of these—the mass matrix approximation— can be seen not so much as a ‘selection’ but as a ‘de- parture’ from the algorithm defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' As previously described, however, a simplified treatment of mass matrices occurs at the level of the Hamilto- nian H∆ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (9), and does not affect the Poisson bracket {·, ·}∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' As a result, Yee’s method remains symplectic and—though it loses accuracy—retains all of the advantages of a structure-preserving algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Yee’s method thereby preserves the geomet- ric structure of Maxwell’s equations while employing a parsimonious description of electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Moreover, its sacrifice of accuracy in mass matrices is arguably compensated by the scalability the Yee method achieves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Here, we propose to relax all four of the above se- lections made for the Yee scheme, and in so doing we define a new class of algorithms, scalable FEEC (SFEEC) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In particular, Table 2 enumer- ates the generalizations of Yee’s method afforded by SFEEC methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' As defined, SFEEC methods offer considerable flexibility in generalizing the Yee scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' They have the full flexibility of FEEC behind them, and can therefore handle quite general meshes and high order finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Moreover, the splitting scheme used for SFEEC methods can be chosen to be of arbitrarily high order accuracy in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' As we shall see, however, this improvement in accu- racy is constrained by the requirement that SFEEC methods remain scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In particular, while mass matrices will no longer be maximally simplified into diagonal form (as in Yee’s approach), M−1 1 will re- quire sparse approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Even with this caveat, we shall demonstrate that SFEEC methods offer a significant improvement upon the accuracy of Yee’s method, at limited additional cost in computational effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Let us consider in greater detail the need for sim- plified mass matrices in large scale implementations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We note that essentially any choice of FEEC finite element has only local support, so that the mass matrices M1 and M2 are generally sparse by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' However, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (15) requires the use of M−1 1 , and the inverse of a sparse matrix is generally dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Given M−1 1 dense, however, a computation that evolves subsystem He of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (15) would require every node of a simulation to pass its local e data to every other node, so that a could be evolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' This communication would clearly spoil the efficacy of par- allelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' On the other hand, we may leverage the spirit of Yee’s algorithm as seen through an FEEC lens, and consider more parsimonious approximations of the matrix M−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We emphasize from our discussion above that we can ‘prune’ M−1 1 as minimally or as maximally as desired: Any approximation of M−1 1 will produce, as Yee’s method produces, a symplectic algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The choice of approximation should be guided, there- 8 fore, strictly by the trade-off between its scalability and its accuracy in approximating the electromag- netic Hamiltonian, H∆ ≈ HEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Before proceeding to describe our numerical results in the next section, we first describe our method to find sparse approximations of M−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Our general strat- egy (following [26],[39]) is to identify a desirable spar- sity pattern a priori, and find a matrix Q ≈ M−1 1 of the desired sparsity pattern that best approximates M−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We shall denote the best fit for M−1 1 with sparsity pat- tern S(A) as QS(A), where S(A) denotes the sparsity pattern of an appropriate matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The goodness of fit may be determined by the Froebenius norm ∥·∥F , in which case Q can be solved for column-by-column, since ∥M1Q − 1∥2 F = N1 � ℓ=1 ∥M1qℓ − 1ℓ∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (32) Here, qℓ is the ℓth column of Q and 1ℓ is the ℓth col- umn of the identity matrix, (a standard basis vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Thus, qℓ can be found ∀ ℓ independently—in parallel, if desired—as the solution to a least squares problem, in which the only entries of qℓ permitted to vary are those that fit the desired sparsity pattern for the ℓth column of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' More explicitly, consider the problem of solving for the entries of one such column qℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We may define an index Iℓ ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' , N1} corresponding to the row numbers of nonzero entries permitted in qℓ according to the desired sparsity pattern S(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Denote the car- dinality of such an index |Iℓ| ≤ N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Then let M1(:, Iℓ) denote the N1 × |Iℓ| submatrix of M1 comprised of the subset of its columns indicated by Iℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' For each 1 ≤ ℓ ≤ N1 we minimize ∥M1(:, Iℓ)qℓ(Iℓ) − 1ℓ∥, that is, arg min qℓ(Iℓ) � 1≤i≤N1 j∈Iℓ � (M1)ij(qℓ)j − δiℓ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (33) In the numerical examples we describe below, we solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (33) directly via qℓ(Iℓ) = [M1(:, Iℓ)T M1(:, Iℓ)]−1M1(:, Iℓ)T 1ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (34) Most implementations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (33) will be sparse enough to be solved this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Denser choices of the sparsity pattern (|Iℓ| ≫ 1), however, for which the in- version in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (34) is more difficult, can also be read- ily solved using, for example, the conjugate gradient method (see [40]) or, if desired, constrained versions thereof (see [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Numerical Results To measure the accuracy of an approximation Q ≈ M−1 1 in a manner relevant to our electromag- netic problem, we examine the curl-of-curl oper- ator ∇ × ∇×, discretized by M−1 1 CT M2C in the FEEC setting, as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' For simplic- ity, we work with simplicial finite elements in 2-D, and generate Delaunay triangulations Th of a 2-torus domain |Th| = [0, Lx] × [0, Ly] ⊂ R2 with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' On |Th|, we con- sider a sinusoidal vector potential A = sin(kny)dx for kn = 2πn/Ly and n ∈ N, and canonically project it onto some choice of discrete FEEC basis [6], a · Λ1 ∈ Λ1(Th).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We measure the L2Λ1 error be- tween ∇ × ∇ × A = k2 n sin(kny)dx (exactly repre- senting ˙E/c2 in the continuum) and its FEEC dis- cretization, M−1 1 CT M2Ca (representing M−1 1 ˙e/c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Fi- nally, we investigate how the error of approximating M−1 1 with Q—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=', of approximating ∇ × ∇ × A with QCT M2Ca—varies with the sparsity pattern S(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We repeat this procedure over a range of frequen- cies kn = 2πn/Ly and mesh diameters h, (the latter achieved by varying the number of vertices in T2 and generating a Delaunay triangulation Th for each).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' We examine two different FEEC bases on this 2-D sim- plicial mesh: P− 1 Λp(Th), the Whitney 1-forms (de- scribed above for a cubical mesh), and their counter- parts at the next order of accuracy, P− 2 Λp(Th)—the second-order trimmed polynomial family of finite el- ements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' For each basis, we compare four different sparsity patterns for our approximation of M−1 1 , in- cluding: diagonal (Yee’s implicit pattern), M1 spar- sity, (M1)2 = M1 · M1 sparsity,1 and dense (the exact 1(M1)ij is nonzero whenever basis elements Λ1 i and Λ1 j of Λ1 have nontrivial ‘overlap,’ in the sense of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (M1)2 ij is generally nonzero whenever Λ1 i and Λ1 j each has nontrivial overlap with a third basis finite element Λ1 k (where k ∈ {i, j} 9 Figure 3: The figures above depict the L2Λ1 log relative error in an FEEC approximation of ˙E = c2∇ × ∇ × A vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' log cell size h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The x-axis measures the number of cells (of size h) per wavelength λA = 2π/kn of the vector potential A = sin(kny)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The left plot uses the first order (Whitney form) P− 1 Λp(Th) FEEC basis, while the right plot uses a more accurate second order basis, P− 2 Λp(Th).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' For each basis, we plot the relative errors of four approximations of the FEEC curl-of-curl operator, ∇ × ∇× ≈ QiCT M2C, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' , 4}, and fit their power scalings against h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Here, Qi is an approximation of the inverse mass matrix M−1 1 with i indicating one of four sparsity patterns: diagonal, M1, (M1)2, and dense (which recovers the exact FEEC operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The left plot demonstrates that the accuracy of Yee’s method, (which effectively uses Whitney forms and diagonal mass matrices), is meaningfully improved upon by an approximation of M−1 1 that has the sparsity pattern of M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The right plot demonstrates that approximating M−1 1 by a matrix with the sparsity pattern of (M1)2 achieves much of the improved accuracy possible with second order finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' It is seen that the power scalings extend to a finite resolution, at which point relative error saturates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' inverse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The results of this investigation are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' With the setup described above, the two subplots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 3 display the log relative L2Λ1 errors in our approximations of ˙E = c2∇ × ∇ × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (For brevity, we use ˙E merely as a shorthand notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=') In partic- is also permitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In our context, therefore, (M1)2 is strictly denser than M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' By extension, the Neumann representation of M−1 1 demonstrates that S((M1)n) → S(M−1 1 ) as n approaches N1 [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' ular, given the L2Λ1 norm (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (4)) || ˙E||L2Λ1 = � |Th| ( ˙E, ˙E)1dx, (35) we compute ||ˆ˙E − ˙E||L2Λ1/|| ˙E||L2Λ1 (36) where ˆ˙E = c2QCT M2C denotes the relevant finite element approximation of the continuum 1-form ˙E = c2k2 n sin(kny)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 10 Pi-A→ (Whitney 1-forms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='2 0r 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='2 IL2△1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='5 二 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='6 log10 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='8 Diagonal (“"Yee")△α h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='56 Diagonal （“"Yee"）△ α h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='56 M1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='5 △ α h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='73 M1 △ α h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='19 1 (M1)2 △ α h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='81 (Mi)2 △ α h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='7 Dense (FEEC) △ α h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='82 Dense (FEEC) △ α h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='2 3 20 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='46 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='47 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='11 1 20 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='46 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='47 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='11 入A/h 入A/hThe left plot demonstrates that the FEEC curl- of-curl operator M−1 1 CT M2C is well approximated for a Whitney form basis using QS(M1)CT M2C (where QS(M1) denotes the approximation of M−1 1 with a spar- sity pattern of M1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In particular, the relative error of QS(M1)CT M2C scales with cell size as h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='73, while the exact FEEC operator M−1 1 CT M2C error scales com- parably as h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Remarkably, the first plot demonstrates that a substantial improvement in accuracy over a diago- nal sparsity pattern analogous to Yee’s method is achieved by using only a slightly less parsimonious estimate of M−1 1 in the curl-of-curl FEEC operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In particular, while QS(M1)CT M2C scales as h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='73 (as noted just above), the accuracy of QS(1)CT M2C (with QS(1) diagonal-patterned) scales as h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' More important than this modest improvement in power scaling, however, is its applicability to finer meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The first plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 3 shows that the power scaling of the diagonal mass matrix curl-of- curl approximation can only be assumed for meshes of relatively low resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The relative error in the QS(1) discrete differential operator is seen to saturate when the number of cells per signal wavelength climbs above 5—that is λA/h ≳ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' This is in contrast to the QS(M1) power scaling, which continues to hold even at the higher resolution of λA/h ∼ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' This could be a particular advantage in simulations of small-scale electromagnetic structure, such as in turbulent plas- mas or nanomaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' It must be further emphasized that the additional computational cost introduced by this less parsimo- nious QS(M1) approximation of M−1 1 will generally be fairly modest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The pairs of nodes that communicate in a simulation with an M1-sparsity approximation are typically the same as those that communicate with a diagonal-sparsity approximation, such that no ‘new channels’ of communication are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Each communication between nodes will pass more data, however—in our 2-D example, roughly 5 times as much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Nevertheless, the amount of data passed is small to begin with, as it scales only with the number of boundary cells of a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 3 demonstrates that, to achieve the accuracy of higher order finite elements, denser approximations of M−1 1 must be used in the curl-of-curl operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Nevertheless, greater accuracy is readily attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In our 2-D simplicial example, we see that much of the improved accuracy possi- ble with second order finite elements is achieved by using a curl-of-curl operator QS(M2 1)CT M2C, which ap- proximates M−1 1 with a sparsity pattern of (M1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In- deed, this approximate FEEC operator achieves an error scaling with cell size of h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='70, while the exact FEEC curl-of-curl operator error scales in our results as h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' It is worth noting, however, that the approx- imation’s scaling holds only until a mesh resolution of roughly λA/h ≳ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Nevertheless, due to the higher order finite element, this saturation occurs at a sig- nificantly higher accuracy overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Discussion We have demonstrated that Yee’s method can be interpreted as a structure-preserving splitting method using an FEEC formalism on a cubical mesh with simplified mass matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' In so doing, we have identified Yee’s algorithm to be a special case of a larger family of methods we call SFEEC (scal- able finite element exterior calculus) methods, sum- marized by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' (15) and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' These methods are structure-preserving, as Yee’s method is, and are therefore accurate in long-time numerical simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' They also respect the topological and geo- metric properties of Maxwell’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Crucially, SFEEC methods are also scalable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' they adopt the strategy implicit in Yee’s method of using sparse approximations of finite element mass matrices to achieve computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The classification of SFEEC methods enabled us to identify higher order extensions of Yee’s method that preserve its scalability nevertheless (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Depending on the computing architecture employed and the desired accuracy of the problem of interest, these alternative methods may enable greater long- time accuracy than Yee’s method provides, with lit- tle additional computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' The applicability of SFEEC methods on general meshes may also be of particular benefit in problems with irregular geome- tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 11 The saturation of the relative errors plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 3 suggest that it would be fruitful to investigate in future work whether improved sparse approxima- tions of the discrete codifferential operator, M−1 1 CT M2 can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Such efforts might be geared to pre- serve higher order finite element error scaling at ar- bitrarily high mesh resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Acknowledgments Thank you to Josh Burby and Tyrus Berry for helpful discussions, and to Phil Morrison for his sup- port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' This research was further supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Department of Energy (DE-AC02-09CH11466), as well as the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAzT4oBgHgl3EQfyf4o/content/2301.01753v1.pdf'} +page_content=' Department of Energy Fusion En- ergy Sciences Postdoctoral Research Program admin- istered by the Oak Ridge Institute for Science and Education (ORISE) for the DOE.' metadata={'source': 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Wozniak +maciejw@kth.se +KTH Royal Institute of Technology +Stockholm, Sweden +Rebecca Stower +stower@kth.se +KTH Royal Institute of Technology +Stockholm, Sweden +Patric Jensfelt +patric@kth.se +KTH Royal Institute of Technology +Stockholm, Sweden +Andre Pereira +atap@kth.se +KTH Royal Institute of Technology +Stockholm, Sweden +ABSTRACT +Many solutions tailored for intuitive visualization or teleoperation +of virtual, augmented and mixed (VAM) reality systems are not +robust to robot failures, such as the inability to detect and recognize +objects in the environment or planning unsafe trajectories. In this +paper, we present a novel virtual reality (VR) framework where +users can (i) recognize when the robot has failed to detect a real- +world object, (ii) correct the error in VR, (iii) modify proposed object +trajectories and, (iv) implement behaviors on a real-world robot. +Finally, we propose a user study aimed at testing the efficacy of our +framework. Project materials can be found in the OSF repository1. +ACM Reference Format: +Maciej K. Wozniak, Rebecca Stower, Patric Jensfelt, and Andre Pereira. 2023. +What you see is (not) what you get: A VR Framework for Correcting Robot +Errors. In ACM/IEEE International Conference on Human-Robot Interaction, +March 13–16, 2023, Stockholm, SE. ACM, New York, NY, USA, 5 pages. https: +//doi.org/10.1145/nnnnnnn.nnnnnnn +1 +INTRODUCTION +Robots designed for human-robot-interaction (HRI), such as cobots, +can increasingly be seen in homes [21], offices [5] or common +areas like restaurants or bars [3]. Nevertheless, such robots are not +perfect and can suffer from many potential failures. One of the +most common points of failure is in the robot perception module. +Although deep learning models used in the perception modules +can perform well on the datasets they are trained and evaluated on, +they often fail when deployed in the real world [6]. Simple failures, +such as not detecting an object, can have severe consequences. First +of all, if the robot fails a simple task, the user may get angry and +annoyed, resulting in lower trust and motivation to collaborate [9, +1https://osf.io/c8wap/?view_only=e1c799b564b043d0aeaac289513ebff0. The code will +be added after user studies are completed. +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +HRI ’23, March 13–16, 2023, Stockholm, SE +© 2023 Association for Computing Machinery. +ACM ISBN 978-1-4503-XXXX-X/18/06...$15.00 +https://doi.org/10.1145/nnnnnnn.nnnnnnn +Figure 1: Example of the framework with Franka Panda robotic arm. +From the left: (1) the VR user interface; (2) user interacting with the +robot through Oculus Quest 2 headset; (3) real-world environment. +13]. Secondly, a robot may collide or crash with objects if it fails to +detect them. +Another potential issue relates to how the robot plans and exe- +cutes the trajectories. Users may have different preferences about +how the robot should complete a specific task. A robot reaching +for an object in a collaborative task (e.g., joint assembly) may be +perceived as unsafe from the user’s perspective, even if the move- +ment is executed successfully from the robot point of view (i.e., +no collision occurs). Allowing users to see the robot’s actions in +advance, and potentially modify them, could then help improve the +overall user experience. +In our framework, we therefore consider two main sources of +failure that can occur in human-robot-interactions; failure to de- +tect an object (potentially causing a collision in the real world), or +failure to plan an acceptable trajectory for the user. Rather than +preventing these failures entirely, we aim to provide users with the +opportunity to directly correct the robot themselves when such +failures occur. We provide two main functions for (i) correcting the +robot’s understanding about its surroundings by modifying the vir- +tual environment created from its perception module (ii) modifying +the robot’s trajectory to adjust to the users’ preferences or to avoid +obstacles. In doing so, we contribute a virtual reality framework +for human-robot collaboration that considers that robots are not +perfect and makes it easy for the user to correct their mistakes. +arXiv:2301.04919v1 [cs.RO] 12 Jan 2023 + +eRealRobot +SendMeHome +Go To real robot pose +through2 +RELATED WORK +This section reviews recent projects focused on improving human- +robot collaboration using virtual/augmented reality (VR/AR) de- +vices. Teleoperation (remotely controlling the robot) is one of the +most common forms of human-robot interaction [17, 18, 22]. Os- +tatin [20] and Togias [24] both use VR to plan the robotic arm’s +trajectory. Others, like Chandan et al. [2] use AR to visualize the +states and intentions of the robots. Xu et al. [31] used VR to steer the +robot by moving its end effector in the VR space. Kennel-Maushart +et al. [8] and Barentine et al. [1] also focused on steering the robot, +however, the former did so by applying force in AR by pressing +virtual objects held by the virtual robot, making the real world robot +move. The latter uses VR for steering the robot from a first-person +view. Lastly, Wozniak and Jensfelt [29] presented a VR simulation +framework for interacting with the virtual representation of a robot +and modifying its trajectory. +Many of these projects focus only on the execution of planned +robot trajectories and do not consider the perception module (and +associated potential failures). In addition, the visualization and +manipulation of the robot’s movements are limited to either only +choosing the final position of the robot’s end-effector or adding no- +go zones. Finally, none of these projects follow the whole pipeline of +first visualizing, then correcting and testing the planned trajectory +in simulation, followed by deploying it to a real-world robot. +Our framework could also be classified as a digital twin project, +which is a virtual representation of a robot’s actions and abili- +ties [11]. Modern physics engines can imitate reality in great detail, +allowing a digital twin to be an accurate test-bed for real-world ap- +plications [14]. There are various applications and benefits of using +VR interfaces for digital twin projects, such as improved factory +safety or workers’ training [7, 26], collaborative tasks [19] or as- +sembly process [4]. However, these methods so far focus primarily +on personnel training for the manufacturing industry. +The projects described above are a significant contribution to +the VAM and HRI research communities. However, what is still +missing is an explicit understanding of the robot’s perception of +the working space and planned course of action. Our framework +addresses this challenge by allowing the user to (i) assess how the +robot perceives the environment, (ii) modify and correct the robot’s +actions and understanding of the surroundings, and (iii) verify the +proposed actions and trajectories before deploying them and testing +in the real world. +3 +TECHNICAL IMPLEMENTATION +Our proposed framework integrates several systems; the cameras +and associated perception module, the VR headset and user inter- +face, and the real-world robot. +3.1 +Camera views and perception module +The images detected by the main and secondary cameras are fed into +the deep learning model [30] for scene segmentation and object +detection to obtain segmentation masks and bounding boxes of +the detected objects2. Classes and bounding boxes of the detected +objects tell us what the object is and where it is located. Our system +2Any other object detection architectures could be used as well. +Sensor +data +Sensors +Motors +Deep Learning +module +Data +transfer +node +VR +environment +User +Robot +controller +Figure 2: Proposed system architecture. The elements within the or- +ange box correspond to nodes and parts connected with the robot’s +motion and perception, whereas the ones in the green box are ex- +plicitly corresponding to its hardware. Nodes inside the blue box +are connected to the user and VR setting. +then uses this information to correctly position and add objects to +the scene in VR. +The VR environment is then built from the output of the percep- +tion module. The user can only ask the robot to interact with objects +shown in the VR environment. In our setup, we use two cameras +to create and examine the environment: one fixed, pointing at the +table (main camera) and the other one on the robot’s end effector +(secondary camera). In the VR environment, the user can switch +between the VR environment view and the two distinct camera +views to verify whether the robot has successfully detected all the +objects in the scene. +In case an object is occluded, the robot’s arm with the secondary +camera can be moved (since from another angle the object may +be more clearly visible) to detect it and add it to the environment. +Alternatively, if the object is still not detectable or the perception +module itself fails, the user can add the missing object to the VR +environment by bringing up a menu and adding the object to the +scene (as shown in the supplementary video1). +3.2 +Robot and data transfer +We have so far tested our framework with the Franka Panda and +Niryo One robotic arms; however our framework can accommodate +most robotic arms, provided URDF files of that robotic arm. We use +the MoveIt library3 for planning and Robotic Operating System +(ROS Melodic)4 for transferring data and running different tasks +on separate nodes. +Since the user interface (UI) of our framework is in the VR head- +set, other operations, such as planning and object detection, must be +run on separate machines. This creates the challenge of efficiently +transferring the data between different nodes of the system and the +VR headset itself. +In order to facilitate the exchange of information, improve the +user experience, and minimize the necessary bandwidth, we only +transfer the output of a detection network between the robot and +3https://moveit.ros.org/ +4https://www.ros.org/ + +the virtual environment. When the robot detects and classifies an +object, it sends its class, location, and estimated size to the VR UI +application. In the virtual environment, we have multiple prefabs +(3D models of the objects we built into the project) corresponding +to the detected classes, from which we can quickly create a 3D +representation of the environment when the message from the +perception module (containing detected object classes and poses) +is received. +3.3 +VR Interface +The VR interface is built in Unity (2020.3 version). We use the +Oculus Quest 2 headset; however, our framework can be adapted +to other headsets by changing the target device while building the +Unity project. +In our UI design (shown in Fig. 3d and the supplementary video1) +the user can bring up two different menus with primary and sec- +ondary buttons on the left controller and choose an option by point- +ing at the specific button with the right-hand controller. The first +menu has options for creating and modifying the environment. The +second menu contains commands for the robot (such as selecting +the object or sending trajectories to the real robot). +4 +INTERACTION SCENARIO +In this section we show an example scenario of our framework +with a 7 DoF Franka Panda robotic arm. While we use a real-world +robot, we want to ensure that the framework is available to labs +without access to a physical robot. In such a case, communication +with a real-world robot can be simulated by running the same Unity +project on another machine1. +There are three main actions that users take to arrive at real- +world object manipulation with the robot. First, the VR environment +depicting both the robot and the objects detected and recognized +by the perception module is created (and corrected by the users, +if necessary). Next, the users ask the virtual robot to perform an +action in VR. If the users are satisfied with how the virtual robot +acts, they can send the trajectory to the real–world robot. If not, +they can modify the trajectory beforehand. All the steps described +in this section can be found in the video in attached materials1. +4.1 +Common environment understanding +The VR environment corresponds to what the robot understands +about the real-world environment. Objects detected by the robot’s +perception module appear in the UI. Users can examine whether +the objects correspond to what is in the real-world environment in +two ways. First, as shown in Fig. 3c, they can activate passthrough +where they will see the main camera image projected on the table +where the objects are located. Secondly, they can go to camera +view, where they can see and compare the VR environment with +the output from both cameras’ deep learning modules as shown in +Fig. 3a and Fig. 3b. +However, the perception module is imperfect and often partially +fails, as described in Section 3.1. There are two ways to recover +from these failures. The users can move the robotic arm around +and try to detect the objects using the camera on the robot’s arm. +This can help with, e.g., partially occluded objects or objects that +cannot be recognized from the main camera perspective (Fig. 3a vs. +(a) Main camera view +(b) Secondary camera view +(c) Passthrough view +(d) Adding missing objects to the +environment +Figure 3: Different ways to see the real world view in the VR. Users +can verify if the environment was correctly created and correct po- +tential flaws of the robot’s perception module by comparing these +views with the state of the VR environment. +Fig. 3b). Another way is to insert virtual objects into the environ- +ment. Users can choose the object from the UI and grab and place +it in the VR environment where it belongs, as shown in Fig. 3d. The +passthrough view on the table allows the user to see where the ob- +jects are in the real world and place a matching virtual object in the +correct position. Objects in the environment are then considered +for collision avoidance while planning the trajectory. +4.2 +Trajectory manipulation +When the users are satisfied that the VR environment’s current +state is an accurate representation of reality, they can proceed to +ask the virtual robot to move the objects. Whilst the virtual robot +is executing a movement, it generates a trajectory and waypoints +in real-time in the VR environment. After the robot finishes its +movement, the user then has the option to either send the action +to the real robot (by clicking a button), or bring the virtual robot +and the selected object back to their initial positions and edit the +trajectory. While the planner considers objects in the scene and +plans the trajectory so that the robotic arm will not collide with +them (except for the object we want to pick up), this might not +always align with user’s preferences. +To accommodate this, we first show the user the planned trajec- +tory containing multiple waypoints in VR. If the user is not satisfied +with the proposed trajectory, they can edit it by moving the way- +points. Now, the user can ask the robot to perform the task again, +following the modified trajectory (see video for demonstration1). +This additional functionality could be used to e.g., help home robots +learn users’ preferences. +4.3 +Real robot +When the users are satisfied with the state of the environment +and the way the robot performs the task, they can translate the + +cube79% +banana71%apple54% +Come +SwitchCame +banana99%Insert +Object +Orange +Cube +Bottle +Banana +Applamovement to the real robot. After the real robot finishes the task, +they can ask it to start another interaction in the same manner as +before: by planning and verifying it first in VR and then sending it +to the real robot. +5 +PROPOSED USER STUDY +In order to test the efficacy of our framework, we plan to conduct a +user study comparing our VR system with a screen-based interface +using a keyboard and mouse (hereon referred to as screen interface). +In this study, we will focus only on failures within the perception +module, rather than editing robot trajectories. Our goal is to in- +vestigate whether VR systems are more intuitive than screen-based +systems for perceiving and correcting robot perceptual errors. +As our independent variable, we will manipulate the type of +interface participants use to interact with the robot (VR, screen). +We chose the screen interface as our point of comparison because +(i) screen interfaces are often used to teleoperate robots [12, 23, 28], +and (ii) there are very few direct VR - screen comparisons in the +HRI literature to date [10, 16]. We plan to use a within-groups study +design, where all participants will be exposed to both conditions. +Whether participants start with the VR system first or screen first +will be randomly assigned. Before beginning the study, participants +will be given a pre-questionnaire assessing their familiarity with +robots and VAM reality systems. +Participants will first be given a training phase where they can +familiarise themselves with the system setup and controls. In par- +ticular, participants will be shown the two different cameras (main +and secondary) with which they can view the real-world scene, +as well as the passthrough option. Participants will then have 5 +experimental trials where they interact with the robot. Each trial +will consist of the robot failing to recognize an object. That is, the +user can see that the object exists in the real world (via the external +cameras/passthrough), but it is not represented in the virtual world +(either in VR for the VR condition, or on the screen in the screen +condition). The user can then correct this error in one of two differ- +ent ways. First, they can move the secondary camera in an effort to +detect the missing object. Alternatively, they can manually add the +object to the environment by selecting the missing object from an +array of different possible objects and placing it (on the screen / in +VR) as close to the real-life position of the object as possible. Once +the user is satisfied with their object positioning, they can submit +their feedback to the virtual robot, view the planned trajectory, +and finally, execute the movement in the real world (as described +in section 4.2) before moving on to the next trial. Although the +overall framework also allows users to view and edit the planned +trajectory, for this study we will disable this function, as we aim to +focus first on how users correct the robot when an object detection +failure occurs. +As dependent variables, we will measure how often users choose +each correction method (moving the secondary camera, or adding +virtual objects), the time taken per trial, and the user’s subjective +perceptions of each interface. We will also measure the accuracy of +the object placement by looking at where users choose to manually +add the object to the environment. For the subjective perceptions, +we will use the Technology Acceptance Model (TAM) [25] and the +performance trust subscale from the Multi-Dimensional Measure of +Trust (MDMT) scale [15]. Participants will complete the question- +naires directly after interacting with each system. After participants +have seen both conditions, we will also ask a binary forced-choice +question about which system they preferred to use. Finally, we +will interview participants about the perceived strengths and weak- +nesses of each system and to identify areas of improvement. +We aim to recruit 50 participants, based on an a-priori power +analysis with 𝛼 = 0.05, 𝛽 = 0.8 and 𝑑 = 0.4. All hypotheses +and planned analyses will be pre-registered on the Open Science +Framework1 prior to data collection. +5.1 +Hypotheses +Based on previous studies which show that VR systems lead to +better task performance [12] and have higher perceived usability +[27] than 2D interactions, we formulate the following preliminary +hypotheses: +H1 The VR interface will be preferred over the screen interface +for interacting with the robot +H2 Participants will have better task performance with the VR +system in terms of: +2a More accurate object placement when adding virtual ob- +jects to the environment +2b Shorter average time taken per trial +H3 Subjective ratings on the self–report questionnaires (technol- +ogy acceptance and trust) will be higher for the VR system +compared to the screen interface. +6 +CONCLUSIONS AND FUTURE WORK +This paper presents a virtual reality human-robot collaboration +framework for object detection and interaction accounting for ro- +bot perceptual failures. Although we are interested in using our +framework to enhance human robot collaboration, it could also be +used for other tasks. +As briefly mentioned in Section 4.2, trajectory manipulation +can be adapted to the preference learning task. While future home +robots might be designed and programmed to perform many differ- +ent tasks, the users’ satisfaction will vary, even if the robot always +completes its tasks. The satisfaction may differ because one group +of users can have different preferences about how to do certain +things. It can also be connected with how comfortable particular +people feel being close to the moving robot. A similar tool to what +we present can be used in the transition when a user buys a new +robot and wants to collect data on how users correct a robot’s tra- +jectory so that the system learns to perform tasks in the way that +this particular user likes. +The framework can also be used to collect data and improve +deep learning models responsible for robot perception. When the +users realize a specific object is not detected, they could, e.g., draw a +bounding box around that object or even just place it in the correct +place in the VR, triggering bounding box generation. Then, such +images could be collected together, and when enough of them are +collected, the model could be retrained to perform better on the +tasks it failed before. Such a solution can help the users improve the +robot’s functionality, even if they do not have any programming +experience. + +In sum, in this paper we presented a technical implementation +and interaction scenario of a VR framework for correcting robot +perception and planning errors, as well as proposed a future user +study. Future work will continue to develop and test this framework +further. +REFERENCES +[1] Christian Barentine, Andrew McNay, Ryan Pfaffenbichler, Addyson Smith, Eric +Rosen, and Elizabeth Phillips. A vr teleoperation suite with manipulation assist. +In Companion of the 2021 ACM/IEEE International Conference on Human-Robot +Interaction, HRI ’21 Companion, page 442–446, New York, NY, USA, 2021. Associ- +ation for Computing Machinery. +[2] Kishan Chandan, Vidisha Kudalkar, Xiang Li, and Shiqi Zhang. Arroch: Aug- +mented reality for robots collaborating with a human. In 2021 IEEE International +Conference on Robotics and Automation (ICRA), pages 3787–3793. 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Shared-control robotic manipulation +in virtual reality. arXiv preprint arXiv:2205.10564, 2022. + diff --git a/KdE4T4oBgHgl3EQfJQxU/content/tmp_files/load_file.txt b/KdE4T4oBgHgl3EQfJQxU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4baf32a4fbc908ad51d58cd87facbe82f66a60f4 --- /dev/null +++ b/KdE4T4oBgHgl3EQfJQxU/content/tmp_files/load_file.txt @@ -0,0 +1,335 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf,len=334 +page_content='What you see is (not) what you get: A VR Framework for Correcting Robot Errors Maciej K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Wozniak maciejw@kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='se KTH Royal Institute of Technology Stockholm, Sweden Rebecca Stower stower@kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='se KTH Royal Institute of Technology Stockholm, Sweden Patric Jensfelt patric@kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='se KTH Royal Institute of Technology Stockholm, Sweden Andre Pereira atap@kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='se KTH Royal Institute of Technology Stockholm, Sweden ABSTRACT Many solutions tailored for intuitive visualization or teleoperation of virtual, augmented and mixed (VAM) reality systems are not robust to robot failures, such as the inability to detect and recognize objects in the environment or planning unsafe trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In this paper, we present a novel virtual reality (VR) framework where users can (i) recognize when the robot has failed to detect a real- world object, (ii) correct the error in VR, (iii) modify proposed object trajectories and, (iv) implement behaviors on a real-world robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Finally, we propose a user study aimed at testing the efficacy of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Project materials can be found in the OSF repository1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' ACM Reference Format: Maciej K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Wozniak, Rebecca Stower, Patric Jensfelt, and Andre Pereira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' What you see is (not) what you get: A VR Framework for Correcting Robot Errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In ACM/IEEE International Conference on Human-Robot Interaction, March 13–16, 2023, Stockholm, SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' ACM, New York, NY, USA, 5 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='nnnnnnn 1 INTRODUCTION Robots designed for human-robot-interaction (HRI), such as cobots, can increasingly be seen in homes [21], offices [5] or common areas like restaurants or bars [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Nevertheless, such robots are not perfect and can suffer from many potential failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' One of the most common points of failure is in the robot perception module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Although deep learning models used in the perception modules can perform well on the datasets they are trained and evaluated on, they often fail when deployed in the real world [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Simple failures, such as not detecting an object, can have severe consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' First of all, if the robot fails a simple task, the user may get angry and annoyed, resulting in lower trust and motivation to collaborate [9, 1https://osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='io/c8wap/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='view_only=e1c799b564b043d0aeaac289513ebff0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' The code will be added after user studies are completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' HRI ’23, March 13–16, 2023, Stockholm, SE © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' ACM ISBN 978-1-4503-XXXX-X/18/06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='nnnnnnn Figure 1: Example of the framework with Franka Panda robotic arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' From the left: (1) the VR user interface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' (2) user interacting with the robot through Oculus Quest 2 headset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' (3) real-world environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Secondly, a robot may collide or crash with objects if it fails to detect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Another potential issue relates to how the robot plans and exe- cutes the trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Users may have different preferences about how the robot should complete a specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' A robot reaching for an object in a collaborative task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=', joint assembly) may be perceived as unsafe from the user’s perspective, even if the move- ment is executed successfully from the robot point of view (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=', no collision occurs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Allowing users to see the robot’s actions in advance, and potentially modify them, could then help improve the overall user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In our framework, we therefore consider two main sources of failure that can occur in human-robot-interactions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' failure to de- tect an object (potentially causing a collision in the real world), or failure to plan an acceptable trajectory for the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Rather than preventing these failures entirely, we aim to provide users with the opportunity to directly correct the robot themselves when such failures occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' We provide two main functions for (i) correcting the robot’s understanding about its surroundings by modifying the vir- tual environment created from its perception module (ii) modifying the robot’s trajectory to adjust to the users’ preferences or to avoid obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In doing so, we contribute a virtual reality framework for human-robot collaboration that considers that robots are not perfect and makes it easy for the user to correct their mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='04919v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='RO] 12 Jan 2023 eRealRobot SendMeHome Go To real robot pose through2 RELATED WORK This section reviews recent projects focused on improving human- robot collaboration using virtual/augmented reality (VR/AR) de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Teleoperation (remotely controlling the robot) is one of the most common forms of human-robot interaction [17, 18, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Os- tatin [20] and Togias [24] both use VR to plan the robotic arm’s trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Others, like Chandan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' [2] use AR to visualize the states and intentions of the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' [31] used VR to steer the robot by moving its end effector in the VR space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Kennel-Maushart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' [8] and Barentine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' [1] also focused on steering the robot, however, the former did so by applying force in AR by pressing virtual objects held by the virtual robot, making the real world robot move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' The latter uses VR for steering the robot from a first-person view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Lastly, Wozniak and Jensfelt [29] presented a VR simulation framework for interacting with the virtual representation of a robot and modifying its trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Many of these projects focus only on the execution of planned robot trajectories and do not consider the perception module (and associated potential failures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In addition, the visualization and manipulation of the robot’s movements are limited to either only choosing the final position of the robot’s end-effector or adding no- go zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Finally, none of these projects follow the whole pipeline of first visualizing, then correcting and testing the planned trajectory in simulation, followed by deploying it to a real-world robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Our framework could also be classified as a digital twin project, which is a virtual representation of a robot’s actions and abili- ties [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Modern physics engines can imitate reality in great detail, allowing a digital twin to be an accurate test-bed for real-world ap- plications [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' There are various applications and benefits of using VR interfaces for digital twin projects, such as improved factory safety or workers’ training [7, 26], collaborative tasks [19] or as- sembly process [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' However, these methods so far focus primarily on personnel training for the manufacturing industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' The projects described above are a significant contribution to the VAM and HRI research communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' However, what is still missing is an explicit understanding of the robot’s perception of the working space and planned course of action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Our framework addresses this challenge by allowing the user to (i) assess how the robot perceives the environment, (ii) modify and correct the robot’s actions and understanding of the surroundings, and (iii) verify the proposed actions and trajectories before deploying them and testing in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 3 TECHNICAL IMPLEMENTATION Our proposed framework integrates several systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' the cameras and associated perception module, the VR headset and user inter- face, and the real-world robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='1 Camera views and perception module The images detected by the main and secondary cameras are fed into the deep learning model [30] for scene segmentation and object detection to obtain segmentation masks and bounding boxes of the detected objects2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Classes and bounding boxes of the detected objects tell us what the object is and where it is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Our system 2Any other object detection architectures could be used as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Sensor data Sensors Motors Deep Learning module Data transfer node VR environment User Robot controller Figure 2: Proposed system architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' The elements within the or- ange box correspond to nodes and parts connected with the robot’s motion and perception, whereas the ones in the green box are ex- plicitly corresponding to its hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Nodes inside the blue box are connected to the user and VR setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' then uses this information to correctly position and add objects to the scene in VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' The VR environment is then built from the output of the percep- tion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' The user can only ask the robot to interact with objects shown in the VR environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In our setup, we use two cameras to create and examine the environment: one fixed, pointing at the table (main camera) and the other one on the robot’s end effector (secondary camera).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In the VR environment, the user can switch between the VR environment view and the two distinct camera views to verify whether the robot has successfully detected all the objects in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In case an object is occluded, the robot’s arm with the secondary camera can be moved (since from another angle the object may be more clearly visible) to detect it and add it to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Alternatively, if the object is still not detectable or the perception module itself fails, the user can add the missing object to the VR environment by bringing up a menu and adding the object to the scene (as shown in the supplementary video1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='2 Robot and data transfer We have so far tested our framework with the Franka Panda and Niryo One robotic arms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' however our framework can accommodate most robotic arms, provided URDF files of that robotic arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' We use the MoveIt library3 for planning and Robotic Operating System (ROS Melodic)4 for transferring data and running different tasks on separate nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Since the user interface (UI) of our framework is in the VR head- set, other operations, such as planning and object detection, must be run on separate machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' This creates the challenge of efficiently transferring the data between different nodes of the system and the VR headset itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In order to facilitate the exchange of information, improve the user experience, and minimize the necessary bandwidth, we only transfer the output of a detection network between the robot and 3https://moveit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='ros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='org/ 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='ros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='org/ the virtual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' When the robot detects and classifies an object, it sends its class, location, and estimated size to the VR UI application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In the virtual environment, we have multiple prefabs (3D models of the objects we built into the project) corresponding to the detected classes, from which we can quickly create a 3D representation of the environment when the message from the perception module (containing detected object classes and poses) is received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='3 VR Interface The VR interface is built in Unity (2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='3 version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' We use the Oculus Quest 2 headset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' however, our framework can be adapted to other headsets by changing the target device while building the Unity project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In our UI design (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 3d and the supplementary video1) the user can bring up two different menus with primary and sec- ondary buttons on the left controller and choose an option by point- ing at the specific button with the right-hand controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' The first menu has options for creating and modifying the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' The second menu contains commands for the robot (such as selecting the object or sending trajectories to the real robot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 4 INTERACTION SCENARIO In this section we show an example scenario of our framework with a 7 DoF Franka Panda robotic arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' While we use a real-world robot, we want to ensure that the framework is available to labs without access to a physical robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In such a case, communication with a real-world robot can be simulated by running the same Unity project on another machine1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' There are three main actions that users take to arrive at real- world object manipulation with the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' First, the VR environment depicting both the robot and the objects detected and recognized by the perception module is created (and corrected by the users, if necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Next, the users ask the virtual robot to perform an action in VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' If the users are satisfied with how the virtual robot acts, they can send the trajectory to the real–world robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' If not, they can modify the trajectory beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' All the steps described in this section can be found in the video in attached materials1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='1 Common environment understanding The VR environment corresponds to what the robot understands about the real-world environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Objects detected by the robot’s perception module appear in the UI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Users can examine whether the objects correspond to what is in the real-world environment in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' First, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 3c, they can activate passthrough where they will see the main camera image projected on the table where the objects are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Secondly, they can go to camera view, where they can see and compare the VR environment with the output from both cameras’ deep learning modules as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 3a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' However, the perception module is imperfect and often partially fails, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' There are two ways to recover from these failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' The users can move the robotic arm around and try to detect the objects using the camera on the robot’s arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' This can help with, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=', partially occluded objects or objects that cannot be recognized from the main camera perspective (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 3a vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' (a) Main camera view (b) Secondary camera view (c) Passthrough view (d) Adding missing objects to the environment Figure 3: Different ways to see the real world view in the VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Users can verify if the environment was correctly created and correct po- tential flaws of the robot’s perception module by comparing these views with the state of the VR environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Another way is to insert virtual objects into the environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Users can choose the object from the UI and grab and place it in the VR environment where it belongs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' The passthrough view on the table allows the user to see where the ob- jects are in the real world and place a matching virtual object in the correct position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Objects in the environment are then considered for collision avoidance while planning the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='2 Trajectory manipulation When the users are satisfied that the VR environment’s current state is an accurate representation of reality, they can proceed to ask the virtual robot to move the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Whilst the virtual robot is executing a movement, it generates a trajectory and waypoints in real-time in the VR environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' After the robot finishes its movement, the user then has the option to either send the action to the real robot (by clicking a button), or bring the virtual robot and the selected object back to their initial positions and edit the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' While the planner considers objects in the scene and plans the trajectory so that the robotic arm will not collide with them (except for the object we want to pick up), this might not always align with user’s preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' To accommodate this, we first show the user the planned trajec- tory containing multiple waypoints in VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' If the user is not satisfied with the proposed trajectory, they can edit it by moving the way- points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Now, the user can ask the robot to perform the task again, following the modified trajectory (see video for demonstration1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' This additional functionality could be used to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=', help home robots learn users’ preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='3 Real robot When the users are satisfied with the state of the environment and the way the robot performs the task, they can translate the cube79% banana71%apple54% Come SwitchCame banana99%Insert Object Orange Cube Bottle Banana Applamovement to the real robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' After the real robot finishes the task, they can ask it to start another interaction in the same manner as before: by planning and verifying it first in VR and then sending it to the real robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 5 PROPOSED USER STUDY In order to test the efficacy of our framework, we plan to conduct a user study comparing our VR system with a screen-based interface using a keyboard and mouse (hereon referred to as screen interface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In this study, we will focus only on failures within the perception module, rather than editing robot trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Our goal is to in- vestigate whether VR systems are more intuitive than screen-based systems for perceiving and correcting robot perceptual errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' As our independent variable, we will manipulate the type of interface participants use to interact with the robot (VR, screen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' We chose the screen interface as our point of comparison because (i) screen interfaces are often used to teleoperate robots [12, 23, 28], and (ii) there are very few direct VR - screen comparisons in the HRI literature to date [10, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' We plan to use a within-groups study design, where all participants will be exposed to both conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Whether participants start with the VR system first or screen first will be randomly assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Before beginning the study, participants will be given a pre-questionnaire assessing their familiarity with robots and VAM reality systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Participants will first be given a training phase where they can familiarise themselves with the system setup and controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In par- ticular, participants will be shown the two different cameras (main and secondary) with which they can view the real-world scene, as well as the passthrough option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Participants will then have 5 experimental trials where they interact with the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Each trial will consist of the robot failing to recognize an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' That is, the user can see that the object exists in the real world (via the external cameras/passthrough), but it is not represented in the virtual world (either in VR for the VR condition, or on the screen in the screen condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' The user can then correct this error in one of two differ- ent ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' First, they can move the secondary camera in an effort to detect the missing object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Alternatively, they can manually add the object to the environment by selecting the missing object from an array of different possible objects and placing it (on the screen / in VR) as close to the real-life position of the object as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Once the user is satisfied with their object positioning, they can submit their feedback to the virtual robot, view the planned trajectory, and finally, execute the movement in the real world (as described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='2) before moving on to the next trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Although the overall framework also allows users to view and edit the planned trajectory, for this study we will disable this function, as we aim to focus first on how users correct the robot when an object detection failure occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' As dependent variables, we will measure how often users choose each correction method (moving the secondary camera, or adding virtual objects), the time taken per trial, and the user’s subjective perceptions of each interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' We will also measure the accuracy of the object placement by looking at where users choose to manually add the object to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' For the subjective perceptions, we will use the Technology Acceptance Model (TAM) [25] and the performance trust subscale from the Multi-Dimensional Measure of Trust (MDMT) scale [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Participants will complete the question- naires directly after interacting with each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' After participants have seen both conditions, we will also ask a binary forced-choice question about which system they preferred to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Finally, we will interview participants about the perceived strengths and weak- nesses of each system and to identify areas of improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' We aim to recruit 50 participants, based on an a-priori power analysis with 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='05, 𝛽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='8 and 𝑑 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' All hypotheses and planned analyses will be pre-registered on the Open Science Framework1 prior to data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='1 Hypotheses Based on previous studies which show that VR systems lead to better task performance [12] and have higher perceived usability [27] than 2D interactions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' we formulate the following preliminary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='hypotheses: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='H1 The VR interface will be preferred over the screen interface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='for interacting with the robot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='H2 Participants will have better task performance with the VR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='system in terms of: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='2a More accurate object placement when adding virtual ob- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='jects to the environment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='2b Shorter average time taken per trial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='H3 Subjective ratings on the self–report questionnaires (technol- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='ogy acceptance and trust) will be higher for the VR system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='compared to the screen interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' 6 CONCLUSIONS AND FUTURE WORK This paper presents a virtual reality human-robot collaboration framework for object detection and interaction accounting for ro- bot perceptual failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Although we are interested in using our framework to enhance human robot collaboration, it could also be used for other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' As briefly mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='2, trajectory manipulation can be adapted to the preference learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' While future home robots might be designed and programmed to perform many differ- ent tasks, the users’ satisfaction will vary, even if the robot always completes its tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' The satisfaction may differ because one group of users can have different preferences about how to do certain things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' It can also be connected with how comfortable particular people feel being close to the moving robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' A similar tool to what we present can be used in the transition when a user buys a new robot and wants to collect data on how users correct a robot’s tra- jectory so that the system learns to perform tasks in the way that this particular user likes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' The framework can also be used to collect data and improve deep learning models responsible for robot perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' When the users realize a specific object is not detected, they could, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=', draw a bounding box around that object or even just place it in the correct place in the VR, triggering bounding box generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Then, such images could be collected together, and when enough of them are collected, the model could be retrained to perform better on the tasks it failed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Such a solution can help the users improve the robot’s functionality, even if they do not have any programming experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In sum, in this paper we presented a technical implementation and interaction scenario of a VR framework for correcting robot perception and planning errors, as well as proposed a future user study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' Future work will continue to develop and test this framework further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' REFERENCES [1] Christian Barentine, Andrew McNay, Ryan Pfaffenbichler, Addyson Smith, Eric Rosen, and Elizabeth Phillips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' A vr teleoperation suite with manipulation assist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE4T4oBgHgl3EQfJQxU/content/2301.04919v1.pdf'} +page_content=' In Companion of the 2021 ACM/IEEE International Conference on Human-Robot Interaction, 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100644 index 0000000000000000000000000000000000000000..8562b17a733b91ccd63108fb2dcc3556e84d9494 --- /dev/null +++ b/NdFAT4oBgHgl3EQfyB5j/content/tmp_files/2301.08690v1.pdf.txt @@ -0,0 +1,1597 @@ +SHAPE OPTIMISATION IN THE W 1,∞ TOPOLOGY WITH THE +ADMM ALGORITHM∗ +KLAUS DECKELNICK†, PHILIP J. HERBERT‡, AND MICHAEL HINZE§ +Abstract. We present a general shape optimisation framework based on the method of mappings +in the W 1,∞ topology. We propose steepest descent and Newton-like minimisation algorithms for the +numerical solution of the respective shape optimisation problems. Our work is built upon previous +work of the authors in Deckelnick, Herbert, and Hinze, ESAIM: COCV 28 (2022), where a W 1,∞ +framework for star-shaped domains is proposed. To illustrate our approach we present a selection +of PDE constrained shape optimisation problems and compare our findings to results from so far +classical Hilbert space methods and recent p-approximations. +Key words. PDE constrained shape optimisation, Lipschitz functions, W 1,∞-descent +MSC codes. 35Q93, 49Q10, 49J20 +1. Introduction. In this work, we are interested in the numerical solutions of a +number of shape optimisation problems +(1.1) +min J (Ω), Ω ∈ S, +where S is a collection of admissible domains. +This collection and the functional +J will vary depending on the application. To find, at least local, minima of this +problem, we will consider a descent method. By this, we mean that, given Ω ∈ S, +we seek V ∗ : Rn → Rn such that J ′(Ω)[V ∗] < 0 and set Ωnew = (id + αV ∗)(Ω) for +some suitably chosen α > 0. To ensure that the map id +αV ∗ is a homeomorphism, +it is sufficient to restrict to α to be small enough that α|DV ∗| < 1 a.e., where, | · | +is pointwise the spectral (operator) norm. +While it is sufficient to take any sub- +multiplicative norm, the spectral norm is convenient as it relates the Lipschitz and +W 1,∞ semi-norms. +In the literature, it is common to seek V ∗ in a Hilbert space H which represents +the negative gradient i.e. +(1.2) +(V ∗, η)H = (−∇HJ (Ω), η)H := −J ′(Ω)[η] +for all η ∈ H, or equivalently, one might seek +(1.3) +V ∗ ∈ arg min +�1 +2∥V ∥2 +H + J ′(Ω)[V ] : V ∈ H +� +. +A crucial issue in this context is the regularity of the solution V ∗ to problem (1.3), +which strongly depends on the regularity of the current domain Ω as well as the choice +of H. It for example is not clear whether id +αV ∗ defines a Lipschitz transformation +for many frequent choices of H. In order to avoid these issues it was suggested in +∗Submitted to the editors January 23, 2023. +Funding: +This work is part of the project P8 of the German Research Foundation Priority +Programme 1962, whose support is gratefully acknowledged by the second and the third author. +†Otto-von-Guericke-University Magdeburg, Institut f¨ur Analysis und Numerik, Universit¨atsplatz +2, 39106 Magdeburg (klaus.deckelnick@ovgu.de) +‡Maxwell Institute for Mathematical Sciences, Department of Mathematics, Heriot-Watt Univer- +sity, Edinburgh EH14 4AS, UK (p.herbert@hw.ac.uk) +§Mathematical Institute, University of Koblenz, Universit¨atsstr. 1, D-56070 Koblenz (hinze@uni- +koblenz.de) +1 +arXiv:2301.08690v1 [math.OC] 20 Jan 2023 + +2 +K. DECKELNICK, P.J. HERBERT, AND M. HINZE +[DHH22] to work directly in the space W 1,∞(Ω, Rd) and to consider the following +problem +(1.4) +V ∗ ∈ arg min +� +J ′(Ω)[V ] : V ∈ W 1,∞(Ω, Rd), |DV | ≤ 1 a.e. in Ω +� +. +In [DHH22] this idea was analyzed and implemented for shape optimization problems +involving star-shaped domains. It is the purpose of this paper to extend this approach +to more general domains including the use of Newton–type methods. +Literature. There continues to be rapid development in the mathematical and +numerical analysis of shape optimisation. +The seminal works of Delfour and +Zol´esio [DZ11], Sokolowski and Zol´esio, and the recent overview by Allaire, +Dapogny, and Jouve [ADJ21] and the comprehensive bibliographies within provide +an extensive overview of the topic of shape optimisation. The analysis, both mathe- +matical and numerical, of shape optimisation problems has an extensive history, see +e.g. [Bel+97; GM94; MS76; Sim80]. +While computational power has increased in +recent years, it has encouraged further development of shape optimisation [SSW15; +SSW16a; SW17], particularly fluid dynamical applications [Ben+15; Fis+17; Gar+15; +Gar+18; RCP16; HUU20; HSU21; K¨uh+19; Sch+13]. Many articles have considered +different choices of inner products on Hilbert spaces. A variety of choices are presented +in [HPS15]. One particularly interesting example is [ISW18] which uses a penalty to +weakly enforce the Cauchy-Riemann equations however it only appears applicable +in two dimensions. Another category of interesting choices are reproducing kernel +Hilbert spaces [ES18], which for certain kernels, one may provide an explicit shape +gradient. While in a Hilbertian setting, the work [OS21] considers non-smooth terms +to ensure that a mesh does not become degenerate. Some methods very much tar- +get having a particularly good mesh, a particular example is the so-called pre-shape +calculus [LS21b; LS21a]. +The utilisation of Banach spaces for shape optimisation is gathering attention. +To the best of our knowledge this was introduced in [DHH22] and considered W 1,∞ +perturbations for a star-shaped setting. The direction of steepest descent in a star- +shaped setting has been linked to optimal transport [Her23]. The star-shaped setting +is frequently exploited [EHS07; BCS21] to allow for a deeper analysis at the expense of +generality. A p-approximation to the infinity problem (1.4) is utilised in [M¨ul+21] to +optimise a fluid dynamic problem using a p-Laplace relaxation. Such a fluid problem +is frequently discussed in shape optimisation as it is known [Pir74] that, for Stokes +flow, the optimal shape should have a tip. In [M¨ul+21], experiments demonstrate that +the p-method will form a tip as opposed to in more classical Hilbertian methods. The +article [M¨ul+22] develops upon [M¨ul+21] to consider the computational scalability of +a method closely related to a p-Laplace relaxation of (1.4). +Higher order methods are also of interest and will be considered in this work; +second order methods have been considered in [SS22], utilising a linear version of the +second shape derivative. +Outline. We begin in Section 2 by outlining some necessary definitions and +results for shape optimisation, mentioning the Lagrange approach from optimisation +to write down first and second derivatives and providing examples which we will +consider. +We then move onto a discussion about the discretisation of the infinity +method in Section 3. Section 4 then provides numerical experiments of the previously +described numerical experiments using the novel W 1,∞ method we discuss. +2. Shape derivatives and Lagrangian calculus. + +APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION +3 +2.1. Preliminaries. In what follows we denote by D ⊂ Rd a convex hold-all +domain. We consider the shape optimisation problem +(2.1) +min J (Ω), Ω ∈ S, +where S is a collection of admissible domains such that Ω ⋐ D for all Ω ∈ S. Here, +we use the symbol ⋐ to denote compactly contained. It is not difficult to see that +id +V is a bi-Lipschitz transformation from D to D provided that V ∈ W 1,∞ +0 +(D, Rd) +with ∥DV ∥L∞ < 1. Assuming that (id +V )(Ω) ∈ S for such V we say that J is shape +differentiable at Ω if (cf. [ADJ21, Definition 4.1]) V �→ J +� +(id +V )(Ω) +� +is Fr´echet– +differentiable at V = 0 as a mapping from W 1,∞ +0 +(D, Rd) into R. An update step in +a descent algorithm based on the Fr´echet derivative of J will then seek a direction +V ∈ W 1,∞ +0 +(D, Rd) such that J ′(Ω)[V ] < 0. In order to determine the direction of +steepest descent we are led to the problem of finding V ∗ ∈ W 1,∞ +0 +(D, Rd) with +(2.2) +V ∗ ∈ arg min +� +J ′(Ω)[V ] : V ∈ W 1,∞ +0 +(D, Rd), |DV | ≤ 1 a.e. in D +� +. +Let us note that we are including the hold-all domain within this minimisation prob- +lem for the determination of a direction of steepest descent, along with a Dirichlet +boundary condition on the boundary of the hold-all domain. Note that the fact that +D is convex ensures that V is a Lipschitz-1 function. Using the direction (2.2) within +a descent algorithm hence requires the solution of a highly nontrivial constrained min- +imisation problem which can be approximated at the discrete level with the help of +an alternating direction method of multipliers (ADMM). +The above approach will lead to a first order method. If J is twice shape differ- +entiable, it is worthwhile considering a Newton–type approach as well. This can be +achieved by replacing the minimisation problem (2.2) by +(2.3) +min +� t +2J ′′(Ω)[V, V ] + J ′(Ω)[V ] : V ∈ W 1,∞ +0 +(D; Rd), |DV | ≤ 1 a.e. in D +� +, +where t > 0 is a given damping factor. +Here, the evaluation of J ′′(Ω) is by no +means straightforward and we will use the Lagrangian calculus described in the next +subsection to carry out the calculations for the class of problems that we are interested +in. +2.2. Lagrangian framework for PDE–constrained optimization. For the +ease of exposition, let us consider a shape functional of the form +(2.4) +J (Ω) = +� +Ω +j(·, yΩ) dx, +where j : D ×R → R is assumed to be sufficiently smooth and yΩ denotes the solution +of a PDE posed in Ω. We shall adapt the Lagrangian framework developed in Sections +1.6.4 and 1.6.5 of [Hin+08] in order to compute J ′(ˆΩ) and J ′′(ˆΩ) at a fixed domain +ˆΩ ⋐ D. The main aspect of the Lagriangian method is to, in effect, decouple the +state, yΩ, from, in the setting we consider, the shape, Ω. Denoting by B a small open +neighbourhood of 0 in W 1,∞ +0 +(D, Rd) we associate with V ∈ B the perturbed domain +ΩV := (id +V )(ˆΩ). By transforming to ˆΩ we find that, for the choice made in (2.4), +J (ΩV ) = J(V, yΩV ◦ (id +V )), where +(2.5) +J(V, y) := +� +ˆΩ +j(id +V, y) det(I +DV ) dˆx, + +4 +K. DECKELNICK, P.J. HERBERT, AND M. HINZE +and we note that | det(I +DV )| = det(I +DV ) if |DV | is sufficiently small. +The +derivatives of this choice of J may be found in Appendix A.1. In order to incorporate +the PDE constraint we let y = yΩV ◦ (id +V ) and suppose that yΩV solves the given +PDE problem on ΩV if and only if e(V, y) = 0 for some mapping e: B × X → Z. +Here, X, Z are suitable function spaces on ˆΩ and we assume in what follows that +ey(0, ˆy): X → Z is invertible, where ˆy = yˆΩ. After choosing B smaller if necessary +to apply an Implicit Function Theorem, there exists for every V ∈ B a unique y = +y(V ) ∈ X such that e(V, y(V )) = 0, so that we may write +J (ΩV ) = J(V, y(V )) +where, in the context of optimal control, the map V �→ J (ΩV ) takes the role of a +reduced cost functional. In order to calculate the derivatives of J it is convenient to +introduce the Lagrange functional L: X × B × Z∗ → R +(2.6) +L(y, V, p) = J(V, y) + ⟨p, e(V, y)⟩, +so that +J (ΩV ) = L(y(V ), V, p) +for any p ∈ Z∗. +If we denote by p(V ) the solution of Ly(y(V ), V, p(V )) = 0, one immediately obtains +that +(2.7) +J ′(ˆΩ)[V ] = LV (ˆy, 0, ˆp)[V ], +where ˆp = p(0). In a similar way one finds for the second derivative +J ′′(ˆΩ)[V, W] = Lyy(ˆy, 0, ˆp)[y′(0)[V ], y′(0)[W]] + LyV (ˆy, 0, ˆp)[V, y′(0)[W]] ++LV y(ˆy, 0, ˆp)[y′(0)[V ], W] + LV V (ˆy, 0, ˆp)[V, W], +where y′(0)[V ] is the derivative of W �→ y(W) at W = 0 in direction V ∈ W 1,∞ +0 +(D, Rd), +which satisfies +(2.8) +⟨p, ey(0, ˆy)[y′(0)[V ]]⟩ = −⟨p, eV (0, ˆy)[V ]⟩, +for all p ∈ Z∗. +For the implementation of the Newton-like method in (2.3), it is necessary to evaluate +J ′′(ˆΩ)[V, W] many times. In order to carry out the corresponding calculations as +efficiently as possible we would like to avoid the frequent evaluation of y′(0)[W]. To +do, let us write +J ′′(ˆΩ)[V, W] = ⟨h1, y′(0)[W]⟩ + ⟨h2, W⟩, +where +⟨h1, y⟩ = Lyy(ˆy, 0, ˆp)[y′(0)[V ], y] + LyV (ˆy, 0, ˆp)[V, y], +⟨h2, W⟩ = LV y(ˆy, 0, ˆp)[y′(0)[V ], W] + LV V (ˆy, 0, ˆp)[V, W]. +We then first define g ∈ Z∗ as the solution of +(2.9) +⟨g, ey(0, ˆy)[y]⟩ = ⟨h1, y⟩ +∀y ∈ X +and then set +⟨h3, W⟩ = −⟨g, eV (0, ˆy)[W]⟩, +W ∈ W 1,∞ +0 +(D, Rd). +This gives +J ′′(ˆΩ)[V, W] = ⟨h1, y′(0)[W]⟩ + ⟨h2, W⟩ = ⟨g, ey(0, ˆy)[y′(0)[W]]⟩ + ⟨h2, W⟩ += −⟨g, eV (0, ˆy)[W]⟩ + ⟨h2, W⟩ = ⟨h2 + h3, W⟩. +The evaluation of J ′′(ˆΩ)[V, ·] hence essentially requires the solutions of (2.8) and of +the adjoint problem (2.9). + +APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION +5 +2.2.1. Poisson problem. As a first PDE constraint we here consider the Pois- +son problem. We set X = H1 +0(ˆΩ) and Z = X∗. Since we are in a reflexive setting, we +use the canonical injection and identify Z∗ with X. By yΩ we denote the solution of +(2.10) +− ∆yΩ = F in Ω, +yΩ = 0 on ∂Ω +for a given F ∈ L2(D). In particular, we find that yΩV ◦ (id +V ) is a solution of +e(V, y) = 0 where e: B × H1 +0(ˆΩ) → H−1(ˆΩ) is given by +(2.11) ⟨e(V, y), p⟩ = +� +ˆΩ +(A(V )∇y · ∇p − F ◦ (id +V )p) det(I +DV )dˆx, +p ∈ H1 +0(ˆΩ), +and A(V ) := (I +DV )−1(I +DV )−T . +Derivatives of the map e may be found in +Appendix A.2.1. With the Lagrange functional +L(V, y, p) = +� +ˆΩ +j(id +V, y) det(I +DV ) dˆx + ⟨e(V, y), p⟩ +we deduce from (2.7) the well–known formula +J ′(ˆΩ)[V ] = LV (ˆy, 0, ˆp)[V ] = +� +ˆΩ +(j(·, ˆy) div V + jx(·, ˆy) · V + A[V ]∇ˆy · ∇ˆp − ˆp div(FV )) dˆx, +where +(2.12) +A[V ] := I div V − DV − DV T , +ˆy ∈ H1 +0(ˆΩ) satisfies e(0, ˆy) = 0, and the adjoint ˆp ∈ H1 +0(ˆΩ) satisfies Ly(ˆy, 0, ˆp) = 0, +i.e. +(2.13) +� +ˆΩ +∇ˆp · ∇η dˆx = − +� +ˆΩ +jy(·, ˆy)η dˆx +for all η ∈ H1 +0(ˆΩ). +2.2.2. Bi-Laplace-type equation. Let us next consider the minimsation of J +as in (2.4) subject to the linear PDE of fourth order +(2.14) +∆2yΩ = F in Ω, +yΩ = ∆yΩ = 0 on ∂Ω. +If the boundary of Ω is sufficiently regular the above problem can be split into two +second order Poisson problems by introducing −∆yΩ as an additional variable. Let us +note that this splitting is analytically useful to ensure that the shape derivative exists +in the sense of [ADJ21, Definition 4.1], due to the fourth order nature of the problem. +Let us comment that this need not be necessary since the shape differentiability, +particularly boundedness in Lipschitz functions, with a fourth order constraint was +demonstrated in [EH22] in a surface context, while [Las17] shows this for a fourth +order eigenvalue problem. +On the fixed domain, we set X = +� +H1 +0(ˆΩ) +�2 +and Z = X∗. Again we will use the +canonical injection to identify Z∗ with X. Posing the split formulation of (2.14) on +ΩV and transforming it back onto ˆΩ in the same way as above we write the map e +which represents the PDE constraint as, +⟨e(V, y), p⟩ = +� +ˆΩ +� +A(V )∇y1 · ∇p2 − y2p2 +� +det(I +DV ) dˆx ++ +� +ˆΩ +� +A(V )∇y2 · ∇p1 − F ◦ (id +V )p1 +� +det(I +DV ) dˆx, +(2.15) + +6 +K. DECKELNICK, P.J. HERBERT, AND M. HINZE +for all y = (y1, y2), p = (p1, p2) ∈ +� +H1 +0(ˆΩ) +�2 +. Derivatives of the map e may be found +in Appendix A.2.2. Similar to (2.12) we obtain for the shape derivative +J ′(ˆΩ)[V ] = +� +ˆΩ +(j(·, ˆy1) div V + jx(·, ˆy1) · V + A[V ]∇ˆy1 · ∇ˆp2 − ˆy2ˆp2 div V ) dˆx ++ +� +ˆΩ +(A[V ]∇ˆy2 · ∇ˆp1 − div(FV )ˆp1) dˆx, +(2.16) +where ˆy = (ˆy1, ˆy2) ∈ +� +H1 +0(ˆΩ) +�2 +satisfies e(0, ˆy) = 0 and the adjoint ˆp = (ˆp1, ˆp2) ∈ +(H1 +0(ˆΩ))2 satisfies +� +ˆΩ +∇ˆp2 · ∇η1 dˆx = − +� +ˆΩ +jy(·, ˆy1)η1 dˆx +∀η1 ∈ H1 +0(ˆΩ), +(2.17) +� +ˆΩ +∇ˆp1 · ∇η2 dˆx = +� +ˆΩ +ˆp2η2 dˆx +∀η2 ∈ H1 +0(ˆΩ). +(2.18) +2.3. Optimisation of the first eigenvalue for the Laplacian. Our aim is to +apply the above Lagrangian framework also for the optimisation of the first Dirichlet +eigenvalue of the Laplacian, i.e. +(2.19) +J (Ω) = λ1(Ω), +where λ1(Ω) is defined by +(2.20) +λ1(Ω) := inf +�� +Ω +|∇z|2 : z ∈ H1 +0(Ω), +� +Ω +z2 = 1 +� +. +With the notation of Section 2.2 we again fix a ˆΩ ⋐ D which we now assume to be +connected and set ΩV = (id +V )(ˆΩ). We transform the eigenvalue relation +−∆zΩV = λzΩV in ΩV , +zΩV = 0 on ∂ΩV +together with the condition +� +ΩV z2 +ΩV = 1 onto ˆΩ and write it in the form e(V, y) = 0, +where e: B × X → Z, with X = H1 +0(ˆΩ) × R, Z = X∗, and +⟨e(V, y), p⟩ = +� +ˆΩ +(A(V )∇z · ∇q − λzq) det(I +DV ) dˆx ++µ +� +1 − +� +ˆΩ +z2 det(I +DV ) dˆx +� +, +(2.21) +for y = (z, λ), p = (q, µ) ∈ H1 +0(ˆΩ) × R. Derivatives of the map e may be found in +Appendix A.2.3. Let ˆz ∈ H1 +0(ˆΩ) be an eigenfunction to the first Dirichlet eigenvalue +ˆλ with +� +ˆΩ ˆz2 = 1. Then we have for all p = (q, µ) ∈ H1 +0(ˆΩ) × R +⟨eV (0, ˆy)[V ], p⟩ = +� +ˆΩ +� +A[V ]∇ˆz · ∇q − ˆλ div V ˆzq − µ div V ˆz2� +dˆx, +(2.22) +⟨ey(0, ˆy)[(η, ˜η)], p⟩ = +� +ˆΩ +� +∇η · ∇q − ˆληq − ˜ηˆzq − 2µˆzη +� +dˆx, +(2.23) + +APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION +7 +where ˆy = (ˆz, ˆλ). Since ˆλ is simple, cf. ˆΩ is connected and ˆλ is the first Dirichlet +eigenvalue [Gil+77, Theorem 8.38], it can be shown that ey(0, ˆy): H1 +0(ˆΩ) × R → +H−1(ˆΩ) × R is invertible. Thus we can write for V ∈ B +J (ΩV ) = J(V, y(V )), +where J(V, (z, λ)) = λ. +The Lagrange functional is given by L(y, V, p) = λ+⟨e(V, y), p⟩ so that we derive with +the help of (2.12) +(2.24) +J ′(ˆΩ)[V ] = LV (ˆy, 0, ˆp)[V ] = ⟨eV (0, ˆy)V, ˆp⟩. +The adjoint ˆp = (ˆq, ˆµ) is given by the relation Ly(ˆy, 0, ˆp) = 0, i.e. +˜η + +� +ˆΩ +� +∇η · ∇ˆq − ˆληˆq − ˜ηˆzˆq − 2ˆµˆzη +� +dˆx = 0 +∀(η, ˜η) ∈ H1 +0(ˆΩ) × R. +We infer that +� +ˆΩ ˆzˆq dˆx = 1 as well as +� +ˆΩ +� +∇η · ∇ˆq − ˆληˆq +� +dˆx = 2ˆµ +� +ˆΩ +ˆzη dˆx +∀η ∈ H1 +0(ˆΩ). +Choosing η = ˆz we deduce that ˆµ = 0, so that ˆq is an eigenfunction for the eigenvalue +ˆλ. Since ˆλ is simple and +� +ˆΩ ˆzˆq dˆx = 1 we infer that ˆq = ˆz and hence by (2.24) that +(cf. [Hen06; HP06]) +(2.25) +λ′ +1(ˆΩ)[V ] = +� +ˆΩ +� +A[V ]∇ˆz · ∇ˆz − ˆλ div V ˆz2� +dˆx. +It is known that the first eigenvalue scales with volume, as such we are interested in +fixing the volume of Ω. While it is known that the minimiser of the first eigenvalue +is a ball, the methodology is interesting and can be applied to more complicated +eigenvalue problems. +3. Discretisation. Our aim is to formulate a descent algorithm which produces +in each step a polygonal domain and which replaces a possible PDE constraint with +a corresponding finite element approximation. To begin, let T 0 +h be a triangulation of +the hold–all domain D. We look for discrete directions of descent in the finite element +spaces +(3.1) +Vn +h := +� +Vh ∈ C0(D; Rd) : Vh|T ∈ P 1(T; Rd), ∀T ∈ T n +h , Vh = 0 on ∂D +� +, +where P 1(T; Rd) denotes polynomials of degree at most one on T with values in Rd +and T n +h is to be determined for n ≥ 1. With a polygonal initial domain, Ω0 which is +a union of the triangles in the triangulation T 0 +h , we will set Ωn+1 = (id +tnVn)(Ωn) +for n ≥ 1, where tn ∈ (0, 1) is a step size and we will shortly explain how to choose +Vn ∈ Vn +h . As well as updating the domain, the triangulation will also be updated, +T n+1 +h += {(id +tnVn)(T) : T ∈ T n +h }. By the choice of Vn +h and the fact that Vn will +satisfy |DVn| ≤ 1, it holds that the updated mesh will be admissible since tn ∈ (0, 1). +3.1. Choice of descent direction. Let n ≥ 0 be fixed and let us denote the +polygonal domain ˆΩ = Ωn ⋐ D which is a union of triangles in T n +h . For simplicity we +will henceforth neglect the dependence on n. We aim to find V ∗ +h ∈ Vh such that +V ∗ +h ∈ arg min +� +J ′(ˆΩ)h[Vh] : Vh ∈ Vh, |DVh| ≤ 1 a.e. in D +� += arg min +�� +D +φ(DVh)dx + J ′(ˆΩ)h[Vh] : Vh ∈ Vh +� +, + +8 +K. DECKELNICK, P.J. HERBERT, AND M. HINZE +where J ′(ˆΩ)h is a suitable approximation of J ′(ˆΩ) and +(3.2) +φ(A) := +� +0, +|A| ≤ 1, +∞, +|A| > 1. +We use an alternating direction method of multipliers (ADMM) approach in order +to solve the above problem. To do so, we set +(3.3) +Qh := +� +qh ∈ L2(D; Rd×d) : qh|T ∈ P 0(T; Rd×d), ∀T ∈ Th +� +and consider for a given τ > 0 the functional Lτ : Vh × Qh × Qh → R with +(3.4) +Lτ(Vh, qh; λh) := +� +D +φ(qh) + λh : (DVh − qh) + J ′(ˆΩ)h[Vh] + τ +2∥DVh − qh∥2 +L2(D;Rd×d). +The idea of ADMM is to alternatively minimise Lτ over qh and Vh, then perform an +update to λh and repeat this until a certain quantity is small. More precisely, the +algorithm has the following form: +Algorithm 3.1 ADMM +Choose V 0 +h and λ0 +h such that J ′(ˆΩ)h[V 0 +h ] < ∞ +Set R = ∞, j = 1 +while R > tol do +Find qj +h ∈ arg min{Lτ(V j−1 +h +, qh; λj−1 +h +) : qh ∈ Qh, |qh| ≤ 1} +Find V j +h ∈ arg min{Lτ(Vh, qj +h; λj−1 +h +) : Vh ∈ Vh} +Set λj +h = λj−1 +h ++ τ(DV j +h − qj +h) +Set R = +� +∥λj +h − λj−1 +h +∥2 +L2(D;Rd×d) + τ 2∥DV j +h − DV j−1 +h +∥2 +L2(D;Rd×d) +� 1 +2 +Update j = j+1 +end while +Let us also mention [BM20] which considers more general ADMM methods with +variable τ. Particularly, in our experiments we make use of such an algorithm with +variable τ, namely [BM20, Algorithm 3.19]. We note that Algorithm 3.1 can also be +applied to find a solution to (2.3) using the modified Lagrangian +(3.5) +LNewton +τ +(Vh, qh; λh) := Lτ(Vh, qh; λh) + t +2J ′′(ˆΩ)h[Vh, Vh], +where J ′′(ˆΩ)h is a suitable approximation of J ′′(ˆΩ). +3.2. Evaluation of J ′(ˆΩ)h. +3.2.1. PDE–constrained shape optimisation. Let us formulate suitable ap- +proximations of the shape derivatives derived in 2.2. Given the polygonal domain ˆΩ +we denote by Sh(ˆΩ) the space of linear finite elements on ˆΩ (resolved by a subtrian- +gulation of Th) which vanish on ∂ ˆΩ. If the constraint is given by (2.10) we set +(3.6) +J ′(ˆΩ)h[Vh] = +� +ˆΩ +(j(·, ˆyh) div Vh + jx(·, ˆyh) · Vh + A[Vh]∇ˆyh · ∇ˆph − ˆph div(FVh)) dˆx, + +APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION +9 +for Vh ∈ Vh, where ˆyh, ˆph ∈ Sh(ˆΩ) satisfy +� +ˆΩ +∇ˆyh · ∇ηh dˆx = +� +ˆΩ +Fηh dˆx +∀ηh ∈ Sh(ˆΩ), +� +ˆΩ +∇ˆph · ∇ηh dˆx = − +� +ˆΩ +jy(·, ˆyh)ηh dˆx +∀ηh ∈ Sh(ˆΩ). +(3.7) +On the other hand, if the constraint is given by (2.14) then we let +J ′(ˆΩ)h[Vh] = +� +ˆΩ +(j(·, ˆyh,1) div Vh + jx(·, ˆyh,1) · Vh + A[V ]∇ˆyh,1 · ∇ˆph,2 − ˆyh,2ˆph,2 div Vh) dˆx ++ +� +ˆΩ +(A[Vh]∇ˆyh,2 · ∇ˆph,1 − div(VhF)ˆph,1) dˆx, +(3.8) +for Vh ∈ Vh. Here, ˆyh = (ˆyh,1, ˆyh,2) ∈ (Sh(ˆΩ))2 satisfies +� +ˆΩ +∇ˆyh,1 · ∇ηh dˆx = +� +ˆΩ +ˆyh,2ηh dˆx +∀ηh ∈ Sh(ˆΩ), +(3.9) +� +ˆΩ +∇ˆyh,2 · ∇ηh dˆx = +� +ˆΩ +Fηh dˆx +∀ηh ∈ Sh(ˆΩ) +(3.10) +while ˆph = (ˆph,1, ˆph,2) ∈ (Sh(ˆΩ))2 satisfies +� +ˆΩ +∇ˆph,2 · ∇ηh dˆx = − +� +ˆΩ +jy(·, yh,1)ηh dˆx +∀ηh ∈ Sh(ˆΩ), +(3.11) +� +ˆΩ +∇ˆph,1 · ∇ηh dˆx = +� +ˆΩ +ˆph,2ηh dˆx +∀ηh ∈ Sh(ˆΩ). +(3.12) +3.2.2. Optimisation of the first eigenvalue for the Laplacian. For a given +polygonal domain we determine ˆzh ∈ Sh(ˆΩ) and ˆλh > 0 such that +� +ˆΩ ˆy2 +h dx = 1 and +ˆλh = inf +�� +ˆΩ +|∇ˆzh|2 dˆx : zh ∈ Sh(ˆΩ), +� +ˆΩ +ˆz2 +h dˆx = 1 +� += +� +ˆΩ +|∇ˆzh|2 dˆx. +Supposing that the eigenvalue λh is simple we let, recalling (2.25) +(3.13) +J ′(ˆΩ)h[Vh] = +� +ˆΩ +� +A[Vh]∇ˆzh · ∇ˆzh − ˆλhˆz2 +h div Vh +� +dˆx for Vh ∈ Vh. +4. Numerical experiments. We now provide numerical experiments for the +applications we have described. In the integrals for the energy we use quadrature of +order 2, while for the shape derivatives, we are using the order which is automatically +decided by the software. +As mentioned above we will solve the state and adjoint PDEs with a finite element +approximation. The finite element approximation is performed with DUNE [Bas+21], +making particular use of the DUNE Python bindings [DNK20; DN18]. We consider a +construction of update direction using four different approaches. Our approaches will +be: +• The direction of steepest descent method using the W 1,∞-topology, construc- +ted with an adaptive ADMM method, as mentioned after Algorithm 3.1. This +will be referred to as p = ∞. + +10 +K. DECKELNICK, P.J. HERBERT, AND M. HINZE +• A Newton-type direction, which will be a discrete minimiser of (2.3) for a +given t > 0, referred to as the Newton method. Much like the p = ∞ case +above, this will be constructed with an adaptive ADMM method. +• To compare against existing approaches, for p = 2, 4, we will consider the +minimiser of Vh ∋ Vh �→ J ′(Ω)h + 1 +p +� +D |DVh|p. In the case that p = 2, this +is seen to coincide with the discrete case of (1.3) with H = H1 +0(D; Rd) with +the Dirichlet inner product. We will refer to these cases by their p value. +The discrete functions produced by the p = 2 and p = 4 methods will be normalised +so that they have a W 1,∞(D; Rd) semi-norm of 1. This normalisation is performed so +that we need not check whether the mesh has overlapped. For each of the experiments, +we will set the hold-all domain to be the box D = (−2, 2)2. With these directions, +we will move the vertices of our mesh according to an Armijo step rule. We will stop +after 20 shape updates have been made. In most cases the domain has become close +to stationary at this point and the Armijio step-size has become rather small. +The energy along the iterations will be plotted. In the case that the minimiser is +known, the origin will be offset by the known value, when the minimiser is not known, +the origin will be offset by the smallest value attained in the experiment. +4.1. Minimisation without a PDE constraint. Here we will consider that +there is no PDE constraint, so that the map e need not be included. We comment that +the no PDE example may be derived as an example from the following Section 4.2 +where one chooses right hand side data F = 0 so that the state constraint guarantees +y = 0. +For this experiment, the main contributions to the error is that induced by the +quadrature rules when calculating the energy and the shape derivative, as well as the +direction of descent with the chosen method. +4.1.1. No PDE experiment 1. For this problem, we consider +(4.1) +j(x, y) = −Z(x) +where +(4.2) +Z(x) = +� +� +� +� +� +� +� +� +� +cos(0.5πx1) cos(0.5πx2) +|x1| and |x2| ≤ 1, +π +4 (1 − x2 +1) +|x1| > 1 and |x2| < 1, +π +4 (1 − x2 +2) +|x1| < 1 and |x2| > 1, +π +4 (2 − x2 +1 − x2 +2) +otherwise. +For the Newton direction we take t = 0.0625. One expects the square (−1, 1)2 to be a +minimiser of J . It holds that J +� +(−1, 1)2� += − 16 +π2 . We start with the initial domain +(−1.5, −1) × (−1, 1). The triangulation of the domain and hold-all is displayed in +Figure 1a. In Figure 1b, the energy of shapes along the minimising sequences we +produce are given. In Figure 2, the meshes of the final domains Ω for each of the +methods are given. +4.1.2. No PDE experiment 2. For this problem, we consider +(4.3) +j(x, y) = 1 +2Z(x)2 +where for given ϵ > 0, +(4.4) +Z(x) = +� +(x1 + x2)2 + ϵ + +� +(x1 − x2)2 + ϵ + +APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION +11 +(a) Initial domain for the first No PDE ex- +periment in Section 4.1.1, with (−1.5, −1) × +(−1, 1) in red and the hold-all, (−2, 2)2 in +blue. +0 +5 +10 +15 +20 +Iterations +10−5 +10−4 +10−3 +10−2 +10−1 +100 +Energy + 16 +π2 +p = 2 +p = 4 +p = ∞ first order +p = ∞ second order +(b) Graph of the energy for the iterates in +the first No PDE experiment in Section 4.1.1. +It is seen that the Newton-type method is +energetically performing the best while the +first order infinity method appears to struggle +compared to the traditional p = 2 method. +Fig. 1: Initial mesh and graph of the energy for the experiment in Section 4.1.1. +is a smooth approximation to |x1 + x2| + |x1 − x2|. For the Newton direction we take +t = 0.125. A very similar experiment with the non-smooth energy was considered +in [DHH22]. This smooth approximation is used because we intend to employ the +Newton method for which it would be useful to have (weak) second derivatives of +Z. Without any constraint, we know that the theoretical minimiser is degenerate, a +measure zero set. To avoid this, we will fix the area to be constrained equal to 4. We +expect this to have minimiser close to the square (−1, 1)2 which, for ϵ = 0, has energy +4. Our directions of descent will only preserve the area constraint in a linear sense by +restricting to V with +� +Ω div V = 0. We will perform a projection step to fix the area +after each update. +We take ϵ = 10−4 and start with an approximation of a ball of radius +2 +√π at the +origin. The triangulation of the domain and hold-all is displayed in Figure 3a. In +Figure 3b, the energy of shapes along the minimising sequences we produce are given. +In Figure 4, the meshes for the final domains Ω for each of the methods are given. +4.2. A Poisson problem. +4.2.1. Poisson experiment 1. For our first experiment we consider j(x, y) = y +and F(x) = 2.5(x1 + 0.5 − x2 +2)2 + x2 +1 + x2 +2 − 1. This has appeared in [Etl+20; HL21], +for example, as a benchmark for the comparisons of shape optimisation algorithms. +For the Newton direction we take t = 0.125. The minimising shape is not explicitly +known, however it appears to be a shape not so dissimilar to a kidney. Similarly, the +energy of a minimiser is not known. +We start with an approximation of an ellipse with semiaxes +2 +√π and +1 +√π centred +at the origin. The triangulation of the domain and hold-all is displayed in Figure 5a. +In Figure 5b, the energy of shapes along the minimising sequences we produce are +given. In Figure 6, the meshes for the final domains Ω for each of the methods are +given. + +12 +K. DECKELNICK, P.J. HERBERT, AND M. HINZE +Fig. 2: The meshes of Ω for the final domains produced in the first No PDE experiment +in Section 4.1.1: top left to bottom, p = 2, 4, ∞ and second order. Due to symmetry +of the result, we show only a half of each mesh. It is seen that none of the methods +correctly captures the corners which are expected from the minimising shape. This +is unsurprising as there is not a particularly large influence of the energy around the +corners, where Z is quite close to zero. The Newton-type method is the closest to +forming corners. +4.2.2. Poisson experiment 2. For this experiment we consider j(x, y) = 1 +2(y− +yd(x))2 where yd(x) = 4 +π − |x|2 and F = 1. Let us note that −∆yd = 4F. For the +Newton direction we take t = 0.125. This experiment will be equipped with an area +constraint that the domain has fixed area 4 - we will use the same linear constraint +on the update direction and projection as in Section 4.1.2. In this setting, we expect +the minimiser to be given by the ball of radius +2 +√π at the origin which has energy +6 +π2 . +We start with the square (−1, 1)2. The triangulation of the domain and hold-all +is displayed in Figure 7a. In Figure 7b, the energy of shapes along the minimising +sequences we produce are given. In Figure 8, the meshes for the final domains Ω for +each of the methods are given. +It is worth noting that when larger values of t were taken during testing, the +Newton-type method struggled to perform well. With the Newton method, once the +shape was sufficiently close to a ball, the directions generated by ADMM would rotate +the almost-ball by large angles which caused large deformations of the mesh in the +hold-all. +4.3. A coupled Poisson problem. For this experiment we will consider j(x, y) = +1 +2(y1−yd(x))2 where yd(x) = 0.05+(1−x2 +1)3(1−x2 +2)3 and F(x) = ∆2 � +(1 − x2 +1)3(1 − x2 +2)3� +. +For the Newton direction we take t = 0.0625. This experiment will be equipped with +an area constraint that the domain has fixed area 4 - we will use the same linear con- +straint on the update direction and projection as in Section 4.1.2. One would expect +the minimiser to be relatively close to the square (−1, 1)2 which should have energy +0.005. +We start with the square (−1, 1)2. The triangulation of the domain and hold-all +is displayed in Figure 9a. In Figure 9b, the energy of shapes along the minimising +sequences we produce are given. In Figure 10, the meshes for the final domains Ω for +each of the methods are given. + +APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION +13 +(a) Initial domain for the second No PDE ex- +periment in Section 4.1.2, an approximation +of the ball of radius +2 +√π at the origin, is in red +and the hold-all, (−2, 2)2 in blue. +0 +5 +10 +15 +20 +Iterations +10−4 +10−3 +10−2 +10−1 +|Energy - 4| +p = 2 +p = 4 +p = ∞ first order +p = ∞ second order +(b) Graph of the energy for the iterates in the +second No PDE experiment in Section 4.1.2. +Let us note that the bumps in the graphs are +due to the absolute value we are using. It is +seen that the Newton method has the closest +energy. +Fig. 3: Initial mesh and graph of the energy for the experiment in Section 4.1.2. +Fig. 4: The meshes of Ω for the final domains produced in the second No PDE +experiment in Section 4.1.2: left to right, p = 2, 4, ∞ and second order. +Due to +symmetry of the result, we show only a half of each mesh. Generally, the first order +methods provide a rather good approximation of the expected minimising shape of a +square with p = ∞ having the most regular triangles around the corners. The Newton +method does not quite form the corners we expect. +4.4. Optimisation of the first eigenvalue for the Laplacian. We use the +function eigs from the module sparse.linalg in scipy [Vir+20] to find pairs (v, λh) ∈ +RN × R such that Bv = λhMv, where B ∈ RN×N is the stiffness matrix and +M ∈ RN×N is the mass matrix. +This experiment will be equipped with an area +constraint that the domain has fixed area 4 - we will use the same linear constraint + +14 +K. DECKELNICK, P.J. HERBERT, AND M. HINZE +(a) Initial domain for the first Poisson exper- +iment in Section 4.2.1, an approximation of +the ellipse with semiaxes +2 +√π and +1 +√π at the +origin, is in red and the hold-all, (−2, 2)2 in +blue. +0 +5 +10 +15 +20 +Iterations +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +Energy + 0.09377080539770344 +p = 2 +p = 4 +p = ∞ first order +p = ∞ second order +(b) Graph of the energy for the iterates in the +first Poisson experiment in Section 4.2.1. We +see that p = ∞ does not perform as well as +the traditional p = 2 method. The Newton +method performs well energetically. +Fig. 5: Initial mesh and graph of the energy for the experiment in Section 4.2.1. +Fig. 6: The meshes of Ω for the final domains produced in the first Poisson experiment +in Section 4.2.1: top left to bottom, p = 2, 4, ∞ and second order. Due to symmetry +of the result, we show only a half of each mesh. We see that all of the shapes are +rather similar, with the p = ∞ being a slight outlier. The triangles for both p = ∞ +and the Newton method appear to be more regularly spaced at the boundary. +on the update direction and projection as in Section 4.1.2. For the Newton direction +we take t = 0.125. In this setting, the minimiser is known to be the ball of radius +2 +√π +which has λ1(B +2 +√π ) = β2 +0,1 +π +4 ≈ 4.542103633884308, where β0,1 is the first zero of the +0th Bessel function. +We start with the square (−1, 1)2. The triangulation of the domain and hold-all + +APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION +15 +(a) Initial domain for the second Poisson ex- +periment in Section 4.2.2, (−1, 1)2 is in red +and the hold-all, (−2, 2)2 in blue. +0 +5 +10 +15 +20 +Iterations +10−4 +10−3 +10−2 +Energy - +6 +π2 +p = 2 +p = 4 +p = ∞ first order +p = ∞ second order +(b) Graph of the energy for the iterates in the +second Poisson experiment in Section 4.2.2. +We see that all the methods give roughly the +same energy. +Fig. 7: Initial mesh and graph of the energy for the experiment in Secetion 4.2.2. +is displayed in Figure 11a. In Figure 11b, the energy of shapes along the minimising +sequences we produce are given. In Figure 12, the meshes for the final domains Ω for +each of the methods are given. Let us further elaborate on Figure 12, particularly +the mesh produced by the infinity method. We note that the mesh in the centre for +the infinity method appears less regular than the other methods. This appears to +occur due to a somewhat undesirable approximation of the shape derivative, which +should be concentrated at the boundary. The undesirable approximation is due to the +FEM approximation giving interior contributions to the shape derivatives, so-called +spurious contributions, in the limit of the mesh becoming infinitely fine this should +disappear. Investigation suggests that these are more apparent in the infinity method +but disappear when using shape derivatives with only boundary contributions. This +sort of expert knowledge is often exploited in the literature and appeared in e.g. +[SSW16b] to ’delete’ contributions to the shape derivative from nodes on the interior. +5. Outlook: Convergence of the infinity method. Here we consider the ex- +perimental convergence of the example provided in Section 4.2.2 as h → 0. In this ex- +periment, we will compute a quantity which we will refer to as the discrete complemen- +tary Hausdorff distance. The Hausdorff complementary distance is utilised in [DZ11] +to give a metric on a space of shapes and is given by ρc +H(Ω1, Ω2) := supx′∈D |d∁Ω1(x′)− +d∁Ω2(x′)|, where the function d∁Ω : D → R is given by d∁Ω(x) = infx′∈D\Ω |x − x′|. +To calculate a discrete Hausdorff complementary distance, we consider that the exact +minimiser, Ω∗, is known to be the ball of radius +2 +√π at the origin and +(5.1) +d∁Ω∗(x) = min +� +0, 2 +√π − |x| +� +. +For each h > 0, let us denote by Ωh the final domain produced in each experiment. +Let {xi}N +i=1 be the vertices of the triangulation of D and let {yi}n +i=1 be the vertices +which lie on the boundary of Ωh, then we set +(5.2) +dh +∁Ωh(xi) = +inf +j=1,...,n |xi − yj| for xi ∈ Ωh, otherwise 0. + +16 +K. DECKELNICK, P.J. HERBERT, AND M. HINZE +Fig. 8: The meshes of Ω for the final domains produced in the second Poisson ex- +periment in Section 4.2.2: top left to bottom, p = 2, 4, ∞ and second order. Due to +symmetry of the result, we show only a quarter of each mesh. We see that both p = 2 +and p = 4 have very degenerate elements where the corners of the original domain +were. Both p = ∞ and Newton methods do not have these degenerate triangles. We +notice that the triangles which previously made up the corners of the original domain +are rather regular in the Newton method. For p = ∞, there are many triangles which +have obtuse angles. +(a) Initial domain for the coupled Poisson ex- +periment in Section 4.3, (−1, 1)2 is in red and +the hold-all, (−2, 2)2 in blue. +0 +5 +10 +15 +20 +Iterations +10−7 +10−6 +10−5 +10−4 +10−3 +Energy - 0.0014111218894029705 +p = 2 +p = 4 +p = ∞ +p = ∞ second order +(b) Graph of the energy for the iterates in the +coupled Poisson experiment in Section 4.3. +We see that p = ∞ outperforms the finite +p experiments, but the Newton method is en- +ergetically performing the best. +Fig. 9: Initial mesh and graph of the energy for the experiment in Section 4.3. + +APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION +17 +Fig. 10: The meshes of Ω for the final domains produced in the coupled Poisson +experiment in Section 4.3: top left to bottom, p = 2, 4, ∞ and second order. Due +to symmetry of the result, we show only a quarter of each mesh. To the eye, the +first order methods are all seemingly the same. The Newton method appears to have +found a more pronounced shape than the others. +(a) Initial domain for the Eigenvalue experi- +ment in Section 4.4, (−1, 1)2 is in red and the +hold-all, (−2, 2)2 in blue. +0 +5 +10 +15 +20 +Iterations +10−2 +10−1 +Energy - π +4 β2 +0,1 +p = 2 +p = 4 +p = ∞ first order +p = ∞ second order +(b) Graph of the energy for the iterates in the +Eigenvalue experiment in Section 4.4. We see +that all the methods are performing roughly +the same. +Fig. 11: Initial mesh and graph of the energy for the experiment in Section 4.4. +We then calculate our discrete Hausdorff complementary distance to be given by +(5.3) +dh(Ωh, Ω∗) = +sup +i=1,...,N +|d∁Ω∗(xi) − dh +∁Ωh(xi)|. +We start with Ω = (−1, 1)2 and D = (−2, 2)2. The mesh for the coarsest refine- +ment is given in Figure 13. For each subsequent refinement, each triangle in the mesh + +18 +K. DECKELNICK, P.J. HERBERT, AND M. HINZE +Fig. 12: The meshes of Ω for the final domains produced in the Eigenvalue experiment +in Section 4.4: top left to bottom, p = 2, 4, ∞ and second order. Due to symmetry +of the result, we show only a quarter of each mesh. We see that spikes appear for +p = 2 and p = 4 where the corners were for the original shape. The mesh for p = ∞ +appears rather regular. We comment that the Newton method has lead to the mesh +rotating when compared to its original orientation. +Fig. 13: Initial domain with coarsest mesh for the experimental convergence experi- +ment for the Poisson problem in Section 5, in red is the initial domain (−1, 1)2 and +the hold all (−2, 2)2 is in blue. +is replaced with four equivalent triangles. In Figure 14, the energy of the shapes along +the minimising sequence is given along with a graph plotting the final energy against +the mesh size (as measured on the red part of the initial mesh) along with a reference +line. While the convergence of the energy appears to be quadratic, it is also interest- +ing to consider some convergence of the shape itself. To measure this convergence we +consider the discrete counterpart to the Hausdorff complementary distance which is +detailed above. We see this convergence to be essentially linear. Also in Figure 14 we + +APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION +19 +0 +5 +10 +15 +20 +Iterations +10−4 +10−3 +10−2 +10−1 +|Energy - +6 +π2 | +p = ∞ refines = 0 +p = ∞ refines = 1 +p = ∞ refines = 2 +p = ∞ refines = 3 +p = ∞ refines = 4 +10−1 +Initial h +10−4 +10−3 +10−2 +Energy - +6 +π2 +Experiment +Reference h2 +10−1 +Initial h +10−2 +Discrete Hausdorff Complementary Distance +Experiment +Reference h +Fig. 14: Graphs of the energy (left and middle) and discrete Hausdorff complementary +distance for the final domains (right) in the experimental convergence experiment for +the Poisson problem in Section 5. It is seen that the energies converge like h2. With +the discretisation uses, one expects that the state would converge like order h2 in +the L2 norm, however this convergence need not directly translate to the energy as +the domain is also approximated. For the discrete Hausdorff complementary distance +convergence of order approximately h is observed. +Fig. 15: The meshes of Ω for the final domains produced with the first order method +in the experimental convergence experiment for the Poisson problem in Section 5: top +left to bottom are further refinements. Due to symmetry of the result, we show only +a quarter of each mesh. +show this discrete Hausdorff complementary distance against the (initial) mesh size. +In Figure 15, the meshes for the final domains Ω for each refinement is given. +Conclusion. In this work we have introduced two new frameworks in which to +perform shape optimisation. These methods are more applicable to physical scenarios +than the previous works involving W 1,∞ as they are not restricted to star-shaped +domains. While our examples had star-shaped final domains, our framework allows +for more general shapes and is more readily applicable to industrial problems. From +the experiments, it was seen that our introduced first order method did not necessarily +perform well energetically, however the meshes the method produced are regular. The +second order method we introduced performs well both energetically and in terms of + +20 +K. DECKELNICK, P.J. HERBERT, AND M. HINZE +the regularity of the mesh, a downside however is the need to tune the damping +parameter t. An analysis of the steepest descent method using the W 1,∞–topology is +work in progress. +Acknowledgements. The authors wish to extend their gratitude to Andreas +Dedner for providing useful insight into the use of the Python bindings for DUNE. +This work is part of the project P8 of the German Research Foundation Priority +Programme 1962, whose support is gratefully acknowledged by the second and the +third author. P.J.H acknowledges the support of EPSRC (grant EP/W005840/1). +Appendix A. Calculations for the second shape derivatives. +Here we +collect the derivatives of J and e for the examples in Section 4. These derivatives are +particularly useful for the calculation of the second shape derivative. All of the maps +and functions we consider are sufficiently smooth that we may exchange the order of +differentiation. It will be convenient to define +D[V, W] := (div(V ) div(W) − Tr(DV DW)) . +A.1. Derivatives for the energy functionals. Let us consider J as in (2.5), +that is J(V, y) := +� +ˆΩ j(id +V, y) det(I +DV ) for some fixed ˆΩ ⋐ D. When j is twice +differentiable, it holds that +JV (0, y)[V ] = +� +ˆΩ +div V j(·, y) + jx(·, y) · V, +Jy(0, y)[η] = +� +ˆΩ +jy(·, y) η, +JyV (0, y)[η, V ] = +� +ˆΩ +div V jy(·, y) η + jyx(·, y) · V η, +Jyy(0, y)[η, ξ] = +� +ˆΩ +jyy(·, y)ηξ, +JV V (0, y)[V, W] = +� +ˆΩ +D[V, W]j(·, y) + div V jx(·, y) · W + div Wjx(·, y) · V + jxx(·, y)V · W. +A.2. Derivatives for PDE constraints. Here, we collect the derivatives for +the maps e which appear in Sections 4.2, 4.3, and 2.3. Recall that we define A(V ) := +(I +DV )−1(I +DV )−T and A[V ] := I div V − DV − DV T . We furthermore define +A[V, W] :=D[V, W] I − div(V )DW − div(V )DW T +− div(W)DV + (DWDV + DV DW) + DV DW T +− div(W)DV T + DWDV T + (DWDV + DV DW)T . +A.2.1. Derivatives for the Poisson Problem. In the case that +⟨e(V, y), p⟩ = +� +ˆΩ +(A(V )∇y · ∇p − F ◦ (id +V )p) det(I +DV ) +as in Section 4.2, then it holds that +⟨eV (0, y)[V ], p⟩ = +� +ˆΩ +A[V ]∇y · ∇p − div(V F)p, +⟨ey(0, y)[η], p⟩ = +� +ˆΩ +∇η · ∇p, +⟨eyV (0, y)[η, V ], p⟩ = +� +ˆΩ +A[V ]∇η · ∇p, +⟨eyy(0, y)[η, ξ], p⟩ = 0, +⟨eV V (0, y)[V, W], p⟩ = +� +ˆΩ +A[V, W]∇y · ∇p − D[V, W]Fp − div(V )pW · ∇F +− div(W)pV · ∇F − W ⊗ V : pD2F. + +APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION +21 +A.2.2. Derivatives for the coupled Poisson Problem. In the case that +⟨e(V, y), p⟩ = +� +ˆΩ +(A(V )∇y1 · ∇p2 − y2p2) det(I +DV ) ++ (A(V )∇y2 · ∇p1 − p1F ◦ (id +V )) det(I +DV ), +as in Section 4.3, it holds that +⟨eV (0, y)[V ], p⟩ = +� +ˆΩ +A[V ]∇y1 · ∇p2 − div V y2p2 + A[V ]∇y2 · ∇p1 − div(V F)p1, +⟨ey(0, y)[η], p⟩ = +� +ˆΩ +∇η1 · ∇p2 − η2p2 + ∇η2 · ∇p1, +⟨eyV (0, y)[η, V ], p⟩ = +� +ˆΩ +A[V ]∇η1 · ∇p2 − div V η2p1 + A[V ]∇η2 · ∇p2, +⟨eyy(0, y)[η, ξ], p⟩ =0, +⟨eV V (0, y)[V, W], p⟩ = +� +ˆΩ +A[V, W]∇y1 · ∇p2 − D[V, W]y2p2 ++ A[V, W]∇y2 · ∇p1 − D[V, W]Fp1 +− div(V )W · ∇Fp1 − div(W)V · ∇Fp1 − W ⊗ V : D2Fp1. +A.2.3. Derivatives for the Eigenvalue Problem. In the case that +⟨e(V, (z, λ)), (q, µ)⟩ = +� +ˆΩ +(A(V )∇z · ∇q − λzq) det(I +DV ) ++ µ +� +1 − +� +ˆΩ +det(I +DV )z2 +� +, +as in section 2.3, it holds that +⟨eV (0, (z, λ))[V ], (q, µ)⟩ = +� +ˆΩ +A[V ]∇z · ∇q − λ div V zq − µ div V z2, +⟨ey(0, (z, λ))[(η, ˜η)], (q, µ)⟩ = +� +ˆΩ +∇η · ∇q − ληq − ˜ηzq − µzη, +⟨eV y(0, (z, λ))[V, (η, ˜η)], (q, µ)⟩ = +� +ˆΩ +A[V ]∇q · ∇η − div V ληq − div V ˜ηzq +− 2µ +� +ˆΩ +div V zη, +⟨eyy(0, (z, λ))[(η, ˜η), (ζ, ˜ζ)], (q, µ)⟩ = +� +ˆΩ +−˜ζηq − ˜ηζq − µζη, +⟨eV V (0, (z, λ))[V, W], (q, µ)⟩ = +� +ˆΩ +A[V, W]∇z · ∇q − D[V, W] +� +λzq − µz2� +. +For the energy, J(V, (z, λ)) = λ, it is clear that only the derivative in the second +component is non-vanishing and one has that +Jy(0, (z, λ)[(η, ˜η)] =˜η. +References. + +22 +K. DECKELNICK, P.J. HERBERT, AND M. HINZE +[ADJ21] +G. Allaire, C. Dapogny, and F. Jouve. “Shape and topology optimiza- +tion”. In: Differential Geometric Partial Differential Equations: Part II. +Vol. 22. Handbook of Numerical Analysis. 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In: Nature Methods 17 +(2020), pp. 261–272. + diff --git a/NdFAT4oBgHgl3EQfyB5j/content/tmp_files/load_file.txt b/NdFAT4oBgHgl3EQfyB5j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..18d84c62d63cb265efefec217e07d65eca16ab1e --- /dev/null +++ b/NdFAT4oBgHgl3EQfyB5j/content/tmp_files/load_file.txt @@ -0,0 +1,1129 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf,len=1128 +page_content='SHAPE OPTIMISATION IN THE W 1,∞ TOPOLOGY WITH THE ADMM ALGORITHM∗ KLAUS DECKELNICK†, PHILIP J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HERBERT‡, AND MICHAEL HINZE§ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We present a general shape optimisation framework based on the method of mappings in the W 1,∞ topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We propose steepest descent and Newton-like minimisation algorithms for the numerical solution of the respective shape optimisation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Our work is built upon previous work of the authors in Deckelnick, Herbert, and Hinze, ESAIM: COCV 28 (2022), where a W 1,∞ framework for star-shaped domains is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' To illustrate our approach we present a selection of PDE constrained shape optimisation problems and compare our findings to results from so far classical Hilbert space methods and recent p-approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' PDE constrained shape optimisation, Lipschitz functions, W 1,∞-descent MSC codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 35Q93, 49Q10, 49J20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In this work, we are interested in the numerical solutions of a number of shape optimisation problems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1) min J (Ω), Ω ∈ S, where S is a collection of admissible domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' This collection and the functional J will vary depending on the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' To find, at least local, minima of this problem, we will consider a descent method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' By this, we mean that, given Ω ∈ S, we seek V ∗ : Rn → Rn such that J ′(Ω)[V ∗] < 0 and set Ωnew = (id + αV ∗)(Ω) for some suitably chosen α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' To ensure that the map id +αV ∗ is a homeomorphism, it is sufficient to restrict to α to be small enough that α|DV ∗| < 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=', where, | · | is pointwise the spectral (operator) norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' While it is sufficient to take any sub- multiplicative norm, the spectral norm is convenient as it relates the Lipschitz and W 1,∞ semi-norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In the literature, it is common to seek V ∗ in a Hilbert space H which represents the negative gradient i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2) (V ∗, η)H = (−∇HJ (Ω), η)H := −J ′(Ω)[η] for all η ∈ H, or equivalently, one might seek (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3) V ∗ ∈ arg min �1 2∥V ∥2 H + J ′(Ω)[V ] : V ∈ H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' A crucial issue in this context is the regularity of the solution V ∗ to problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3), which strongly depends on the regularity of the current domain Ω as well as the choice of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' It for example is not clear whether id +αV ∗ defines a Lipschitz transformation for many frequent choices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In order to avoid these issues it was suggested in ∗Submitted to the editors January 23, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Funding: This work is part of the project P8 of the German Research Foundation Priority Programme 1962, whose support is gratefully acknowledged by the second and the third author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' †Otto-von-Guericke-University Magdeburg, Institut f¨ur Analysis und Numerik, Universit¨atsplatz 2, 39106 Magdeburg (klaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='deckelnick@ovgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='de) ‡Maxwell Institute for Mathematical Sciences, Department of Mathematics, Heriot-Watt Univer- sity, Edinburgh EH14 4AS, UK (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='herbert@hw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='uk) §Mathematical Institute, University of Koblenz, Universit¨atsstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 1, D-56070 Koblenz (hinze@uni- koblenz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='de) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='08690v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='OC] 20 Jan 2023 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' DECKELNICK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HERBERT, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HINZE [DHH22] to work directly in the space W 1,∞(Ω, Rd) and to consider the following problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='4) V ∗ ∈ arg min � J ′(Ω)[V ] : V ∈ W 1,∞(Ω, Rd), |DV | ≤ 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' in Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In [DHH22] this idea was analyzed and implemented for shape optimization problems involving star-shaped domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' It is the purpose of this paper to extend this approach to more general domains including the use of Newton–type methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' There continues to be rapid development in the mathematical and numerical analysis of shape optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The seminal works of Delfour and Zol´esio [DZ11], Sokolowski and Zol´esio, and the recent overview by Allaire, Dapogny, and Jouve [ADJ21] and the comprehensive bibliographies within provide an extensive overview of the topic of shape optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The analysis, both mathe- matical and numerical, of shape optimisation problems has an extensive history, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' [Bel+97;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' GM94;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' MS76;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Sim80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' While computational power has increased in recent years, it has encouraged further development of shape optimisation [SSW15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' SSW16a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' SW17], particularly fluid dynamical applications [Ben+15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Fis+17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Gar+15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Gar+18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' RCP16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HUU20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HSU21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' K¨uh+19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Sch+13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Many articles have considered different choices of inner products on Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' A variety of choices are presented in [HPS15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' One particularly interesting example is [ISW18] which uses a penalty to weakly enforce the Cauchy-Riemann equations however it only appears applicable in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Another category of interesting choices are reproducing kernel Hilbert spaces [ES18], which for certain kernels, one may provide an explicit shape gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' While in a Hilbertian setting, the work [OS21] considers non-smooth terms to ensure that a mesh does not become degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Some methods very much tar- get having a particularly good mesh, a particular example is the so-called pre-shape calculus [LS21b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' LS21a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The utilisation of Banach spaces for shape optimisation is gathering attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' To the best of our knowledge this was introduced in [DHH22] and considered W 1,∞ perturbations for a star-shaped setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The direction of steepest descent in a star- shaped setting has been linked to optimal transport [Her23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The star-shaped setting is frequently exploited [EHS07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' BCS21] to allow for a deeper analysis at the expense of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' A p-approximation to the infinity problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='4) is utilised in [M¨ul+21] to optimise a fluid dynamic problem using a p-Laplace relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Such a fluid problem is frequently discussed in shape optimisation as it is known [Pir74] that, for Stokes flow, the optimal shape should have a tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In [M¨ul+21], experiments demonstrate that the p-method will form a tip as opposed to in more classical Hilbertian methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The article [M¨ul+22] develops upon [M¨ul+21] to consider the computational scalability of a method closely related to a p-Laplace relaxation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Higher order methods are also of interest and will be considered in this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' second order methods have been considered in [SS22], utilising a linear version of the second shape derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We begin in Section 2 by outlining some necessary definitions and results for shape optimisation, mentioning the Lagrange approach from optimisation to write down first and second derivatives and providing examples which we will consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We then move onto a discussion about the discretisation of the infinity method in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Section 4 then provides numerical experiments of the previously described numerical experiments using the novel W 1,∞ method we discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Shape derivatives and Lagrangian calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In what follows we denote by D ⊂ Rd a convex hold-all domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We consider the shape optimisation problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1) min J (Ω), Ω ∈ S, where S is a collection of admissible domains such that Ω ⋐ D for all Ω ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Here, we use the symbol ⋐ to denote compactly contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' It is not difficult to see that id +V is a bi-Lipschitz transformation from D to D provided that V ∈ W 1,∞ 0 (D, Rd) with ∥DV ∥L∞ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Assuming that (id +V )(Ω) ∈ S for such V we say that J is shape differentiable at Ω if (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' [ADJ21, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1]) V �→ J � (id +V )(Ω) � is Fr´echet– differentiable at V = 0 as a mapping from W 1,∞ 0 (D, Rd) into R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' An update step in a descent algorithm based on the Fr´echet derivative of J will then seek a direction V ∈ W 1,∞ 0 (D, Rd) such that J ′(Ω)[V ] < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In order to determine the direction of steepest descent we are led to the problem of finding V ∗ ∈ W 1,∞ 0 (D, Rd) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2) V ∗ ∈ arg min � J ′(Ω)[V ] : V ∈ W 1,∞ 0 (D, Rd), |DV | ≤ 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' in D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Let us note that we are including the hold-all domain within this minimisation prob- lem for the determination of a direction of steepest descent, along with a Dirichlet boundary condition on the boundary of the hold-all domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Note that the fact that D is convex ensures that V is a Lipschitz-1 function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Using the direction (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2) within a descent algorithm hence requires the solution of a highly nontrivial constrained min- imisation problem which can be approximated at the discrete level with the help of an alternating direction method of multipliers (ADMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The above approach will lead to a first order method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' If J is twice shape differ- entiable, it is worthwhile considering a Newton–type approach as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' This can be achieved by replacing the minimisation problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2) by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3) min � t 2J ′′(Ω)[V, V ] + J ′(Ω)[V ] : V ∈ W 1,∞ 0 (D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Rd), |DV | ≤ 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' in D � , where t > 0 is a given damping factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Here, the evaluation of J ′′(Ω) is by no means straightforward and we will use the Lagrangian calculus described in the next subsection to carry out the calculations for the class of problems that we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Lagrangian framework for PDE–constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For the ease of exposition, let us consider a shape functional of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='4) J (Ω) = � Ω j(·, yΩ) dx, where j : D ×R → R is assumed to be sufficiently smooth and yΩ denotes the solution of a PDE posed in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We shall adapt the Lagrangian framework developed in Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='5 of [Hin+08] in order to compute J ′(ˆΩ) and J ′′(ˆΩ) at a fixed domain ˆΩ ⋐ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The main aspect of the Lagriangian method is to, in effect, decouple the state, yΩ, from, in the setting we consider, the shape, Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Denoting by B a small open neighbourhood of 0 in W 1,∞ 0 (D, Rd) we associate with V ∈ B the perturbed domain ΩV := (id +V )(ˆΩ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' By transforming to ˆΩ we find that, for the choice made in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='4), J (ΩV ) = J(V, yΩV ◦ (id +V )), where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='5) J(V, y) := � ˆΩ j(id +V, y) det(I +DV ) dˆx, 4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' DECKELNICK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HERBERT, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HINZE and we note that | det(I +DV )| = det(I +DV ) if |DV | is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The derivatives of this choice of J may be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In order to incorporate the PDE constraint we let y = yΩV ◦ (id +V ) and suppose that yΩV solves the given PDE problem on ΩV if and only if e(V, y) = 0 for some mapping e: B × X → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Here, X, Z are suitable function spaces on ˆΩ and we assume in what follows that ey(0, ˆy): X → Z is invertible, where ˆy = yˆΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' After choosing B smaller if necessary to apply an Implicit Function Theorem, there exists for every V ∈ B a unique y = y(V ) ∈ X such that e(V, y(V )) = 0, so that we may write J (ΩV ) = J(V, y(V )) where, in the context of optimal control, the map V �→ J (ΩV ) takes the role of a reduced cost functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In order to calculate the derivatives of J it is convenient to introduce the Lagrange functional L: X × B × Z∗ → R (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='6) L(y, V, p) = J(V, y) + ⟨p, e(V, y)⟩, so that J (ΩV ) = L(y(V ), V, p) for any p ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' If we denote by p(V ) the solution of Ly(y(V ), V, p(V )) = 0, one immediately obtains that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='7) J ′(ˆΩ)[V ] = LV (ˆy, 0, ˆp)[V ], where ˆp = p(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In a similar way one finds for the second derivative J ′′(ˆΩ)[V, W] = Lyy(ˆy, 0, ˆp)[y′(0)[V ], y′(0)[W]] + LyV (ˆy, 0, ˆp)[V, y′(0)[W]] +LV y(ˆy, 0, ˆp)[y′(0)[V ], W] + LV V (ˆy, 0, ˆp)[V, W], where y′(0)[V ] is the derivative of W �→ y(W) at W = 0 in direction V ∈ W 1,∞ 0 (D, Rd), which satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='8) ⟨p, ey(0, ˆy)[y′(0)[V ]]⟩ = −⟨p, eV (0, ˆy)[V ]⟩, for all p ∈ Z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For the implementation of the Newton-like method in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3), it is necessary to evaluate J ′′(ˆΩ)[V, W] many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In order to carry out the corresponding calculations as efficiently as possible we would like to avoid the frequent evaluation of y′(0)[W].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' To do, let us write J ′′(ˆΩ)[V, W] = ⟨h1, y′(0)[W]⟩ + ⟨h2, W⟩, where ⟨h1, y⟩ = Lyy(ˆy, 0, ˆp)[y′(0)[V ], y] + LyV (ˆy, 0, ˆp)[V, y], ⟨h2, W⟩ = LV y(ˆy, 0, ˆp)[y′(0)[V ], W] + LV V (ˆy, 0, ˆp)[V, W].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We then first define g ∈ Z∗ as the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='9) ⟨g, ey(0, ˆy)[y]⟩ = ⟨h1, y⟩ ∀y ∈ X and then set ⟨h3, W⟩ = −⟨g, eV (0, ˆy)[W]⟩, W ∈ W 1,∞ 0 (D, Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' This gives J ′′(ˆΩ)[V, W] = ⟨h1, y′(0)[W]⟩ + ⟨h2, W⟩ = ⟨g, ey(0, ˆy)[y′(0)[W]]⟩ + ⟨h2, W⟩ = −⟨g, eV (0, ˆy)[W]⟩ + ⟨h2, W⟩ = ⟨h2 + h3, W⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The evaluation of J ′′(ˆΩ)[V, ·] hence essentially requires the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='8) and of the adjoint problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Poisson problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' As a first PDE constraint we here consider the Pois- son problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We set X = H1 0(ˆΩ) and Z = X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Since we are in a reflexive setting, we use the canonical injection and identify Z∗ with X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' By yΩ we denote the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='10) − ∆yΩ = F in Ω, yΩ = 0 on ∂Ω for a given F ∈ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In particular, we find that yΩV ◦ (id +V ) is a solution of e(V, y) = 0 where e: B × H1 0(ˆΩ) → H−1(ˆΩ) is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='11) ⟨e(V, y), p⟩ = � ˆΩ (A(V )∇y · ∇p − F ◦ (id +V )p) det(I +DV )dˆx, p ∈ H1 0(ˆΩ), and A(V ) := (I +DV )−1(I +DV )−T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Derivatives of the map e may be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' With the Lagrange functional L(V, y, p) = � ˆΩ j(id +V, y) det(I +DV ) dˆx + ⟨e(V, y), p⟩ we deduce from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='7) the well–known formula J ′(ˆΩ)[V ] = LV (ˆy, 0, ˆp)[V ] = � ˆΩ (j(·, ˆy) div V + jx(·, ˆy) · V + A[V ]∇ˆy · ∇ˆp − ˆp div(FV )) dˆx, where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='12) A[V ] := I div V − DV − DV T , ˆy ∈ H1 0(ˆΩ) satisfies e(0, ˆy) = 0, and the adjoint ˆp ∈ H1 0(ˆΩ) satisfies Ly(ˆy, 0, ˆp) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='13) � ˆΩ ∇ˆp · ∇η dˆx = − � ˆΩ jy(·, ˆy)η dˆx for all η ∈ H1 0(ˆΩ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Bi-Laplace-type equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Let us next consider the minimsation of J as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='4) subject to the linear PDE of fourth order (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='14) ∆2yΩ = F in Ω, yΩ = ∆yΩ = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' If the boundary of Ω is sufficiently regular the above problem can be split into two second order Poisson problems by introducing −∆yΩ as an additional variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Let us note that this splitting is analytically useful to ensure that the shape derivative exists in the sense of [ADJ21, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1], due to the fourth order nature of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Let us comment that this need not be necessary since the shape differentiability, particularly boundedness in Lipschitz functions, with a fourth order constraint was demonstrated in [EH22] in a surface context, while [Las17] shows this for a fourth order eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' On the fixed domain, we set X = � H1 0(ˆΩ) �2 and Z = X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Again we will use the canonical injection to identify Z∗ with X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Posing the split formulation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='14) on ΩV and transforming it back onto ˆΩ in the same way as above we write the map e which represents the PDE constraint as, ⟨e(V, y), p⟩ = � ˆΩ � A(V )∇y1 · ∇p2 − y2p2 � det(I +DV ) dˆx + � ˆΩ � A(V )∇y2 · ∇p1 − F ◦ (id +V )p1 � det(I +DV ) dˆx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='15) 6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' DECKELNICK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HERBERT, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HINZE for all y = (y1, y2), p = (p1, p2) ∈ � H1 0(ˆΩ) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Derivatives of the map e may be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Similar to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='12) we obtain for the shape derivative J ′(ˆΩ)[V ] = � ˆΩ (j(·, ˆy1) div V + jx(·, ˆy1) · V + A[V ]∇ˆy1 · ∇ˆp2 − ˆy2ˆp2 div V ) dˆx + � ˆΩ (A[V ]∇ˆy2 · ∇ˆp1 − div(FV )ˆp1) dˆx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='16) where ˆy = (ˆy1, ˆy2) ∈ � H1 0(ˆΩ) �2 satisfies e(0, ˆy) = 0 and the adjoint ˆp = (ˆp1, ˆp2) ∈ (H1 0(ˆΩ))2 satisfies � ˆΩ ∇ˆp2 · ∇η1 dˆx = − � ˆΩ jy(·, ˆy1)η1 dˆx ∀η1 ∈ H1 0(ˆΩ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='17) � ˆΩ ∇ˆp1 · ∇η2 dˆx = � ˆΩ ˆp2η2 dˆx ∀η2 ∈ H1 0(ˆΩ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='18) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Optimisation of the first eigenvalue for the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Our aim is to apply the above Lagrangian framework also for the optimisation of the first Dirichlet eigenvalue of the Laplacian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='19) J (Ω) = λ1(Ω), where λ1(Ω) is defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='20) λ1(Ω) := inf �� Ω |∇z|2 : z ∈ H1 0(Ω), � Ω z2 = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' With the notation of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2 we again fix a ˆΩ ⋐ D which we now assume to be connected and set ΩV = (id +V )(ˆΩ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We transform the eigenvalue relation −∆zΩV = λzΩV in ΩV , zΩV = 0 on ∂ΩV together with the condition � ΩV z2 ΩV = 1 onto ˆΩ and write it in the form e(V, y) = 0, where e: B × X → Z, with X = H1 0(ˆΩ) × R, Z = X∗, and ⟨e(V, y), p⟩ = � ˆΩ (A(V )∇z · ∇q − λzq) det(I +DV ) dˆx +µ � 1 − � ˆΩ z2 det(I +DV ) dˆx � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='21) for y = (z, λ), p = (q, µ) ∈ H1 0(ˆΩ) × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Derivatives of the map e may be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Let ˆz ∈ H1 0(ˆΩ) be an eigenfunction to the first Dirichlet eigenvalue ˆλ with � ˆΩ ˆz2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Then we have for all p = (q, µ) ∈ H1 0(ˆΩ) × R ⟨eV (0, ˆy)[V ], p⟩ = � ˆΩ � A[V ]∇ˆz · ∇q − ˆλ div V ˆzq − µ div V ˆz2� dˆx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='22) ⟨ey(0, ˆy)[(η, ˜η)], p⟩ = � ˆΩ � ∇η · ∇q − ˆληq − ˜ηˆzq − 2µˆzη � dˆx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='23) APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION 7 where ˆy = (ˆz, ˆλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Since ˆλ is simple, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' ˆΩ is connected and ˆλ is the first Dirichlet eigenvalue [Gil+77, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='38], it can be shown that ey(0, ˆy): H1 0(ˆΩ) × R → H−1(ˆΩ) × R is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Thus we can write for V ∈ B J (ΩV ) = J(V, y(V )), where J(V, (z, λ)) = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The Lagrange functional is given by L(y, V, p) = λ+⟨e(V, y), p⟩ so that we derive with the help of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='12) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='24) J ′(ˆΩ)[V ] = LV (ˆy, 0, ˆp)[V ] = ⟨eV (0, ˆy)V, ˆp⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The adjoint ˆp = (ˆq, ˆµ) is given by the relation Ly(ˆy, 0, ˆp) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' ˜η + � ˆΩ � ∇η · ∇ˆq − ˆληˆq − ˜ηˆzˆq − 2ˆµˆzη � dˆx = 0 ∀(η, ˜η) ∈ H1 0(ˆΩ) × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We infer that � ˆΩ ˆzˆq dˆx = 1 as well as � ˆΩ � ∇η · ∇ˆq − ˆληˆq � dˆx = 2ˆµ � ˆΩ ˆzη dˆx ∀η ∈ H1 0(ˆΩ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Choosing η = ˆz we deduce that ˆµ = 0, so that ˆq is an eigenfunction for the eigenvalue ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Since ˆλ is simple and � ˆΩ ˆzˆq dˆx = 1 we infer that ˆq = ˆz and hence by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='24) that (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' [Hen06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HP06]) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='25) λ′ 1(ˆΩ)[V ] = � ˆΩ � A[V ]∇ˆz · ∇ˆz − ˆλ div V ˆz2� dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' It is known that the first eigenvalue scales with volume, as such we are interested in fixing the volume of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' While it is known that the minimiser of the first eigenvalue is a ball, the methodology is interesting and can be applied to more complicated eigenvalue problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Discretisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Our aim is to formulate a descent algorithm which produces in each step a polygonal domain and which replaces a possible PDE constraint with a corresponding finite element approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' To begin, let T 0 h be a triangulation of the hold–all domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We look for discrete directions of descent in the finite element spaces (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1) Vn h := � Vh ∈ C0(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Rd) : Vh|T ∈ P 1(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Rd), ∀T ∈ T n h , Vh = 0 on ∂D � , where P 1(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Rd) denotes polynomials of degree at most one on T with values in Rd and T n h is to be determined for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' With a polygonal initial domain, Ω0 which is a union of the triangles in the triangulation T 0 h , we will set Ωn+1 = (id +tnVn)(Ωn) for n ≥ 1, where tn ∈ (0, 1) is a step size and we will shortly explain how to choose Vn ∈ Vn h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' As well as updating the domain, the triangulation will also be updated, T n+1 h = {(id +tnVn)(T) : T ∈ T n h }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' By the choice of Vn h and the fact that Vn will satisfy |DVn| ≤ 1, it holds that the updated mesh will be admissible since tn ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Choice of descent direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Let n ≥ 0 be fixed and let us denote the polygonal domain ˆΩ = Ωn ⋐ D which is a union of triangles in T n h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For simplicity we will henceforth neglect the dependence on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We aim to find V ∗ h ∈ Vh such that V ∗ h ∈ arg min � J ′(ˆΩ)h[Vh] : Vh ∈ Vh, |DVh| ≤ 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' in D � = arg min �� D φ(DVh)dx + J ′(ˆΩ)h[Vh] : Vh ∈ Vh � , 8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' DECKELNICK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HERBERT, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HINZE where J ′(ˆΩ)h is a suitable approximation of J ′(ˆΩ) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2) φ(A) := � 0, |A| ≤ 1, ∞, |A| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We use an alternating direction method of multipliers (ADMM) approach in order to solve the above problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' To do so, we set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3) Qh := � qh ∈ L2(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Rd×d) : qh|T ∈ P 0(T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Rd×d), ∀T ∈ Th � and consider for a given τ > 0 the functional Lτ : Vh × Qh × Qh → R with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='4) Lτ(Vh, qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' λh) := � D φ(qh) + λh : (DVh − qh) + J ′(ˆΩ)h[Vh] + τ 2∥DVh − qh∥2 L2(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='Rd×d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The idea of ADMM is to alternatively minimise Lτ over qh and Vh, then perform an update to λh and repeat this until a certain quantity is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' More precisely, the algorithm has the following form: Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1 ADMM Choose V 0 h and λ0 h such that J ′(ˆΩ)h[V 0 h ] < ∞ Set R = ∞, j = 1 while R > tol do Find qj h ∈ arg min{Lτ(V j−1 h , qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' λj−1 h ) : qh ∈ Qh, |qh| ≤ 1} Find V j h ∈ arg min{Lτ(Vh, qj h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' λj−1 h ) : Vh ∈ Vh} Set λj h = λj−1 h + τ(DV j h − qj h) Set R = � ∥λj h − λj−1 h ∥2 L2(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='Rd×d) + τ 2∥DV j h − DV j−1 h ∥2 L2(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='Rd×d) � 1 2 Update j = j+1 end while Let us also mention [BM20] which considers more general ADMM methods with variable τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Particularly, in our experiments we make use of such an algorithm with variable τ, namely [BM20, Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We note that Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1 can also be applied to find a solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3) using the modified Lagrangian (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='5) LNewton τ (Vh, qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' λh) := Lτ(Vh, qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' λh) + t 2J ′′(ˆΩ)h[Vh, Vh], where J ′′(ˆΩ)h is a suitable approximation of J ′′(ˆΩ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Evaluation of J ′(ˆΩ)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' PDE–constrained shape optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Let us formulate suitable ap- proximations of the shape derivatives derived in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Given the polygonal domain ˆΩ we denote by Sh(ˆΩ) the space of linear finite elements on ˆΩ (resolved by a subtrian- gulation of Th) which vanish on ∂ ˆΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' If the constraint is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='10) we set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='6) J ′(ˆΩ)h[Vh] = � ˆΩ (j(·, ˆyh) div Vh + jx(·, ˆyh) · Vh + A[Vh]∇ˆyh · ∇ˆph − ˆph div(FVh)) dˆx, APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION 9 for Vh ∈ Vh, where ˆyh, ˆph ∈ Sh(ˆΩ) satisfy � ˆΩ ∇ˆyh · ∇ηh dˆx = � ˆΩ Fηh dˆx ∀ηh ∈ Sh(ˆΩ), � ˆΩ ∇ˆph · ∇ηh dˆx = − � ˆΩ jy(·, ˆyh)ηh dˆx ∀ηh ∈ Sh(ˆΩ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='7) On the other hand, if the constraint is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='14) then we let J ′(ˆΩ)h[Vh] = � ˆΩ (j(·, ˆyh,1) div Vh + jx(·, ˆyh,1) · Vh + A[V ]∇ˆyh,1 · ∇ˆph,2 − ˆyh,2ˆph,2 div Vh) dˆx + � ˆΩ (A[Vh]∇ˆyh,2 · ∇ˆph,1 − div(VhF)ˆph,1) dˆx, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='8) for Vh ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Here, ˆyh = (ˆyh,1, ˆyh,2) ∈ (Sh(ˆΩ))2 satisfies � ˆΩ ∇ˆyh,1 · ∇ηh dˆx = � ˆΩ ˆyh,2ηh dˆx ∀ηh ∈ Sh(ˆΩ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='9) � ˆΩ ∇ˆyh,2 · ∇ηh dˆx = � ˆΩ Fηh dˆx ∀ηh ∈ Sh(ˆΩ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='10) while ˆph = (ˆph,1, ˆph,2) ∈ (Sh(ˆΩ))2 satisfies � ˆΩ ∇ˆph,2 · ∇ηh dˆx = − � ˆΩ jy(·, yh,1)ηh dˆx ∀ηh ∈ Sh(ˆΩ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='11) � ˆΩ ∇ˆph,1 · ∇ηh dˆx = � ˆΩ ˆph,2ηh dˆx ∀ηh ∈ Sh(ˆΩ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='12) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Optimisation of the first eigenvalue for the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For a given polygonal domain we determine ˆzh ∈ Sh(ˆΩ) and ˆλh > 0 such that � ˆΩ ˆy2 h dx = 1 and ˆλh = inf �� ˆΩ |∇ˆzh|2 dˆx : zh ∈ Sh(ˆΩ), � ˆΩ ˆz2 h dˆx = 1 � = � ˆΩ |∇ˆzh|2 dˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Supposing that the eigenvalue λh is simple we let, recalling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='25) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='13) J ′(ˆΩ)h[Vh] = � ˆΩ � A[Vh]∇ˆzh · ∇ˆzh − ˆλhˆz2 h div Vh � dˆx for Vh ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We now provide numerical experiments for the applications we have described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In the integrals for the energy we use quadrature of order 2, while for the shape derivatives, we are using the order which is automatically decided by the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' As mentioned above we will solve the state and adjoint PDEs with a finite element approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The finite element approximation is performed with DUNE [Bas+21], making particular use of the DUNE Python bindings [DNK20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' DN18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We consider a construction of update direction using four different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Our approaches will be: The direction of steepest descent method using the W 1,∞-topology, construc- ted with an adaptive ADMM method, as mentioned after Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' This will be referred to as p = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' DECKELNICK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HERBERT, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HINZE A Newton-type direction, which will be a discrete minimiser of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3) for a given t > 0, referred to as the Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Much like the p = ∞ case above, this will be constructed with an adaptive ADMM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' To compare against existing approaches, for p = 2, 4, we will consider the minimiser of Vh ∋ Vh �→ J ′(Ω)h + 1 p � D |DVh|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In the case that p = 2, this is seen to coincide with the discrete case of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3) with H = H1 0(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Rd) with the Dirichlet inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We will refer to these cases by their p value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The discrete functions produced by the p = 2 and p = 4 methods will be normalised so that they have a W 1,∞(D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Rd) semi-norm of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' This normalisation is performed so that we need not check whether the mesh has overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For each of the experiments, we will set the hold-all domain to be the box D = (−2, 2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' With these directions, we will move the vertices of our mesh according to an Armijo step rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We will stop after 20 shape updates have been made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In most cases the domain has become close to stationary at this point and the Armijio step-size has become rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The energy along the iterations will be plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In the case that the minimiser is known, the origin will be offset by the known value, when the minimiser is not known, the origin will be offset by the smallest value attained in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Minimisation without a PDE constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Here we will consider that there is no PDE constraint, so that the map e need not be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We comment that the no PDE example may be derived as an example from the following Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2 where one chooses right hand side data F = 0 so that the state constraint guarantees y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For this experiment, the main contributions to the error is that induced by the quadrature rules when calculating the energy and the shape derivative, as well as the direction of descent with the chosen method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' No PDE experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For this problem, we consider (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1) j(x, y) = −Z(x) where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2) Z(x) = � � � � � � � � � cos(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='5πx1) cos(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='5πx2) |x1| and |x2| ≤ 1, π 4 (1 − x2 1) |x1| > 1 and |x2| < 1, π 4 (1 − x2 2) |x1| < 1 and |x2| > 1, π 4 (2 − x2 1 − x2 2) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For the Newton direction we take t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='0625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' One expects the square (−1, 1)2 to be a minimiser of J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' It holds that J � (−1, 1)2� = − 16 π2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We start with the initial domain (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='5, −1) × (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The triangulation of the domain and hold-all is displayed in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In Figure 1b, the energy of shapes along the minimising sequences we produce are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In Figure 2, the meshes of the final domains Ω for each of the methods are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' No PDE experiment 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For this problem, we consider (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3) j(x, y) = 1 2Z(x)2 where for given ϵ > 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='4) Z(x) = � (x1 + x2)2 + ϵ + � (x1 − x2)2 + ϵ APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION 11 (a) Initial domain for the first No PDE ex- periment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1, with (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='5, −1) × (−1, 1) in red and the hold-all, (−2, 2)2 in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 0 5 10 15 20 Iterations 10−5 10−4 10−3 10−2 10−1 100 Energy + 16 π2 p = 2 p = 4 p = ∞ first order p = ∞ second order (b) Graph of the energy for the iterates in the first No PDE experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' It is seen that the Newton-type method is energetically performing the best while the first order infinity method appears to struggle compared to the traditional p = 2 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 1: Initial mesh and graph of the energy for the experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' is a smooth approximation to |x1 + x2| + |x1 − x2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For the Newton direction we take t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' A very similar experiment with the non-smooth energy was considered in [DHH22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' This smooth approximation is used because we intend to employ the Newton method for which it would be useful to have (weak) second derivatives of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Without any constraint, we know that the theoretical minimiser is degenerate, a measure zero set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' To avoid this, we will fix the area to be constrained equal to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We expect this to have minimiser close to the square (−1, 1)2 which, for ϵ = 0, has energy 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Our directions of descent will only preserve the area constraint in a linear sense by restricting to V with � Ω div V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We will perform a projection step to fix the area after each update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We take ϵ = 10−4 and start with an approximation of a ball of radius 2 √π at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The triangulation of the domain and hold-all is displayed in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In Figure 3b, the energy of shapes along the minimising sequences we produce are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In Figure 4, the meshes for the final domains Ω for each of the methods are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' A Poisson problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Poisson experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For our first experiment we consider j(x, y) = y and F(x) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='5(x1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='5 − x2 2)2 + x2 1 + x2 2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' This has appeared in [Etl+20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HL21], for example, as a benchmark for the comparisons of shape optimisation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For the Newton direction we take t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The minimising shape is not explicitly known, however it appears to be a shape not so dissimilar to a kidney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Similarly, the energy of a minimiser is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We start with an approximation of an ellipse with semiaxes 2 √π and 1 √π centred at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The triangulation of the domain and hold-all is displayed in Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In Figure 5b, the energy of shapes along the minimising sequences we produce are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In Figure 6, the meshes for the final domains Ω for each of the methods are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 12 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' DECKELNICK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HERBERT, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HINZE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 2: The meshes of Ω for the final domains produced in the first No PDE experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1: top left to bottom, p = 2, 4, ∞ and second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Due to symmetry of the result, we show only a half of each mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' It is seen that none of the methods correctly captures the corners which are expected from the minimising shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' This is unsurprising as there is not a particularly large influence of the energy around the corners, where Z is quite close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The Newton-type method is the closest to forming corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Poisson experiment 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For this experiment we consider j(x, y) = 1 2(y− yd(x))2 where yd(x) = 4 π − |x|2 and F = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Let us note that −∆yd = 4F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For the Newton direction we take t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' This experiment will be equipped with an area constraint that the domain has fixed area 4 - we will use the same linear constraint on the update direction and projection as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In this setting, we expect the minimiser to be given by the ball of radius 2 √π at the origin which has energy 6 π2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We start with the square (−1, 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The triangulation of the domain and hold-all is displayed in Figure 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In Figure 7b, the energy of shapes along the minimising sequences we produce are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In Figure 8, the meshes for the final domains Ω for each of the methods are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' It is worth noting that when larger values of t were taken during testing, the Newton-type method struggled to perform well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' With the Newton method, once the shape was sufficiently close to a ball, the directions generated by ADMM would rotate the almost-ball by large angles which caused large deformations of the mesh in the hold-all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' A coupled Poisson problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For this experiment we will consider j(x, y) = 1 2(y1−yd(x))2 where yd(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='05+(1−x2 1)3(1−x2 2)3 and F(x) = ∆2 � (1 − x2 1)3(1 − x2 2)3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For the Newton direction we take t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='0625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' This experiment will be equipped with an area constraint that the domain has fixed area 4 - we will use the same linear con- straint on the update direction and projection as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' One would expect the minimiser to be relatively close to the square (−1, 1)2 which should have energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We start with the square (−1, 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The triangulation of the domain and hold-all is displayed in Figure 9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In Figure 9b, the energy of shapes along the minimising sequences we produce are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In Figure 10, the meshes for the final domains Ω for each of the methods are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION 13 (a) Initial domain for the second No PDE ex- periment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2, an approximation of the ball of radius 2 √π at the origin, is in red and the hold-all, (−2, 2)2 in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 0 5 10 15 20 Iterations 10−4 10−3 10−2 10−1 |Energy - 4| p = 2 p = 4 p = ∞ first order p = ∞ second order (b) Graph of the energy for the iterates in the second No PDE experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Let us note that the bumps in the graphs are due to the absolute value we are using.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' It is seen that the Newton method has the closest energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 3: Initial mesh and graph of the energy for the experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 4: The meshes of Ω for the final domains produced in the second No PDE experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2: left to right, p = 2, 4, ∞ and second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Due to symmetry of the result, we show only a half of each mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Generally, the first order methods provide a rather good approximation of the expected minimising shape of a square with p = ∞ having the most regular triangles around the corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The Newton method does not quite form the corners we expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Optimisation of the first eigenvalue for the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We use the function eigs from the module sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='linalg in scipy [Vir+20] to find pairs (v, λh) ∈ RN × R such that Bv = λhMv, where B ∈ RN×N is the stiffness matrix and M ∈ RN×N is the mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' This experiment will be equipped with an area constraint that the domain has fixed area 4 - we will use the same linear constraint 14 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' DECKELNICK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HERBERT, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HINZE (a) Initial domain for the first Poisson exper- iment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1, an approximation of the ellipse with semiaxes 2 √π and 1 √π at the origin, is in red and the hold-all, (−2, 2)2 in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 0 5 10 15 20 Iterations 10−7 10−6 10−5 10−4 10−3 10−2 Energy + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='09377080539770344 p = 2 p = 4 p = ∞ first order p = ∞ second order (b) Graph of the energy for the iterates in the first Poisson experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We see that p = ∞ does not perform as well as the traditional p = 2 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The Newton method performs well energetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 5: Initial mesh and graph of the energy for the experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 6: The meshes of Ω for the final domains produced in the first Poisson experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1: top left to bottom, p = 2, 4, ∞ and second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Due to symmetry of the result, we show only a half of each mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We see that all of the shapes are rather similar, with the p = ∞ being a slight outlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The triangles for both p = ∞ and the Newton method appear to be more regularly spaced at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' on the update direction and projection as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For the Newton direction we take t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In this setting, the minimiser is known to be the ball of radius 2 √π which has λ1(B 2 √π ) = β2 0,1 π 4 ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='542103633884308, where β0,1 is the first zero of the 0th Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We start with the square (−1, 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The triangulation of the domain and hold-all APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION 15 (a) Initial domain for the second Poisson ex- periment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2, (−1, 1)2 is in red and the hold-all, (−2, 2)2 in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 0 5 10 15 20 Iterations 10−4 10−3 10−2 Energy - 6 π2 p = 2 p = 4 p = ∞ first order p = ∞ second order (b) Graph of the energy for the iterates in the second Poisson experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We see that all the methods give roughly the same energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 7: Initial mesh and graph of the energy for the experiment in Secetion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' is displayed in Figure 11a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In Figure 11b, the energy of shapes along the minimising sequences we produce are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In Figure 12, the meshes for the final domains Ω for each of the methods are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Let us further elaborate on Figure 12, particularly the mesh produced by the infinity method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We note that the mesh in the centre for the infinity method appears less regular than the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' This appears to occur due to a somewhat undesirable approximation of the shape derivative, which should be concentrated at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The undesirable approximation is due to the FEM approximation giving interior contributions to the shape derivatives, so-called spurious contributions, in the limit of the mesh becoming infinitely fine this should disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Investigation suggests that these are more apparent in the infinity method but disappear when using shape derivatives with only boundary contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' This sort of expert knowledge is often exploited in the literature and appeared in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' [SSW16b] to ’delete’ contributions to the shape derivative from nodes on the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Outlook: Convergence of the infinity method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Here we consider the ex- perimental convergence of the example provided in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2 as h → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In this ex- periment, we will compute a quantity which we will refer to as the discrete complemen- tary Hausdorff distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The Hausdorff complementary distance is utilised in [DZ11] to give a metric on a space of shapes and is given by ρc H(Ω1, Ω2) := supx′∈D |d∁Ω1(x′)− d∁Ω2(x′)|, where the function d∁Ω : D → R is given by d∁Ω(x) = infx′∈D\\Ω |x − x′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' To calculate a discrete Hausdorff complementary distance, we consider that the exact minimiser, Ω∗, is known to be the ball of radius 2 √π at the origin and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1) d∁Ω∗(x) = min � 0, 2 √π − |x| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For each h > 0, let us denote by Ωh the final domain produced in each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Let {xi}N i=1 be the vertices of the triangulation of D and let {yi}n i=1 be the vertices which lie on the boundary of Ωh, then we set (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2) dh ∁Ωh(xi) = inf j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=',n |xi − yj| for xi ∈ Ωh, otherwise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 16 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' DECKELNICK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HERBERT, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HINZE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 8: The meshes of Ω for the final domains produced in the second Poisson ex- periment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2: top left to bottom, p = 2, 4, ∞ and second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Due to symmetry of the result, we show only a quarter of each mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We see that both p = 2 and p = 4 have very degenerate elements where the corners of the original domain were.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Both p = ∞ and Newton methods do not have these degenerate triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We notice that the triangles which previously made up the corners of the original domain are rather regular in the Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For p = ∞, there are many triangles which have obtuse angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' (a) Initial domain for the coupled Poisson ex- periment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3, (−1, 1)2 is in red and the hold-all, (−2, 2)2 in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 0 5 10 15 20 Iterations 10−7 10−6 10−5 10−4 10−3 Energy - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='0014111218894029705 p = 2 p = 4 p = ∞ p = ∞ second order (b) Graph of the energy for the iterates in the coupled Poisson experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We see that p = ∞ outperforms the finite p experiments, but the Newton method is en- ergetically performing the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 9: Initial mesh and graph of the energy for the experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION 17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 10: The meshes of Ω for the final domains produced in the coupled Poisson experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3: top left to bottom, p = 2, 4, ∞ and second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Due to symmetry of the result, we show only a quarter of each mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' To the eye, the first order methods are all seemingly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The Newton method appears to have found a more pronounced shape than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' (a) Initial domain for the Eigenvalue experi- ment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='4, (−1, 1)2 is in red and the hold-all, (−2, 2)2 in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 0 5 10 15 20 Iterations 10−2 10−1 Energy - π 4 β2 0,1 p = 2 p = 4 p = ∞ first order p = ∞ second order (b) Graph of the energy for the iterates in the Eigenvalue experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We see that all the methods are performing roughly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 11: Initial mesh and graph of the energy for the experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We then calculate our discrete Hausdorff complementary distance to be given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3) dh(Ωh, Ω∗) = sup i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=',N |d∁Ω∗(xi) − dh ∁Ωh(xi)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We start with Ω = (−1, 1)2 and D = (−2, 2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The mesh for the coarsest refine- ment is given in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For each subsequent refinement, each triangle in the mesh 18 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' DECKELNICK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HERBERT, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HINZE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 12: The meshes of Ω for the final domains produced in the Eigenvalue experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='4: top left to bottom, p = 2, 4, ∞ and second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Due to symmetry of the result, we show only a quarter of each mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We see that spikes appear for p = 2 and p = 4 where the corners were for the original shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The mesh for p = ∞ appears rather regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We comment that the Newton method has lead to the mesh rotating when compared to its original orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 13: Initial domain with coarsest mesh for the experimental convergence experi- ment for the Poisson problem in Section 5, in red is the initial domain (−1, 1)2 and the hold all (−2, 2)2 is in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' is replaced with four equivalent triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In Figure 14, the energy of the shapes along the minimising sequence is given along with a graph plotting the final energy against the mesh size (as measured on the red part of the initial mesh) along with a reference line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' While the convergence of the energy appears to be quadratic, it is also interest- ing to consider some convergence of the shape itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' To measure this convergence we consider the discrete counterpart to the Hausdorff complementary distance which is detailed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We see this convergence to be essentially linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Also in Figure 14 we APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION 19 0 5 10 15 20 Iterations 10−4 10−3 10−2 10−1 |Energy - 6 π2 | p = ∞ refines = 0 p = ∞ refines = 1 p = ∞ refines = 2 p = ∞ refines = 3 p = ∞ refines = 4 10−1 Initial h 10−4 10−3 10−2 Energy - 6 π2 Experiment Reference h2 10−1 Initial h 10−2 Discrete Hausdorff Complementary Distance Experiment Reference h Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 14: Graphs of the energy (left and middle) and discrete Hausdorff complementary distance for the final domains (right) in the experimental convergence experiment for the Poisson problem in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' It is seen that the energies converge like h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' With the discretisation uses, one expects that the state would converge like order h2 in the L2 norm, however this convergence need not directly translate to the energy as the domain is also approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For the discrete Hausdorff complementary distance convergence of order approximately h is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 15: The meshes of Ω for the final domains produced with the first order method in the experimental convergence experiment for the Poisson problem in Section 5: top left to bottom are further refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Due to symmetry of the result, we show only a quarter of each mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' show this discrete Hausdorff complementary distance against the (initial) mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In Figure 15, the meshes for the final domains Ω for each refinement is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In this work we have introduced two new frameworks in which to perform shape optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' These methods are more applicable to physical scenarios than the previous works involving W 1,∞ as they are not restricted to star-shaped domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' While our examples had star-shaped final domains, our framework allows for more general shapes and is more readily applicable to industrial problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' From the experiments, it was seen that our introduced first order method did not necessarily perform well energetically, however the meshes the method produced are regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The second order method we introduced performs well both energetically and in terms of 20 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' DECKELNICK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HERBERT, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' HINZE the regularity of the mesh, a downside however is the need to tune the damping parameter t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' An analysis of the steepest descent method using the W 1,∞–topology is work in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' The authors wish to extend their gratitude to Andreas Dedner for providing useful insight into the use of the Python bindings for DUNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' This work is part of the project P8 of the German Research Foundation Priority Programme 1962, whose support is gratefully acknowledged by the second and the third author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='H acknowledges the support of EPSRC (grant EP/W005840/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Calculations for the second shape derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Here we collect the derivatives of J and e for the examples in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' These derivatives are particularly useful for the calculation of the second shape derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' All of the maps and functions we consider are sufficiently smooth that we may exchange the order of differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' It will be convenient to define D[V, W] := (div(V ) div(W) − Tr(DV DW)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Derivatives for the energy functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Let us consider J as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='5), that is J(V, y) := � ˆΩ j(id +V, y) det(I +DV ) for some fixed ˆΩ ⋐ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' When j is twice differentiable, it holds that JV (0, y)[V ] = � ˆΩ div V j(·, y) + jx(·, y) · V, Jy(0, y)[η] = � ˆΩ jy(·, y) η, JyV (0, y)[η, V ] = � ˆΩ div V jy(·, y) η + jyx(·, y) · V η, Jyy(0, y)[η, ξ] = � ˆΩ jyy(·, y)ηξ, JV V (0, y)[V, W] = � ˆΩ D[V, W]j(·, y) + div V jx(·, y) · W + div Wjx(·, y) · V + jxx(·, y)V · W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Derivatives for PDE constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Here, we collect the derivatives for the maps e which appear in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Recall that we define A(V ) := (I +DV )−1(I +DV )−T and A[V ] := I div V − DV − DV T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' We furthermore define A[V, W] :=D[V, W] I − div(V )DW − div(V )DW T − div(W)DV + (DWDV + DV DW) + DV DW T − div(W)DV T + DWDV T + (DWDV + DV DW)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Derivatives for the Poisson Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In the case that ⟨e(V, y), p⟩ = � ˆΩ (A(V )∇y · ∇p − F ◦ (id +V )p) det(I +DV ) as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2, then it holds that ⟨eV (0, y)[V ], p⟩ = � ˆΩ A[V ]∇y · ∇p − div(V F)p, ⟨ey(0, y)[η], p⟩ = � ˆΩ ∇η · ∇p, ⟨eyV (0, y)[η, V ], p⟩ = � ˆΩ A[V ]∇η · ∇p, ⟨eyy(0, y)[η, ξ], p⟩ = 0, ⟨eV V (0, y)[V, W], p⟩ = � ˆΩ A[V, W]∇y · ∇p − D[V, W]Fp − div(V )pW · ∇F − div(W)pV · ∇F − W ⊗ V : pD2F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' APPLICATIONS OF A NOVEL W 1,∞ APPROACH TO SHAPE OPTIMISATION 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Derivatives for the coupled Poisson Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In the case that ⟨e(V, y), p⟩ = � ˆΩ (A(V )∇y1 · ∇p2 − y2p2) det(I +DV ) + (A(V )∇y2 · ∇p1 − p1F ◦ (id +V )) det(I +DV ), as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3, it holds that ⟨eV (0, y)[V ], p⟩ = � ˆΩ A[V ]∇y1 · ∇p2 − div V y2p2 + A[V ]∇y2 · ∇p1 − div(V F)p1, ⟨ey(0, y)[η], p⟩ = � ˆΩ ∇η1 · ∇p2 − η2p2 + ∇η2 · ∇p1, ⟨eyV (0, y)[η, V ], p⟩ = � ˆΩ A[V ]∇η1 · ∇p2 − div V η2p1 + A[V ]∇η2 · ∇p2, ⟨eyy(0, y)[η, ξ], p⟩ =0, ⟨eV V (0, y)[V, W], p⟩ = � ˆΩ A[V, W]∇y1 · ∇p2 − D[V, W]y2p2 + A[V, W]∇y2 · ∇p1 − D[V, W]Fp1 − div(V )W · ∇Fp1 − div(W)V · ∇Fp1 − W ⊗ V : D2Fp1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Derivatives for the Eigenvalue Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In the case that ⟨e(V, (z, λ)), (q, µ)⟩ = � ˆΩ (A(V )∇z · ∇q − λzq) det(I +DV ) + µ � 1 − � ˆΩ det(I +DV )z2 � , as in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='3, it holds that ⟨eV (0, (z, λ))[V ], (q, µ)⟩ = � ˆΩ A[V ]∇z · ∇q − λ div V zq − µ div V z2, ⟨ey(0, (z, λ))[(η, ˜η)], (q, µ)⟩ = � ˆΩ ∇η · ∇q − ληq − ˜ηzq − µzη, ⟨eV y(0, (z, λ))[V, (η, ˜η)], (q, µ)⟩ = � ˆΩ A[V ]∇q · ∇η − div V ληq − div V ˜ηzq − 2µ � ˆΩ div V zη, ⟨eyy(0, (z, λ))[(η, ˜η), (ζ, ˜ζ)], (q, µ)⟩ = � ˆΩ −˜ζηq − ˜ηζq − µζη, ⟨eV V (0, (z, λ))[V, W], (q, µ)⟩ = � ˆΩ A[V, W]∇z · ∇q − D[V, W] � λzq − µz2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' For the energy, J(V, (z, λ)) = λ, it is clear that only the derivative in the second component is non-vanishing and one has that Jy(0, (z, λ)[(η, ˜η)] =˜η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' References.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 22 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' DECKELNICK, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Cimrman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Henriksen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Quintero, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Harris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Archibald, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Ribeiro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' Pedregosa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' van Mulbregt, and SciPy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='0 Contributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' “SciPy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content='0: Fundamental Algorithms for Scientific Computing in Python”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' In: Nature Methods 17 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} +page_content=' 261–272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFAT4oBgHgl3EQfyB5j/content/2301.08690v1.pdf'} diff --git a/NtAyT4oBgHgl3EQfUPfO/content/2301.00123v1.pdf b/NtAyT4oBgHgl3EQfUPfO/content/2301.00123v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4f16a6f1a6a248bccdfcc73fe43c914dbe92208a --- /dev/null +++ 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Biswal1*, Sushree S. Mishra1† and K. Sridhar2‡ +1. Department of Physics, Ravenshaw University, +Cuttack, 753003, India. +2. School of Arts and Sciences, Azim Premji University, +Sarjapura, Bangalore, 562125, India. +Abstract +Due to the heavy-quark symmetry of Non-Relativistic Quantum Chromodynamics +(NRQCD), the cross-section for the production of ηc can be predicted. This NRQCD +prediction when confronted with data from the LHCb is seen to fail miserably. We +address this LHCb ηc anomaly in this paper using a new approach called modified +NRQCD, an approach that has been shown to work extremely well for studying +J/ψ, ψ′ and χc production at the LHC. We show, in the present paper, that the +predictions for ηc production agrees very well with LHCb measurements at the +three different values of energy that the experiment has presented data for. Modi- +fied NRQCD also explains the intriguing agreement of the LHCb ηc data with the +colour-singlet prediction. The remarkable agreement of the theoretical predictions +with the LHCb data suggests that modified NRQCD is closer to apprehending the +true dynamics of quarkonium production. +One of the most important and daunting problems in Quantum Chromodynamics +(QCD) is the understanding of how quarks form physical bound states – the hadrons. +One small corner of this unnavigated terrain where one can make some headway is in +the study of quarkonia – when a heavy quark and anti-quark come together to form +a neutral meson. A very heavy quark like the top decays before it can form a bound- +state so when we study quarkonia we are interested in charmonium and bottomonium +systems. In these systems, the relative velocity, v, of the Q ¯Q pair is small and, there- +fore, the bound state can be studied in a non-relativistic approximation. The effective +field theory that has been formulated to study such systems is called Non-Relativistic +Quantum Chromodynamics (NRQCD) [1] which is derived from the QCD Lagrangian +by neglecting all states of momenta much larger than the heavy quarkonium mass, M +*E-mail: sudhansu.biswal@gmail.com +†Email: sushreesimran.mishra97@gmail.com +‡E-mail: sridhar.k@apu.edu.in +1 +arXiv:2301.03158v1 [hep-ph] 9 Jan 2023 + +and accounting for this exclusion by adding new interaction terms yielding the effec- +tive Lagrangian. +The quarkonium state admits of a Fock-state expansion in orders of v. At leading +order, the QQ state is in a colour-singlet state but at O(v), it can be in a colour-octet +state which is connected to the physical J/ψ state through a non-perturbative gluon +emission. +The cross section for production of a quarkonium state H of mass M in NRQCD +can be factorised as: +σ(H) = +� +n={α,S,L,J} +Fn +M dn−4⟨OH +n (2S+1LJ)⟩, +(1) +where Fn’s are the short-distance coefficients and On are operators of naive dimension +dn, describing the long-distance effects. These non-perturbative matrix elements are +guaranteed to be energy-independent due to the NRQCD factorization formula, so that +they may be extracted at a given energy and used to predict quarkonium cross-sections +at other energies. +NRQCD found much success in explaining the systematics of charmonium produc- +tion at the Fermilab Tevatron [2] in contrast to the then existing model of quarkonium +production – the colour-singlet model [3]. But NRQCD does not predict the normali- +sation of the pT distributions because of the unknown non-perturbative parameters so +other tests of NRQCD were needed to validate it [4]. Of these, the polarisation of the +J/ψ provides an important test: NRQCD predicts [5, 6] a fully transversely polarised +J/ψ at large pT. The Tevatron experiments found no evidence for this [7]. +Another important test of NRQCD comes from the study of ηc production. The +heavy-quark symmetry of NRQCD provides a set of relations which connect non- +perturbative parameters of different resonances so a measurement of a given state +yields information on the non-perturbative parameters of another state related to the +former by heavy-quark symmetry. In particular, the non-perturbative parameters re- +quired for ηc production can be obtained, using heavy quark symmetry, from the pa- +rameters of J/ψ production. This approach has been used to predict the ηc production +cross-section at the Tevatron [8] and at the LHC [9]. 1. +Like the prediction of polarisation, the prediction of the ηc cross-section is a defini- +tive test of NRQCD. Just as NRQCD fails miserably in predicting the J/ψ polarisation, +it also gets the ηc cross-section completely wrong [11]: the NRQCD prediction is com- +pletely at variance with the cross-section measured by the LHCb experiment at three +different values of centre- of-mass energy. It is this LHCb ηc anomaly that we address +in this paper. +1hc production at the Tevatron [10] and at the LHC [9] has also been studied using this approach +2 + +The Fock space expansion of the physical ηc, which is a 1S0 (JPC = 0−+) state, is: +|ηc⟩ = O(1) +���QQ[1S[1] +0 ] +� ++ O(v2) +���QQ[1P [8] +1 ] g +� ++ O(v4) +���QQ[3S[8] +1 ] g +� ++ · · · . +(2) +In the above expansion the colour-singlet 1S0 state contributes at O(1). As the P-state +production is itself down by factor of O(v2) both the colour-octet 1P1 and 3S1 channels +effectively contribute at the same order. The colour-octet state 1P [8] +1 +(3S[8] +1 ) becomes a +physical ηc by emitting a gluon in an E1 (M1) transition. Keeping terms up-to O(α3 +sv7) +the ηc production cross section can be written as: +σ(ηc) += +F1[1S0] ⟨0| Oηc +1 [1S0] |0⟩ ++F8[1P1] +M 2 +⟨0| Oηc +8 [1P1] |0⟩ + F8[3S1] ⟨0| Oηc +8 [3S1] |0⟩ , +(3) +where the coefficients, Fn’s, are the cross sections for the production of cc pair in the +respective angular momentum and colour states. +To predict the ηc cross-section, the values of ⟨Oηc +n ⟩’s obtained from the experimen- +tally predicted values ⟨OJ/ψ +n +⟩’s using relations given by heavy-quark symmetry: +⟨0| Oηc +1 [1S0] |0⟩ += +1 +3 ⟨0| OJ/ψ +1 +[3S1] |0⟩ (1 + O(v2)), +⟨0| Oηc +8 [1P1] |0⟩ += +⟨0| OJ/ψ +8 +[3P0] |0⟩ (1 + O(v2)), +⟨0| Oηc +8 [3S1] |0⟩ += +⟨0| OJ/ψ +8 +[1S0] |0⟩ (1 + O(v2)). +(4) +With the values of the non-perturbative parameters determined as above, the ηc +cross-section is a prediction of NRQCD that can be directly tested in experiments. The +pT-dependence of the ηc cross-section has been measured at the LHCb at three different +energies. In Fig. 1, we compare the predictions of NRQCD obtained, using the heavy- +quark symmetry relations, with the ηc pT distribution at √s = 13 TeV. The disagreement +between NRQCD predictions and the experimental values of the cross-section is huge. +The other conundrum is that the colour-singlet prediction alone seems to be well in +agreement with the data in total contrast to the situation with J/ψ production. The +results presented in Fig. 1 are not new: this anomaly, as we mentioned earlier, has +been known and noted in the literature for several years now [11]; we have presented +the results in Fig. 1 to draw attention to the magnitude of the discrepancy and to also +bring to the fore the mysterious agreement of the colour-singlet prediction with the +data. +In a recently proposed modification of NRQCD [12], which we have named mod- +ified NRQCD, we had suggested that the colour-octet c¯c state can radiate several soft +perturbative gluons – each emission taking away little energy but carrying away units +3 + +10-3 +10-1 + 10 +103 +105 +107 +109 + 6 + 8 + 10 + 12 + 14 + 16 +2.0 < y < 4.5 +√s = 13 TeV +dσ/dPT [nb/GeV] +PT[GeV] +NRQCD +LHCb +Figure 1: NRQCD predicted differential cross section compared with the data on ηc +production from the LHCb experiment at √s = 13 TeV. +of angular momentum. In the multiple emissions that the colour-octet state can make +before it makes the final NRQCD transition to a quarkonium state, the angular mo- +mentum and spin assignments of the c¯c state changes constantly. Consequently, the +Fock expansion for J/ψ (analogous to the one given in Eq. 2 for ηc) is no longer valid in +modified NRQCD and the equation for J/ψ corresponding to Eq. 3 also gets changed. +In Refs. [12] and [13], we have studied J/ψ and χc production, respectively, in mod- +ified NRQCD. Fitting the non-perturbative parameters from the Tevatron J/ψ and χc +data, we have made predictions for the cross-sections of these charmonium states at +the LHC and find excellent agreement with the LHC data. +In this paper, we confront the LHCb ηc cross-section using modified NRQCD and +we present the details in the following. +For ηc, for example, the NRQCD cross-section formula which was given as follows +when written down explicitly in terms of the octet and singlet states +σηc = ˆF1S[1] +0 × ⟨O(1S0)[1])⟩ + +1 +M 2 +� +ˆF1P [8] +1 × ⟨O(1P1)[8]⟩ +� ++ ˆF3S[8] +1 × ⟨O(3S1)[8])⟩ +(5) +gets modified to the following in the modified NRQCD with perturbative soft gluon +emission: +σηc += +� +ˆF1S[1] +0 × (⟨OJ/ψ(3S[1] +1 )⟩ +3 +) +� ++ +� +ˆF3S[8] +1 + ˆF1P [8] +1 + ˆF1S[8] +0 + ˆF3P [8] +J +� +× (⟨OJ/ψ(3S[1] +1 )⟩ +8 +) ++ +� +ˆF3S[8] +1 + ˆF1P [8] +1 + ˆF1S[8] +0 + ˆF3P [8] +J +� +× ⟨Oηc⟩, +(6) +4 + +where +⟨Oηc⟩ = × +� +⟨O(3S[8] +1 )⟩ + ⟨O(1S[8] +0 )⟩ + ⟨O(3P [8] +J )⟩ +M 2 +� +. +(7) +It is important to pay attention to the second line in Eq. 6. This term arises because +of the following physical situation in modified NRQCD: in emitting soft gluons, the +colour-octet state can every once in a while make a transition to a colour-singlet state +and this colour-singlet-state can no longer emit any soft-gluons but will eventually +make a non-perturbative transition to a physical charmonium state. This is a mixed +octet-singlet contribution and there is nothing like this in NRQCD: it is a novel feature +of modified NRQCD. +The non-perturbative parameters for ηc are not obtained from any independent fits +but from the J/ψ case, using the heavy-quark symmetry relations alluded to below. +These are: +� +Oηc +1 [1S0] +� += +1 +3 +� +OJ/ψ +1 +[3S1] +� +, +(8) +⟨Oηc⟩ += +� +OJ/ψ� +, +(9) +where +� +OJ/ψ� +is the fitted parameter for J/ψ. We have taken +� +OJ/ψ� += −0.161 GeV3, +which was obtained earlier using modified NRQCD [12]. +The cross-section kinematics are the usual and, even at the risk of pedantry, we +write the expression for the cross-section explicitly: +dσ +dpT +(p¯p → cc [2S+1L[1,8] +J +] X) = +� � +dy +� +dx1 x1 Ga/p(x1) x2 Gb/p(x2) +4pT +2x1 − xT ey +dˆσ +dˆt (ab → cc[2S+1L[1,8] +J +] d), +(10) +where the summation is over the partons (a and b), Ga/p, Gb/p are the distributions of +partons a and b in the protons and x1, x2 are the respective momentum they carry. In +the above formula, xT = +� +x2 +T + 4τ ≡ 2MT/√s with xT = 2pT /√s and τ = M 2/s. +√s is the center-of-mass energy, M is the mass of the resonance and y is the rapidity +at which the resonance is produced. The matrix elements for the subprocesses can be +found in Refs. [14, 15, 8]. +In Fig. 2 we compare the predictions for the ηc cross-section in modified NRQCD +with the experimental measurements at different energies from the LHC experiment. +The remarkable agreement of the predictions with LHCb cross-sections are obvious +and the agreement is seen at all the energies. We have also shown the colour-singlet +5 + + 10 +103 +105 +107 + 6 + 8 + 10 12 14 +2.0 < y < 4.5 +√s = 7 TeV +dσ/dPT [nb/GeV] +PT[GeV] +NRQCD +Modified NRQCD +LHCb +singlet + 10 +103 +105 +107 + 6 + 8 + 10 12 14 +2.0 < y < 4.5 +√s = 8 TeV +PT[GeV] +NRQCD +Modified NRQCD +LHCb +singlet + 10 +103 +105 +107 + 6 + 8 + 10 12 14 +2.0 < y < 4.5 +√s = 13 TeV +PT[GeV] +NRQCD +Modified NRQCD +LHCb +singlet +Figure 2: Predicted differential cross sections for ηc production at the LHC running at +different center-of-mass energies compared with the data from the LHCb experiment. +prediction separately in Fig. 2. As can be seen, the modified NRQCD prediction and +the colour-singlet prediction are very similar. In other words, in modified NRQCD, +there is a delicate contribution between the octet contribution and the mixed octet- +singlet contribution and that is why the total contribution is essentially given by the +colour-singlet contribution. we should add that this cancellation materialises only in +the forward kinematic region (where LHCb measures the ηc cross-section) and not in +the central region where the LHC experiments or even the Tevatron experiments mea- +sured their J/ψ cross-section. +Finally, we present in Fig. 3 the ratio of the ηc to J/ψ cross-section at √s = 13 TeV +and in the same forward region of kinematics, a measurement that may be undertaken +in the LHCb experiment in the near future. +In summary, modified NRQCD provides a neat solution to the LHCb ηc anomaly +and provides an understanding of all the features of the ηc data. It is important to reit- +erate that the ηc cross-section in modified NRQCD is a prediction and not a fit and the +remarkable agreement with the LHCb data suggests that modified NRQCD is closer to +apprehending the true dynamics of quarkonium production. +6 + + 1 + 10 +102 +103 +104 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 +2.0 < y < 4.5 +√s = 13 TeV + dσ/dPT [nb/GeV] +PT [GeV] +ηc +J/ψ +Ratio +Figure 3: Ratio of ηc to J/ψ differential production cross-sections at the LHC for √s = +13 TeV. +Acknowledgments +One of us (K.S.) is grateful to members of the LHCb collaboration – Monica Pepe- +Altarelli, Sergey Barsuk, Valeriia Zhovkovska and Andrii Usachov for valuable dis- +cussions. Discussions with Vaia Papadimitriou are also gratefully acknowledged. +References +[1] G. T. Bodwin, E. Braaten and G. P. Lepage, Phys. Rev. D 51, 1125 (1995) [Erratum- +ibid. D 55, 5853 (1997)] [arXiv:hep-ph/9407339]. +[2] F. Abe et al. [CDF], Phys. Rev. Lett. 79, 572 (1997); F. Abe et al. [CDF], Phys. 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B 291, 731 (1987). +8 + diff --git a/O9E1T4oBgHgl3EQfaATv/content/tmp_files/load_file.txt b/O9E1T4oBgHgl3EQfaATv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e115c3bca93d37b605b8a464e8bf902364531e4b --- /dev/null +++ b/O9E1T4oBgHgl3EQfaATv/content/tmp_files/load_file.txt @@ -0,0 +1,378 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf,len=377 +page_content='Resolution of the LHCb ηc anomaly Sudhansu S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Biswal1*, Sushree S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Mishra1† and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Sridhar2‡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Department of Physics, Ravenshaw University, Cuttack, 753003, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' School of Arts and Sciences, Azim Premji University, Sarjapura, Bangalore, 562125, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Abstract Due to the heavy-quark symmetry of Non-Relativistic Quantum Chromodynamics (NRQCD), the cross-section for the production of ηc can be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' This NRQCD prediction when confronted with data from the LHCb is seen to fail miserably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' We address this LHCb ηc anomaly in this paper using a new approach called modified NRQCD, an approach that has been shown to work extremely well for studying J/ψ, ψ′ and χc production at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' We show, in the present paper, that the predictions for ηc production agrees very well with LHCb measurements at the three different values of energy that the experiment has presented data for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Modi- fied NRQCD also explains the intriguing agreement of the LHCb ηc data with the colour-singlet prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The remarkable agreement of the theoretical predictions with the LHCb data suggests that modified NRQCD is closer to apprehending the true dynamics of quarkonium production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' One of the most important and daunting problems in Quantum Chromodynamics (QCD) is the understanding of how quarks form physical bound states – the hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' One small corner of this unnavigated terrain where one can make some headway is in the study of quarkonia – when a heavy quark and anti-quark come together to form a neutral meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' A very heavy quark like the top decays before it can form a bound- state so when we study quarkonia we are interested in charmonium and bottomonium systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' In these systems, the relative velocity, v, of the Q ¯Q pair is small and, there- fore, the bound state can be studied in a non-relativistic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The effective field theory that has been formulated to study such systems is called Non-Relativistic Quantum Chromodynamics (NRQCD) [1] which is derived from the QCD Lagrangian by neglecting all states of momenta much larger than the heavy quarkonium mass, M E-mail: sudhansu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='biswal@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='com †Email: sushreesimran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='mishra97@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='com ‡E-mail: sridhar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='k@apu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='in 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='03158v1 [hep-ph] 9 Jan 2023 and accounting for this exclusion by adding new interaction terms yielding the effec- tive Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The quarkonium state admits of a Fock-state expansion in orders of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' At leading order, the QQ state is in a colour-singlet state but at O(v), it can be in a colour-octet state which is connected to the physical J/ψ state through a non-perturbative gluon emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The cross section for production of a quarkonium state H of mass M in NRQCD can be factorised as: σ(H) = � n={α,S,L,J} Fn M dn−4⟨OH n (2S+1LJ)⟩, (1) where Fn’s are the short-distance coefficients and On are operators of naive dimension dn, describing the long-distance effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' These non-perturbative matrix elements are guaranteed to be energy-independent due to the NRQCD factorization formula, so that they may be extracted at a given energy and used to predict quarkonium cross-sections at other energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' NRQCD found much success in explaining the systematics of charmonium produc- tion at the Fermilab Tevatron [2] in contrast to the then existing model of quarkonium production – the colour-singlet model [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' But NRQCD does not predict the normali- sation of the pT distributions because of the unknown non-perturbative parameters so other tests of NRQCD were needed to validate it [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Of these, the polarisation of the J/ψ provides an important test: NRQCD predicts [5, 6] a fully transversely polarised J/ψ at large pT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The Tevatron experiments found no evidence for this [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Another important test of NRQCD comes from the study of ηc production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The heavy-quark symmetry of NRQCD provides a set of relations which connect non- perturbative parameters of different resonances so a measurement of a given state yields information on the non-perturbative parameters of another state related to the former by heavy-quark symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' In particular, the non-perturbative parameters re- quired for ηc production can be obtained, using heavy quark symmetry, from the pa- rameters of J/ψ production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' This approach has been used to predict the ηc production cross-section at the Tevatron [8] and at the LHC [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Like the prediction of polarisation, the prediction of the ηc cross-section is a defini- tive test of NRQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Just as NRQCD fails miserably in predicting the J/ψ polarisation, it also gets the ηc cross-section completely wrong [11]: the NRQCD prediction is com- pletely at variance with the cross-section measured by the LHCb experiment at three different values of centre- of-mass energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' It is this LHCb ηc anomaly that we address in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' 1hc production at the Tevatron [10] and at the LHC [9] has also been studied using this approach 2 The Fock space expansion of the physical ηc, which is a 1S0 (JPC = 0−+) state, is: |ηc⟩ = O(1) ���QQ[1S[1] 0 ] � + O(v2) ���QQ[1P [8] 1 ] g � + O(v4) ���QQ[3S[8] 1 ] g � + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' (2) In the above expansion the colour-singlet 1S0 state contributes at O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' As the P-state production is itself down by factor of O(v2) both the colour-octet 1P1 and 3S1 channels effectively contribute at the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The colour-octet state 1P [8] 1 (3S[8] 1 ) becomes a physical ηc by emitting a gluon in an E1 (M1) transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Keeping terms up-to O(α3 sv7) the ηc production cross section can be written as: σ(ηc) = F1[1S0] ⟨0| Oηc 1 [1S0] |0⟩ +F8[1P1] M 2 ⟨0| Oηc 8 [1P1] |0⟩ + F8[3S1] ⟨0| Oηc 8 [3S1] |0⟩ , (3) where the coefficients, Fn’s, are the cross sections for the production of cc pair in the respective angular momentum and colour states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' To predict the ηc cross-section, the values of ⟨Oηc n ⟩’s obtained from the experimen- tally predicted values ⟨OJ/ψ n ⟩’s using relations given by heavy-quark symmetry: ⟨0| Oηc 1 [1S0] |0⟩ = 1 3 ⟨0| OJ/ψ 1 [3S1] |0⟩ (1 + O(v2)), ⟨0| Oηc 8 [1P1] |0⟩ = ⟨0| OJ/ψ 8 [3P0] |0⟩ (1 + O(v2)), ⟨0| Oηc 8 [3S1] |0⟩ = ⟨0| OJ/ψ 8 [1S0] |0⟩ (1 + O(v2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' (4) With the values of the non-perturbative parameters determined as above, the ηc cross-section is a prediction of NRQCD that can be directly tested in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The pT-dependence of the ηc cross-section has been measured at the LHCb at three different energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' 1, we compare the predictions of NRQCD obtained, using the heavy- quark symmetry relations, with the ηc pT distribution at √s = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The disagreement between NRQCD predictions and the experimental values of the cross-section is huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The other conundrum is that the colour-singlet prediction alone seems to be well in agreement with the data in total contrast to the situation with J/ψ production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The results presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' 1 are not new: this anomaly, as we mentioned earlier, has been known and noted in the literature for several years now [11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' we have presented the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' 1 to draw attention to the magnitude of the discrepancy and to also bring to the fore the mysterious agreement of the colour-singlet prediction with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' In a recently proposed modification of NRQCD [12], which we have named mod- ified NRQCD, we had suggested that the colour-octet c¯c state can radiate several soft perturbative gluons – each emission taking away little energy but carrying away units 3 10-3 10-1 10 103 105 107 109 6 8 10 12 14 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='0 < y < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='5 √s = 13 TeV dσ/dPT [nb/GeV] PT[GeV] NRQCD LHCb Figure 1: NRQCD predicted differential cross section compared with the data on ηc production from the LHCb experiment at √s = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' of angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' In the multiple emissions that the colour-octet state can make before it makes the final NRQCD transition to a quarkonium state, the angular mo- mentum and spin assignments of the c¯c state changes constantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Consequently, the Fock expansion for J/ψ (analogous to the one given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' 2 for ηc) is no longer valid in modified NRQCD and the equation for J/ψ corresponding to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' 3 also gets changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' [12] and [13], we have studied J/ψ and χc production, respectively, in mod- ified NRQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Fitting the non-perturbative parameters from the Tevatron J/ψ and χc data, we have made predictions for the cross-sections of these charmonium states at the LHC and find excellent agreement with the LHC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' In this paper, we confront the LHCb ηc cross-section using modified NRQCD and we present the details in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' For ηc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' the NRQCD cross-section formula which was given as follows ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='when written down explicitly in terms of the octet and singlet states ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='σηc = ˆF1S[1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='0 × ⟨O(1S0)[1])⟩ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='ˆF1P [8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='1 × ⟨O(1P1)[8]⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='+ ˆF3S[8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='1 × ⟨O(3S1)[8])⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='(5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='gets modified to the following in the modified NRQCD with perturbative soft gluon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='emission: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='σηc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='ˆF1S[1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='0 × (⟨OJ/ψ(3S[1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='1 )⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='ˆF3S[8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='1 + ˆF1P [8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='1 + ˆF1S[8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='0 + ˆF3P [8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='× (⟨OJ/ψ(3S[1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='1 )⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='ˆF3S[8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='1 + ˆF1P [8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='1 + ˆF1S[8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='0 + ˆF3P [8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='× ⟨Oηc⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' (6) 4 where ⟨Oηc⟩ = × � ⟨O(3S[8] 1 )⟩ + ⟨O(1S[8] 0 )⟩ + ⟨O(3P [8] J )⟩ M 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' (7) It is important to pay attention to the second line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' This term arises because of the following physical situation in modified NRQCD: in emitting soft gluons, the colour-octet state can every once in a while make a transition to a colour-singlet state and this colour-singlet-state can no longer emit any soft-gluons but will eventually make a non-perturbative transition to a physical charmonium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' This is a mixed octet-singlet contribution and there is nothing like this in NRQCD: it is a novel feature of modified NRQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The non-perturbative parameters for ηc are not obtained from any independent fits but from the J/ψ case, using the heavy-quark symmetry relations alluded to below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' These are: � Oηc 1 [1S0] � = 1 3 � OJ/ψ 1 [3S1] � , (8) ⟨Oηc⟩ = � OJ/ψ� , (9) where � OJ/ψ� is the fitted parameter for J/ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' We have taken � OJ/ψ� = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='161 GeV3, which was obtained earlier using modified NRQCD [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The cross-section kinematics are the usual and, even at the risk of pedantry, we write the expression for the cross-section explicitly: dσ dpT (p¯p → cc [2S+1L[1,8] J ] X) = � � dy � dx1 x1 Ga/p(x1) x2 Gb/p(x2) 4pT 2x1 − xT ey dˆσ dˆt (ab → cc[2S+1L[1,8] J ] d), (10) where the summation is over the partons (a and b), Ga/p, Gb/p are the distributions of partons a and b in the protons and x1, x2 are the respective momentum they carry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' In the above formula, xT = � x2 T + 4τ ≡ 2MT/√s with xT = 2pT /√s and τ = M 2/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' √s is the center-of-mass energy, M is the mass of the resonance and y is the rapidity at which the resonance is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The matrix elements for the subprocesses can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' [14, 15, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' 2 we compare the predictions for the ηc cross-section in modified NRQCD with the experimental measurements at different energies from the LHC experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' The remarkable agreement of the predictions with LHCb cross-sections are obvious and the agreement is seen at all the energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' We have also shown the colour-singlet 5 10 103 105 107 6 8 10 12 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='0 < y < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='5 √s = 7 TeV dσ/dPT [nb/GeV] PT[GeV] NRQCD Modified NRQCD LHCb singlet 10 103 105 107 6 8 10 12 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='0 < y < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='5 √s = 8 TeV PT[GeV] NRQCD Modified NRQCD LHCb singlet 10 103 105 107 6 8 10 12 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='0 < y < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='5 √s = 13 TeV PT[GeV] NRQCD Modified NRQCD LHCb singlet Figure 2: Predicted differential cross sections for ηc production at the LHC running at different center-of-mass energies compared with the data from the LHCb experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' prediction separately in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' As can be seen, the modified NRQCD prediction and the colour-singlet prediction are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' In other words, in modified NRQCD, there is a delicate contribution between the octet contribution and the mixed octet- singlet contribution and that is why the total contribution is essentially given by the colour-singlet contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' we should add that this cancellation materialises only in the forward kinematic region (where LHCb measures the ηc cross-section) and not in the central region where the LHC experiments or even the Tevatron experiments mea- sured their J/ψ cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Finally, we present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' 3 the ratio of the ηc to J/ψ cross-section at √s = 13 TeV and in the same forward region of kinematics, a measurement that may be undertaken in the LHCb experiment in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' In summary, modified NRQCD provides a neat solution to the LHCb ηc anomaly and provides an understanding of all the features of the ηc data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' It is important to reit- erate that the ηc cross-section in modified NRQCD is a prediction and not a fit and the remarkable agreement with the LHCb data suggests that modified NRQCD is closer to apprehending the true dynamics of quarkonium production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' 6 1 10 102 103 104 6 8 10 12 14 16 18 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='0 < y < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='5 √s = 13 TeV dσ/dPT [nb/GeV] PT [GeV] ηc J/ψ Ratio Figure 3: Ratio of ηc to J/ψ differential production cross-sections at the LHC for √s = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Acknowledgments One of us (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=') is grateful to members of the LHCb collaboration – Monica Pepe- Altarelli, Sergey Barsuk, Valeriia Zhovkovska and Andrii Usachov for valuable dis- cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Discussions with Vaia Papadimitriou are also gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Bodwin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' Braaten and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9E1T4oBgHgl3EQfaATv/content/2301.03158v1.pdf'} +page_content=' P.' metadata={'source': 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0000000000000000000000000000000000000000..cca40cdf465b6b74a19c43817a0aaa13e3b06d01 --- /dev/null +++ b/OtE3T4oBgHgl3EQfCAnu/content/tmp_files/2301.04273v1.pdf.txt @@ -0,0 +1,2151 @@ +Draft version January 12, 2023 +Typeset using LATEX twocolumn style in AASTeX62 +Heat transport and convective velocities in compositionally-driven convection in +neutron star and white dwarf interiors +J. R. Fuentes1, 2, Andrew Cumming2, Matias Castro-Tapia2, and Evan H. Anders3 +1Department of Applied Mathematics, University of Colorado Boulder, Boulder, CO 80309-0526, USA +2Department of Physics and Trottier Space Institute, McGill University, Montreal, QC H3A 2T8, Canada +3Center for Interdisciplinary Exploration and Research in Astrophysics, Northwestern University, Evanston, Illinois 60201, USA +Abstract +We investigate heat transport associated with compositionally-driven convection driven by crystallization at +the ocean-crust interface in accreting neutron stars, or growth of the solid core in cooling white dwarfs. We study +the effect of thermal diffusion and rapid rotation on the convective heat transport, using both mixing length theory +and numerical simulations of Boussinesq convection. We determine the heat flux, composition gradient and +Péclet number (the ratio of thermal diffusion time to convective turnover time) as a function of the composition +flux. We find that the ratio between the heat flux and composition flux is independent of Péclet number, because +the loss of heat from convecting fluid elements due to thermal diffusion is offset by the smaller composition +gradient needed to overcome the reduced thermal buoyancy. We find two regimes of convection with a rapid +transition between them as the composition flux increases. We discuss the implications for neutron star and white +dwarf cooling. Convection in neutron stars spans both regimes. We find rapid mixing of neutron star oceans, +with a convective turnover time of order weeks to minutes depending on rotation. Except during the early stages +of core crystallization, white dwarf convection is in the thermal-diffusion-dominated fingering regime. We find +convective velocities much smaller than recent estimates for crystallization-driven dynamos. The small fraction +of energy carried as kinetic energy calls into question the effectiveness of crystallization-driven dynamos as an +explanation for observed white dwarf magnetic fields. +Key words: convection – stars:neutron – stars: white dwarfs – X-rays: binaries +1. Introduction +When a multicomponent plasma freezes, the composition +of the solid is typically different from the composition of +the liquid. If the solid preferentially retains heavy elements, +the liquid left behind is lighter and buoyant, driving con- +vection. The compositionally-driven convection transports +light elements outwards and mixes the liquid region. This +process has been studied in the context of dense interiors of +white dwarfs (Stevenson 1980; Mochkovitch 1983; Isern et al. +1997) and accreting neutron stars (Medin & Cumming 2011, +2014, 2015), and also occurs in planetary interiors, e.g. Earth +(Fearn & Loper 1981), the Moon (Laneuville et al. 2014; +Scheinberg et al. 2015) and Mercury (Manglik et al. 2010). +Depending on the phase diagram, another possibility is that +heavy elements preferentially go into the liquid phase, so that +solid crystals float upwards. This distillation process has re- +cently been suggested to be occurring in white dwarfs, driven +by chemical separation of 22Ne between the liquid and solid +phases (Blouin et al. 2021) (see also Mochkovitch 1983). +Redistribution of elements in white dwarf interiors is im- +portant because the gravitational energy released can pro- +jofu5477@colorado.edu +long white dwarf cooling. The large increase in the number +of white dwarfs with well-determined distances from Gaia +(Gentile Fusillo et al. 2021) has enabled the cooling delay +associated with crystallization to be definitely detected. The +slowed cooling is visible as an increased density of white +dwarfs in the HR diagram or luminosity function (Tremblay +et al. 2019). One puzzling feature in the HR diagram known +as the Q-branch indicates an additional cooling delay in a +small fraction of massive white dwarfs (Cheng et al. 2019). +Explanations for the delay have focused on the redistribution +of elements (22Ne in particular) within the white dwarf (Bauer +et al. 2020; Blouin et al. 2021; Camisassa et al. 2021; Caplan +et al. 2020). +Neutron stars in low mass X-ray binaries accrete enough +mass over their lifetimes to replace the entire neutron star +crust (eg. Suleiman et al. 2022). The accreted light elements +first undergo thermonuclear burning in the surface layers, +generating a complex mixture of heavy elements that forms +a liquid ocean (Bildsten & Cutler 1995). At the base of the +ocean, compressed matter continuously freezes and forms +solid crust as accretion continues (Brown & Bildsten 1998). +In sources that undergo transient accretion outbursts, the neu- +tron star cools in quiescence and the liquid ocean refreezes +arXiv:2301.04273v1 [astro-ph.SR] 11 Jan 2023 + +2 +Fuentes et al. +(see Wijnands et al. 2017 for a review of transiently accreting +neutron stars). Horowitz et al. (2007) showed that chemical +separation between liquid and solid phases is expected for the +mixtures found in neutron star oceans, with lighter elements +typically left behind in the liquid phase (see Mckinven et al. +2016 and Caplan et al. 2018 for a survey of different com- +positions). Medin & Cumming (2011, 2014, 2015) studied +the compositional changes and heat transport in the ocean in +these different scenarios. +Unlike in many planetary interiors, where convection +is driven by both compositional and thermal buoyancy, +crystallization-driven convection in white dwarf and neutron +star interiors occurs in a part of the star that is thermally-stable +to convection, i.e. has a sub-adiabatic temperature gradient. +This is because of the large thermal conductivity from degen- +erate electrons which can transport the cooling luminosity and +latent heat of crystallization with only a small temperature +gradient. In this case, when compositionally-driven convec- +tion occurs in a thermally-stratified background, convection +transports heat in the opposite direction to the composition +flux (Loper 1978; Medin & Cumming 2011). Rising fluid el- +ements adiabatically expand and cool down to a temperature +that is lower than their surroundings. This cools the surround- +ings, giving an effective heat flow that is directed downwards. +By transporting heat towards the liquid/solid interface, con- +vection acts in a similar way to the latent heat. Medin & +Cumming (2014) showed that this changes the cooling rate +of neutron stars following accretion outbursts, an observable +signature of the newly-forming crust and its composition. +When calculating the convective heat flux in neutron star +oceans, Medin & Cumming (2011, 2015) assumed that the +fluid motions would be adiabatic. However, a large enough +thermal conductivity could cause rising parcels of fluid to +lose a significant amount of energy by thermal diffusion, re- +ducing the effective heat flux. The likely importance of ther- +mal diffusion in white dwarf convection was pointed out by +Stevenson (1980) and included in estimates of the convective +velocities by Mochkovitch (1983) and Isern et al. (1997). The +Péclet number, the ratio of thermal diffusion time to convec- +tive turnover time was estimated to be ≈ 0.3 by Isern et al. +(1997), implying that this effect is important. The transport +of heat by compositionally-driven convection does not appear +to have been considered in white dwarfs; instead, it is usu- +ally assumed that the liquid region above the crystallization +front mixes rapidly, and the resulting change in energy is put +directly into the model as a localized heat source (Isern et al. +1997, 2000). +Interest in compositionally-driven convection in white +dwarfs has also been recently revived with the suggestion of +Isern et al. (2017) that it leads to a magnetic dynamo in crys- +tallizing white dwarfs (Schreiber et al. 2021a,b; Belloni et al. +2021; Camisassa et al. 2022; Ginzburg et al. 2022; Schreiber +et al. 2022). Using the scaling of Christensen et al. (2009) +for a saturated dynamo, Isern et al. (2017) found that fields +up to ∼ 1 MG could be generated. However, whether the +dynamo is in the saturated regime depends on the convective +turnover time, and estimates of the convective velocity differ +significantly. Isern et al. (1997) found 𝑣𝑐 ≈ 30 km s−1 by +considering rising carbon-enriched liquid bubbles released at +the crystallization front, whereas Ginzburg et al. (2022) ar- +gued that the velocity should be much lower, ∼ 100 cm s−1, +based on the available convective energy flux. Both of these +velocity estimates are significantly larger than previous esti- +mates for (non-magnetic) compositionally-driven convection. +Mochkovitch (1983) found 𝑣𝑐 ∼ 10−6 cm s−1 for non-rotating +or ∼ 0.1 cm s−1 for rapidly-rotating white dwarfs. +In this paper, we revisit compositionally-driven convec- +tion in dense stellar interiors. Our goal is to determine the +expected convective velocities and convective heat flux for ac- +creting neutron stars and cooling white dwarfs. We apply stel- +lar mixing length theory to the case of compositionally-driven +convection, and use numerical simulations to demonstrate that +heat is indeed transported inwards and test the mixing length +theory predictions. The mixing length theory is presented in +section 2, where we derive expressions for the heat flux and +convective velocities, and discuss the steady-state outcome in +which the inwards heat flux due to convection is balanced by +an outwards conductive heat flux. In section 3, we present +our numerical simulations of Boussinesq convection in the +non-rotating case and compare with mixing length theory. +We conclude in section 4 with a discussion of how our results +apply to white dwarfs and neutron stars. +2. Mixing length theory for compositionally-driven +convection +In this section, we use mixing length theory to investi- +gate the size of the heat flux associated with compositionally- +driven convection, and the expected convective velocities. We +first write down mixing length theory including thermal diffu- +sion (§2.1), and then discuss the expected heat flux (§2.2) and +convective velocities in the non-rotating and rapidly-rotating +limits (§2.3). +We then investigate the steady-state in which the inwards +convective flux is balanced by outwards conduction (§2.4). +2.1. Mixing length theory including thermal diffusion +In mixing length theory, the heat and composition fluxes are +written in terms of the excess temperature 𝐷𝑇 or composition +𝐷𝑋 carried by a fluid element, 𝐹𝐻 = 𝜌𝑣𝑐𝑐𝑃𝐷𝑇 and 𝐹𝑋 = +𝜌𝑣𝑐𝐷𝑋, where 𝑣𝑐 is the convective velocity, 𝑐𝑃 is the specific +heat capacity at constant pressure, and 𝜌 the density. For +simplicity, we assume a mixture of two elements, so that +the composition can be described by only one variable, here +chosen to be 𝑋, the mass fraction of the lighter component1. +We include the effect of thermal diffusion following the +formulation of mixing length theory discussed by Kippenhahn +et al. (2012), which is based on Böhm-Vitense (1958) (see also +Henyey et al. 1965 and Gough 1977). The temperature excess +1 The results can be easily generalized to more complex mixtures, +e.g. Medin & Cumming (2015). We also make the approximation that the +excess entropy carried by fluid elements is 𝐷𝑆 ≈ 𝑐𝑃𝐷𝑇 /𝑇 , ignoring any +contribution to the entropy from compositional differences. Again, this can +be included in a straightforward way, writing the heat flux as 𝜌𝑣𝑐𝑇 𝐷𝑆, but +is typically a small correction (Medin & Cumming 2015). + +Heat transport by compositionally-driven convection +3 +is written as +𝐷𝑇 = (∇ − ∇𝑒) 𝑇 +ℓ +2𝐻𝑃 +, +(1) +where ∇ = 𝑑 ln𝑇/𝑑 ln 𝑃|★ is the temperature gradient in the +star, ∇𝑒 is the rate of change of temperature with pressure +experienced by the fluid element, ℓ is the mixing length, +and 𝐻𝑃 the pressure scale height. Similarly, we can write +𝐷𝑋/𝑋 = ∇𝑋 (ℓ/2𝐻𝑃), where ∇𝑋 = 𝑑 ln 𝑋/𝑑 ln 𝑃|★ is the +composition gradient in the star. +The heat and composition fluxes are then given by +𝐹𝐻 = 𝜌𝑣𝑐𝑐𝑃𝐷𝑇 = 𝜌𝑣𝑐𝑐𝑃𝑇 (∇ − ∇𝑒) +ℓ +2𝐻𝑃 +, +(2) +and +𝐹𝑋 = 𝜌𝑣𝑐𝑋∇𝑋 +ℓ +2𝐻𝑃 +. +(3) +The sign of these fluxes is such that a positive flux is in the +upwards direction. For example, an outwards flux of light +elements is associated with a gradient ∇𝑋 > 0, i.e. +the +mass fraction of light elements increases with pressure. Note +that the composition flux gives the mass of light elements +crossing unit area per unit time (i.e. in cgs the units of 𝐹𝑋 are +g cm−2 s−1). +By considering the exchange of energy by thermal diffusion +with the surroundings as the fluid element moves, Kippenhahn +et al. (2012) derive an expression for ∇𝑒 +∇𝑒 − ∇ad +∇ − ∇𝑒 += 9 +2 +𝐾 +𝜌𝑐𝑃ℓ𝑣𝑐 += 9 +2 +𝜅𝑇 +ℓ𝑣𝑐 +≡ 9 +2 +1 +Pe , +(4) +where 𝜅𝑇 is the thermal diffusivity and we define the dimen- +sionless Péclet number Pe ≡ ℓ𝑣𝑐/𝜅𝑇 . The numerical pref- +actor of 9/2 in equation (4) depends on assumptions about +the shape of the fluid element and the temperature distribu- +tion (see discussion in Henyey et al. 1965). For example, +Hubeny & Mihalas (2014) following Böhm-Vitense (1958) +give a prefactor of 3 instead, whereas Henyey et al. (1965) +have a prefactor of 2𝜋2 ≈ 20. Here, instead of adopting any +particular value, we keep in mind that it is model-dependent +and treat it as a free parameter 𝐶. Replacing the 9/2 by 𝐶 in +equation (4) gives +∇ − ∇𝑒 = +� +Pe +𝐶 + Pe +� +(∇ − ∇ad) . +(5) +When the convective motions are rapid, Pe ≫ 1 and ∇𝑒 → +∇ad as expected since the motions become adiabatic. In the +opposite limit in which the convective motions are slow and +thermal diffusion can act, Pe ≪ 1 and ∇𝑒 → ∇, so that the +fluid element is able to adjust its temperature to follow the +background temperature gradient. +2.2. The heat flux in compositionally-driven convection +Taking the ratio of equations (2) and (3), the convective +velocity and mixing length drop out, giving the heat flux in +terms of the composition flux, +𝐹𝐻 +𝐹𝑋 += −𝑐𝑃𝑇 +𝑋 +∇𝑒 − ∇ +∇𝑋 +. +(6) +This shows that transport of composition is associated also +with a transport of heat, provided the fluid elements expe- +rience a different temperature evolution with pressure com- +pared to the background. Equation (5) shows that ∇𝑒 ranges +from ∇ to ∇ad as Pe goes from small to large values. In a back- +ground that is stably-stratified thermally, ie. with ∇ad > ∇, +this means that ∇𝑒 ≥ ∇, giving a heat flux oppositely-directed +to the composition flux. +The fact that ∇𝑒 approaches ∇ for Pe ≪ 1 (eq. [5]) acts to +reduce the heat flux. However, the composition gradient in the +convection zone also depends on Pe, since the effective ther- +mal stratification is reduced at low Pe when thermal diffusion +is efficient, which means that a smaller composition gradient +is needed to maintain the convective motions. To see this, +consider the typical density contrast in the convection zone, +𝐷𝜌 +𝜌 +≈ − 𝜒𝑇 +𝜒𝜌 +𝐷𝑇 +𝑇 +− 𝜒𝑋 +𝜒𝜌 +𝐷𝑋 +𝑋 , +(7) +where 𝜒𝑇 = 𝜕 ln 𝑃/𝜕 ln𝑇|𝜌,𝑋, 𝜒𝜌 = 𝜕 ln 𝑃/𝜕 ln 𝜌|𝑇 ,𝑋, and +𝜒𝑋 = 𝜕 ln 𝑃/𝜕 ln 𝑋|𝜌,𝑇 . +The density contrast determines +the buoyant acceleration ∝ −𝐷𝜌/𝜌. Written in terms of the +gradients, +𝐷𝜌 +𝜌 ≈− +ℓ +2𝐻𝑃 +� 𝜒𝑇 +𝜒𝜌 +(∇ − ∇𝑒) + 𝜒𝑋 +𝜒𝜌 +∇𝑋 +� +≈− +ℓ +2𝐻𝑃 +𝜒𝑋 +𝜒𝜌 +� +∇𝑋 − ∇𝑋,crit +� +, +(8) +where we define the critical composition gradient +∇𝑋,crit = 𝜒𝑇 +𝜒𝑋 +(∇𝑒 − ∇) = 𝜒𝑇 +𝜒𝑋 +(∇ad − ∇) +� +Pe +𝐶 + Pe +� +. +(9) +For adiabatic displacements (large Pe), where ∇𝑒 → ∇ad, +𝐷𝜌 < 0 in equation (8) is equivalent to the Ledoux criterion +for convection, 𝜒𝑇 (∇ − ∇ad) + 𝜒𝑋∇𝑋 > 0, and so ∇𝑋,crit in +this limit is the composition gradient needed to be unstable +to convection according to the Ledoux criterion. At small Pe, +thermal diffusion lowers the effective thermal stratification, +reducing ∇𝑋,crit, and allowing convection to occur for smaller +composition gradients. This is the regime of fingering or ther- +mohaline convection2. A similar expression to equation (8) +was previously written down by Mochkovitch (1983) for the +case 𝐶 = 1. +If the convection is efficient in the sense that only small den- +sity perturbations are required to drive the required convective +velocities and fluxes, then 𝐷𝜌/𝜌 ≪ 1 ⇒ ∇𝑋 ≈ ∇𝑋,crit3. In +this case, the reduction in ∇𝑒 − ∇ at small Pe is exactly offset +2 In the limit Pe ≪ 1 and assuming ∇𝑋 ≈ ∇𝑋,crit, equation (9) agrees with +the prescription for convection in the MESA code (Paxton et al. 2013) based +on Ulrich (1972) and Kippenhahn et al. (1980). To see this, write the diffusion +coefficient in eq. (14) of Paxton et al. (2013) as 𝐷th = 𝑣𝑐ℓ, in which case +their expression reduces to eq. (9). The efficiency parameter for thermohaline +convection 𝛼th is related to our shape parameter by 𝛼th = 2𝐶/3. +3 This is analagous to efficient thermal convection where ∇ − ∇ad ≪ ∇ad. + +4 +Fuentes et al. +by the reduction in ∇𝑋, so the ratio 𝐹𝐻/𝐹𝑋 is actually inde- +pendent of Pe. To see this more explicitly, we can write the +heat flux in terms of ∇𝑋,crit, giving +𝐹𝐻 +𝐹𝑋 += −𝑐𝑃𝑇 +𝑋 +𝜒𝑋 +𝜒𝑇 +� ∇𝑋,crit +∇𝑋 +� +. +(10) +This relation between 𝐹𝐻 and 𝐹𝑋 is the same as derived by +Medin & Cumming (2011) under the assumption that fluid +elements move adiabatically (the only difference is that ∇𝑋,crit +in that case is given by the large Pe limit of eq. [9]). +2.3. Convective velocity and effect of rotation +We can estimate the extent to which ∇𝑋 exceeds ∇𝑋,crit by +writing the expression for the convective velocity +𝑣2 +𝑐 ≈ 𝑔ℓ +4 +𝐷𝜌 +𝜌 +≈ 𝑔ℓ2 +8𝐻𝑃 +𝜒𝑋 +𝜒𝜌 +�∇𝑋 − ∇𝑋,crit +� , +(11) +where we take the numerical prefactors 1/4 and 1/8 from the +particular formulation of mixing length theory we are using +(Kippenhahn et al. 2012). Using the definition Pe = ℓ𝑣𝑐/𝜅𝑇 +and defining a Rayleigh number +RaT = +𝑔𝐻3 +𝑃 𝜒𝑇 ∇ad +𝜒𝜌𝜅2 +𝑇 +, +(12) +we obtain +𝜒𝑋 +𝜒𝑇 ∇ad +(∇𝑋 − ∇𝑋,crit) = +8 +RaT +� 𝐻𝑃 +ℓ +�4 +Pe2. +(13) +For the large RaT in astrophysical applications, the term on +the right hand side will be small as long as Pe is not too large, +so that taking ∇𝑋 ≈ ∇𝑋,crit should be a good approximation. +However, for a large enough composition flux, this term can +become important as we will see below. +Equation (11) assumes that the velocity of fluid elements is +set by the buoyant acceleration acting over a mixing length. In +rapidly-rotating convection, Coriolis forces modify the force +balance and change the convective velocity. We estimate the +effect of rapid rotation following the scaling relations of Au- +rnou et al. (2020), who considered the balance between Corio- +lis, inertial and buoyancy terms in rapidly-rotating convection +(CIA balance). Rewriting their eq. [24] in our notation, this +balance can be expressed as +𝑣2 +𝑐 +𝐿2 ∼ 2Ω𝑣𝑐 +𝐻𝑃 +∼ +𝑔 +𝐻𝑃 +𝜒𝑋 +𝜒𝜌 +� +∇𝑋 − ∇𝑋,crit +� +(14) +(compare eq. [24] of Aurnou et al. 2020). We assume that +the lengthscale associated with convective motions in the di- +rection of the rotation vector is the pressure scale height 𝐻𝑃, +while 𝐿 is the lengthscale associated with motions perpen- +dicular to the rotation vector. +The first and last terms of +equation (14) give an expression for the convective velocity +that has the same functional form as equation (11) but with +the replacement ℓ → 𝐿. The first and second terms in equa- +tion (14) give the ratio between perpendicular and parallel +scales as +𝐿 +𝐻𝑃 +≈ +� +𝑣𝑐 +2Ω𝐻𝑃 +�1/2 +≈ Ro1/2, +(15) +where we define the Rossby number Ro ≡ 𝑣𝑐/2Ω𝐻𝑃. Simu- +lations of non-magnetic rapidly-rotating convection in plane- +tary cores give support to this scaling (Guervilly et al. 2019). +These scalings suggest that we can estimate the effect of +rapid rotation by making the substitution ℓ → 𝐿 ≈ Ro1/2𝐻𝑃 +in the non-rotating results. The Rossby number is given in +terms of Pe (which is now defined as Pe ≡ 𝑣𝑐𝐿/𝜅𝑇 ) by +Ro ≡ +𝑣𝑐 +2Ω𝐻𝑃 += Pe2/3Ta−1/3, +(16) +where we define the Taylor number +Ta ≡ 4Ω2𝐻4 +𝑃 +𝜅2 +𝑇 +. +(17) +With these scalings, we find +𝜒𝑋 +𝜒𝑇 ∇ad +(∇𝑋 − ∇𝑋,crit) = +8 +RaT +Pe2 +Ro2 = +8 +RaT +Ta2/3Pe2/3. +(18) +Comparing to equation (13), we see that rapid rotation (Ro ≪ +1) acts to steepen the composition gradient. Even so, the large +value of RaT in astrophysical scenarios means that ∇𝑋 will +remain very close to ∇𝑋,crit in many cases. +2.4. The steady-state balance with thermal conduction +We now consider the consequences of the mixing length +theory outlined above in a situation with a specified outwards +flux of light elements 𝐹𝑋. +As the compositionally-driven +convection transports heat inwards, the temperature gradient +will steepen until the outwards conductive heat flux balances +the inwards convective heat flux4, +𝜌𝑐𝑃𝜅𝑇 +𝑇∇ +𝐻𝑃 += −𝜌𝑣𝑐𝑐𝑃𝑇 (∇ − ∇𝑒) +ℓ +2𝐻𝑃 +. +(19) +Solving for the steady-state temperature gradient gives ∇ = +∇𝑒Pe/(2 + Pe), or using equation (5), +∇ = ∇ad +Pe2 +Pe2 + 2Pe + 2𝐶 +. +(20) +When Pe ≪ 1, conduction acts efficiently on the timescale +of convection, so that a small temperature gradient ∇ ≈ +4 The outwards conductive flux and inwards convective flux will not exactly +cancel. For example, in a cooling white dwarf there must be a net outwards +cooling luminosity. In neutron star envelopes, Medin & Cumming (2015) +considered a steady-state in which the net heat flux was inwards, carrying +nuclear energy released in a low density H/He burning shell into the neutron +star interior. In both cases, the latent heat needs to be removed from the +crystallization front. For simplicity here we assume that any net flux is small +compared to the convective heat flux. + +Heat transport by compositionally-driven convection +5 +∇adPe2/2𝐶 is sufficient for conduction to balance the con- +vective heat flux. However, when convection is driven very +strongly and the convective velocities become large, Pe ≫ 1, +the steady-state temperature gradient approaches the adia- +batic gradient. The reason for this is that the heat flux due to +convection (∝ ∇ad − ∇) is then reduced to a level where it can +be balanced by the conductive flux along the adiabat5. +We can write an expression for Pe in terms of the com- +position flux using equation (3), which gives the convective +velocity 𝑣𝑐 ≈ (𝐹𝑋/𝜌𝑋∇𝑋)(2𝐻𝑃/ℓ), or +Pe = 𝑣𝑐ℓ +𝜅𝑇 +≈ +� +𝐻2 +𝑃 +𝜅𝑇 +� � 2𝐹𝑋 +𝜌𝐻𝑃𝑋 +� +∇−1 +𝑋 . +(21) +The first term is the thermal diffusion time across the pressure +scale height 𝑡therm = 𝐻2 +𝑃/𝜅𝑇 . The second term is related to the +timescale on which the light elements are being injected into +the layer. For example, consider a region of a star with mass +Δ𝑀 ∼ 4𝜋𝑟2𝜌𝐻𝑃. If light elements are being injected at a rate +�𝑀𝑋 = Δ𝑀 �𝑋, the composition flux is 𝐹𝑋 ∼ Δ𝑀 �𝑋/4𝜋𝑟2 = +𝜌𝐻𝑃 �𝑋, and the second term in equation (21) is 𝐹𝑋/𝜌𝐻𝑃𝑋 ∼ +�𝑋/𝑋. We therefore define the timescale 𝑡𝑋 = 𝜌𝐻𝑃𝑋/𝐹𝑋, +giving6 +Pe ≈ 𝑡therm +𝑡𝑋 +2 +∇𝑋 +. +(22) +We see that the Péclet number is set by both the ratio of ther- +mal and injection timescales and the composition gradient. +For fixed timescales, a smaller composition gradient requires +a larger velocity to transport the composition. +Equations (20), and (22) both relate Pe to one of the gradi- +ents ∇ or ∇𝑋. Adding a third relation, either equation (13) for +no rotation or equation (18) for rapid rotation, we can solve +for Pe, ∇ and ∇𝑋. Before presenting the solution, it is useful +to define the dimensionless parameter +𝜏 = +�𝑡therm +𝑡𝑋 +� � +𝜒𝑋 +𝜒𝑇 ∇ad +� +. +(23) +which is a measure of the composition flux driving convec- +tion. This can also be written explicitly in terms of 𝐹𝑋 as +𝜏 = +𝐹𝑋 +𝐹𝐻,ad +� 𝑐𝑃𝑇 +𝑋 +𝜒𝑋 +𝜒𝑇 +� +, +(24) +where 𝐹𝐻,ad = 𝜌𝑐𝑃𝜅𝑇 𝑇∇ad/𝐻𝑃 is the heat flux conducted +along the thermal adiabat. Comparing with equation (10) we +see that 𝜏 is a measure of the effect of the convection on the +temperature gradient: when 𝜏 = 1, the value of 𝐹𝑋 is such +5 In reality, the temperature gradient may saturate below ∇ad. Once the +temperature gradient reaches the slope of the liquidus curve ∇𝐿 ≈ 1/4 < +∇ad ≈ 1/3, large portions of liquid will freeze, shutting down convec- +tion. Medin & Cumming (2014, 2015) found that the system then becomes +time-dependent with periodic freezing and melting of large regions near the +liquid/solid boundary, on average maintaining a gradient ∇ ≈ ∇𝐿. +For +simplicity, we ignore this effect in this section. +6 A similar expression for Pe was previously obtained by Mochkovitch +(1983) (their eq. [23]) and Isern et al. (1997) (their eq. [32]) for the case +where ∇𝑋 = ∇𝑋,crit and assuming 𝐶 = 1. +10 +3 +10 +2 +10 +1 +100 +101 +102 +103 +=( +ad X/ T)/(ttherm/tX) +10 +1 +100 +101 +102 +103 +104 +Pe +Pe=1 +=1 +(2C )1/2 +A(2 )1/3 +10 +3 +10 +2 +10 +1 +100 +101 +102 +103 +=( +ad X/ T)/(ttherm/tX) +10 +3 +10 +2 +10 +1 +100 +Gradients +ad +X +ad T/ X +Figure 1. The steady-state Péclet number (top panel) and temperature and +composition gradients (bottom panel) as a function of the driving parameter +𝜏 ∝ 𝐹𝑋 (eq. 23). Near 𝜏 = 1, Pe rapidly transitions from the small 𝜏 solution +of eq. (27) (lower dotted line) to the large 𝜏 solution given by eq. (28) (upper +dotted line). For this example, we set 𝐶 = 9/2 and 𝐴 = 103 (RaT ∼ 1010). +that the associated convective heat flux for efficient convection +(∇𝑋 ≈ ∇𝑋,ad) is equal to 𝐹𝐻,ad. This means that for 𝜏 ≪ 1, +the heat flux can be balanced by a small temperature gradient +∇ = 𝜏∇ad. The temperature gradient is much shallower than +the adiabat, thermal diffusion is efficient, and Pe is small. For +𝜏 ≫ 1, the heat flux for efficient convection exceeds 𝐹𝐻,ad, +the composition gradient steepens ∇𝑋 > ∇𝑋,crit to reduce the +heat flux to ≈ 𝐹𝐻,ad, the temperature gradient is close to the +adiabat (∇ ≈ ∇ad) with inefficient thermal diffusion and large +Pe. +The full solution for the non-rotating case can be written as +𝜏 = +Pe2 +Pe2 + 2Pe + 2𝐶 ++ 1 +2 +�Pe +𝐴 +�3 +, +(25) +where +𝐴 = 1 +2RaT1/3 +� ℓ +𝐻𝑃 +�4/3 +. +(26) + +6 +Fuentes et al. +100 +102 +104 +Pe +No rotation +Ta=108 +Ta=1010 +Ta=1012 +10 +5 +10 +3 +10 +1 +Ro +10 +2 +10 +1 +100 +/ +ad +10 +3 +10 +2 +10 +1 +100 +101 +102 +103 +=( +ad X/ T)(ttherm/tX) +10 +3 +10 +2 +10 +1 +100 +101 +X/( +ad T/ X) +Figure 2. The effect of rapid rotation on the steady-state solutions. We +show the non-rotating solution from Fig. 1 as the dashed black line. The +other curves show the effect of increasing rotation on this model, with +Ta = 108, 1010, and 1012. Rapid rotation has only a small effect on the +temperature gradient/heat flux, but leads to smaller convective velocities and +larger composition gradients. +Equation (25) gives Pe(𝜏) which can then be used to obtain the +gradients ∇ and ∇𝑋 using equations (20) and (22) respectively. +An example for particular choices of 𝐴 and 𝐶 is shown in +Figure 1. The solution for Pe(𝜏) (top panel) has two branches: +at small 𝜏, ∇𝑋 ≈ ∇𝑋,crit ∝ Pe and ∇ ≪ ∇ad, so that equation +(22) gives +Pe ≈ (2𝐶𝜏)1/2 +(𝜏 small), +(27) +while at large 𝜏, the composition gradient is set by the right +hand term in equation (13), giving +Pe ≈ 𝐴(2𝜏)1/3, +(𝜏 large). +(28) +The value of Pe makes a rapid transition between these two +branches at 𝜏 = 1. The lower panel of Figure 1 shows the +gradients. +The temperature gradient closely follows ∇ = +∇ad𝜏 for 𝜏 < 1 and ∇ = ∇ad for 𝜏 > 1. The composition +gradient shows a more complicated behaviour. For 𝜏 < 1, +it is very close to ∇𝑋 = ∇𝑋,crit. At small values of 𝜏, this +gives ∇𝑋 increasing with 𝜏, ∇𝑋 ≈ (∇ad𝜒𝑇 /𝜒𝑋)(2𝜏/𝐶)1/2. +As 𝜏 → 1, ∇ → ∇ad, decreasing the thermal buoyancy and +therefore ∇𝑋,crit, which leads to the rapid decrease in ∇𝑋 +near 𝜏 = 1 in Figure 1. For 𝜏 > 1, ∇𝑋 increases with 𝜏 +again as it starts to significantly exceed ∇𝑋,crit. For large 𝜏, +∇𝑋 ≈ (∇ad𝜒𝑇 /𝜒𝑋)(2𝜏)2/3/𝐴. +For rapid rotation, we use equation (18) instead of (13). +The solution is +𝜏 = +Pe2 +Pe2 + 2Pe + 2𝐶 ++ +�Ta2/3 +2𝐴3 +� +Pe5/3. +(29) +An example is shown in Figure 2 which shows the effect of +increasing rotation on the model from Figure 1. As long as +Ta2/3 ≲ 𝐴3 (corresponding approximately to Ta2/3 ≲ RaT), +then the first term in equation (29) dominates for 𝜏 < 1. +The results for Pe and the gradients are therefore the same as +without rotation7. The convective velocities are significantly +increased, by a factor of Ro−1/2 (since Pe ≡ 𝑣𝑐𝐿/𝜅𝑇 is un- +changed by rotation, and 𝐿/𝐻𝑃 = Ro1/2)8. For 𝜏 > 1, the +last term in equation (29) dominates, giving +Pe ≈ 𝐴9/5 +Ta2/5 (2𝜏)3/5, +(𝜏 large) +(30) +and +∇𝑋 ≈ 𝜒𝑇 ∇ad +𝜒𝑋 +Ta2/5 +𝐴9/5 (2𝜏)2/5. +(𝜏 large) +(31) +Comparing equations (28) and (30), we see that the effect +of rapid rotation is to reduce Pe (increase ∇𝑋) for 𝜏 > 1, +multiplying (dividing) it by a factor ≈ (𝐴4/5/Ta2/5)(2𝜏)4/15 ∝ +RaT4/15/Ta2/5 (see Fig. 2). +3. Numerical simulations +The mixing length theory in the previous section makes a +number of approximations and assumptions, in particular for +how thermal diffusion acts to suppress the thermal buoyancy +(eq. [5] and eq. [9]). +In this section, we compare against +numerical simulations of compositionally-driven convection. +We first check that indeed there is an inwards directed heat +flux associated with an outwards composition flux. Then, +we allow the thermal gradient to come into steady-state and +investigate the relation between the composition and thermal +gradients and the value of the Péclet number that characterizes +the flow. +7 In the limit of very rapid rotation, when Ta2/3 > 𝐴3, the last term in +equation (29) dominates for all 𝜏, giving Pe ≈ (𝐴3/Ta2/3)3/5(2𝜏)3/5. The +largest value of Ta shown in Figure 2 is just large enough to enter this regime, +where Pe is reduced by rotation at 𝜏 < 1. +However, this regime is not +relevant for the parameter values appropriate for white dwarf and neutron +star interiors, and so we do not focus on it here. +8 If using the Rossby number defined with the non-rotating value of 𝑣𝑐, +the factor by which rotation increases the velocity is Ro−1/3 . + +Heat transport by compositionally-driven convection +7 +3.1. Model and simulation setup +We conduct simulations for a binary fluid within a 3D spher- +ical shell of depth Δ𝑟. For this first numerical investigation, +and to simplify comparison with mixing length theory, we +consider a non-rotating system. We express the fluid quan- +tities as the sum of a constant background (denoted by the +subscript 0) and a dynamic perturbation to the background +(denoted by the prime symbol), e.g., the density 𝜌 = 𝜌0 + 𝜌′. +We use the Boussinesq approximation (Spiegel & Veronis +1960), where density perturbations satisfy 𝜌′/𝜌0 ≪ 1, and are +related to perturbations in temperature 𝑇 ′ and mass fraction +of the lighter component 𝑋′ through 𝜌′ = −𝜌0(𝛽𝑋′ + 𝛼𝑇 ′), +where 𝛽 and 𝛼 are the coefficients of compositional and ther- +mal contraction/expansion (both assumed positive constants), +respectively. Convection is driven by imposing a constant flux +of light elements across the domain, such that light elements +are injected (removed) at the inner (outer) boundary. +We non-dimensionalize the fluid equations using as units +of length and time the shell depth, Δ𝑟, and the diffusion time +for solute, 𝑡diff = Δ𝑟2/𝜅𝑋, where 𝜅𝑋 is the solute diffusivity. +The temperature scale is [𝑇] = |𝜕𝑟𝑇0 − 𝜕𝑟𝑇ad|Δ𝑟, where 𝜕𝑟𝑇0 +is the radial temperature gradient of the background (zero in +this problem), and 𝜕𝑟𝑇ad is the adiabatic temperature gradient +(equal to −1 given our choice of units). For solute, we use +[𝑋] = (𝛼/𝛽)|𝜕𝑟𝑇0 − 𝜕𝑟𝑇ad|Δ𝑟. By this choice, a unit of pres- +sure corresponds to [𝑃] = 𝜌0(𝜅𝑋/Δ𝑟)2. The dimensionless +equations are +∇ · u = 0 , +(32) +𝜕u +𝜕𝑡 + (u · ∇)u = −∇𝑃′ + ScR (𝑋′ + 𝑇 ′) ˆr + Sc∇2u ,(33) +𝜕𝑋′ +𝜕𝑡 + (u · ∇)𝑋′ = ∇2𝑋′ , +(34) +𝜕𝑇 ′ +𝜕𝑡 + (u · ∇)𝑇 ′ + (𝜕𝑟𝑇0 − 𝜕𝑟𝑇ad)𝑢𝑟 = Le∇2𝑇 ′ , +(35) +where we have assumed constant gravity, and u = (𝑢𝑟,𝑢𝜃,𝑢𝜙) +is the velocity field (with 𝑢𝑟, 𝑢𝜃 and 𝑢𝜙, the radial, polar, +and azimuthal components of the velocity, respectively). In +the equations above, there are 3 dimensionless numbers that +characterize the evolution of the flow. These are the Rayleigh, +Schmidt, and Lewis number, which are defined respectively +as +R = 𝑔𝛼Δ𝑟4|𝜕𝑟𝑇0 − 𝜕𝑟𝑇ad| +𝜅𝑋𝜈 +, +Sc = 𝜈 +𝜅𝑋 +, +Le = 𝜅𝑇 +𝜅𝑋 +, +(36) +where 𝜅𝑇 is the solute diffusivity, and 𝜈 is the kinematic +viscosity. +Note that Sc = Pr Le, where Pr = 𝜈/𝜅𝑇 is the +Prandtl number. +We set the inner and outer radius of the shell to 𝑟i = 7/3, +and 𝑟o = 10/3, respectively. Note that for this choice, the shell +depth is Δ𝑟 = 𝑟o − 𝑟i = 1, and the aspect ratio is 𝑟i/𝑟o = 0.7. +For the dimensionless numbers above, we use R = 106, Le = +3.3, Pr = 0.5, and Sc = 1.6, and the strength of the convective +flow is controlled by changing the flux of light elements at the +boundaries. The boundary conditions are zero gradient for +temperature, and impenetrable and stress-free for velocity. +We specify the composition flux by setting the value of +the composition gradient at each boundary. In our dimen- +sionless variables, this is 𝜕𝑟 𝑋′|𝑟=𝑟i,𝑟o = −𝐹0/𝐹crit, where +the desired composition flux 𝐹0 is normalized by 𝐹crit = +𝜌0𝜅𝑋 (𝛼/𝛽)|𝜕𝑟𝑇0 − 𝜕𝑟𝑇ad|, the flux of light elements that, +if carried by molecular diffusion, would result in a com- +position gradient that is marginally stable against convection +(ie. 𝛽𝜕𝑟 𝑋′ = 𝛼|𝜕𝑟𝑇0 −𝜕𝑟𝑇ad|). We consider values of 𝐹0/𝐹crit +between 0.5 and 30. +We solve the governing equations and boundary conditions +presented above using the pseudo-spectral solver Dedalus +(Burns et al. 2020; Vasil et al. 2019; Lecoanet et al. 2019). The +variables are represented in spherical harmonics for the an- +gular directions and Chebyshev polynomials for the radial di- +rection. All the simulations have 𝐿max = 𝑁max = 255, where +𝐿max is the maximum spherical harmonic degree, and 𝑁max is +the maximal degree of the Chebyshev polynomials used in the +radial expansion. Therefore, the number of radial, latitudinal, +and longitudinal points are (𝑁𝑟, 𝑁𝜃, 𝑁𝜙) = (256, 256, 512), +respectively. For time-stepping, we use a second order semi- +implicit BDF scheme (SBDF2, Wang & Ruuth 2008), where +the linear and nonlinear terms are treated implicitly and ex- +plicitly, respectively. We use a CFL safety factor of 0.35 and +dealias factor of 3/2. To start our simulations, we add random +noise perturbations to the background composition. +3.2. Qualitative description of the flow +We first present results for the runs using 𝐹0/𝐹crit = 1 +and 𝐹0/𝐹crit = 25 as fiducial cases for low and high Pe, +respectively. However, we find that the behavior is qualita- +tively similar for all values of 𝐹0/𝐹crit: once the fluxes at +the boundaries are turned on, an excess (deficit) of light el- +ements develops at the inner (outer) boundary of the shell. +Eventually, the fluid becomes compositionally-buoyant and +suddenly overturns, driving convection. All the simulations +reach a statistically stationary state where the fluxes and the +flow velocities fluctuate around a constant value (see top pan- +els in Fig. 3). The time to reach steady-state depends on the +value of 𝐹0/𝐹crit. For the fiducial cases here, at low Pe the +steady state is achieved at 𝑡 ≈ 0.5, whereas at high Pe it is +achieved at 𝑡 ≈ 0.05, an order of magnitude difference. This +is expected since the convective motions are much slower at +low Pe. We also see differences in the flow structure between +the two cases. This can be seen in the 3D snapshots of the +composition field in the top panels of Fig. 3. We find that the +structure of the flow is more diffusive at low Pe, and more +turbulent at high Pe. +Our simulations confirm the expected inwards convective +heat flux. We find that for a given composition flux, there +is an oppositely directed heat flux that is larger when the +composition flux that drives convection is larger (see the green +curves in Fig. 3). Further, as heat is transported inward, a +temperature gradient develops over time until the associated +flux carried by diffusion balances the convective heat flux +(see bottom panels in Fig. 3). This cancellation means that +once the simulation reaches steady-state, the total heat flux + +8 +Fuentes et al. +Figure 3. Top panels: Time series of the volume-averaged Péclet number, Pe = 𝑢rms/Le (where 𝑢rms = +�√︃ +𝑢2𝑟 + 𝑢2 +𝜃 + 𝑢2 +𝜙 +� +, with the brackets denoting the average +over the shell), total composition flux in the radial direction, 𝐹𝑋,tot = ⟨ ˆr · (u𝑋 − ∇𝑋)⟩ (where the first term corresponds to the convective flux, and the second +term to the diffusion flux), and the magnitude of the convective heat flux in the radial direction, −𝐹𝐻,conv = −⟨ ˆr · u𝑇 ⟩. Note that the total composition flux +converges to 𝐹0/𝐹crit once the fluid reaches steady state. Bottom panels: Radial temperature profile at different times. Results are shown for the fiducial cases at +low and high Pe, i.e., 𝐹0/𝐹crit = 1 (left panels) and 25 (right panels), respectively. To show the differences in the structure of the flow, we overplot 3D snapshots +of the composition and temperature fields, once the simulation reaches steady state. +across the fluid is zero, as expected from our choice of zero +flux boundary conditions. +3.3. Gradients and Péclet number in the convective region +As discussed in §2.4, the properties of the flow in the con- +vection zone are expected to change as a function of the driv- +ing parameter 𝜏. In particular, mixing-length theory predicts +a transition when 𝜏 = 1. To check whether the simulations +support this transition, we measure the shell-averaged con- +vective velocities and radial fluxes as a function of time, and +then for each quantity we take the time-average value over +an interval for which the system is statistically stationary. +When computing volume averages, we exclude regions near +the diffusive boundary layers and confine our measurements +to the convective region. We evaluate 𝜏 using the convective +composition flux in equation (A8), giving 𝜏 = Le−1(𝑢𝑟 𝑋′). +Figure 4 shows the numerical results. We show the mea- +sured gradients as a function of 𝜏 in the left panel, and the +Péclet number as a function of 𝜏 in the right panel. Note that +whereas Pe is defined elsewhere in the paper in terms of the +mixing length as 𝑣𝑐ℓ/𝜅𝑇 , the measured Pe we show in Figure +4 is the quantity 𝑣𝑐Δ𝑟/𝜅𝑇 , ie. using Δ𝑟 as the lengthscale + +Fo/Fcrit = 1 +Fo/Fcr +crit = 25 +-0.4 +102 +102 +Pe +X' +X,tot +-0.9 +101 +101 +Time +X' +-5 +-2.5 +Pe +Fx,tot +100 +100 +H,conv +0.0 +0.2 +0.4 +0.6 +0.8 +0.0 +0.2 +0.4 +0.6 +0.8Fo/Fcrit = 1 +Fo/Fcrit = 25 +-0.1 +0.4 +0.4 +T' +0.2 +0.2 +Temperature ++0.1 +T +±0.5 +0.0 +0.0 +-0.2 +-0.2 +Temperature gradient +Temperature gradient +develops over time +develops over time +-0.4L +-0.4 +2.4 +2.6 +2.8 +3.0 +3.2 +2.4 +2.6 +2.8 +3.0 +3.2 +rHeat transport by compositionally-driven convection +9 +Figure 4. Absolute value of the volume-averaged temperature and composition gradient (left panel) and Péclet number (right panel) measured from the simulations +as a function of the driving parameter 𝜏 = Le−1 ⟨𝑢𝑟 𝑋′⟩. Results are shown for simulations using 𝐹0/𝐹crit = 0.5–30. Lines on each panel are predictions from +the steady-state mixing-length theory solution (Appendix A) using ℓ = 1, R𝑇 = RSc/Le2 ≈ 1.5 × 105, and different values of 𝐶. +since mixing length is not a measured quantity. For 𝑣𝑐, we +measure the rms convective velocity from the simulations. +The solid curves in Figure 4 are the mixing length theory +predictions (which we rewrite for Boussinesq convection in +Appendix A). We set ℓ = Δ𝑟 and show results for three dif- +ferent values of 𝐶 reported in the literature (see discussion in +§2.1). We find that the data supports the predicted transition +at 𝜏 = 1, and the general shape of the curves match well. +The transition is smoother and shows less of a jump than the +example shown in Figure 1 because of the lower value of +Rayleigh number in our simulations. The measured temper- +ature gradient agrees well with the prediction, showing that +the magnitude of the convective heat flux is also as predicted. +We also see the expected inflection in the dependence of the +composition gradient with 𝜏. +There are some differences between the measured values +and the predictions. We find a better agreement for the gradi- +ents as a function of 𝜏, than for the Péclet number as a function +of 𝜏. The numerical values of Pe are larger than the predicted +values for all 𝜏. Fitting separately a power-law to the data +gives Pe ∝ 𝜏0.75 for 𝜏 < 1 (compared to the analytic predic- +tion Pe ∝ 𝜏1/2), and Pe ∝ 𝜏0.65 for 𝜏 > 1 (compared to the +analytic prediction Pe ∝ 𝜏1/3). We find that the composition +gradient approaches 𝜕𝑟 𝑋′ ≈ 1/Le ≈ 0.3 at small 𝜏, which +is consistent with the expected threshold for double-diffusive +instabilities (eg. Traxler et al. 2011a), whereas the analytic +model assumes that Le is large enough that the threshold can +be neglected. We were not able to find values of 𝐶 and ℓ +that fit all the data points. For example, for the choice ℓ = 𝐻 +used in Figure 4, we find that smaller values of 𝐶 are pre- +ferred when fitting ∇𝑋 (left panel), whereas a larger value +of 𝐶 is preferred when fitting Pe (right panel). Nonetheless, +the overall general agreement is encouraging especially given +the approximate nature of mixing length theory (particularly +the approximations made in deriving eq. [4] for the thermal +leakage during convection). +4. Discussion +4.1. Summary of our results +We have used both mixing length theory and numerical sim- +ulations to investigate the heat transport in compositionally- +driven convection. Our results show that there are two dif- +ferent convection regimes, depending on the value of the +parameter 𝜏 defined in equation (23). When thermal diffu- +sion is very efficient, 𝜏 ≪ 1, the convective motions have a +small Péclet number and only a small composition gradient is +needed in the convection zone to overcome the reduced ther- +mal buoyancy (∇𝑋 ≈ ∇𝑋,crit; eq. [9]). A small temperature +gradient ∇ ≈ 𝜏∇ad develops in the convection zone to bal- +ance the inwards transport of heat due to convection. When +thermal diffusion is inefficient, 𝜏 ≫ 1, the behavior is very +different. The temperature gradient steepens to approach the +adiabatic gradient, ∇ → ∇ad, reducing the convective heat +flux to a level where it can be balanced by outwards conduc- +tion along the adiabat9. Depending on the size of the com- +position flux driving convection, the composition gradient in +the convection zone can significantly exceed the critical gra- +dient, ∇𝑋 > ∇𝑋,crit. There is rapid change from one regime +to another as 𝜏 crosses unity. In both cases, the effect of rapid +rotation is to increase the convective velocity and reduce the +composition gradient, with only a minor effect on the heat +flux or temperature gradient unless the rotation is extremely +strong. +With the particular assumptions of equations (5) and (9) +giving the reduction in the temperature excess of a convect- +ing fluid element and the thermal buoyancy, we find that the +ratio of heat flux to composition flux is independent of Péclet +number (eq. [10]). Rising fluid elements lose heat due to ther- +mal diffusion, but a smaller composition gradient is needed to +9 This regime in which inwards heat transport by convection almost bal- +ances the outwards conductive flux along the adiabat has been discussed for +the Earth’s core, eg. Loper (1978) and Labrosse et al. (1997). + +100 +- C = 2元² +I<0,T>1 +102 +.. C = 9/2 +I<0,X)I +-- C=3 +adients +/Le +.0.65 +αT +10- +-1 +Gra( +II +-0.75 +100 +10-2 +10-2 +10-1 +100 +101 +10-2 +10-1 +100 +101 +T=Le-l(urX'> +t=Le-l10 +Fuentes et al. +overcome the thermal buoyancy, requiring a larger convective +velocity and compensating for the heat loss. Our numerical +results give support to this scaling. After an initial build up of +composition at the boundaries, convection starts and evolves +to a state in which, at small Péclet number, the gradients in +the convection zone take on values that would be stable to the +(adiabatic) Ledoux criterion, indicating that thermal diffusion +significantly reduces the stratification. This can be seen by +the fact that |𝜕𝑟 𝑋| < 1 − |𝜕𝑟𝑇| for small 𝜏 in the left panel +of Figure 4. This ordering of gradients (Ledoux stable with +an unstable composition gradient and stable thermal gradi- +ent) corresponds to the regime of fingering or thermohaline +convection driven by double-diffusive instabilities (eg. Ga- +raud 2021). Often investigated as the outcome of unstable +imposed gradients, in our case the convection is maintained +by the continuous injection of elements at the lower boundary, +and the gradients develop as a result of the convection. +4.2. Implications for accreting neutron stars +The lack of dependence of 𝐹𝐻/𝐹𝑋 on Pe means that the cal- +culations of Medin & Cumming (2011, 2014, 2015) for accret- +ing neutron star oceans used a correct expression for the heat +flux even though they assumed adiabatic motions. However, +the composition gradient is overestimated and convective ve- +locity underestimated in those calculations. +For example, +whereas the composition gradient that is marginally stable to +the Ledoux criterion is given by ∇𝑋/(∇ad𝜒𝑇 /𝜒𝑋) = 1, Fig- +ure 1 for example shows that ∇𝑋/(∇ad𝜒𝑇 /𝜒𝑋) ranges from +≈ 10−2–0.3 for 𝜏 in the range 10−3–1, and can be much smaller +for 𝜏 > 1. +The case of accreting neutron stars is interesting because +𝜏 spans a range of values from small to large, covering both +convective regimes. The factor 𝜒𝑋/𝜒𝑇 ∇ad is ∼ 30–100 un- +der the degenerate ocean conditions and depends only on the +composition at the crystallization depth (see Appendix A), +so that 𝜏 ∼ (30–100)(𝑡therm/𝑡𝑋). For cooling following an +accretion outburst, the crystallization timescale is compara- +ble to the cooling time, 𝑡𝑋 ∼ 𝑡therm, so 𝜏 ∼ 30–100. This is +consistent with the rapid steepening of the temperature pro- +file seen by Medin & Cumming (2014, 2015). For steady +accretion, new crust forms on the accretion timescale, which +is ∼ 30 yr for typical parameters (taking an ocean depth +≈ 1013 g cm−2 and accretion rate 104 g cm−2 s−1), whereas +the thermal timescale is a few days at these depths (Bildsten & +Cutler 1995). Therefore 𝑡therm/𝑡𝑋 ∼ 3×10−4, giving 𝜏 ∼ 10−2 +for steady accretion. +Even though the neutron star ocean takes years to ac- +crete, it mixes much more rapidly when chemical separa- +tion is happening. +For a non-rotating star, Figure 1 gives +∇𝑋/(∇ad𝜒𝑇 /𝜒𝑋) ≈ 0.1 for 𝜏 ∼ 10−2, implying that the con- +vective velocity is ≈ 10 times larger than under the adiabatic +assumption. With Pe ≈ 0.3, the convective turnover timescale +𝑡conv = 𝐻𝑃/𝑣𝑐 = 𝑡therm/Pe at the base of the ocean is a few +thermal times (∼ 10 days). Rapid rotation reduces this dra- +matically. Using equation (16) for the Rossby number, the +convective turnover time in the rapidly-rotating limit can be +written +𝑡conv,rot = +�𝑡therm +Pe +�2/3 +(2Ω)−1/3 . +(37) +With a rotation period of a few milliseconds, the convective +turnover time is ≈ 10 min (for a scale height ≈ 3000 cm +this corresponds to a convective velocity 𝑣𝑐 ≈ 5 cm s−1). +For cooling neutron stars with 𝜏 > 1, the convective ve- +locities are even larger. +Evaluating the Rayleigh number +with the help of Bildsten & Cutler (1995) equation (3.9), +we find RaT ≈ 𝑡2 +therm(𝑔/𝐻𝑃)(3/2𝑍)(𝑘𝐵𝑇/𝐸𝐹) ≈ 1018. For +non-rotating convection with 𝜏 > 1, equation (28) gives +Pe ≈ (RaT𝜏)1/3 ≈ 106𝜏1/3. The convective turnover time +is therefore ≈ 0.3 s (velocity ≈ 0.1 km/s). +For 𝜏 > 1, +rapid rotation decreases the convective velocity. With Ta = +(2Ω𝑡therm)2 ≈ 4 × 1017, equation (30) gives Pe ≈ 6000 𝜏3/5, +or a turnover time ≈ 1 min and velocity ≈ 60 cm s−1. +Further calculations of the evolution of accreting neutron +star oceans would be interesting taking into account our re- +vised estimates of the composition gradients and convective +velocities. Mixing on a rapid timescale should have impli- +cations for superbursts. +These long thermonuclear flashes +are thought to be the result of unstable ignition of carbon +in the ocean, although significant problems remain in making +enough carbon and getting it to ignition temperature (in’t Zand +2017). For example, mixing in the ocean could transport car- +bon to greater depths where it can burn (stably or unstably). It +would also be interesting to revisit the calculations of Medin +& Cumming (2014) for neutron stars cooling after accretion +outbursts. Recently, Parikh et al. (2020) reported observa- +tions of two accreting neutron stars in quiescence that showed +a late time (≈ 2000 days after outburst) decrease in temper- +ature, followed by a temperature increase. They pointed out +that this behaviour is similar to the models of Medin & Cum- +ming (2014) that include compositionally-driven convection. +Further investigations are needed to compare against the ob- +servations for these two sources and explore the constraints +on ocean composition and temperature needed to fit the data. +4.3. Implications for white dwarf cooling and dynamos +To investigate the parameters for cystallization-driven con- +vection in white dwarfs, we ran an example 0.6 𝑀⊙ white +dwarf model using the MESA stellar evolution code (Paxton +et al. 2011, 2013, 2015, 2018, 2019; Jermyn et al. 2022) (us- +ing the default wd_cool_0.6M test suite in MESA version +22.11.1). Note that although the code follows the solid-liquid +transition and includes the latent heat, it does not include +chemical separation and so the composition profile does not +evolve in this calculation. +Instead, we estimate the com- +position flux due to chemical separation by measuring the +rate of growth of the solid core �𝑀𝑐 and assuming a value +Δ𝑋melt = 0.1 for the carbon enhancement in the liquid phase +relative to the solid (approximately the liquid-solid compo- +sition difference for the C/O phase diagram; Horowitz et al. +2010). The composition flux is then 𝐹𝑋 = �𝑀𝑐Δ𝑋melt/4𝜋𝑅2 +𝑐, +where 𝑅𝑐 is the core radius. +Figure 5 shows different parameters associated with the liq- +uid region just above the crystallization front as a function of + +Heat transport by compositionally-driven convection +11 +time. The top panel shows the mass of the solid core and +the composition at the freezing point. The white dwarf has +an oxygen-rich inner core surrounded by a carbon-rich outer +core; growth of the core pauses at ≈ 3 Gyr when the crys- +tallization front reaches the edge of the inner core; it takes +≈ 0.5 Gyr of further cooling before the outer core begins to +freeze. The values of 𝑡𝑋, 𝑡therm, 𝜏 and Pe and the temperature +gradient ∇ are shown in the middle two panels of Figure 5. +As in the neutron star case, 𝜒𝑋/𝜒𝑇 ∇ad ∼ 10 is relatively large +(see Appendix A), but 𝑡therm in the conductive interior is short +enough compared to the evolution time 𝑡𝑋 that 𝜏 is small +for much of the evolution. We find 𝜏 > 1 for a short time +at the beginning of crystallization, but it quickly drops and +stabilizes at a value of 𝜏 ≈ 0.01. The corresponding Péclet +numbers are Pe ≈ 0.3, in good agreement with the estimates +of Mochkovitch (1983) and Isern et al. (1997). The bottom +panel of Figure 5 shows the convective velocity. For the non- +rotating case, this is given by 𝑣𝑐 = 𝜅𝑇 Pe/𝐻𝑃, and for the ro- +tating case, we use the convective turnover time from equation +(37). The velocities we obtain are in reasonable agreement +with Mochkovitch (1983) who, using a similar formulation +of mixing length theory, estimated 𝑣𝑐 ≲ 10−6 cm s−1 for no +rotation and ≈ 0.2 cm s−1 for a 1 hour rotation period. +Our convective velocities are much smaller than the recent +estimates of Isern et al. (2017) and Ginzburg et al. (2022) for +crystallization-driven dynamos in white dwarfs. The initial +estimates of Isern et al. (2017) considered the acceleration +of carbon-rich parcels of fluid released at the phase transi- +tion, finding 𝑣𝑐 ≈ 30 km s−1. Ginzburg et al. (2022) argued +that this was an overestimate and instead obtain a velocity +∼ (𝑞𝑐/𝜌)1/3, where 𝑞𝑐 is the gravitational energy flux associ- +ated with the redistribution of elements across the crystalliza- +tion front. This estimate actually corresponds to the situation +where 𝜏 ≫ 1 and ∇𝑋 ≫ ∇𝑋,crit. In that case, equation (11) +gives 𝑣3 +𝑐 ≈ 𝑔𝐻𝑃(𝜒𝑋/𝜒𝜌)𝑣𝑐∇𝑋 ≈ 𝑔𝐻𝑃(𝜒𝑋/𝜒𝜌)(𝐹𝑋/𝜌𝑋) ≈ +𝑞𝑐/𝜌 (eg. compare eq. [C23]). However, we find that white +dwarf interiors are in the 𝜏 ≪ 1 regime, as previously found +by Mochkovitch (1983), with much lower accelerations and +velocities since ∇𝑋 ≈ ∇𝑋,crit. The change of regime has a +huge effect on the velocities: even our rotating convective +turnover times are thousands of years, compared to turnover +times of months in Ginzburg et al. (2022). Even with this +lower velocity, the magnetic Reynolds number Rm is likely +to be large enough to support a dynamo once rotation is +taken into account. With the electrical conductivity in the +range 𝜎 ∼ 1021–1022 s−1 (eg. see Fig. 1 of Cumming 2002), +Rm = 𝐻𝑃𝑣𝑐/𝜂 = 4𝜋𝜎𝑣𝑐𝐻𝑃/𝑐2 ∼ 106–107 for 𝐻𝑃 ≈ 108 cm +and 𝑣𝑐 ≈ 10−3 cm s−1 appropriate for the rapidly-rotating +case (bottom panel of Fig. 5). The threshold value of Rm for +a dynamo is uncertain, but with 𝑣𝑐 ∝ Ω1/3, considering even +slower rotation does not reduce Rm significantly. +However, another major issue for dynamos is the energy +reservoir available to grow the field. In Appendix B, we show +that the kinetic energy flux is a small fraction of the available +gravitational energy (𝐹𝐾/𝑞𝑐 ∼ (∇𝑋 − ∇𝑋,crit)/∇𝑋,crit ≪ 1 +for small 𝜏). The saturated dynamo scaling used by Isern +et al. (2017) is 𝐵2/4𝜋 ∼ 𝜌𝑣2 with 𝑣 ∼ (𝐹/𝜌)1/3 (Chris- +2 +4 +6 +8 +0.0 +0.2 +0.4 +0.6 +Solid core mass (M ) +2 +4 +6 +8 +10 +3 +10 +2 +10 +1 +100 +101 +/ +ad +Pe +2 +4 +6 +8 +107 +108 +108 +109 +1010 +1011 +Time scale (yr) +ttherm +tconv, non +rot +tX +2 +4 +6 +8 +Age (Gyr) +10 +8 +10 +7 +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +vc (cm s 1) +Prot =1 hour +Prot =1 day +No rotation +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Xi +12C +16O +Figure 5. Convection parameters just above the crystallization front as a +function of time for a 0.6 𝑀⊙ white dwarf, evolved with the MESA stellar +evolution code. +Chemical separation and convection are not included in +this model, but we use the rate of crystallization of the core to calculate the +expected properties of compositionally-driven convection. +tensen et al. 2009), where they assumed that the energy flux +𝐹 available to drive the dynamo was the gravitational en- +ergy flux 𝑞𝑐. The mechanism by which dynamo saturation +occurs and the force balance in the saturated state is still an +area of active study (Christensen & Aubert 2006; Schaeffer +et al. 2017; Orvedahl et al. 2021). +However, since mag- +netic field generation occurs as a result of induction by fluid +motions, it seems unlikely that the magnetic energy den- +sity could be many orders of magnitude larger than the ki- +netic energy of the flow. In the context of the Earth’s core, +Loper (1978) also pointed out that much less kinetic energy is +available to drive the dynamo when compositionally-driven +convection occurs in a thermally-stable background. To es- + +12 +Fuentes et al. +timate how small this is, we can use equation (18). +As- +suming a solid core mass ∼ 0.1 𝑀⊙ and 𝐻𝑃 ∼ 108 cm, +we find RaT ∼ 1028 and Ta ∼ 1024 (𝑃rot/h)−2), giving +𝐹𝐾/𝑞𝑐 ∼ (∇𝑋 − ∇𝑋,crit)/∇𝑋,crit ∼ Ta2/3/RaT ∼ 10−12. Us- +ing 𝐵2/4𝜋 ∼ 𝜌𝑣2 with 𝑣𝑐 ∼ 10−3 cm s−1 gives 𝐵 ∼ 3 G 𝜌1/2 +6 , +much smaller than needed to explain observed magnetic fields +in white dwarfs. +The results in Figure 5 show that the temperature gradient +needed to balance the inwards convective transport of heat +(≈ 𝜏∇ad for small 𝜏) is larger or comparable in size to the +existing temperature gradient in the cooling model for much +of the early evolution. This can be seen in the third panel of +Figure 5 where, between ≈ 2–6 Gyr, the temperature gradi- +ent in the white dwarf normalized to the adiabatic gradient, +∇/∇ad, is comparable to the value of 𝜏. This is consistent with +the significant contribution that chemical separation makes to +white dwarf cooling curves. Chemical separation is typically +included in white dwarf cooling codes by assuming that the +cooling is slow enough that the liquid region is well-mixed +(Isern et al. 1997, 2000; Salaris et al. 1997; Montgomery et al. +1999). The energy change due to the changing composition +profile is then added to the latent heat, and distributed in a +small region around the crystallization front (Althaus et al. +2010; Camisassa et al. 2019; Bédard et al. 2022). This ad- +ditional energy will lead to a steepening of the temperature +gradient (to conduct the extra heat to the surface), and in- +deed we estimate in Appendix B that the magnitude of the +convective heat flux is comparable in magnitude to the over- +all energy release due to chemical separation. This suggests +that the temperature profile including the detailed transport +of heat associated with mixing above the crystallization front +may not be that different from current models, but further +calculations are needed to check this in detail. Of particu- +lar interest is the beginning of crystallization, when 𝜏 > 10 +and there is the possibility of significant steepening of the +temperature gradient in the central regions of the star. +4.4. Future work on compositionally-driven convection +The agreement between our numerical simulations and the +mixing-length theory predictions shown in §3 is encourag- +ing. There are many interesting questions to address with +further numerical simulations. The value of Rayleigh num- +ber that we used in §3 gives a relatively smooth transition +between the small and large 𝜏 regimes (Fig. 4). Simulations +at larger Rayleigh number would be interesting to check the +rapid transition predicted at 𝜏 = 1 for large RaT. Our equation +(9) provides a convenient interpolation between the fingering +and overturning convection regimes that at low Pe agrees +with earlier analytic prescriptions for thermohaline convec- +tion (Ulrich 1972 and Kippenhahn et al. 1980 as implemented +in the MESA code for example, Paxton et al. 2013). However, +more recent results are available which provide composition +and heat fluxes for fingering convection that are measured +directly from numerical simulations (Traxler et al. 2011a,b; +Brown et al. 2013). It would improve the modelling to incor- +porate these results at low Pe. +Even more important is that our simulations do not include +rotation, and also adopt the Bousinessq approximation which +limits the vertical scale to be much less than a pressure scale +height. Rapid rotation should greatly reduce the lengthscale +of convection perpendicular to the rotation vector, and is im- +portant to check numerically. Similarly, stratification over +many pressure scale heights would be expected to limit the +vertical transport. Dynamos in fingering convection are be- +ginning to be addressed with numerical simulations. Mather +& Simitev (2021) simulated dynamos with internal volumet- +ric sources or sinks of both thermal and compositional buoy- +ancy, and did not find dynamo action in the fingering con- +vection regime, although Guervilly (2022) argues that finger- +ing convection could support a dynamo at larger Rayleigh +numbers. Numerical simulations of compositionally-driven +dynamos with a thermally-stable background are needed for +application to white dwarfs. It will also be interesting to inves- +tigate other sources of compositional buoyancy, for example +the distillation process involving production of light crystals +proposed by Blouin et al. (2021) for white dwarfs, or electron +captures in neutron star oceans that produce heavy crystals +within the liquid layer that then sink (Medin & Cumming +2014) (an analagous case in planetary dynamos is the iron +snow in Ganymede’s core; Rückriemen et al. 2015). These +improvements in numerical modelling are needed to interpret +the rich set of observations of both white dwarfs and neutron +stars now available. +We thank Simon Blouin whose question about the effect +of thermal diffusion on compositionally-driven convection +sparked this investigation, Brad Hindman and Nick Feather- +stone for useful conversations on rotating convection. We +also thank Thomas Villeneuve and Charles Wilson for pre- +liminary work on this problem. This work was supported +by NSERC Discovery Grant RGPIN-2017-04780. J.R.F. ac- +knowledges support from a McGill Space Institute (MSI) +Fellowship. +A. C., J. R. F. and M. C.-T. are members +of the Centre de Recherche en Astrophysique du Québec +(CRAQ) and the Institut de recherche sur les exoplanètes +(iREx). EHA was supported by a CIERA Postdoctoral Fel- +lowship. This research was enabled in part by support pro- +vided by Calcul Québec (calculquebec.ca), and Compute +Canada (www.computecanada.ca). Computations were per- +formed on Graham and Béluga. +References +Althaus, L. G., García-Berro, E., Renedo, I., et al. 2010, ApJ, 719, 612 +Aurnou, J. M., Horn, S., & Julien, K. 2020, Physical Review Research, 2, +043115 +Bauer, E. B., Schwab, J., Bildsten, L., & Cheng, S. 2020, ApJ, 902, 93 +Bédard, A., Brassard, P., Bergeron, P., & Blouin, S. 2022, ApJ, 927, 128 +Belloni, D., Schreiber, M. 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Mixing length theory for Boussinesq convection +In this Appendix, we give the mixing-length theory results from §2 in a form appropriate for comparison with our numerical +results in §3, ie. in terms of the spatial gradients and using the Boussinesq equation of state. The convective fluxes are +𝐹𝐻 ≈ 1 +2 𝜌0𝑣𝑐𝑐𝑃ℓ (𝜕𝑟𝑇ad − 𝜕𝑟𝑇) +Pe +𝐶 + Pe , +(A1) +𝐹𝑋 ≈ −1 +2 𝜌0𝑣𝑐ℓ𝜕𝑟 𝑋 , +(A2) +with +𝑣2 +𝑐 ≈ 𝑔ℓ2 +8 +� +𝛼 (𝜕𝑟𝑇ad − 𝜕𝑟𝑇) +Pe +𝐶 + Pe − 𝛽𝜕𝑟 𝑋 +� +. +(A3) +The minus sign in the definition of 𝐹𝑋 takes into account the fact that decreasing composition with radius, 𝜕𝑟 𝑋 < 0, leads to an +outwards composition flux, 𝐹𝑋 > 0. Then, the equivalent to equations (9), (13), (20) and (21) are +𝜕𝑟 𝑋crit = 𝛼 +𝛽 (𝜕𝑟𝑇ad − 𝜕𝑟𝑇) +Pe +𝐶 + Pe, +(A4) +𝜕𝑟 𝑋 − 𝜕𝑟 𝑋crit = +8 +R𝑇 +𝛼𝜕𝑟𝑇ad +𝛽 +Pe2 +�Δ𝑟 +ℓ +�4 +, +(A5) +𝜕𝑟𝑇 = 𝜕𝑟𝑇ad +� +Pe2 +Pe2 + 2Pe + 2𝐶 +� +(A6) +Pe = 𝑡therm +𝑡𝑋 +� −2𝑋 +𝜕𝑟 𝑋Δ𝑟 +� +, +(A7) +where we have defined R𝑇 = 𝛼𝑔(−𝜕𝑟𝑇ad)Δ𝑟4/𝜅2 +𝑇 , 𝑡therm = Δ𝑟2/𝜅𝑇 , and 𝑡𝑋 = 𝜌0𝑋Δ𝑟/𝐹𝑋. Following the same argument as in §2, +we solve the system of equations above in terms of the driving parameter +𝜏 = +�𝑡therm +𝑡𝑋 +� � +−𝛽𝑋 +𝛼𝜕𝑟𝑇adΔ𝑟 +� += 𝐹𝑋 +𝐹crit +Le−1, +(A8) +where 𝐹crit is defined in §3. The expressions above are used to generate the analytic curves in Figure 4. +B. Microphysics of white dwarf interiors and neutron star oceans +In this Appendix, we estimate the expected size of the ratio 𝜒𝑋/𝜒𝑇 ∇ad that enters into the parameter 𝜏 (eq. 23). For simplicity, +as in the main text we consider a mixture of two species only, although it is straightforward to generalize to additional species +if needed. The pressure has a contribution from electrons and ions, 𝑃 = 𝑃𝑒 + �2 +𝑖=1 𝑃𝑖, where the terms with 𝑖 = 1 and 𝑖 = 2 +are the ion contributions from each species. Under the degenerate conditions in white dwarf and neutron star interiors, the +degenerate electrons dominate the pressure, with Fermi momentum 𝑝𝐹 = ℏ(3𝜋2𝑛𝑒)1/3 = 𝑥𝑚𝑒𝑐 given by 𝑥 = 1.01 (𝜌6𝑌𝑒)1/3 where +𝜌6 = 𝜌/106 g cm−3. For non-relativistic electrons (𝑥 ≪ 1), the pressure is 𝑃𝑒 = (2/5)𝑛𝑒𝐸𝐹, with 𝐸𝐹 = 𝑝2 +𝐹/2𝑚𝑒. Therefore, +𝑃𝑒 ∝ (𝜌𝑌𝑒)5/3, so that 𝜒𝜌 = 5/3. For relativistic electrons (𝑥 ≫ 1), 𝐸𝐹 = 𝑝𝐹𝑐, 𝑃𝑒 ∝ (𝜌𝑌𝑒)4/3, giving 𝜒𝜌 = 4/3. For the +temperature-dependence of the pressure, we can take the ideal gas pressure as the leading temperature-dependent pressure term +for the ions, i.e. 𝑃𝑖 = 𝑛𝑖𝑘𝐵𝑇 = 𝜌𝑋𝑖𝑘𝐵𝑇/𝐴𝑖𝑚 𝑝. For degenerate electrons, 𝜕 ln 𝑃𝑒/𝜕 ln𝑇 ∼ (𝑘𝐵𝑇/𝐸𝐹)2, which is much smaller +than the ion contribution. Therefore, 𝜒𝑇 ≈ 𝜌𝑘𝐵𝑇/𝜇𝑖𝑚 𝑝𝑃 where 𝜇−1 +𝑖 += � +𝑖 𝑋𝑖/𝐴𝑖, giving +𝜒𝑇 ≈ 5 +2 +𝑘𝐵𝑇 +𝐸𝐹 +𝑌−1 +𝑒 +∑︁ +𝑖 +𝑋𝑖 +𝐴𝑖 +≈ 8.4 × 10−4 𝑇6(𝑌𝑒𝜌6)−2/3𝑌−1 +𝑒 +∑︁ +𝑖 +𝑋𝑖 +𝐴𝑖 +𝑥 ≪ 1 +(B9) +and +𝜒𝑇 ≈ 4 𝑘𝐵𝑇 +𝐸𝐹 +𝑌−1 +𝑒 +∑︁ +𝑖 +𝑋𝑖 +𝐴𝑖 +≈ 6.7 × 10−3 𝑇8𝜌−1/3 +9 +𝑌−4/3 +𝑒 +∑︁ +𝑖 +𝑋𝑖 +𝐴𝑖 +𝑥 ≫ 1 +(B10) +For the compositional dependence of the pressure, we must look at both the electron and ion contributions. The dominant +contribution is from the 𝑇 = 0 terms. For a two-component mixture, +𝑌𝑒 = 𝑋𝑍1 +𝐴1 ++ (1 − 𝑋)𝑍2 +𝐴2 +, +(B11) + +Heat transport by compositionally-driven convection +15 +where 𝑋 = 𝑋1 is the mass fraction of the lighter species, and 1 − 𝑋 = 𝑋2 is the mass fraction of the heavier species. This gives +𝜕𝑃𝑒 +𝜕𝑋 = 𝜕𝑃𝑒 +𝜕𝑌𝑒 +𝜕𝑌𝑒 +𝜕𝑋 = 𝜕𝑃𝑒 +𝜕𝑌𝑒 +� 𝑍1 +𝐴1 +− 𝑍2 +𝐴2 +� +, +(B12) +where 𝜕 ln 𝑃𝑒/𝜕 ln𝑌𝑒 = 5/3 and 4/3 in the non-relativistic and relativistic limits respectively, and the partial derivatives are +taken at constant temperature and density. For ions in the liquid phase, the leading order term in the Helmholtz free-energy at +zero-temperature is 𝐹𝑖 = −𝐶𝑀 Γ𝑖𝑁𝑖𝑘𝐵𝑇, for 𝑁𝑖 ions in a volume 𝑉, where Γ𝑖 = 𝑍5/3 +𝑖 +Γ𝑒 and Γ𝑒 = 𝑒2/𝑎𝑒𝑘𝐵𝑇 with 4𝜋𝑎3 +𝑒𝑛𝑒/3 = 1 +defines the mean electron separation 𝑎𝑒, and 𝐶𝑀 ≈ 0.90 is related to the Madelung constant (Dewitt & Slattery 1999; Farouki & +Hamaguchi 1993; Potekhin & Chabrier 2000; Medin & Cumming 2010). This gives 𝐹𝑖 ∝ 𝑉−1/3 leading to the Coulomb pressure +𝑃𝑖 = (1/3)(𝐹𝑖/𝑉), or +𝑃𝑖 = −1 +3𝐶𝑀𝑛𝑖𝑍5/3 +𝑖 +𝑒2 +�4𝜋𝑛𝑒 +3 +�1/3 += −1 +3𝐶𝑀𝑒2 +�4𝜋 +3 +�1/3 � 𝜌 +𝑚 𝑝 +�4/3 +𝑍5/3 +𝑖 +� 𝑋𝑖 +𝐴𝑖 +� +𝑌1/3 +𝑒 +. +(B13) +Therefore, +𝜕𝑃𝑖 +𝜕𝑋𝑖 += 𝑃𝑖 +𝑋𝑖 ++ 𝜕𝑌𝑒 +𝜕𝑋𝑖 +𝜕𝑃𝑖 +𝜕𝑌𝑒 += 𝑃𝑖 +𝑋𝑖 +� +1 + 𝑋𝑖𝑍𝑖 +3𝐴𝑖𝑌𝑒 +� +, +(B14) +so that +𝜕(𝑃1 + 𝑃2) +𝜕𝑋 += 𝑃1 +𝑋 − +𝑃2 +1 − 𝑋 + 1 +3 +𝑃1 + 𝑃2 +𝑌𝑒 +𝜕𝑌𝑒 +𝜕𝑋 = 𝑃1 +𝑋 − +𝑃2 +1 − 𝑋 + 1 +3 +𝑃1 + 𝑃2 +𝑌𝑒 +� 𝑍1 +𝐴1 +− 𝑍2 +𝐴2 +� +. +(B15) +Now adding the ion and electron contributions (eqs. [B12] and [B15]) gives +𝜕𝑃 +𝜕𝑋 = 𝑃1 +𝑋 − +𝑃2 +1 − 𝑋 + +�1 +3 +𝑃1 + 𝑃2 +𝑌𝑒 ++ 𝜕𝑃𝑒 +𝜕𝑌𝑒 +� � 𝑍1 +𝐴1 +− 𝑍2 +𝐴2 +� +. +(B16) +Noting that |𝑃1 + 𝑃2| ≪ 𝑃𝑒, we can drop the 𝑃1 + 𝑃2 term relative to the 𝜕𝑃𝑒/𝜕𝑌𝑒 term, and take 𝑃 ≈ 𝑃𝑒, giving +𝜒𝑋 ≈ −𝑃1 +𝑃 +�� 𝑍2 +𝑍1 +�5/3 � 𝐴1 +𝐴2 +� +− 1 +� ++ 𝑋 +𝑌𝑒 +𝜕 ln 𝑃𝑒 +𝜕 ln𝑌𝑒 +� 𝑍1 +𝐴1 +− 𝑍2 +𝐴2 +� +(B17) +as our final expression for 𝜒𝑋. A similar expression for the internal energy per gram 𝐸 was derived by Isern et al. (1997, 2000); +as a check, calculating 𝜒𝑋 as (𝑋/𝑃)(𝜌2𝜕/𝜕𝜌)(𝜕𝐸/𝜕𝑋) using their results for 𝜕𝐸/𝜕𝑋 gives agreement with equation (B17). +In neutron star oceans, the second term in equation (B17) dominates, since 𝑃𝑖 ≪ 𝑃 and we typically have species with different +ratios 𝑍/𝐴. With 𝑍2/𝐴2 < 𝑍1/𝐴1 (since species 1 is the lighter species), this term is positive. For example, for the neutron star +ocean with a mixture of O (𝑍 = 8, 𝐴 = 16) and Se (𝑍 = 34, 𝐴 = 79) considered by Medin & Cumming (2011), Δ(𝑍/𝐴) = 0.070, +and the second term gives 𝜒𝑋 ≈ 0.2𝑋. In this case, taking ∇ad ≈ 1/3 and using equation (B10), we find +𝜒𝑋 +𝜒𝑇 ∇ad +≈ 270 +� 𝑋 +0.1 +� � 𝜇𝑖 +79 +� �Δ(𝑍/𝐴) +0.07 +� 𝜌1/3 +9 +𝑇8 +≈ 69 +� 𝑋 +0.1 +� � 𝜇𝑖 +79 +� �Δ(𝑍/𝐴) +0.07 +� � ⟨𝑍5/3⟩ +345/3 +�−1 � Γ +178 +� +, +neutron star +(B18) +where 𝜇−1 +𝑖 += � +𝑖(𝑋𝑖/𝐴𝑖) and Γ is the Coulomb coupling parameter with ⟨𝑍5/3⟩ averaged by number (see Medin & Cumming 2015 +eq. [2]). Note that at fixed Γ, 𝜒𝑋/𝜒𝑇 ∇ad depends only on composition and is independent of temperature and density. +In white dwarf interiors, however, the electron term in equation (B17) is small or vanishing (as pointed out by Isern et al. 1997, +2000). For a mixture of C and O for example, 𝑌𝑒, and therefore the electron pressure, is independent of the C/O ratio, since +both species have 𝐴 = 2𝑍. In that case, 𝜒𝑋 is set by the ion term. Using 𝑃𝑒 for non-relativistic electrons and 𝑃𝑖 from equation +(B13), we find 𝑃𝑖/𝑃𝑒 ≈ −0.0057 (𝑌𝑒𝜌6)−1/3(𝑍𝑖/𝑌𝑒𝐴𝑖)𝑋𝑖𝑍2/3 +𝑖 +. For a mixture of C/O, 𝑃𝑖/𝑃𝑒 ≈ 0.019𝑋(𝑌𝑒𝜌6)−1/3 and the factor +(𝑍2/𝑍1)5/3(𝐴1/𝐴2) − 1 ≈ 0.21, giving 𝜒𝑋 ≈ 4.0 × 10−3 𝑋(𝑌𝑒𝜌6)−1/3, approximately two orders of magnitude smaller than in the +neutron star case. Again taking ∇ad ≈ 1/3, and using equation (B9), we find +𝜒𝑋 +𝜒𝑇 ∇ad +≈ 40 +� 𝑋 +0.5 +� � 𝜇𝑖 +14 +� 𝜌1/3 +6 +𝑇6 +≈ 14 +� 𝑋 +0.5 +� � 𝜇𝑖 +14 +� � ⟨𝑍5/3⟩ +75/3 +�−1 � Γ +178 +� +. +white dwarf +(B19) +We see that 𝜒𝑋/𝜒𝑇 ∇ad is about an order of magnitude smaller than in the neutron star ocean case at the same value of 𝑋, but still +larger than one. We apply these values of 𝜒𝑋/𝜒𝑇 ∇ad in our estimates in §4. +For the MESA simulation results shown in §4, we take 𝜒𝑇 directly from the code, and compute 𝜒𝑋 by perturbing 𝑋 and calling +the equation-of-state directly to compute 𝜕𝑃/𝜕𝑋. The analytic formulae above agree well with the numerical results, as can be +seen in Figure 6. + +16 +Fuentes et al. +2 +4 +6 +8 +Age (Gyr) +10 +3 +10 +2 +T +T analytic +X +X analytic +2 +4 +6 +8 +Age (Gyr) +10 +1 +100 +101 +ad +X +T +ad +X +T +ad analytic +Figure 6. 𝜒𝑋, 𝜒𝑇 , ∇ad, and the ratio 𝜒𝑋/𝜒𝑇 ∇ad at the crystallization front for the white dwarf models shown in Figure 5. We compare against the analytic +results given by equation (B9), the first term of equation (B17), and equation (B19). +C. Energetics of chemical separation in white dwarfs +By considering the change of internal energy with composition across the white dwarf, Isern et al. (1997, 2000) found the extra +luminosity generated by the redistribution of elements in the convection zone is given by +𝐿chem = �𝑀𝑐Δ𝑋melt +� 𝜕𝐸 +𝜕𝑋 +���� +𝑐 +− +� 𝜕𝐸 +𝜕𝑋 +�� += �𝑀𝑐Δ𝑋melt 𝛼 𝜕𝐸 +𝜕𝑋 +���� +𝑐 +, +(C20) +where the first term in the square brackets is evaluated at the crystallization boundary, the second is an average over the liquid +region, and we introduce the same averaging parameter 𝛼 ≲ 1 as Isern et al. (1997). The partial derivative 𝜕𝐸/𝜕𝑋 is taken at +constant 𝑇 and 𝜌; for clarity we do not indicate this explicity. Note that as elsewhere in this paper, 𝑋 is the mass fraction of the +light element, and define Δ𝑋melt = 𝑋𝑙 − 𝑋𝑠 > 0 as the difference in the light element mass fraction between liquid and solid phases. +Now using equation (8) of Isern et al. (2000) for 𝜕𝐸/𝜕𝑋 and the first term in equation (B17) for 𝜒𝑋, we find 𝜕𝐸/𝜕𝑋 = 3𝑔𝐻𝑃 𝜒𝑋, +and therefore +𝐿chem ≈ �𝑀𝑐𝑔𝐻𝑃 Δ𝑋melt (3𝛼𝜒𝑋) +(C21) +(see Isern et al. 1997 for a similar argument). +We can compare this with Ginzburg et al. (2022), who write the rate of gravitational energy release (see their eq. [6]) as +𝐿grav ≈ �𝑀𝑐𝑔𝐻𝑃 +Δ𝜌 +𝜌 , +(C22) +where �𝑀𝑐 is the growth rate of the mass of the solid core and Δ𝜌 = 𝜌𝑠 − 𝜌𝑙 > 0 is the density contrast between solid and liquid +phases at the crystallization front. Now writing Δ𝜌/𝜌 = −(𝜒𝑋/𝜒𝜌)(−Δ𝑋melt/𝑋) gives +𝐿grav ≈ �𝑀𝑐𝑔𝐻𝑃 Δ𝑋melt +𝜒𝑋 +𝑋 𝜒𝜌 +≈ �𝑀𝑐𝑔𝐻𝑃 Δ𝑋melt +�3𝜒𝑋 +5𝑋 +� +, +(C23) +which is approximately equal to 𝐿chem (depending on the value of 𝛼). +We can compare this with the convective heat flux associated with the flux of light elements using equation (10). Writing +4𝜋𝑅2 +𝑐𝐹𝑋 = �𝑀𝑐Δ𝑋melt and assuming 𝜏 < 1 so that ∇𝑋 ≈ ∇𝑋,crit, gives +𝐿𝐻 = 4𝜋𝑅2 +𝑐𝐹𝐻 = 4𝜋𝑅2 +𝑐𝐹𝑋 +𝑐𝑃𝑇 +𝑋 +𝜒𝑋 +𝜒𝑇 += �𝑀𝑐𝑔𝐻𝑃 Δ𝑋melt +𝜌𝑐𝑃𝑇 +𝑋𝑃 +𝜒𝑋 +𝜒𝑇 +, +(C24) +where we have also used the relation 𝑃 = 𝜌𝑔𝐻𝑃. Now applying the thermodynamic identities ∇ad = 𝜒𝑇 𝑃/𝜌𝑐𝑉 𝑇Γ1 and Γ1 ≈ 𝜒𝜌, +𝑐𝑃 ≈ 𝑐𝑉 for a degenerate gas gives +𝐿𝐻 = �𝑀𝑐𝑔𝐻𝑃 Δ𝑋melt +� +1 +𝑋∇ad +𝜒𝑋 +𝜒𝜌 +� +. +(C25) + +Heat transport by compositionally-driven convection +17 +This shows that the inwards convective luminosity (and compensating outwards luminosity carried by thermal conduction) is of +the same order of magnitude as 𝐿chem. +Also of interest is the kinetic energy flux 𝐹𝐾 ≈ 𝜌𝑣3 +𝑐. Comparing this to 𝐿chem gives +𝐿𝐾 +𝐿chem += 4𝜋𝑅2 +𝑐𝐹𝐾 +𝐿chem += 4𝜋𝑅2 +𝑐𝜌𝑣𝑐 +�𝑀Δ𝑋melt +𝑣2 +𝑐 +𝑔𝐻𝑃 +1 +3𝛼𝜒𝑋 += +𝑣2 +𝑐 +𝑔𝐻𝑃 +1 +3𝛼𝑋 𝜒𝑋 +1 +∇𝑋 +. +(C26) +Using equation (11) to replace 𝑣2 +𝑐 in the non-rotating limit, +𝐿𝐾 +𝐿chem +≈ +1 +24𝛼𝑋 𝜒𝜌 +∇𝑋 − ∇𝑋,crit +∇𝑋 +, +(C27) +where we take the mixing length to be equal to the pressure scale height for simplicity. This shows that the kinetic energy flux is +a small fraction of the total energy flux. This is different from efficient thermal convection in a typical stellar convection zone, +where 𝐹𝐻 ≈ 𝜌𝑣𝑐𝑐𝑃𝑇(∇ − ∇ad), 𝑣2 +𝑐 ≈ 𝑔𝐻𝑃(∇ − ∇ad), and 𝑐𝑃𝑇 ≈ 𝑃/𝜌 ≈ 𝑔𝐻𝑃 gives the standard result 𝐹𝐻 ≈ 𝜌𝑣3 +𝑐. Here we have +𝐹𝐻 ∼ 𝜌𝑣𝑐𝑃𝑇∇adPe at small Pe and 𝑣2 ∼ 𝑔𝐻𝑃(∇𝑋 − ∇𝑋,crit)(𝜒𝑋/𝜒𝜌), giving 𝜌𝑣3 ∼ 𝐹𝐻 (𝜒𝑋/𝜒𝑇 )(∇𝑋 − ∇𝑋,crit)Pe−1 ≪ 1. + diff --git a/OtE3T4oBgHgl3EQfCAnu/content/tmp_files/load_file.txt b/OtE3T4oBgHgl3EQfCAnu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2874f1c1c0cad4d7cd595a354f99f0f57713a3f --- /dev/null +++ b/OtE3T4oBgHgl3EQfCAnu/content/tmp_files/load_file.txt @@ -0,0 +1,1109 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf,len=1108 +page_content='Draft version January 12, 2023 Typeset using LATEX twocolumn style in AASTeX62 Heat transport and convective velocities in compositionally-driven convection in neutron star and white dwarf interiors J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Fuentes1, 2, Andrew Cumming2, Matias Castro-Tapia2, and Evan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Anders3 1Department of Applied Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' CO 80309-0526,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' USA 2Department of Physics and Trottier Space Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' McGill University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Montreal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' QC H3A 2T8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Canada 3Center for Interdisciplinary Exploration and Research in Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Northwestern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Evanston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Illinois 60201,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' USA Abstract We investigate heat transport associated with compositionally-driven convection driven by crystallization at the ocean-crust interface in accreting neutron stars,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' or growth of the solid core in cooling white dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We study the effect of thermal diffusion and rapid rotation on the convective heat transport, using both mixing length theory and numerical simulations of Boussinesq convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We determine the heat flux, composition gradient and Péclet number (the ratio of thermal diffusion time to convective turnover time) as a function of the composition flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We find that the ratio between the heat flux and composition flux is independent of Péclet number, because the loss of heat from convecting fluid elements due to thermal diffusion is offset by the smaller composition gradient needed to overcome the reduced thermal buoyancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We find two regimes of convection with a rapid transition between them as the composition flux increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We discuss the implications for neutron star and white dwarf cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Convection in neutron stars spans both regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We find rapid mixing of neutron star oceans, with a convective turnover time of order weeks to minutes depending on rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Except during the early stages of core crystallization, white dwarf convection is in the thermal-diffusion-dominated fingering regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We find convective velocities much smaller than recent estimates for crystallization-driven dynamos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The small fraction of energy carried as kinetic energy calls into question the effectiveness of crystallization-driven dynamos as an explanation for observed white dwarf magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Key words: convection – stars:neutron – stars: white dwarfs – X-rays: binaries 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Introduction When a multicomponent plasma freezes, the composition of the solid is typically different from the composition of the liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' If the solid preferentially retains heavy elements, the liquid left behind is lighter and buoyant, driving con- vection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The compositionally-driven convection transports light elements outwards and mixes the liquid region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This process has been studied in the context of dense interiors of white dwarfs (Stevenson 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Mochkovitch 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 1997) and accreting neutron stars (Medin & Cumming 2011, 2014, 2015), and also occurs in planetary interiors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Earth (Fearn & Loper 1981), the Moon (Laneuville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Scheinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2015) and Mercury (Manglik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Depending on the phase diagram, another possibility is that heavy elements preferentially go into the liquid phase, so that solid crystals float upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This distillation process has re- cently been suggested to be occurring in white dwarfs, driven by chemical separation of 22Ne between the liquid and solid phases (Blouin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2021) (see also Mochkovitch 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Redistribution of elements in white dwarf interiors is im- portant because the gravitational energy released can pro- jofu5477@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='edu long white dwarf cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The large increase in the number of white dwarfs with well-determined distances from Gaia (Gentile Fusillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2021) has enabled the cooling delay associated with crystallization to be definitely detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The slowed cooling is visible as an increased density of white dwarfs in the HR diagram or luminosity function (Tremblay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' One puzzling feature in the HR diagram known as the Q-branch indicates an additional cooling delay in a small fraction of massive white dwarfs (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Explanations for the delay have focused on the redistribution of elements (22Ne in particular) within the white dwarf (Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Blouin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Camisassa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Caplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Neutron stars in low mass X-ray binaries accrete enough mass over their lifetimes to replace the entire neutron star crust (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Suleiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The accreted light elements first undergo thermonuclear burning in the surface layers, generating a complex mixture of heavy elements that forms a liquid ocean (Bildsten & Cutler 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' At the base of the ocean, compressed matter continuously freezes and forms solid crust as accretion continues (Brown & Bildsten 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In sources that undergo transient accretion outbursts, the neu- tron star cools in quiescence and the liquid ocean refreezes arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='04273v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='SR] 11 Jan 2023 2 Fuentes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (see Wijnands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2017 for a review of transiently accreting neutron stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Horowitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2007) showed that chemical separation between liquid and solid phases is expected for the mixtures found in neutron star oceans, with lighter elements typically left behind in the liquid phase (see Mckinven et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2016 and Caplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2018 for a survey of different com- positions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Medin & Cumming (2011, 2014, 2015) studied the compositional changes and heat transport in the ocean in these different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Unlike in many planetary interiors, where convection is driven by both compositional and thermal buoyancy, crystallization-driven convection in white dwarf and neutron star interiors occurs in a part of the star that is thermally-stable to convection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' has a sub-adiabatic temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This is because of the large thermal conductivity from degen- erate electrons which can transport the cooling luminosity and latent heat of crystallization with only a small temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In this case, when compositionally-driven convec- tion occurs in a thermally-stratified background, convection transports heat in the opposite direction to the composition flux (Loper 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Medin & Cumming 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Rising fluid el- ements adiabatically expand and cool down to a temperature that is lower than their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This cools the surround- ings, giving an effective heat flow that is directed downwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' By transporting heat towards the liquid/solid interface, con- vection acts in a similar way to the latent heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Medin & Cumming (2014) showed that this changes the cooling rate of neutron stars following accretion outbursts, an observable signature of the newly-forming crust and its composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' When calculating the convective heat flux in neutron star oceans, Medin & Cumming (2011, 2015) assumed that the fluid motions would be adiabatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' However, a large enough thermal conductivity could cause rising parcels of fluid to lose a significant amount of energy by thermal diffusion, re- ducing the effective heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The likely importance of ther- mal diffusion in white dwarf convection was pointed out by Stevenson (1980) and included in estimates of the convective velocities by Mochkovitch (1983) and Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The Péclet number, the ratio of thermal diffusion time to convec- tive turnover time was estimated to be ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='3 by Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (1997), implying that this effect is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The transport of heat by compositionally-driven convection does not appear to have been considered in white dwarfs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' instead, it is usu- ally assumed that the liquid region above the crystallization front mixes rapidly, and the resulting change in energy is put directly into the model as a localized heat source (Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 1997, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Interest in compositionally-driven convection in white dwarfs has also been recently revived with the suggestion of Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2017) that it leads to a magnetic dynamo in crys- tallizing white dwarfs (Schreiber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2021a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Belloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Camisassa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Schreiber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Using the scaling of Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2009) for a saturated dynamo, Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2017) found that fields up to ∼ 1 MG could be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' However, whether the dynamo is in the saturated regime depends on the convective turnover time, and estimates of the convective velocity differ significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (1997) found 𝑣𝑐 ≈ 30 km s−1 by considering rising carbon-enriched liquid bubbles released at the crystallization front, whereas Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2022) ar- gued that the velocity should be much lower, ∼ 100 cm s−1, based on the available convective energy flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Both of these velocity estimates are significantly larger than previous esti- mates for (non-magnetic) compositionally-driven convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Mochkovitch (1983) found 𝑣𝑐 ∼ 10−6 cm s−1 for non-rotating or ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1 cm s−1 for rapidly-rotating white dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In this paper, we revisit compositionally-driven convec- tion in dense stellar interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Our goal is to determine the expected convective velocities and convective heat flux for ac- creting neutron stars and cooling white dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We apply stel- lar mixing length theory to the case of compositionally-driven convection, and use numerical simulations to demonstrate that heat is indeed transported inwards and test the mixing length theory predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The mixing length theory is presented in section 2, where we derive expressions for the heat flux and convective velocities, and discuss the steady-state outcome in which the inwards heat flux due to convection is balanced by an outwards conductive heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In section 3, we present our numerical simulations of Boussinesq convection in the non-rotating case and compare with mixing length theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We conclude in section 4 with a discussion of how our results apply to white dwarfs and neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Mixing length theory for compositionally-driven convection In this section, we use mixing length theory to investi- gate the size of the heat flux associated with compositionally- driven convection, and the expected convective velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We first write down mixing length theory including thermal diffu- sion (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1), and then discuss the expected heat flux (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2) and convective velocities in the non-rotating and rapidly-rotating limits (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We then investigate the steady-state in which the inwards convective flux is balanced by outwards conduction (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Mixing length theory including thermal diffusion In mixing length theory, the heat and composition fluxes are written in terms of the excess temperature 𝐷𝑇 or composition 𝐷𝑋 carried by a fluid element, 𝐹𝐻 = 𝜌𝑣𝑐𝑐𝑃𝐷𝑇 and 𝐹𝑋 = 𝜌𝑣𝑐𝐷𝑋, where 𝑣𝑐 is the convective velocity, 𝑐𝑃 is the specific heat capacity at constant pressure, and 𝜌 the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For simplicity, we assume a mixture of two elements, so that the composition can be described by only one variable, here chosen to be 𝑋, the mass fraction of the lighter component1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We include the effect of thermal diffusion following the formulation of mixing length theory discussed by Kippenhahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2012), which is based on Böhm-Vitense (1958) (see also Henyey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 1965 and Gough 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The temperature excess 1 The results can be easily generalized to more complex mixtures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Medin & Cumming (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We also make the approximation that the excess entropy carried by fluid elements is 𝐷𝑆 ≈ 𝑐𝑃𝐷𝑇 /𝑇 , ignoring any contribution to the entropy from compositional differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Again, this can be included in a straightforward way, writing the heat flux as 𝜌𝑣𝑐𝑇 𝐷𝑆, but is typically a small correction (Medin & Cumming 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Heat transport by compositionally-driven convection 3 is written as 𝐷𝑇 = (∇ − ∇𝑒) 𝑇 ℓ 2𝐻𝑃 , (1) where ∇ = 𝑑 ln𝑇/𝑑 ln 𝑃|★ is the temperature gradient in the star, ∇𝑒 is the rate of change of temperature with pressure experienced by the fluid element, ℓ is the mixing length, and 𝐻𝑃 the pressure scale height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Similarly, we can write 𝐷𝑋/𝑋 = ∇𝑋 (ℓ/2𝐻𝑃), where ∇𝑋 = 𝑑 ln 𝑋/𝑑 ln 𝑃|★ is the composition gradient in the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The heat and composition fluxes are then given by 𝐹𝐻 = 𝜌𝑣𝑐𝑐𝑃𝐷𝑇 = 𝜌𝑣𝑐𝑐𝑃𝑇 (∇ − ∇𝑒) ℓ 2𝐻𝑃 , (2) and 𝐹𝑋 = 𝜌𝑣𝑐𝑋∇𝑋 ℓ 2𝐻𝑃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (3) The sign of these fluxes is such that a positive flux is in the upwards direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For example, an outwards flux of light elements is associated with a gradient ∇𝑋 > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' the mass fraction of light elements increases with pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Note that the composition flux gives the mass of light elements crossing unit area per unit time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' in cgs the units of 𝐹𝑋 are g cm−2 s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' By considering the exchange of energy by thermal diffusion with the surroundings as the fluid element moves, Kippenhahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2012) derive an expression for ∇𝑒 ∇𝑒 − ∇ad ∇ − ∇𝑒 = 9 2 𝐾 𝜌𝑐𝑃ℓ𝑣𝑐 = 9 2 𝜅𝑇 ℓ𝑣𝑐 ≡ 9 2 1 Pe , (4) where 𝜅𝑇 is the thermal diffusivity and we define the dimen- sionless Péclet number Pe ≡ ℓ𝑣𝑐/𝜅𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The numerical pref- actor of 9/2 in equation (4) depends on assumptions about the shape of the fluid element and the temperature distribu- tion (see discussion in Henyey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For example, Hubeny & Mihalas (2014) following Böhm-Vitense (1958) give a prefactor of 3 instead, whereas Henyey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (1965) have a prefactor of 2𝜋2 ≈ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Here, instead of adopting any particular value, we keep in mind that it is model-dependent and treat it as a free parameter 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Replacing the 9/2 by 𝐶 in equation (4) gives ∇ − ∇𝑒 = � Pe 𝐶 + Pe � (∇ − ∇ad) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (5) When the convective motions are rapid, Pe ≫ 1 and ∇𝑒 → ∇ad as expected since the motions become adiabatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In the opposite limit in which the convective motions are slow and thermal diffusion can act, Pe ≪ 1 and ∇𝑒 → ∇, so that the fluid element is able to adjust its temperature to follow the background temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The heat flux in compositionally-driven convection Taking the ratio of equations (2) and (3), the convective velocity and mixing length drop out, giving the heat flux in terms of the composition flux, 𝐹𝐻 𝐹𝑋 = −𝑐𝑃𝑇 𝑋 ∇𝑒 − ∇ ∇𝑋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (6) This shows that transport of composition is associated also with a transport of heat, provided the fluid elements expe- rience a different temperature evolution with pressure com- pared to the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Equation (5) shows that ∇𝑒 ranges from ∇ to ∇ad as Pe goes from small to large values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In a back- ground that is stably-stratified thermally, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' with ∇ad > ∇, this means that ∇𝑒 ≥ ∇, giving a heat flux oppositely-directed to the composition flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The fact that ∇𝑒 approaches ∇ for Pe ≪ 1 (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [5]) acts to reduce the heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' However, the composition gradient in the convection zone also depends on Pe, since the effective ther- mal stratification is reduced at low Pe when thermal diffusion is efficient, which means that a smaller composition gradient is needed to maintain the convective motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' To see this, consider the typical density contrast in the convection zone, 𝐷𝜌 𝜌 ≈ − 𝜒𝑇 𝜒𝜌 𝐷𝑇 𝑇 − 𝜒𝑋 𝜒𝜌 𝐷𝑋 𝑋 , (7) where 𝜒𝑇 = 𝜕 ln 𝑃/𝜕 ln𝑇|𝜌,𝑋, 𝜒𝜌 = 𝜕 ln 𝑃/𝜕 ln 𝜌|𝑇 ,𝑋, and 𝜒𝑋 = 𝜕 ln 𝑃/𝜕 ln 𝑋|𝜌,𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The density contrast determines the buoyant acceleration ∝ −𝐷𝜌/𝜌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Written in terms of the gradients, 𝐷𝜌 𝜌 ≈− ℓ 2𝐻𝑃 � 𝜒𝑇 𝜒𝜌 (∇ − ∇𝑒) + 𝜒𝑋 𝜒𝜌 ∇𝑋 � ≈− ℓ 2𝐻𝑃 𝜒𝑋 𝜒𝜌 � ∇𝑋 − ∇𝑋,crit � , (8) where we define the critical composition gradient ∇𝑋,crit = 𝜒𝑇 𝜒𝑋 (∇𝑒 − ∇) = 𝜒𝑇 𝜒𝑋 (∇ad − ∇) � Pe 𝐶 + Pe � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (9) For adiabatic displacements (large Pe), where ∇𝑒 → ∇ad, 𝐷𝜌 < 0 in equation (8) is equivalent to the Ledoux criterion for convection, 𝜒𝑇 (∇ − ∇ad) + 𝜒𝑋∇𝑋 > 0, and so ∇𝑋,crit in this limit is the composition gradient needed to be unstable to convection according to the Ledoux criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' At small Pe, thermal diffusion lowers the effective thermal stratification, reducing ∇𝑋,crit, and allowing convection to occur for smaller composition gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This is the regime of fingering or ther- mohaline convection2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' A similar expression to equation (8) was previously written down by Mochkovitch (1983) for the case 𝐶 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' If the convection is efficient in the sense that only small den- sity perturbations are required to drive the required convective velocities and fluxes, then 𝐷𝜌/𝜌 ≪ 1 ⇒ ∇𝑋 ≈ ∇𝑋,crit3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In this case, the reduction in ∇𝑒 − ∇ at small Pe is exactly offset 2 In the limit Pe ≪ 1 and assuming ∇𝑋 ≈ ∇𝑋,crit, equation (9) agrees with the prescription for convection in the MESA code (Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2013) based on Ulrich (1972) and Kippenhahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' To see this, write the diffusion coefficient in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (14) of Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2013) as 𝐷th = 𝑣𝑐ℓ, in which case their expression reduces to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The efficiency parameter for thermohaline convection 𝛼th is related to our shape parameter by 𝛼th = 2𝐶/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 3 This is analagous to efficient thermal convection where ∇ − ∇ad ≪ ∇ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 4 Fuentes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' by the reduction in ∇𝑋, so the ratio 𝐹𝐻/𝐹𝑋 is actually inde- pendent of Pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' To see this more explicitly, we can write the heat flux in terms of ∇𝑋,crit, giving 𝐹𝐻 𝐹𝑋 = −𝑐𝑃𝑇 𝑋 𝜒𝑋 𝜒𝑇 � ∇𝑋,crit ∇𝑋 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (10) This relation between 𝐹𝐻 and 𝐹𝑋 is the same as derived by Medin & Cumming (2011) under the assumption that fluid elements move adiabatically (the only difference is that ∇𝑋,crit in that case is given by the large Pe limit of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Convective velocity and effect of rotation We can estimate the extent to which ∇𝑋 exceeds ∇𝑋,crit by writing the expression for the convective velocity 𝑣2 𝑐 ≈ 𝑔ℓ 4 𝐷𝜌 𝜌 ≈ 𝑔ℓ2 8𝐻𝑃 𝜒𝑋 𝜒𝜌 �∇𝑋 − ∇𝑋,crit � , (11) where we take the numerical prefactors 1/4 and 1/8 from the particular formulation of mixing length theory we are using (Kippenhahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Using the definition Pe = ℓ𝑣𝑐/𝜅𝑇 and defining a Rayleigh number RaT = 𝑔𝐻3 𝑃 𝜒𝑇 ∇ad 𝜒𝜌𝜅2 𝑇 , (12) we obtain 𝜒𝑋 𝜒𝑇 ∇ad (∇𝑋 − ∇𝑋,crit) = 8 RaT � 𝐻𝑃 ℓ �4 Pe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (13) For the large RaT in astrophysical applications, the term on the right hand side will be small as long as Pe is not too large, so that taking ∇𝑋 ≈ ∇𝑋,crit should be a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' However, for a large enough composition flux, this term can become important as we will see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Equation (11) assumes that the velocity of fluid elements is set by the buoyant acceleration acting over a mixing length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In rapidly-rotating convection, Coriolis forces modify the force balance and change the convective velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We estimate the effect of rapid rotation following the scaling relations of Au- rnou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2020), who considered the balance between Corio- lis, inertial and buoyancy terms in rapidly-rotating convection (CIA balance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Rewriting their eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [24] in our notation, this balance can be expressed as 𝑣2 𝑐 𝐿2 ∼ 2Ω𝑣𝑐 𝐻𝑃 ∼ 𝑔 𝐻𝑃 𝜒𝑋 𝜒𝜌 � ∇𝑋 − ∇𝑋,crit � (14) (compare eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [24] of Aurnou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We assume that the lengthscale associated with convective motions in the di- rection of the rotation vector is the pressure scale height 𝐻𝑃, while 𝐿 is the lengthscale associated with motions perpen- dicular to the rotation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The first and last terms of equation (14) give an expression for the convective velocity that has the same functional form as equation (11) but with the replacement ℓ → 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The first and second terms in equa- tion (14) give the ratio between perpendicular and parallel scales as 𝐿 𝐻𝑃 ≈ � 𝑣𝑐 2Ω𝐻𝑃 �1/2 ≈ Ro1/2, (15) where we define the Rossby number Ro ≡ 𝑣𝑐/2Ω𝐻𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Simu- lations of non-magnetic rapidly-rotating convection in plane- tary cores give support to this scaling (Guervilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' These scalings suggest that we can estimate the effect of rapid rotation by making the substitution ℓ → 𝐿 ≈ Ro1/2𝐻𝑃 in the non-rotating results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The Rossby number is given in terms of Pe (which is now defined as Pe ≡ 𝑣𝑐𝐿/𝜅𝑇 ) by Ro ≡ 𝑣𝑐 2Ω𝐻𝑃 = Pe2/3Ta−1/3, (16) where we define the Taylor number Ta ≡ 4Ω2𝐻4 𝑃 𝜅2 𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (17) With these scalings, we find 𝜒𝑋 𝜒𝑇 ∇ad (∇𝑋 − ∇𝑋,crit) = 8 RaT Pe2 Ro2 = 8 RaT Ta2/3Pe2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (18) Comparing to equation (13), we see that rapid rotation (Ro ≪ 1) acts to steepen the composition gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Even so, the large value of RaT in astrophysical scenarios means that ∇𝑋 will remain very close to ∇𝑋,crit in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The steady-state balance with thermal conduction We now consider the consequences of the mixing length theory outlined above in a situation with a specified outwards flux of light elements 𝐹𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' As the compositionally-driven convection transports heat inwards, the temperature gradient will steepen until the outwards conductive heat flux balances the inwards convective heat flux4, 𝜌𝑐𝑃𝜅𝑇 𝑇∇ 𝐻𝑃 = −𝜌𝑣𝑐𝑐𝑃𝑇 (∇ − ∇𝑒) ℓ 2𝐻𝑃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (19) Solving for the steady-state temperature gradient gives ∇ = ∇𝑒Pe/(2 + Pe), or using equation (5), ∇ = ∇ad Pe2 Pe2 + 2Pe + 2𝐶 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (20) When Pe ≪ 1, conduction acts efficiently on the timescale of convection, so that a small temperature gradient ∇ ≈ 4 The outwards conductive flux and inwards convective flux will not exactly cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For example, in a cooling white dwarf there must be a net outwards cooling luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In neutron star envelopes, Medin & Cumming (2015) considered a steady-state in which the net heat flux was inwards, carrying nuclear energy released in a low density H/He burning shell into the neutron star interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In both cases, the latent heat needs to be removed from the crystallization front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For simplicity here we assume that any net flux is small compared to the convective heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Heat transport by compositionally-driven convection 5 ∇adPe2/2𝐶 is sufficient for conduction to balance the con- vective heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' However, when convection is driven very strongly and the convective velocities become large, Pe ≫ 1, the steady-state temperature gradient approaches the adia- batic gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The reason for this is that the heat flux due to convection (∝ ∇ad − ∇) is then reduced to a level where it can be balanced by the conductive flux along the adiabat5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We can write an expression for Pe in terms of the com- position flux using equation (3), which gives the convective velocity 𝑣𝑐 ≈ (𝐹𝑋/𝜌𝑋∇𝑋)(2𝐻𝑃/ℓ), or Pe = 𝑣𝑐ℓ 𝜅𝑇 ≈ � 𝐻2 𝑃 𝜅𝑇 � � 2𝐹𝑋 𝜌𝐻𝑃𝑋 � ∇−1 𝑋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (21) The first term is the thermal diffusion time across the pressure scale height 𝑡therm = 𝐻2 𝑃/𝜅𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The second term is related to the timescale on which the light elements are being injected into the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For example, consider a region of a star with mass Δ𝑀 ∼ 4𝜋𝑟2𝜌𝐻𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' If light elements are being injected at a rate �𝑀𝑋 = Δ𝑀 �𝑋, the composition flux is 𝐹𝑋 ∼ Δ𝑀 �𝑋/4𝜋𝑟2 = 𝜌𝐻𝑃 �𝑋, and the second term in equation (21) is 𝐹𝑋/𝜌𝐻𝑃𝑋 ∼ �𝑋/𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We therefore define the timescale 𝑡𝑋 = 𝜌𝐻𝑃𝑋/𝐹𝑋, giving6 Pe ≈ 𝑡therm 𝑡𝑋 2 ∇𝑋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (22) We see that the Péclet number is set by both the ratio of ther- mal and injection timescales and the composition gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For fixed timescales, a smaller composition gradient requires a larger velocity to transport the composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Equations (20), and (22) both relate Pe to one of the gradi- ents ∇ or ∇𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Adding a third relation, either equation (13) for no rotation or equation (18) for rapid rotation, we can solve for Pe, ∇ and ∇𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Before presenting the solution, it is useful to define the dimensionless parameter 𝜏 = �𝑡therm 𝑡𝑋 � � 𝜒𝑋 𝜒𝑇 ∇ad � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (23) which is a measure of the composition flux driving convec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This can also be written explicitly in terms of 𝐹𝑋 as 𝜏 = 𝐹𝑋 𝐹𝐻,ad � 𝑐𝑃𝑇 𝑋 𝜒𝑋 𝜒𝑇 � , (24) where 𝐹𝐻,ad = 𝜌𝑐𝑃𝜅𝑇 𝑇∇ad/𝐻𝑃 is the heat flux conducted along the thermal adiabat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Comparing with equation (10) we see that 𝜏 is a measure of the effect of the convection on the temperature gradient: when 𝜏 = 1, the value of 𝐹𝑋 is such 5 In reality, the temperature gradient may saturate below ∇ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Once the temperature gradient reaches the slope of the liquidus curve ∇𝐿 ≈ 1/4 < ∇ad ≈ 1/3, large portions of liquid will freeze, shutting down convec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Medin & Cumming (2014, 2015) found that the system then becomes time-dependent with periodic freezing and melting of large regions near the liquid/solid boundary, on average maintaining a gradient ∇ ≈ ∇𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For simplicity, we ignore this effect in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 6 A similar expression for Pe was previously obtained by Mochkovitch (1983) (their eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [23]) and Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (1997) (their eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [32]) for the case where ∇𝑋 = ∇𝑋,crit and assuming 𝐶 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 10 3 10 2 10 1 100 101 102 103 =( ad X/ T)/(ttherm/tX) 10 1 100 101 102 103 104 Pe Pe=1 =1 (2C )1/2 A(2 )1/3 10 3 10 2 10 1 100 101 102 103 =( ad X/ T)/(ttherm/tX) 10 3 10 2 10 1 100 Gradients ad X ad T/ X Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The steady-state Péclet number (top panel) and temperature and composition gradients (bottom panel) as a function of the driving parameter 𝜏 ∝ 𝐹𝑋 (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Near 𝜏 = 1, Pe rapidly transitions from the small 𝜏 solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (27) (lower dotted line) to the large 𝜏 solution given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (28) (upper dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For this example, we set 𝐶 = 9/2 and 𝐴 = 103 (RaT ∼ 1010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' that the associated convective heat flux for efficient convection (∇𝑋 ≈ ∇𝑋,ad) is equal to 𝐹𝐻,ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This means that for 𝜏 ≪ 1, the heat flux can be balanced by a small temperature gradient ∇ = 𝜏∇ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The temperature gradient is much shallower than the adiabat, thermal diffusion is efficient, and Pe is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For 𝜏 ≫ 1, the heat flux for efficient convection exceeds 𝐹𝐻,ad, the composition gradient steepens ∇𝑋 > ∇𝑋,crit to reduce the heat flux to ≈ 𝐹𝐻,ad, the temperature gradient is close to the adiabat (∇ ≈ ∇ad) with inefficient thermal diffusion and large Pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The full solution for the non-rotating case can be written as 𝜏 = Pe2 Pe2 + 2Pe + 2𝐶 + 1 2 �Pe 𝐴 �3 , (25) where 𝐴 = 1 2RaT1/3 � ℓ 𝐻𝑃 �4/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (26) 6 Fuentes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 100 102 104 Pe No rotation Ta=108 Ta=1010 Ta=1012 10 5 10 3 10 1 Ro 10 2 10 1 100 / ad 10 3 10 2 10 1 100 101 102 103 =( ad X/ T)(ttherm/tX) 10 3 10 2 10 1 100 101 X/( ad T/ X) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The effect of rapid rotation on the steady-state solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We show the non-rotating solution from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 1 as the dashed black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The other curves show the effect of increasing rotation on this model, with Ta = 108, 1010, and 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Rapid rotation has only a small effect on the temperature gradient/heat flux, but leads to smaller convective velocities and larger composition gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Equation (25) gives Pe(𝜏) which can then be used to obtain the gradients ∇ and ∇𝑋 using equations (20) and (22) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' An example for particular choices of 𝐴 and 𝐶 is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The solution for Pe(𝜏) (top panel) has two branches: at small 𝜏, ∇𝑋 ≈ ∇𝑋,crit ∝ Pe and ∇ ≪ ∇ad, so that equation (22) gives Pe ≈ (2𝐶𝜏)1/2 (𝜏 small), (27) while at large 𝜏, the composition gradient is set by the right hand term in equation (13), giving Pe ≈ 𝐴(2𝜏)1/3, (𝜏 large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (28) The value of Pe makes a rapid transition between these two branches at 𝜏 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The lower panel of Figure 1 shows the gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The temperature gradient closely follows ∇ = ∇ad𝜏 for 𝜏 < 1 and ∇ = ∇ad for 𝜏 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The composition gradient shows a more complicated behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For 𝜏 < 1, it is very close to ∇𝑋 = ∇𝑋,crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' At small values of 𝜏, this gives ∇𝑋 increasing with 𝜏, ∇𝑋 ≈ (∇ad𝜒𝑇 /𝜒𝑋)(2𝜏/𝐶)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' As 𝜏 → 1, ∇ → ∇ad, decreasing the thermal buoyancy and therefore ∇𝑋,crit, which leads to the rapid decrease in ∇𝑋 near 𝜏 = 1 in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For 𝜏 > 1, ∇𝑋 increases with 𝜏 again as it starts to significantly exceed ∇𝑋,crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For large 𝜏, ∇𝑋 ≈ (∇ad𝜒𝑇 /𝜒𝑋)(2𝜏)2/3/𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For rapid rotation, we use equation (18) instead of (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The solution is 𝜏 = Pe2 Pe2 + 2Pe + 2𝐶 + �Ta2/3 2𝐴3 � Pe5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (29) An example is shown in Figure 2 which shows the effect of increasing rotation on the model from Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' As long as Ta2/3 ≲ 𝐴3 (corresponding approximately to Ta2/3 ≲ RaT), then the first term in equation (29) dominates for 𝜏 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The results for Pe and the gradients are therefore the same as without rotation7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The convective velocities are significantly increased, by a factor of Ro−1/2 (since Pe ≡ 𝑣𝑐𝐿/𝜅𝑇 is un- changed by rotation, and 𝐿/𝐻𝑃 = Ro1/2)8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For 𝜏 > 1, the last term in equation (29) dominates, giving Pe ≈ 𝐴9/5 Ta2/5 (2𝜏)3/5, (𝜏 large) (30) and ∇𝑋 ≈ 𝜒𝑇 ∇ad 𝜒𝑋 Ta2/5 𝐴9/5 (2𝜏)2/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (𝜏 large) (31) Comparing equations (28) and (30), we see that the effect of rapid rotation is to reduce Pe (increase ∇𝑋) for 𝜏 > 1, multiplying (dividing) it by a factor ≈ (𝐴4/5/Ta2/5)(2𝜏)4/15 ∝ RaT4/15/Ta2/5 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Numerical simulations The mixing length theory in the previous section makes a number of approximations and assumptions, in particular for how thermal diffusion acts to suppress the thermal buoyancy (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [5] and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In this section, we compare against numerical simulations of compositionally-driven convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We first check that indeed there is an inwards directed heat flux associated with an outwards composition flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Then, we allow the thermal gradient to come into steady-state and investigate the relation between the composition and thermal gradients and the value of the Péclet number that characterizes the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 7 In the limit of very rapid rotation, when Ta2/3 > 𝐴3, the last term in equation (29) dominates for all 𝜏, giving Pe ≈ (𝐴3/Ta2/3)3/5(2𝜏)3/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The largest value of Ta shown in Figure 2 is just large enough to enter this regime, where Pe is reduced by rotation at 𝜏 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' However, this regime is not relevant for the parameter values appropriate for white dwarf and neutron star interiors, and so we do not focus on it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 8 If using the Rossby number defined with the non-rotating value of 𝑣𝑐, the factor by which rotation increases the velocity is Ro−1/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Heat transport by compositionally-driven convection 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Model and simulation setup We conduct simulations for a binary fluid within a 3D spher- ical shell of depth Δ𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For this first numerical investigation, and to simplify comparison with mixing length theory, we consider a non-rotating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We express the fluid quan- tities as the sum of a constant background (denoted by the subscript 0) and a dynamic perturbation to the background (denoted by the prime symbol), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=', the density 𝜌 = 𝜌0 + 𝜌′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We use the Boussinesq approximation (Spiegel & Veronis 1960), where density perturbations satisfy 𝜌′/𝜌0 ≪ 1, and are related to perturbations in temperature 𝑇 ′ and mass fraction of the lighter component 𝑋′ through 𝜌′ = −𝜌0(𝛽𝑋′ + 𝛼𝑇 ′), where 𝛽 and 𝛼 are the coefficients of compositional and ther- mal contraction/expansion (both assumed positive constants), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Convection is driven by imposing a constant flux of light elements across the domain, such that light elements are injected (removed) at the inner (outer) boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We non-dimensionalize the fluid equations using as units of length and time the shell depth, Δ𝑟, and the diffusion time for solute, 𝑡diff = Δ𝑟2/𝜅𝑋, where 𝜅𝑋 is the solute diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The temperature scale is [𝑇] = |𝜕𝑟𝑇0 − 𝜕𝑟𝑇ad|Δ𝑟, where 𝜕𝑟𝑇0 is the radial temperature gradient of the background (zero in this problem), and 𝜕𝑟𝑇ad is the adiabatic temperature gradient (equal to −1 given our choice of units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For solute, we use [𝑋] = (𝛼/𝛽)|𝜕𝑟𝑇0 − 𝜕𝑟𝑇ad|Δ𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' By this choice, a unit of pres- sure corresponds to [𝑃] = 𝜌0(𝜅𝑋/Δ𝑟)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The dimensionless equations are ∇ · u = 0 , (32) 𝜕u 𝜕𝑡 + (u · ∇)u = −∇𝑃′ + ScR (𝑋′ + 𝑇 ′) ˆr + Sc∇2u ,(33) 𝜕𝑋′ 𝜕𝑡 + (u · ∇)𝑋′ = ∇2𝑋′ , (34) 𝜕𝑇 ′ 𝜕𝑡 + (u · ∇)𝑇 ′ + (𝜕𝑟𝑇0 − 𝜕𝑟𝑇ad)𝑢𝑟 = Le∇2𝑇 ′ , (35) where we have assumed constant gravity, and u = (𝑢𝑟,𝑢𝜃,𝑢𝜙) is the velocity field (with 𝑢𝑟, 𝑢𝜃 and 𝑢𝜙, the radial, polar, and azimuthal components of the velocity, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In the equations above, there are 3 dimensionless numbers that characterize the evolution of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' These are the Rayleigh, Schmidt, and Lewis number, which are defined respectively as R = 𝑔𝛼Δ𝑟4|𝜕𝑟𝑇0 − 𝜕𝑟𝑇ad| 𝜅𝑋𝜈 , Sc = 𝜈 𝜅𝑋 , Le = 𝜅𝑇 𝜅𝑋 , (36) where 𝜅𝑇 is the solute diffusivity, and 𝜈 is the kinematic viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Note that Sc = Pr Le, where Pr = 𝜈/𝜅𝑇 is the Prandtl number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We set the inner and outer radius of the shell to 𝑟i = 7/3, and 𝑟o = 10/3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Note that for this choice, the shell depth is Δ𝑟 = 𝑟o − 𝑟i = 1, and the aspect ratio is 𝑟i/𝑟o = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For the dimensionless numbers above, we use R = 106, Le = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='3, Pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='5, and Sc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='6, and the strength of the convective flow is controlled by changing the flux of light elements at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The boundary conditions are zero gradient for temperature, and impenetrable and stress-free for velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We specify the composition flux by setting the value of the composition gradient at each boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In our dimen- sionless variables, this is 𝜕𝑟 𝑋′|𝑟=𝑟i,𝑟o = −𝐹0/𝐹crit, where the desired composition flux 𝐹0 is normalized by 𝐹crit = 𝜌0𝜅𝑋 (𝛼/𝛽)|𝜕𝑟𝑇0 − 𝜕𝑟𝑇ad|, the flux of light elements that, if carried by molecular diffusion, would result in a com- position gradient that is marginally stable against convection (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 𝛽𝜕𝑟 𝑋′ = 𝛼|𝜕𝑟𝑇0 −𝜕𝑟𝑇ad|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We consider values of 𝐹0/𝐹crit between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='5 and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We solve the governing equations and boundary conditions presented above using the pseudo-spectral solver Dedalus (Burns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Vasil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Lecoanet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The variables are represented in spherical harmonics for the an- gular directions and Chebyshev polynomials for the radial di- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' All the simulations have 𝐿max = 𝑁max = 255, where 𝐿max is the maximum spherical harmonic degree, and 𝑁max is the maximal degree of the Chebyshev polynomials used in the radial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Therefore, the number of radial, latitudinal, and longitudinal points are (𝑁𝑟, 𝑁𝜃, 𝑁𝜙) = (256, 256, 512), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For time-stepping, we use a second order semi- implicit BDF scheme (SBDF2, Wang & Ruuth 2008), where the linear and nonlinear terms are treated implicitly and ex- plicitly, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We use a CFL safety factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='35 and dealias factor of 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' To start our simulations, we add random noise perturbations to the background composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Qualitative description of the flow We first present results for the runs using 𝐹0/𝐹crit = 1 and 𝐹0/𝐹crit = 25 as fiducial cases for low and high Pe, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' However, we find that the behavior is qualita- tively similar for all values of 𝐹0/𝐹crit: once the fluxes at the boundaries are turned on, an excess (deficit) of light el- ements develops at the inner (outer) boundary of the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Eventually, the fluid becomes compositionally-buoyant and suddenly overturns, driving convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' All the simulations reach a statistically stationary state where the fluxes and the flow velocities fluctuate around a constant value (see top pan- els in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The time to reach steady-state depends on the value of 𝐹0/𝐹crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For the fiducial cases here, at low Pe the steady state is achieved at 𝑡 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='5, whereas at high Pe it is achieved at 𝑡 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='05, an order of magnitude difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This is expected since the convective motions are much slower at low Pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We also see differences in the flow structure between the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This can be seen in the 3D snapshots of the composition field in the top panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We find that the structure of the flow is more diffusive at low Pe, and more turbulent at high Pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Our simulations confirm the expected inwards convective heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We find that for a given composition flux, there is an oppositely directed heat flux that is larger when the composition flux that drives convection is larger (see the green curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Further, as heat is transported inward, a temperature gradient develops over time until the associated flux carried by diffusion balances the convective heat flux (see bottom panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This cancellation means that once the simulation reaches steady-state, the total heat flux 8 Fuentes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Top panels: Time series of the volume-averaged Péclet number, Pe = 𝑢rms/Le (where 𝑢rms = �√︃ 𝑢2𝑟 + 𝑢2 𝜃 + 𝑢2 𝜙 � , with the brackets denoting the average over the shell), total composition flux in the radial direction, 𝐹𝑋,tot = ⟨ ˆr · (u𝑋 − ∇𝑋)⟩ (where the first term corresponds to the convective flux, and the second term to the diffusion flux), and the magnitude of the convective heat flux in the radial direction, −𝐹𝐻,conv = −⟨ ˆr · u𝑇 ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Note that the total composition flux converges to 𝐹0/𝐹crit once the fluid reaches steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Bottom panels: Radial temperature profile at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Results are shown for the fiducial cases at low and high Pe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=', 𝐹0/𝐹crit = 1 (left panels) and 25 (right panels), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' To show the differences in the structure of the flow, we overplot 3D snapshots of the composition and temperature fields, once the simulation reaches steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' across the fluid is zero, as expected from our choice of zero flux boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Gradients and Péclet number in the convective region As discussed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='4, the properties of the flow in the con- vection zone are expected to change as a function of the driv- ing parameter 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In particular, mixing-length theory predicts a transition when 𝜏 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' To check whether the simulations support this transition, we measure the shell-averaged con- vective velocities and radial fluxes as a function of time, and then for each quantity we take the time-average value over an interval for which the system is statistically stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' When computing volume averages, we exclude regions near the diffusive boundary layers and confine our measurements to the convective region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We evaluate 𝜏 using the convective composition flux in equation (A8), giving 𝜏 = Le−1(𝑢𝑟 𝑋′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Figure 4 shows the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We show the mea- sured gradients as a function of 𝜏 in the left panel, and the Péclet number as a function of 𝜏 in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Note that whereas Pe is defined elsewhere in the paper in terms of the mixing length as 𝑣𝑐ℓ/𝜅𝑇 , the measured Pe we show in Figure 4 is the quantity 𝑣𝑐Δ𝑟/𝜅𝑇 , ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' using Δ𝑟 as the lengthscale Fo/Fcrit = 1 Fo/Fcr crit = 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content="4 102 102 Pe X' X,tot 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content="9 101 101 Time X' 5 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='5 Pe Fx,tot 100 100 H,conv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='8Fo/Fcrit = 1 Fo/Fcrit = 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content="4 T' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2 Temperature +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1 T ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2 Temperature gradient Temperature gradient develops over time develops over time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='4L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2 rHeat transport by compositionally-driven convection 9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Absolute value of the volume-averaged temperature and composition gradient (left panel) and Péclet number (right panel) measured from the simulations as a function of the driving parameter 𝜏 = Le−1 ⟨𝑢𝑟 𝑋′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Results are shown for simulations using 𝐹0/𝐹crit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='5–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Lines on each panel are predictions from the steady-state mixing-length theory solution (Appendix A) using ℓ = 1, R𝑇 = RSc/Le2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='5 × 105, and different values of 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' since mixing length is not a measured quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For 𝑣𝑐, we measure the rms convective velocity from the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The solid curves in Figure 4 are the mixing length theory predictions (which we rewrite for Boussinesq convection in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We set ℓ = Δ𝑟 and show results for three dif- ferent values of 𝐶 reported in the literature (see discussion in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We find that the data supports the predicted transition at 𝜏 = 1, and the general shape of the curves match well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The transition is smoother and shows less of a jump than the example shown in Figure 1 because of the lower value of Rayleigh number in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The measured temper- ature gradient agrees well with the prediction, showing that the magnitude of the convective heat flux is also as predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We also see the expected inflection in the dependence of the composition gradient with 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' There are some differences between the measured values and the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We find a better agreement for the gradi- ents as a function of 𝜏, than for the Péclet number as a function of 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The numerical values of Pe are larger than the predicted values for all 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Fitting separately a power-law to the data gives Pe ∝ 𝜏0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='75 for 𝜏 < 1 (compared to the analytic predic- tion Pe ∝ 𝜏1/2), and Pe ∝ 𝜏0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='65 for 𝜏 > 1 (compared to the analytic prediction Pe ∝ 𝜏1/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We find that the composition gradient approaches 𝜕𝑟 𝑋′ ≈ 1/Le ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='3 at small 𝜏, which is consistent with the expected threshold for double-diffusive instabilities (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Traxler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2011a), whereas the analytic model assumes that Le is large enough that the threshold can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We were not able to find values of 𝐶 and ℓ that fit all the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For example, for the choice ℓ = 𝐻 used in Figure 4, we find that smaller values of 𝐶 are pre- ferred when fitting ∇𝑋 (left panel), whereas a larger value of 𝐶 is preferred when fitting Pe (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Nonetheless, the overall general agreement is encouraging especially given the approximate nature of mixing length theory (particularly the approximations made in deriving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [4] for the thermal leakage during convection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Summary of our results We have used both mixing length theory and numerical sim- ulations to investigate the heat transport in compositionally- driven convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Our results show that there are two dif- ferent convection regimes, depending on the value of the parameter 𝜏 defined in equation (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' When thermal diffu- sion is very efficient, 𝜏 ≪ 1, the convective motions have a small Péclet number and only a small composition gradient is needed in the convection zone to overcome the reduced ther- mal buoyancy (∇𝑋 ≈ ∇𝑋,crit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' A small temperature gradient ∇ ≈ 𝜏∇ad develops in the convection zone to bal- ance the inwards transport of heat due to convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' When thermal diffusion is inefficient, 𝜏 ≫ 1, the behavior is very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The temperature gradient steepens to approach the adiabatic gradient, ∇ → ∇ad, reducing the convective heat flux to a level where it can be balanced by outwards conduc- tion along the adiabat9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Depending on the size of the com- position flux driving convection, the composition gradient in the convection zone can significantly exceed the critical gra- dient, ∇𝑋 > ∇𝑋,crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' There is rapid change from one regime to another as 𝜏 crosses unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In both cases, the effect of rapid rotation is to increase the convective velocity and reduce the composition gradient, with only a minor effect on the heat flux or temperature gradient unless the rotation is extremely strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' With the particular assumptions of equations (5) and (9) giving the reduction in the temperature excess of a convect- ing fluid element and the thermal buoyancy, we find that the ratio of heat flux to composition flux is independent of Péclet number (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Rising fluid elements lose heat due to ther- mal diffusion, but a smaller composition gradient is needed to 9 This regime in which inwards heat transport by convection almost bal- ances the outwards conductive flux along the adiabat has been discussed for the Earth’s core, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Loper (1978) and Labrosse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 100 C = 2元² I<0,T>1 102 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='. C = 9/2 I<0,X)I -- C=3 adients /Le .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='65 αT 10- 1 Gra( II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content="75 100 10-2 10-2 10-1 100 101 10-2 10-1 100 101 T=Le-l(urX'> t=Le-l10 Fuentes et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' overcome the thermal buoyancy, requiring a larger convective velocity and compensating for the heat loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Our numerical results give support to this scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' After an initial build up of composition at the boundaries, convection starts and evolves to a state in which, at small Péclet number, the gradients in the convection zone take on values that would be stable to the (adiabatic) Ledoux criterion, indicating that thermal diffusion significantly reduces the stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This can be seen by the fact that |𝜕𝑟 𝑋| < 1 − |𝜕𝑟𝑇| for small 𝜏 in the left panel of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This ordering of gradients (Ledoux stable with an unstable composition gradient and stable thermal gradi- ent) corresponds to the regime of fingering or thermohaline convection driven by double-diffusive instabilities (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Ga- raud 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Often investigated as the outcome of unstable imposed gradients, in our case the convection is maintained by the continuous injection of elements at the lower boundary, and the gradients develop as a result of the convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Implications for accreting neutron stars The lack of dependence of 𝐹𝐻/𝐹𝑋 on Pe means that the cal- culations of Medin & Cumming (2011, 2014, 2015) for accret- ing neutron star oceans used a correct expression for the heat flux even though they assumed adiabatic motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' However, the composition gradient is overestimated and convective ve- locity underestimated in those calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For example, whereas the composition gradient that is marginally stable to the Ledoux criterion is given by ∇𝑋/(∇ad𝜒𝑇 /𝜒𝑋) = 1, Fig- ure 1 for example shows that ∇𝑋/(∇ad𝜒𝑇 /𝜒𝑋) ranges from ≈ 10−2–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='3 for 𝜏 in the range 10−3–1, and can be much smaller for 𝜏 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The case of accreting neutron stars is interesting because 𝜏 spans a range of values from small to large, covering both convective regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The factor 𝜒𝑋/𝜒𝑇 ∇ad is ∼ 30–100 un- der the degenerate ocean conditions and depends only on the composition at the crystallization depth (see Appendix A), so that 𝜏 ∼ (30–100)(𝑡therm/𝑡𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For cooling following an accretion outburst, the crystallization timescale is compara- ble to the cooling time, 𝑡𝑋 ∼ 𝑡therm, so 𝜏 ∼ 30–100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This is consistent with the rapid steepening of the temperature pro- file seen by Medin & Cumming (2014, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For steady accretion, new crust forms on the accretion timescale, which is ∼ 30 yr for typical parameters (taking an ocean depth ≈ 1013 g cm−2 and accretion rate 104 g cm−2 s−1), whereas the thermal timescale is a few days at these depths (Bildsten & Cutler 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Therefore 𝑡therm/𝑡𝑋 ∼ 3×10−4, giving 𝜏 ∼ 10−2 for steady accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Even though the neutron star ocean takes years to ac- crete, it mixes much more rapidly when chemical separa- tion is happening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For a non-rotating star, Figure 1 gives ∇𝑋/(∇ad𝜒𝑇 /𝜒𝑋) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1 for 𝜏 ∼ 10−2, implying that the con- vective velocity is ≈ 10 times larger than under the adiabatic assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' With Pe ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='3, the convective turnover timescale 𝑡conv = 𝐻𝑃/𝑣𝑐 = 𝑡therm/Pe at the base of the ocean is a few thermal times (∼ 10 days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Rapid rotation reduces this dra- matically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Using equation (16) for the Rossby number, the convective turnover time in the rapidly-rotating limit can be written 𝑡conv,rot = �𝑡therm Pe �2/3 (2Ω)−1/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (37) With a rotation period of a few milliseconds, the convective turnover time is ≈ 10 min (for a scale height ≈ 3000 cm this corresponds to a convective velocity 𝑣𝑐 ≈ 5 cm s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For cooling neutron stars with 𝜏 > 1, the convective ve- locities are even larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Evaluating the Rayleigh number with the help of Bildsten & Cutler (1995) equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='9), we find RaT ≈ 𝑡2 therm(𝑔/𝐻𝑃)(3/2𝑍)(𝑘𝐵𝑇/𝐸𝐹) ≈ 1018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For non-rotating convection with 𝜏 > 1, equation (28) gives Pe ≈ (RaT𝜏)1/3 ≈ 106𝜏1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The convective turnover time is therefore ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='3 s (velocity ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1 km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For 𝜏 > 1, rapid rotation decreases the convective velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' With Ta = (2Ω𝑡therm)2 ≈ 4 × 1017, equation (30) gives Pe ≈ 6000 𝜏3/5, or a turnover time ≈ 1 min and velocity ≈ 60 cm s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Further calculations of the evolution of accreting neutron star oceans would be interesting taking into account our re- vised estimates of the composition gradients and convective velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Mixing on a rapid timescale should have impli- cations for superbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' These long thermonuclear flashes are thought to be the result of unstable ignition of carbon in the ocean, although significant problems remain in making enough carbon and getting it to ignition temperature (in’t Zand 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For example, mixing in the ocean could transport car- bon to greater depths where it can burn (stably or unstably).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' It would also be interesting to revisit the calculations of Medin & Cumming (2014) for neutron stars cooling after accretion outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Recently, Parikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2020) reported observa- tions of two accreting neutron stars in quiescence that showed a late time (≈ 2000 days after outburst) decrease in temper- ature, followed by a temperature increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' They pointed out that this behaviour is similar to the models of Medin & Cum- ming (2014) that include compositionally-driven convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Further investigations are needed to compare against the ob- servations for these two sources and explore the constraints on ocean composition and temperature needed to fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Implications for white dwarf cooling and dynamos To investigate the parameters for cystallization-driven con- vection in white dwarfs, we ran an example 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='6 𝑀⊙ white dwarf model using the MESA stellar evolution code (Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2011, 2013, 2015, 2018, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Jermyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2022) (us- ing the default wd_cool_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='6M test suite in MESA version 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Note that although the code follows the solid-liquid transition and includes the latent heat, it does not include chemical separation and so the composition profile does not evolve in this calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Instead, we estimate the com- position flux due to chemical separation by measuring the rate of growth of the solid core �𝑀𝑐 and assuming a value Δ𝑋melt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1 for the carbon enhancement in the liquid phase relative to the solid (approximately the liquid-solid compo- sition difference for the C/O phase diagram;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Horowitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The composition flux is then 𝐹𝑋 = �𝑀𝑐Δ𝑋melt/4𝜋𝑅2 𝑐, where 𝑅𝑐 is the core radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Figure 5 shows different parameters associated with the liq- uid region just above the crystallization front as a function of Heat transport by compositionally-driven convection 11 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The top panel shows the mass of the solid core and the composition at the freezing point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The white dwarf has an oxygen-rich inner core surrounded by a carbon-rich outer core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' growth of the core pauses at ≈ 3 Gyr when the crys- tallization front reaches the edge of the inner core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' it takes ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='5 Gyr of further cooling before the outer core begins to freeze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The values of 𝑡𝑋, 𝑡therm, 𝜏 and Pe and the temperature gradient ∇ are shown in the middle two panels of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' As in the neutron star case, 𝜒𝑋/𝜒𝑇 ∇ad ∼ 10 is relatively large (see Appendix A), but 𝑡therm in the conductive interior is short enough compared to the evolution time 𝑡𝑋 that 𝜏 is small for much of the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We find 𝜏 > 1 for a short time at the beginning of crystallization, but it quickly drops and stabilizes at a value of 𝜏 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The corresponding Péclet numbers are Pe ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='3, in good agreement with the estimates of Mochkovitch (1983) and Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The bottom panel of Figure 5 shows the convective velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For the non- rotating case, this is given by 𝑣𝑐 = 𝜅𝑇 Pe/𝐻𝑃, and for the ro- tating case, we use the convective turnover time from equation (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The velocities we obtain are in reasonable agreement with Mochkovitch (1983) who, using a similar formulation of mixing length theory, estimated 𝑣𝑐 ≲ 10−6 cm s−1 for no rotation and ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2 cm s−1 for a 1 hour rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Our convective velocities are much smaller than the recent estimates of Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2017) and Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2022) for crystallization-driven dynamos in white dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The initial estimates of Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2017) considered the acceleration of carbon-rich parcels of fluid released at the phase transi- tion, finding 𝑣𝑐 ≈ 30 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2022) argued that this was an overestimate and instead obtain a velocity ∼ (𝑞𝑐/𝜌)1/3, where 𝑞𝑐 is the gravitational energy flux associ- ated with the redistribution of elements across the crystalliza- tion front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This estimate actually corresponds to the situation where 𝜏 ≫ 1 and ∇𝑋 ≫ ∇𝑋,crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In that case, equation (11) gives 𝑣3 𝑐 ≈ 𝑔𝐻𝑃(𝜒𝑋/𝜒𝜌)𝑣𝑐∇𝑋 ≈ 𝑔𝐻𝑃(𝜒𝑋/𝜒𝜌)(𝐹𝑋/𝜌𝑋) ≈ 𝑞𝑐/𝜌 (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' compare eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [C23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' However, we find that white dwarf interiors are in the 𝜏 ≪ 1 regime, as previously found by Mochkovitch (1983), with much lower accelerations and velocities since ∇𝑋 ≈ ∇𝑋,crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The change of regime has a huge effect on the velocities: even our rotating convective turnover times are thousands of years, compared to turnover times of months in Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Even with this lower velocity, the magnetic Reynolds number Rm is likely to be large enough to support a dynamo once rotation is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' With the electrical conductivity in the range 𝜎 ∼ 1021–1022 s−1 (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 1 of Cumming 2002), Rm = 𝐻𝑃𝑣𝑐/𝜂 = 4𝜋𝜎𝑣𝑐𝐻𝑃/𝑐2 ∼ 106–107 for 𝐻𝑃 ≈ 108 cm and 𝑣𝑐 ≈ 10−3 cm s−1 appropriate for the rapidly-rotating case (bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The threshold value of Rm for a dynamo is uncertain, but with 𝑣𝑐 ∝ Ω1/3, considering even slower rotation does not reduce Rm significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' However, another major issue for dynamos is the energy reservoir available to grow the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In Appendix B, we show that the kinetic energy flux is a small fraction of the available gravitational energy (𝐹𝐾/𝑞𝑐 ∼ (∇𝑋 − ∇𝑋,crit)/∇𝑋,crit ≪ 1 for small 𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The saturated dynamo scaling used by Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2017) is 𝐵2/4𝜋 ∼ 𝜌𝑣2 with 𝑣 ∼ (𝐹/𝜌)1/3 (Chris- 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='6 Solid core mass (M ) 2 4 6 8 10 3 10 2 10 1 100 101 / ad Pe 2 4 6 8 107 108 108 109 1010 1011 Time scale (yr) ttherm tconv, non rot tX 2 4 6 8 Age (Gyr) 10 8 10 7 10 6 10 5 10 4 10 3 10 2 vc (cm s 1) Prot =1 hour Prot =1 day No rotation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='0 Xi 12C 16O Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Convection parameters just above the crystallization front as a function of time for a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='6 𝑀⊙ white dwarf, evolved with the MESA stellar evolution code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Chemical separation and convection are not included in this model, but we use the rate of crystallization of the core to calculate the expected properties of compositionally-driven convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' tensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2009), where they assumed that the energy flux 𝐹 available to drive the dynamo was the gravitational en- ergy flux 𝑞𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The mechanism by which dynamo saturation occurs and the force balance in the saturated state is still an area of active study (Christensen & Aubert 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Schaeffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Orvedahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' However, since mag- netic field generation occurs as a result of induction by fluid motions, it seems unlikely that the magnetic energy den- sity could be many orders of magnitude larger than the ki- netic energy of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In the context of the Earth’s core, Loper (1978) also pointed out that much less kinetic energy is available to drive the dynamo when compositionally-driven convection occurs in a thermally-stable background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' To es- 12 Fuentes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' timate how small this is, we can use equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' As- suming a solid core mass ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1 𝑀⊙ and 𝐻𝑃 ∼ 108 cm, we find RaT ∼ 1028 and Ta ∼ 1024 (𝑃rot/h)−2), giving 𝐹𝐾/𝑞𝑐 ∼ (∇𝑋 − ∇𝑋,crit)/∇𝑋,crit ∼ Ta2/3/RaT ∼ 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Us- ing 𝐵2/4𝜋 ∼ 𝜌𝑣2 with 𝑣𝑐 ∼ 10−3 cm s−1 gives 𝐵 ∼ 3 G 𝜌1/2 6 , much smaller than needed to explain observed magnetic fields in white dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The results in Figure 5 show that the temperature gradient needed to balance the inwards convective transport of heat (≈ 𝜏∇ad for small 𝜏) is larger or comparable in size to the existing temperature gradient in the cooling model for much of the early evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This can be seen in the third panel of Figure 5 where, between ≈ 2–6 Gyr, the temperature gradi- ent in the white dwarf normalized to the adiabatic gradient, ∇/∇ad, is comparable to the value of 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This is consistent with the significant contribution that chemical separation makes to white dwarf cooling curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Chemical separation is typically included in white dwarf cooling codes by assuming that the cooling is slow enough that the liquid region is well-mixed (Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 1997, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Salaris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Montgomery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The energy change due to the changing composition profile is then added to the latent heat, and distributed in a small region around the crystallization front (Althaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Camisassa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Bédard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This ad- ditional energy will lead to a steepening of the temperature gradient (to conduct the extra heat to the surface), and in- deed we estimate in Appendix B that the magnitude of the convective heat flux is comparable in magnitude to the over- all energy release due to chemical separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This suggests that the temperature profile including the detailed transport of heat associated with mixing above the crystallization front may not be that different from current models, but further calculations are needed to check this in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Of particu- lar interest is the beginning of crystallization, when 𝜏 > 10 and there is the possibility of significant steepening of the temperature gradient in the central regions of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Future work on compositionally-driven convection The agreement between our numerical simulations and the mixing-length theory predictions shown in §3 is encourag- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' There are many interesting questions to address with further numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The value of Rayleigh num- ber that we used in §3 gives a relatively smooth transition between the small and large 𝜏 regimes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Simulations at larger Rayleigh number would be interesting to check the rapid transition predicted at 𝜏 = 1 for large RaT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Our equation (9) provides a convenient interpolation between the fingering and overturning convection regimes that at low Pe agrees with earlier analytic prescriptions for thermohaline convec- tion (Ulrich 1972 and Kippenhahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 1980 as implemented in the MESA code for example, Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' However, more recent results are available which provide composition and heat fluxes for fingering convection that are measured directly from numerical simulations (Traxler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2011a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' It would improve the modelling to incor- porate these results at low Pe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Even more important is that our simulations do not include rotation, and also adopt the Bousinessq approximation which limits the vertical scale to be much less than a pressure scale height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Rapid rotation should greatly reduce the lengthscale of convection perpendicular to the rotation vector, and is im- portant to check numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Similarly, stratification over many pressure scale heights would be expected to limit the vertical transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Dynamos in fingering convection are be- ginning to be addressed with numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Mather & Simitev (2021) simulated dynamos with internal volumet- ric sources or sinks of both thermal and compositional buoy- ancy, and did not find dynamo action in the fingering con- vection regime, although Guervilly (2022) argues that finger- ing convection could support a dynamo at larger Rayleigh numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Numerical simulations of compositionally-driven dynamos with a thermally-stable background are needed for application to white dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' It will also be interesting to inves- tigate other sources of compositional buoyancy, for example the distillation process involving production of light crystals proposed by Blouin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2021) for white dwarfs, or electron captures in neutron star oceans that produce heavy crystals within the liquid layer that then sink (Medin & Cumming 2014) (an analagous case in planetary dynamos is the iron snow in Ganymede’s core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Rückriemen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' These improvements in numerical modelling are needed to interpret the rich set of observations of both white dwarfs and neutron stars now available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We thank Simon Blouin whose question about the effect of thermal diffusion on compositionally-driven convection sparked this investigation, Brad Hindman and Nick Feather- stone for useful conversations on rotating convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We also thank Thomas Villeneuve and Charles Wilson for pre- liminary work on this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This work was supported by NSERC Discovery Grant RGPIN-2017-04780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' ac- knowledges support from a McGill Space Institute (MSI) Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' are members of the Centre de Recherche en Astrophysique du Québec (CRAQ) and the Institut de recherche sur les exoplanètes (iREx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' EHA was supported by a CIERA Postdoctoral Fel- lowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This research was enabled in part by support pro- vided by Calcul Québec (calculquebec.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=', & Page, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2017, Journal of Astrophysics and Astronomy, 38, 49 14 Fuentes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Mixing length theory for Boussinesq convection In this Appendix, we give the mixing-length theory results from §2 in a form appropriate for comparison with our numerical results in §3, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' in terms of the spatial gradients and using the Boussinesq equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The convective fluxes are 𝐹𝐻 ≈ 1 2 𝜌0𝑣𝑐𝑐𝑃ℓ (𝜕𝑟𝑇ad − 𝜕𝑟𝑇) Pe 𝐶 + Pe , (A1) 𝐹𝑋 ≈ −1 2 𝜌0𝑣𝑐ℓ𝜕𝑟 𝑋 , (A2) with 𝑣2 𝑐 ≈ 𝑔ℓ2 8 � 𝛼 (𝜕𝑟𝑇ad − 𝜕𝑟𝑇) Pe 𝐶 + Pe − 𝛽𝜕𝑟 𝑋 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (A3) The minus sign in the definition of 𝐹𝑋 takes into account the fact that decreasing composition with radius, 𝜕𝑟 𝑋 < 0, leads to an outwards composition flux, 𝐹𝑋 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Then, the equivalent to equations (9), (13), (20) and (21) are 𝜕𝑟 𝑋crit = 𝛼 𝛽 (𝜕𝑟𝑇ad − 𝜕𝑟𝑇) Pe 𝐶 + Pe, (A4) 𝜕𝑟 𝑋 − 𝜕𝑟 𝑋crit = 8 R𝑇 𝛼𝜕𝑟𝑇ad 𝛽 Pe2 �Δ𝑟 ℓ �4 , (A5) 𝜕𝑟𝑇 = 𝜕𝑟𝑇ad � Pe2 Pe2 + 2Pe + 2𝐶 � (A6) Pe = 𝑡therm 𝑡𝑋 � −2𝑋 𝜕𝑟 𝑋Δ𝑟 � , (A7) where we have defined R𝑇 = 𝛼𝑔(−𝜕𝑟𝑇ad)Δ𝑟4/𝜅2 𝑇 , 𝑡therm = Δ𝑟2/𝜅𝑇 , and 𝑡𝑋 = 𝜌0𝑋Δ𝑟/𝐹𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Following the same argument as in §2, we solve the system of equations above in terms of the driving parameter 𝜏 = �𝑡therm 𝑡𝑋 � � −𝛽𝑋 𝛼𝜕𝑟𝑇adΔ𝑟 � = 𝐹𝑋 𝐹crit Le−1, (A8) where 𝐹crit is defined in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The expressions above are used to generate the analytic curves in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Microphysics of white dwarf interiors and neutron star oceans In this Appendix, we estimate the expected size of the ratio 𝜒𝑋/𝜒𝑇 ∇ad that enters into the parameter 𝜏 (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For simplicity, as in the main text we consider a mixture of two species only, although it is straightforward to generalize to additional species if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The pressure has a contribution from electrons and ions, 𝑃 = 𝑃𝑒 + �2 𝑖=1 𝑃𝑖, where the terms with 𝑖 = 1 and 𝑖 = 2 are the ion contributions from each species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Under the degenerate conditions in white dwarf and neutron star interiors, the degenerate electrons dominate the pressure, with Fermi momentum 𝑝𝐹 = ℏ(3𝜋2𝑛𝑒)1/3 = 𝑥𝑚𝑒𝑐 given by 𝑥 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='01 (𝜌6𝑌𝑒)1/3 where 𝜌6 = 𝜌/106 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For non-relativistic electrons (𝑥 ≪ 1), the pressure is 𝑃𝑒 = (2/5)𝑛𝑒𝐸𝐹, with 𝐸𝐹 = 𝑝2 𝐹/2𝑚𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Therefore, 𝑃𝑒 ∝ (𝜌𝑌𝑒)5/3, so that 𝜒𝜌 = 5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For relativistic electrons (𝑥 ≫ 1), 𝐸𝐹 = 𝑝𝐹𝑐, 𝑃𝑒 ∝ (𝜌𝑌𝑒)4/3, giving 𝜒𝜌 = 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For the temperature-dependence of the pressure, we can take the ideal gas pressure as the leading temperature-dependent pressure term for the ions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 𝑃𝑖 = 𝑛𝑖𝑘𝐵𝑇 = 𝜌𝑋𝑖𝑘𝐵𝑇/𝐴𝑖𝑚 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For degenerate electrons, 𝜕 ln 𝑃𝑒/𝜕 ln𝑇 ∼ (𝑘𝐵𝑇/𝐸𝐹)2, which is much smaller than the ion contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Therefore, 𝜒𝑇 ≈ 𝜌𝑘𝐵𝑇/𝜇𝑖𝑚 𝑝𝑃 where 𝜇−1 𝑖 = � 𝑖 𝑋𝑖/𝐴𝑖, giving 𝜒𝑇 ≈ 5 2 𝑘𝐵𝑇 𝐸𝐹 𝑌−1 𝑒 ∑︁ 𝑖 𝑋𝑖 𝐴𝑖 ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='4 × 10−4 𝑇6(𝑌𝑒𝜌6)−2/3𝑌−1 𝑒 ∑︁ 𝑖 𝑋𝑖 𝐴𝑖 𝑥 ≪ 1 (B9) and 𝜒𝑇 ≈ 4 𝑘𝐵𝑇 𝐸𝐹 𝑌−1 𝑒 ∑︁ 𝑖 𝑋𝑖 𝐴𝑖 ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='7 × 10−3 𝑇8𝜌−1/3 9 𝑌−4/3 𝑒 ∑︁ 𝑖 𝑋𝑖 𝐴𝑖 𝑥 ≫ 1 (B10) For the compositional dependence of the pressure, we must look at both the electron and ion contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The dominant contribution is from the 𝑇 = 0 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For a two-component mixture, 𝑌𝑒 = 𝑋𝑍1 𝐴1 + (1 − 𝑋)𝑍2 𝐴2 , (B11) Heat transport by compositionally-driven convection 15 where 𝑋 = 𝑋1 is the mass fraction of the lighter species, and 1 − 𝑋 = 𝑋2 is the mass fraction of the heavier species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This gives 𝜕𝑃𝑒 𝜕𝑋 = 𝜕𝑃𝑒 𝜕𝑌𝑒 𝜕𝑌𝑒 𝜕𝑋 = 𝜕𝑃𝑒 𝜕𝑌𝑒 � 𝑍1 𝐴1 − 𝑍2 𝐴2 � , (B12) where 𝜕 ln 𝑃𝑒/𝜕 ln𝑌𝑒 = 5/3 and 4/3 in the non-relativistic and relativistic limits respectively, and the partial derivatives are taken at constant temperature and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For ions in the liquid phase, the leading order term in the Helmholtz free-energy at zero-temperature is 𝐹𝑖 = −𝐶𝑀 Γ𝑖𝑁𝑖𝑘𝐵𝑇, for 𝑁𝑖 ions in a volume 𝑉, where Γ𝑖 = 𝑍5/3 𝑖 Γ𝑒 and Γ𝑒 = 𝑒2/𝑎𝑒𝑘𝐵𝑇 with 4𝜋𝑎3 𝑒𝑛𝑒/3 = 1 defines the mean electron separation 𝑎𝑒, and 𝐶𝑀 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='90 is related to the Madelung constant (Dewitt & Slattery 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Farouki & Hamaguchi 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Potekhin & Chabrier 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Medin & Cumming 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This gives 𝐹𝑖 ∝ 𝑉−1/3 leading to the Coulomb pressure 𝑃𝑖 = (1/3)(𝐹𝑖/𝑉), or 𝑃𝑖 = −1 3𝐶𝑀𝑛𝑖𝑍5/3 𝑖 𝑒2 �4𝜋𝑛𝑒 3 �1/3 = −1 3𝐶𝑀𝑒2 �4𝜋 3 �1/3 � 𝜌 𝑚 𝑝 �4/3 𝑍5/3 𝑖 � 𝑋𝑖 𝐴𝑖 � 𝑌1/3 𝑒 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (B13) Therefore, 𝜕𝑃𝑖 𝜕𝑋𝑖 = 𝑃𝑖 𝑋𝑖 + 𝜕𝑌𝑒 𝜕𝑋𝑖 𝜕𝑃𝑖 𝜕𝑌𝑒 = 𝑃𝑖 𝑋𝑖 � 1 + 𝑋𝑖𝑍𝑖 3𝐴𝑖𝑌𝑒 � , (B14) so that 𝜕(𝑃1 + 𝑃2) 𝜕𝑋 = 𝑃1 𝑋 − 𝑃2 1 − 𝑋 + 1 3 𝑃1 + 𝑃2 𝑌𝑒 𝜕𝑌𝑒 𝜕𝑋 = 𝑃1 𝑋 − 𝑃2 1 − 𝑋 + 1 3 𝑃1 + 𝑃2 𝑌𝑒 � 𝑍1 𝐴1 − 𝑍2 𝐴2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (B15) Now adding the ion and electron contributions (eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [B12] and [B15]) gives 𝜕𝑃 𝜕𝑋 = 𝑃1 𝑋 − 𝑃2 1 − 𝑋 + �1 3 𝑃1 + 𝑃2 𝑌𝑒 + 𝜕𝑃𝑒 𝜕𝑌𝑒 � � 𝑍1 𝐴1 − 𝑍2 𝐴2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (B16) Noting that |𝑃1 + 𝑃2| ≪ 𝑃𝑒, we can drop the 𝑃1 + 𝑃2 term relative to the 𝜕𝑃𝑒/𝜕𝑌𝑒 term, and take 𝑃 ≈ 𝑃𝑒, giving 𝜒𝑋 ≈ −𝑃1 𝑃 �� 𝑍2 𝑍1 �5/3 � 𝐴1 𝐴2 � − 1 � + 𝑋 𝑌𝑒 𝜕 ln 𝑃𝑒 𝜕 ln𝑌𝑒 � 𝑍1 𝐴1 − 𝑍2 𝐴2 � (B17) as our final expression for 𝜒𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' A similar expression for the internal energy per gram 𝐸 was derived by Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (1997, 2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' as a check, calculating 𝜒𝑋 as (𝑋/𝑃)(𝜌2𝜕/𝜕𝜌)(𝜕𝐸/𝜕𝑋) using their results for 𝜕𝐸/𝜕𝑋 gives agreement with equation (B17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In neutron star oceans, the second term in equation (B17) dominates, since 𝑃𝑖 ≪ 𝑃 and we typically have species with different ratios 𝑍/𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' With 𝑍2/𝐴2 < 𝑍1/𝐴1 (since species 1 is the lighter species), this term is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For example, for the neutron star ocean with a mixture of O (𝑍 = 8, 𝐴 = 16) and Se (𝑍 = 34, 𝐴 = 79) considered by Medin & Cumming (2011), Δ(𝑍/𝐴) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='070, and the second term gives 𝜒𝑋 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='2𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In this case, taking ∇ad ≈ 1/3 and using equation (B10), we find 𝜒𝑋 𝜒𝑇 ∇ad ≈ 270 � 𝑋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1 � � 𝜇𝑖 79 � �Δ(𝑍/𝐴) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='07 � 𝜌1/3 9 𝑇8 ≈ 69 � 𝑋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='1 � � 𝜇𝑖 79 � �Δ(𝑍/𝐴) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='07 � � ⟨𝑍5/3⟩ 345/3 �−1 � Γ 178 � , neutron star (B18) where 𝜇−1 𝑖 = � 𝑖(𝑋𝑖/𝐴𝑖) and Γ is the Coulomb coupling parameter with ⟨𝑍5/3⟩ averaged by number (see Medin & Cumming 2015 eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Note that at fixed Γ, 𝜒𝑋/𝜒𝑇 ∇ad depends only on composition and is independent of temperature and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In white dwarf interiors, however, the electron term in equation (B17) is small or vanishing (as pointed out by Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 1997, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For a mixture of C and O for example, 𝑌𝑒, and therefore the electron pressure, is independent of the C/O ratio, since both species have 𝐴 = 2𝑍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' In that case, 𝜒𝑋 is set by the ion term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Using 𝑃𝑒 for non-relativistic electrons and 𝑃𝑖 from equation (B13), we find 𝑃𝑖/𝑃𝑒 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='0057 (𝑌𝑒𝜌6)−1/3(𝑍𝑖/𝑌𝑒𝐴𝑖)𝑋𝑖𝑍2/3 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For a mixture of C/O, 𝑃𝑖/𝑃𝑒 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='019𝑋(𝑌𝑒𝜌6)−1/3 and the factor (𝑍2/𝑍1)5/3(𝐴1/𝐴2) − 1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='21, giving 𝜒𝑋 ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='0 × 10−3 𝑋(𝑌𝑒𝜌6)−1/3, approximately two orders of magnitude smaller than in the neutron star case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Again taking ∇ad ≈ 1/3, and using equation (B9), we find 𝜒𝑋 𝜒𝑇 ∇ad ≈ 40 � 𝑋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='5 � � 𝜇𝑖 14 � 𝜌1/3 6 𝑇6 ≈ 14 � 𝑋 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content='5 � � 𝜇𝑖 14 � � ⟨𝑍5/3⟩ 75/3 �−1 � Γ 178 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' white dwarf (B19) We see that 𝜒𝑋/𝜒𝑇 ∇ad is about an order of magnitude smaller than in the neutron star ocean case at the same value of 𝑋, but still larger than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We apply these values of 𝜒𝑋/𝜒𝑇 ∇ad in our estimates in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' For the MESA simulation results shown in §4, we take 𝜒𝑇 directly from the code, and compute 𝜒𝑋 by perturbing 𝑋 and calling the equation-of-state directly to compute 𝜕𝑃/𝜕𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The analytic formulae above agree well with the numerical results, as can be seen in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 16 Fuentes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 2 4 6 8 Age (Gyr) 10 3 10 2 T T analytic X X analytic 2 4 6 8 Age (Gyr) 10 1 100 101 ad X T ad X T ad analytic Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 𝜒𝑋, 𝜒𝑇 , ∇ad, and the ratio 𝜒𝑋/𝜒𝑇 ∇ad at the crystallization front for the white dwarf models shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We compare against the analytic results given by equation (B9), the first term of equation (B17), and equation (B19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Energetics of chemical separation in white dwarfs By considering the change of internal energy with composition across the white dwarf, Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (1997, 2000) found the extra luminosity generated by the redistribution of elements in the convection zone is given by 𝐿chem = �𝑀𝑐Δ𝑋melt � 𝜕𝐸 𝜕𝑋 ���� 𝑐 − � 𝜕𝐸 𝜕𝑋 �� = �𝑀𝑐Δ𝑋melt 𝛼 𝜕𝐸 𝜕𝑋 ���� 𝑐 , (C20) where the first term in the square brackets is evaluated at the crystallization boundary, the second is an average over the liquid region, and we introduce the same averaging parameter 𝛼 ≲ 1 as Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' The partial derivative 𝜕𝐸/𝜕𝑋 is taken at constant 𝑇 and 𝜌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' for clarity we do not indicate this explicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Note that as elsewhere in this paper, 𝑋 is the mass fraction of the light element, and define Δ𝑋melt = 𝑋𝑙 − 𝑋𝑠 > 0 as the difference in the light element mass fraction between liquid and solid phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Now using equation (8) of Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2000) for 𝜕𝐸/𝜕𝑋 and the first term in equation (B17) for 𝜒𝑋, we find 𝜕𝐸/𝜕𝑋 = 3𝑔𝐻𝑃 𝜒𝑋, and therefore 𝐿chem ≈ �𝑀𝑐𝑔𝐻𝑃 Δ𝑋melt (3𝛼𝜒𝑋) (C21) (see Isern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' 1997 for a similar argument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We can compare this with Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (2022), who write the rate of gravitational energy release (see their eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' [6]) as 𝐿grav ≈ �𝑀𝑐𝑔𝐻𝑃 Δ𝜌 𝜌 , (C22) where �𝑀𝑐 is the growth rate of the mass of the solid core and Δ𝜌 = 𝜌𝑠 − 𝜌𝑙 > 0 is the density contrast between solid and liquid phases at the crystallization front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Now writing Δ𝜌/𝜌 = −(𝜒𝑋/𝜒𝜌)(−Δ𝑋melt/𝑋) gives 𝐿grav ≈ �𝑀𝑐𝑔𝐻𝑃 Δ𝑋melt 𝜒𝑋 𝑋 𝜒𝜌 ≈ �𝑀𝑐𝑔𝐻𝑃 Δ𝑋melt �3𝜒𝑋 5𝑋 � , (C23) which is approximately equal to 𝐿chem (depending on the value of 𝛼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' We can compare this with the convective heat flux associated with the flux of light elements using equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Writing 4𝜋𝑅2 𝑐𝐹𝑋 = �𝑀𝑐Δ𝑋melt and assuming 𝜏 < 1 so that ∇𝑋 ≈ ∇𝑋,crit, gives 𝐿𝐻 = 4𝜋𝑅2 𝑐𝐹𝐻 = 4𝜋𝑅2 𝑐𝐹𝑋 𝑐𝑃𝑇 𝑋 𝜒𝑋 𝜒𝑇 = �𝑀𝑐𝑔𝐻𝑃 Δ𝑋melt 𝜌𝑐𝑃𝑇 𝑋𝑃 𝜒𝑋 𝜒𝑇 , (C24) where we have also used the relation 𝑃 = 𝜌𝑔𝐻𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Now applying the thermodynamic identities ∇ad = 𝜒𝑇 𝑃/𝜌𝑐𝑉 𝑇Γ1 and Γ1 ≈ 𝜒𝜌, 𝑐𝑃 ≈ 𝑐𝑉 for a degenerate gas gives 𝐿𝐻 = �𝑀𝑐𝑔𝐻𝑃 Δ𝑋melt � 1 𝑋∇ad 𝜒𝑋 𝜒𝜌 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (C25) Heat transport by compositionally-driven convection 17 This shows that the inwards convective luminosity (and compensating outwards luminosity carried by thermal conduction) is of the same order of magnitude as 𝐿chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Also of interest is the kinetic energy flux 𝐹𝐾 ≈ 𝜌𝑣3 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Comparing this to 𝐿chem gives 𝐿𝐾 𝐿chem = 4𝜋𝑅2 𝑐𝐹𝐾 𝐿chem = 4𝜋𝑅2 𝑐𝜌𝑣𝑐 �𝑀Δ𝑋melt 𝑣2 𝑐 𝑔𝐻𝑃 1 3𝛼𝜒𝑋 = 𝑣2 𝑐 𝑔𝐻𝑃 1 3𝛼𝑋 𝜒𝑋 1 ∇𝑋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' (C26) Using equation (11) to replace 𝑣2 𝑐 in the non-rotating limit, 𝐿𝐾 𝐿chem ≈ 1 24𝛼𝑋 𝜒𝜌 ∇𝑋 − ∇𝑋,crit ∇𝑋 , (C27) where we take the mixing length to be equal to the pressure scale height for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This shows that the kinetic energy flux is a small fraction of the total energy flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' This is different from efficient thermal convection in a typical stellar convection zone, where 𝐹𝐻 ≈ 𝜌𝑣𝑐𝑐𝑃𝑇(∇ − ∇ad), 𝑣2 𝑐 ≈ 𝑔𝐻𝑃(∇ − ∇ad), and 𝑐𝑃𝑇 ≈ 𝑃/𝜌 ≈ 𝑔𝐻𝑃 gives the standard result 𝐹𝐻 ≈ 𝜌𝑣3 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} +page_content=' Here we have 𝐹𝐻 ∼ 𝜌𝑣𝑐𝑃𝑇∇adPe at small Pe and 𝑣2 ∼ 𝑔𝐻𝑃(∇𝑋 − ∇𝑋,crit)(𝜒𝑋/𝜒𝜌), giving 𝜌𝑣3 ∼ 𝐹𝐻 (𝜒𝑋/𝜒𝑇 )(∇𝑋 − ∇𝑋,crit)Pe−1 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE3T4oBgHgl3EQfCAnu/content/2301.04273v1.pdf'} diff --git a/OtFRT4oBgHgl3EQfHzf5/content/tmp_files/2301.13490v1.pdf.txt b/OtFRT4oBgHgl3EQfHzf5/content/tmp_files/2301.13490v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b92fead77f57ac5d8a2430af4b9c4e835a6b85c --- /dev/null +++ b/OtFRT4oBgHgl3EQfHzf5/content/tmp_files/2301.13490v1.pdf.txt @@ -0,0 +1,1361 @@ +1 + +A study of simulating Raman spectra for alkanes with a +machine learning-based polarizability model + +Mandi Fang,a,b Shi Tang,a Zheyong Fan,c Yao Shi,b Nan Xu,a,b,* Yi He, a,b,d,* + +a Institute of Zhejiang University-Quzhou, Quzhou 324000, China +b College of Chemical and Biological Engineering, Zhejiang University, Hangzhou +310027, China +c College of Physical Science and Technology, Bohai University, Jinzhou 121013, +China +d Department of Chemical Engineering, University of Washington, Seattle, WA 98195, +USA + + + + + +2 + +Abstract +Polarizability is closely related to many fundamental characteristics of molecular +systems and plays an indispensable role in simulating the Raman spectra. However, the +calculations of polarizability for large systems still suffers from the limitations of +processing ability of the quantum mechanical (QM) methods. This work assessed and +compared the accuracy of the bond polarizability model (BPM) and a ML-based atomic +polarizability model (AlphaML) in predicting polarizability of alkanes and then also +investigated the ability of simulating Raman spectra. We found that the AlphaML has +appreciable advantages over the BPM in learning the polarizability in the training data +set and predicting polarizability of molecules that configurational differently from +training structures. In addition, the BPM has inherent disadvantages in predicting +polarizability anisotropy due to many factors including large uncertainties of estimating +bond anisotropy, omitting of off-diagonal parameters in the construction of the model. +As a result, the BPM has larger errors than the AlphaML in the simulation of anisotropic +Raman scattering. Finally, we demonstrated that both the BPM and AlphaML suffer +from transference to alkanes larger than those used in the training data sets, but the +problem for the AlphaML can be circumvented by exploring more proper training +structures. + + + +3 + +1. Introduction +Polarizability describes the tendency of an atom or molecule to adjust its electron +cloud in response to an external electric field,1,2 and therefore plays a vital role in +understanding nonlinear optics and the Raman scattering3,4. It is also an indispensable +ingredient in the development of next-generation polarizable force fields.5,6 +Technically, polarizability can be calculated using the quantum mechanical (QM) +methods such as the density functional theory (DFT).7,8 However, the computational +cost of DFT calculations is proportional to the cube of system size. The small-system- +size limitations hinder the applications of the DFT method in calculating polarizabilities +of systems with more than hundreds of atoms.9–11 It is possible to circumvent the +problem by developing parametric polarizability models, which have a linear scaling of +the computational effort with system size.12 +The bond polarizability model (BPM) is a frequently-used parametric polarizability +model, in which the molecular polarizability is treated as the sum of polarizabilities of +all chemical bonds in the systems.11,13,14 Bougeard and coworkers13 parameterized a +BPM to predict polarizabilities for alkanes using molecular polarizabilities calculated +by the DFT method as training data. The results demonstrate that the trained BPM learn +polarizabilities in the data set well and enables transfers to molecules larger than those +in the training data sets. Based on this line of thought, Milner and coworkers14 retrained +a BPM for alkanes, and transferred the model to polyethylene. By combining the BPM +with molecular dynamics (MD) simulations, the authors simulated Raman spectra of +polyethylene and successfully described the difference between Raman peaks of + +4 + +polyethylene at crystalline and molten phases. +In spite of successfully describing the difference in Raman peaks for polyethylene, +the BPMs still have large errors when predicting the polarizabilities of alkanes,11,13,14 +which seriously questions the quality of the simulated Raman spectra. The preset +functions to map from bonds’ attributes (lengths, orientations etc.) to their +polarizabilities may limit the ability of the BPM to describe polarizability under +complex chemical environments. Machine learning-based polarizability models are +considered as an alternative parametric model for predicting molecular polarizability +with high accuracy. Typical packages for this purpose include the AlphaML15, +embedded atom neural network (EANN)16 and deep neural network (DNN)17. In these +models, chemical environment of each atom within a cutoff is encoded into +mathematical descriptors, and then the descriptors are inputted into a ML model to give +a polarizability tensor. Due to the specially designed descriptors and the powerful fitting +capability of ML, these models are able to retain the accuracy of the underlying QM +methods but can predict molecular polarizability several orders of magnitude +faster.12,15,18 +Alkanes are a class of compounds which are extensively studied in the development +of the BPMs.13,19–21 Two reasons may contribute to the popularity of alkanes. Firstly, +alkanes have a large quantity of spectroscopic data available which serve as +experimental references. Secondly, alkanes and the corresponding polymers (polyolefin) +are well suitable to the study on the transferability of the BPMs.11,22 Although the +machine learning-based polarizability models have achieved great success in predicting + +5 + +polarizabilities of waters and organic compounds15,16,18,23, few researches focus on the +prediction of polarizability, transferability of models and simulation of Raman spectra +for alkanes. +The aim of the present work is to assess a ML-based polarizability model in terms of +the accuracy of predicting molecular polarizability and simulating Raman spectra for +alkanes. We also trained a zero-order BPM for comparisons. Polarizabilities of alkanes +containing no more than 5 carbons were used as the training data set and testing data +set. The performance of the ML-based polarizability model on predicting +polarizabilities were evaluated, and then compared with the results produced by the +BPM. In addition, the Raman spectra of ethane and 2-methylbutane were simulated by +combining the polarizability models with molecular dynamics (MD) simulations, and +the spectra were then compared with those simulated by the combination of the DFT +method and MD simulations. We further investigated the transferability of the AlphaML +model and the BPM by using larger alkanes containing no more than 11 carbons as a +testing data set. +2. Simulation Details +2.1 DFT calculations +Geometries of alkanes were optimized using the B3LYP exchange-correlation +functional24 and the 6-31G(d) basis set.25 Dispersion corrections were considered with +the DFT-D3 method with Becke-Johnson damping (D3BJ).26,27 Based on the optimized +structures, molecular polarizabilities were calculated by solving the CPHF equations +using the B3LYP functional and the 6-311++G(2d,2p) basis set.28 All DFT calculations + +6 + +were carried out using the ORCA-v5.0 package29,30 with the RIJCOSX numerical +integration.31,32 +2.2 Preparations of data sets +The initial training data set consists of methane, ethane, propane, n-butane, n-pentane +and 2-methylbutane. For n-butane and n-pentane, both the trans and gauche +conformations were included, therefore, the initial training data set contains 8 structures +in total, which is the same to the training data set for BPM in the work of Bougeard and +coworkers.13 An additional training data set containing 95 structures was constructed +by stretching the individual C-C and C-H bonds in the data set by ∆푟 = 0.01 Å. +Molecular polarizability of each structure was calculated using the method introduced +in section 2.1. +A testing data set of non-equilibrium molecules was constructed to evaluate the +accuracy of the polarizability models in predicting polarizability of molecules that are +far away from equilibrium. Using structures of 8 alkanes at equilibrium as the initial +structures, MD simulations at 300 K were performed to obtain non-equilibrium +structures. The temperature was controlled using the Berendsen thermostat algorisms +with the NVT ensemble.33 Meanwhile, all C-H bonds were constrained with the +SHAKE algorithm.34 The xTB package35 and the GFN2-xTB36 force field were used +for all MD simulations. Each simulation has a duration of 5 ps with a time step of 1 fs, +and trajectories were saved every 1 ps. In this way, 40 structures were collected. +Examination of transferability was performed for alkanes containing no more than +11 carbons. These structures were firstly generated from SMILES strings in the GDB- + +7 + +11 database37 using the RDKit package.38 Then, geometry optimizations were +performed with the DFT method followed by calculations of molecular polarizability. +2.3 Principle of the ML-based polarizability model +In the present work, we chose the AlphaML15 as a representative of the ML-based +polarizability models. The AlphaML is based on a symmetry-adapted Gaussian process +regression (SA-GPR) scheme, which is designed for the predictions of tensorial +properties.23 We shall introduce the component-wise GPR firstly, which is a +simplification of the SA-GPR scheme.39 In this scheme, each individual polarizability +component 훼￿￿ in the Cartesian frame (푝, 푞 = 푥, 푦, 푧) reads + +훼￿￿(ℬ) = 훼￿￿￿ +￿￿￿ + ∑ +푤￿ +￿￿푘￿ℬ�,�풜￿￿ +￿ +￿￿￿ + +(1) +where 푁 is the number of configurations in the training data set, 훼￿￿￿ +￿￿￿ is the average +of the polarizability component 훼￿￿ over the training data set, 푤￿ +￿￿ are the weights, +and 푘 is a kernel function that measure the similarity between the target system ℬ +and a training structure 풜￿. The kernel function commonly used in the GPR is based +on a Gaussian similarity + +푘￿ℬ�,�풜￿￿ = exp ￿−�￿퐮�(ℬ)�−�퐮�￿풜￿￿￿ +￿ +￿￿￿ +￿ +(2) +where 휎 is the Gaussian width, and 퐮(⋯ ) is a function that maps the atomic +coordinates to a high-dimensional space. +The mapping function 퐮(⋯ ) determines the accuracy and efficiency of the GPR +model. It is often constructed using the atomic density representation, which builds a +Cartesian reference frame centered on an atom and defines a three-dimensional grid +around it. At each grid-point 퐫, the atomic density distribution is calculated in the form + +8 + + +휌￿(퐫) = ∑ +exp ￿−�￿퐫�−�퐫￿￿ +￿ +￿￿￿￿ ￿ +￿∈￿ + +(3) +where 푠 identifies an atom type, 퐫￿ is the coordinate of the central atom, and 훾￿ is a +smearing parameter. 퐮(⋯ ) is given by the set {휌￿(퐫), 푠 = 1, ⋯ , 푁￿}, where 푁￿ is the +number of atomic species in the system. The polarizability tensor predicted in this +manner is not invariant to rotations in Cartesian space, and therefore the atomic density +that uses a Cartesian space representation requires an alignment to a reference +structure.39 +The AlphaML learns the polarizability using a combination of GPR and SA-GPR +scheme23,39. Naturally, the polarizability 훂 is a symmetric rank-2 tensor. Before fitting, +훂 is decomposed into a scalar component 훼(￿) = ￿훼￿￿�+�훼￿￿�+�훼￿￿￿/√3 and a +tensorial component 훼(￿) = √2 ￿훼￿￿�,�훼￿￿�,�훼￿￿�,�￿￿￿￿￿￿￿￿￿￿￿￿ +￿√￿ +�,�￿￿￿￿￿￿￿ +￿ +￿.15 The former is +fitted using the GPR scheme while the latter is fitted using SA-GPR scheme. Within +the SA-GPR scheme, a tensorial generalization of the smooth overlap of atomic position +kernel (λ-SOAP) are employed, which uses the covariant integration of the atomic +density in the mapping functions. By using the λ-SOAP for the tensorial component, +the AlphaML gets rid of the alignment for the atomic density. Detailed descriptions of +the SA-GPR scheme and the λ-SOAP kernels can be found elsewhere.23,39 +The training of the AlphaML model is equivalent to the selection of representative +reference environments and the determination of the corresponding weights 푤￿ +￿￿ , +which is done by minimizing a loss function defining the deviations of predicted +polarizabilities from those given in the training data set.15,39 For 훼(￿), 8 radial functions, +6 angular functions, an environment cutoff of 5 Å and a Gaussian width of 0.25 Å + +9 + +were used. For 훼(￿) , the hyperparameters are the same as those for the scalar +component except that a Gaussian width of 0.35 Å was used. +2.4 Simulations of Raman spectra +The isotropic and anisotropic Raman scattering are closely related to the isotropic +polarizability 훼￿ = ￿훼￿￿�+�훼￿￿�+�훼￿￿￿ 3 +⁄ + and the anisotropic tensor 훂￿ = 훂 − 훼￿퐈 , +respectively. Neglecting the nuclear quantum effects, the differential cross section of +isotropic and anisotropic Raman scattering can be written in terms of the Fourier +transform of the polarizability autocorrelation function (PACF)4,11 + +￿ +￿￿￿ +￿￿￿￿￿ +￿￿￿ = +￿ +￿￿ ∫ +d푡푒￿￿￿￿〈훼￿�(0)�훼￿�(푡)〉 +￿￿ +￿￿ + +(4) + +￿ +￿￿￿ +￿￿￿￿￿ +￿￿￿￿￿ = +￿ +￿￿ ∫ +d푡푒￿￿￿￿〈Tr[훂￿(0)훂￿(푡)]〉 +￿￿ +￿￿ + +(5) +where 휔 is the Raman frequency shift, 훺 is the solid angle range. Tr indicates the +trace and 〈⋯ 〉 indicates an ensemble average. The parallel and perpendicular spectra +that are comparable to the experimental polarized Raman spectra can be obtained +by4,11,18 + +퐼∥(휔) = ￿ +￿￿￿ +￿￿￿￿￿ +￿￿￿ + +￿ +￿￿ ￿ +￿￿￿ +￿￿￿￿￿ +￿￿￿￿￿ +(6) + +퐼￿(휔) = +￿ +￿￿ ￿ +￿￿￿ +￿￿￿￿￿ +￿￿￿￿￿ +(7) +It is worth mentioning that the simulated spectra only record the line shapes of Raman +spectra18, which is independent on the wavelength of the incident light. +To simulate Raman spectra for ethane and 2-methylbutane, we performed a MD +simulation with the GFN2-xTB36 force field. The simulation has a duration of 25 ps +with a time step of 1 fs and trajectories were saved every 2 fs. Polarizabilities of +structures sampled from the MD simulation were calculated by the DFT method, the + +10 + +BPM and the AlphaML model, respectively, and the corresponding PACF along the +evolution of trajectories were then obtained. Finally, we simulated the parallel and +perpendicular spectra according to the formulae in Equation 4~7. The Fourier +transforms of PACF in Equations 4 and 5 were simplified with the use of the discrete +cosine transform (DCT).40 A combination of a sampling interval of 2 fs and a lag time +of 10 ps in DCT produces an increment of 1.67 cm−1 in frequency, which can capture +the highest frequency mode of CH bending with a period of about 23 fs.14 +3. Result and discussion +In the very beginning, we trained the AlphaML model and the zero-order BPM with +the initial training data set containing 8 alkanes at equilibrium and 95 structures with a +stretched C-C or C-H bond. We demonstrated the training process and the fitting quality +of the AlphaML in the following section, while presenting the training process of the +BPM in the supporting material. In the second step, comparisons were performed +between the BPM and the AlphaML model in terms of the accuracy of predicting +polarizabilities for molecules that are far away from equilibrium. Subsequently, Raman +spectra of ethane and 2-methylbutane were simulated by integrating the AlphaML +model, the BPM or the DFT method with MD simulations. The relationship between +the accuracy of polarizability prediction and the quality of simulated Raman spectra +were demonstrated. Finally, we investigate the transferability of the AlphaML model +and the BPM to alkanes larger than those in the training data sets. + + + +11 + +3.1 Training of the AlphaML model +We used the training data set containing a total of 103 structures to train the AlphaML +model. Figure 1a shows explicitly scatter plots of the diagonal and off-diagonal +components of polarizabilities calculated by the DFT method and predicted by the +AlphaML model for the 8 alkanes at equilibrium. The AlphaML model yields +coefficient of determination (R2) near 1.00 for both the diagonal and off-diagonal +components. As presented in the supporting material, the BPM yields R2 of 0.997 and +0.803 for the diagonal and off-diagonal components, indicating that the AlphaML +possesses a better learning capacity for the off-diagonal components of polarizabilities. +Meanwhile, we also examined the ability of predicting the derivatives of polarizability +associated with a bond stretch. The narrow distributions of the derivatives of +polarizability around the 푦 = 푥 line manifest that the AlphaML has enough accuracy +to describe the tiny changes of polarizabilities associated with bond stretches, as shown +in Figure 1b. The R2 for both the diagonal and off-diagonal components of derivatives +of polarizability are greater than 0.993, much higher than the corresponding R2 +produced by the BPM, as shown in Table 1. The result echoes the trend that AlphaML +has advantage over the BPM in learning the polarizability of molecules in the condition +of identical training data sets, which are consistent with what we expected due to the +powerful fitting capability of ML12. + +12 + + +Figure 1. (a) Correlations between diagonal components of polarizabilities of 8 alkanes +at equilibrium calculated by the DFT method and predicted by the AlphaML model. +Correlations for off-diagonal components are shown in the inset. (b) Correlations +between diagonal components of the derivatives of polarizability associated with a bond +stretch calculated by the DFT method and predicted by the AlphaML model. +Correlations for off-diagonal components are shown in the inset. + +Table 1. The R2 for the diagonal and off-diagonal components of derivatives of +polarizability associated with a bond stretch. + +R2 (diagonal/off-diagonal) +Model +C-C stretch +C-H stretch +BPM +0.895/0.879 +0.842/0.909 +AlphaML +0.996/0.993 +0.995/0.997 + + + + + +30 +0.0 +2.S +Q00 +0.0 +C.S +0.S-αDEI (g'n) +αDEI (g'n") +20-5'2 0'0 '2 2'0 1'? +-4'0-S'0 0'0 S'0 4'0J0 +SO +30 +40 +eo +10 +08 +0.0 +c.S +0. +c.1 +0.01 +2.Sr +1O08 +JS'2EαDEI (g'n") +αDEI ('n")10 +C-H pouq 2flefcμ +口、 +C-C pouq 2flefcμ9 +p +0.01.1Q10.2 +40 +'0 +2.113 + +3.2 Performance on predicting polarizability of non-equilibrium structures +The AlphaML model exhibits an extraordinary ability in learning polarizabilities and +derivatives of polarizability from the training data set containing structures at +equilibrium and structures with small bond stretches. A benchmark testing was +performed to the AlphaML model and the BPM to evaluate the accuracy in predicting +polarizability of structures that are far away from equilibrium. +Figure 2a shows that both the BPM and the AlphaML model can predict +polarizabilities of the non-equilibrium structures in agreement with the DFT results, +although the BPM has larger errors in predicting the off-diagonal components than the +AlphaML model. This trend is consistent with the fact that the AlphaML model can +learn the off-diagonal components of polarizabilities of 8 alkanes at equilibrium better +than the BPM as presented in the previous section. To rule out the effects of rotation +operations on molecular polarizabilities, here, we introduced the rotation-invariant +isotropic polarizability (훼￿) and the rotation-invariant polarizability anisotropy41 (∆훼 = +￿￿￿훼￿￿�−�훼￿￿￿ +￿�+�￿훼￿￿�−�훼￿￿￿ +￿�+�(훼￿￿�−�훼￿￿)￿�+�6�￿훼￿￿ +￿ �+�훼￿￿ +￿ �+�훼￿￿ +￿ ￿￿ 2 +⁄ +) to adequately +measure the deviations of polarizability predicted by the AlphaML model and the BPM +from the corresponding polarizabilities calculated by the DFT method. + + +14 + + +Figure 2. (a) Correlations between diagonal components of polarizabilities of non- +equilibrium structures calculated by the DFT method and predicted by the AlphaML +model and the BPM. Correlations for off-diagonal components are shown in the inset. +(b) Correlations between isotropic polarizability (훼￿) calculated by the DFT method and +predicted by the AlphaML model and the BPM. (c) Correlations between polarizability +anisotropy (∆훼) calculated by the DFT method and predicted by the AlphaML model +and the BPM. (d) RMSEs of isotropic polarizability (RMSE￿￿￿) and polarizability +anisotropy (RMSE￿￿￿￿￿) produced by the AlphaML model and the BPM. + +aHso +oib +oib +2.1 +0.0 +0.02 +C4H1O +:H0 +.U +7 +0.0a +0.0a +S0.0S +0.0S +CH +2.5- +ableqicr +CSHe +0.0 +0.08 +0.08 +(gn) +a.s +8H.0 +g +0.2D (g') +αDEI (g'n') +J0'0 50'0 30'0 40'0 20'0 0'0 10'0 80'0 +10'0 50'0 30'0 40'0 20'0 e0'0 10'0 80'0 +0.01 +JO'C +αDEI (g') +.CH4BbW +BbW +JMsdqIA +IMsdqIA +0.0S +0.Stoibg +0.07 +■ +0.0 +(.U.β) +BW +口 +2.0 +2 +口 +m2 +E +0.2 +U.s) +3'0VαDEI (g'n') +CH4 +H H H H +0.0 +0.2 +0.01 +0.0S +0.0 +0'0 +BWS +C +0.01 +0.01 +BbW +BbW +IMsdqlA +MsdqlA +0.08 +0.0815 + +Both the BPM and the AlphaML can predict 훼￿ as good as the DFT method, as +shown in Figure 2b. The results agree with the trend for the diagonal components of +polarizabilities shown in Figure 2a well, as 훼￿ is the average of the three diagonal +components. It is interesting that data in Figure 2b are grouped by their chemical +formula of molecules, namely CH4, C2H6, C3H8, C4H10 and C5H12. Moreover, the +groups of data distribute around the 푦 = 푥 line almost uniformly, which implies that +훼￿ may increase linearly with number of carbons in the molecules. This linear +relationship is confirmed in Figure S2. In other words, the additive assumption of +polarizabilities is roughly valid for the average of diagonal components. Both the BPM +and the AlphaML model assume that the molecular polarizabilities are additive,11,15 +thus, the high consistency between the predicted 훼￿ and 훼￿ calculated by the DFT +method is not surprising. + The AlphaML can predict ∆훼 much better than the BPM for non-equilibrium +structures. Figure 2c clearly shows that the points of ∆훼 predicted by the BPM are far +away from the 푦 = 푥 line, indicating rather higher prediction errors. This trend is +confirmed from high RMSEs of ∆훼 shown in Figure 2d. On the contrary, ∆훼 +predicted by the AlphaML model agree well with DFT values, which is validated by +the low RMSEs of ∆훼 shown in Figure 2d. The points of ∆훼 predicted by the BPM +model are well correlated with those calculated by the DFT method, also having a trend +of slight downward offset, as shown in Figure 2c. The effect of the overall offset on +the quality of Raman spectra will be demonstrated in the following section. By the way, + +16 + +the reasons for the high RMSEs of ∆훼 given by the BPM is deeply discussed in the +supporting material. +3.3 Performance on predicting Raman Spectra +In the previous section, we have demonstrated the performance of the AlphaML +model on predicting the isotropic polarizability and the polarizability anisotropy for +non-equilibrium alkanes. The AlphaML model is proven to have better accuracy than +the BPM in the condition of identical training data sets. In this section, we will evaluate +the quality of simulated Raman spectra for ethane and 2-methylbutane and interpret the +relationship between the accuracy of polarizability prediction and quality of simulated +Raman spectra. +We calculated the time autocorrelation function of isotropic polarizability +( PACF￿￿￿(푡) = 〈훼￿�(0)�훼￿�(푡)〉 ) and anisotropic polarizability tensor ( PACF￿￿￿￿￿(푡) = +〈Tr[훂￿(0)훂￿(푡)]〉) for ethane using the AlphaML model and the BPM, and compared the +time autocorrelation function with the corresponding values calculated by the DFT +method. Figure 3a shows that the wave shape and amplitude of PACFiso predicted by +the AlphaML model and the BPM are close to that calculated by the DFT method, +which agrees with the fact that both the AlphaML model and the BPM can predict the +훼￿ for ethane as good as the DFT method. The PACFaniso predicted by the AlphaML +model also resembles the PACFaniso calculated by the DFT method. However, the +PACFaniso predicted by the BPM has distinctive differences from the PACFaniso +calculated by the DFT method in terms of the wave shape and period of change, as +shown in Figure 3b. Since the Raman scattering intensity is proportional to the Fourier + +17 + +transform of the PACF,4,11 the obvious change of the wave shape and period may cause +the difference in the shifts and the intensity of peaks of the simulated Raman spectra. + +Figure 3. Comparison between (a) PACFiso and (b) PACFaniso for ethane calculated +by the DFT method and predicted by the AlphaML model and the BPM. + +The Raman spectra of ethane simulated from PACFs predicted by the AlphaML +model are consistent with that simulated from PACFs calculated by the DFT method +across a wide range of wavenumbers, as shown in Figure 4. The Raman spectra +simulated from PACFs by the DFT method, the AlphaML model and the BPM can +successfully reproduce an array of features reported in experiments21,42, with coincident +Raman shifts and approximately close intensities. These features include the C-C +stretching peak at about 1057 cm-1, CH3 stretching peaks at about 3054 and 3067 cm-1, +as shown in Table 2. + +0 +S +4 +e +8 +10 +0 +S +4 +8 +1O +0.7-(2q) 9miT +(q) 9miTDEL +DEH +J'OE2.02.0- +2.0IMsdalA +IMsdalA +0. 1 +0.7- +90.70 +U2.0- +2.0-BbW +BbW +0.7- +0.1-0.0 +0'02.0- +2.0-18 + + +Figure 4. Comparisons between parallel and perpendicular Raman spectra of C2H6 +simulated from PACFs by DFT, the AlphaML and the BPM. The latter two Raman +spectra were shifted upward. +Table 2. Wavenumber of the Raman Spectra of C2H6 from experiments and +simulated from PACFs calculated by DFT (cm-1) +skeletal mode +description +experiment +DFT +CH3 stretching +2953.7 +3054 +CH3 stretching +2968.7 +3067 +CH3 deformation +1388.4 +1440 +CH3 deformation +1468.1 +1502 +C-C stretching +994.8 +1057 +CH3 rocking +1195.3 +1202 + +DEI +AI: CH3 2{lercuiua +L: CH3 2flercuiuaBbWJMsdqIADEI3 +1000 +00cT +5000 +3000 +00c8 +4000IVV +Ⅱ +VI IⅡIII: CH3 LoCKIua +I : C-C 2flercpiuaIMsdqA +noitsmioteb sHO :VI +BbW +I: Ch? qetoLsiou19 + +However, both the parallel and perpendicular Raman spectra simulated by the BPM +has produced abnormal signal at about 1202 cm-1, inconsistent with the spectra +simulated from PACFs by DFT and also inconsistent with the fact that the experimental +intensity for the CH rocking peak is extremely weak14,43. This is due to the large errors +in predicting PACFaniso by the BPM. +The most significant discrepancy between the Raman spectra simulated from PACFs +by the AlphaML and DFT is the relative intensities for the CH3 deformation at about +1440 and 1502 cm-1, as shown in Figure 4. We owed the discrepancy to the limited +ability of the AlphaML in predicting polarizability of structures that are far away from +structures in the training data set. Although the AlphaML performs much better than the +BPM in predicting polarizability anisotropy, as shown in Figure 2c, insufficient +samplings44 may still hinder the high-accuracy simulations of Raman spectra. An +attempt of involving 40 structures from MD simulations into the training data set will +produce a right prediction of the relative intensities for the CH3 deformation at about +1440 and 1502 cm-1, as shown in Figure S3. Moreover, Figure S3 also demonstrates +an excellent consistency between Raman spectra simulated from PACFs by the new +trained AlphaML and DFT for 2-methylbutane. +3.5 Discussions on predicting polarizabilities of larger molecules +The additivity assumption for the molecular polarizability endows the AlphaML +model and the BPM the ability with the ability of predicting polarizabilities for larger +molecules. A benchmark testing was performed for alkanes containing no more than 11 +carbons to evaluate the extrapolation ability of the AlphaML model and the BPM. We + +20 + +used the AlphaML model trained with 8 alkanes at equilibrium and additional 95 +structures, with no non-equilibrium structures involved. +For alkanes containing no more than 5 carbons, which is already included in the +training data sets, both the AlphaML and the BPM can predict 훼￿ and ∆훼 as accurate +as the DFT method, as shown in Figure 5. When it comes to alkanes containing more +than 5 carbons, which is beyond the training data sets, large deviations happen for both +the AlphaML model and the BPM. Especially, Figure 5b manifests that the points of +∆훼 predicted by the BPM distributed irregularly and were far away from the y=x line, +indicating that the BPM trained with small alkanes has very poor extrapolation ability. +On the contrary, the points of ∆훼 predicted by the AlphaML model deviated from the +y=x line gradually. At the meantime, the deviations produced by the AlphaML model +are much lower than those produced by the BPM. + +Figure 5. (a) Correlations between 훼￿ of alkanes containing no more than 11 carbons +calculated by the DFT method and predicted by the AlphaML model and the BPM. (b) +Correlations between ∆훼 calculated by the DFT method and predicted by the + +IMcdalA +<-opo +10.00 +C"HS4VαDEI (g'n') +αDEI (S')S +口口 +BbW +■口 +BbW +.C°HS0 +40'06 +300 +6 +口 +8rHg0CH +ibg +C +0. +V10 +0.0S +C'H/S10.02 +C4H10H +:H0 +10.01CH40.0 +0.01 +0.0S +0.08 +40'0 +0.02 +0.2S +0.0221 + +AlphaML model and the BPM. Solid makers stand for alkanes containing no more than +5 carbons, while hollow markers stand for alkanes containing more than 5 carbons. + +The lack of sampling certain local environment is responsible for the large deviations +of ∆훼 produced by the AlphaML model for alkanes containing more than 5 carbons. +As shown in Figure 6, number of neighbors of the central carbon in n-C5H12 and n- +C11H24 within the environment cutoff of 5 Å are 16 and 24, respectively. Therefore, the +AlphaML model trained with alkanes containing no more than 5 carbons must be +unfamiliar with the local environment of n-C11H24 and thus gives unsatisfied +predictions of polarizabilities. Figure 6 also suggests that a maximum of 9 carbons will +be included within environment cutoff of 5 Å for n-alkanes. We inferred that the +inclusion of structures containing 9 carbons in the training data set may be beneficial +to the transferability of the AlphaML model to larger molecules such as n-C11H24. The +consistency between 훼￿ and ∆훼 calculated by DFT and predicted by the retrained +AlphaML model in Figure 7 clearly manifests that the complement of certain local +environment with more neighbors will greatly improve the accuracy of predicting 훼￿ +and ∆훼, and thus endow the AlphaML with rational extrapolation ability. +Having shown the powerful ability of learning and predicting polarizability of the +AlphaML model, future work will focus on the accurate prediction of polarizability of +large systems such as polymers. This work clearly demonstrates the dependence of the +accuracy of the AlphaML model on the diversity of the training data sets. Therefore, to +guarantee the successful transference to polymers, sampling those local structures that + +22 + +suffused with atoms is indispensable12. + +Figure 6. Number of neighbors of the central carbon in n-C5H12 and n-C11H24 within +the environment cutoff of 5 Å. + +Figure 7. (a) Correlations between 훼￿ of alkanes containing no more than 11 carbons +calculated by the DFT method and predicted by the retrained AlphaML model. (b) +Correlations between ∆훼 calculated by the DFT method and predicted by the retrained +AlphaML model. Solid makers stand for alkanes containing no more than 9 carbons, +while hollow markers stand for alkanes containing more than 9 carbons. +4. Conclusion + +H +0.0010.0240'0 +COHSSC°H +3 +0.00 +HSO加 +C-Hle +0.08 +6CH +Leq!0.00 +C"H/0EUAILoUwGuf cnfoH0.0AHO +0.2S +aHso"0.0 +0.01 +0.0S +0.08 +40'0 +0.00 +0.25 +0.0 +0.0Meappor = Je +Meiappo2 = S4VDI (s"n") +D (g'n')23 + +In this work, we constructed and compared the zero-order BPM and the ML-based +AlphaML model for the prediction of polarizability and simulation of Raman spectra +of alkanes. First, the BPM and the AlphaML were trained with polarizability of 8 +alkanes at equilibrium together with the derivatives of polarizability associated with a +bond stretch, respectively. The accuracy of these two models in the prediction of +polarizability for molecules that are far away from equilibrium were compared. Then +the time autocorrelation function of polarizabilities for C2H6 was calculated by these +two models and the corresponding Raman spectra were simulated and compared with +the Raman spectra by DFT. Finally, the extrapolation ability of these two models in +predicting polarizability of alkanes larger than those in the training data sets were +compared, and discussions were made for the AlphaML on the transference to large +systems such as polymers. +We found that the AlphaML has appreciable advantages over the BPM in learning +the polarizability and the derivative of polarizability of alkanes using the same training +data set. Both the BPM and AlphaML can appropriately predict the isotropic +polarizability for structures that are configurational different from those used in the +training data sets. However, the BPM has inherent disadvantages in predicting +polarizability anisotropy due to many factors including large uncertainties of estimating +bond anisotropy, omitting of off-diagonal parameters in the expression of bond +polarizability tensors. As a result, the BPM has large errors in the simulation of +anisotropic Raman scattering. Finally, we demonstrated that both the BPM and +AlphaML suffer from transference to alkanes larger than those used in the training data + +24 + +sets, but the problem for the AlphaML can be circumvented by enhancing samplings +properly. +AUTHOR INFORMATION +Corresponding Author +*E-mail: tamas@zju.edu.cn; yihezj@zju.edu.cn +NOTES +The authors declare that there is no conflict of interest. +ACKNOWLEDGEMENT +This work is supported by the National Key Research and Development Program of +China (grant number 2022YFE0106100), and the National Natural Science Foundation +of China (grant number 22178299). Nan Xu would like to thank the financial support +provided by the Startup Funds of the Institute of Zhejiang University-Quzhou. +REFERENCES +(1) Wang, J.; Xie, X. Q.; Hou, T.; Xu, X. Fast Approaches for Molecular Polarizability +Calculations. The Journal of Physical Chemistry A 2007, 111 (20), 4443–4448. +https://doi.org/10.1021/jp068423w. +(2) Mei, Y.; Yang, N.; Yang, W. 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The Laser- +Raman Spectrum of Polyethylene: The Assignment of the Spectrum to Fundamental +Modes of Vibration. Spectrochimica Acta Part A: Molecular Spectroscopy 1972, 28 (8), +1485–1496. https://doi.org/10.1016/0584-8539(72)80118-1. +(44) Plazinski, W.; Plazinska, A.; Brzyska, A. Efficient Sampling of High-Energy States +by Machine Learning Force Fields. Physical Chemistry Chemical Physics 2020, 22 (25), +14364–14374. https://doi.org/10.1039/D0CP01399D. + + +1 + +Supplementary information for: “A study of simulating Raman +spectra for alkanes with a machine learning-based polarizability +model” + +Mandi Fang,a,b Shi Tang,a Zheyong Fan,c Yao Shi,b Nan Xu,a,b,* Yi He, a,b,d,* + +a Institute of Zhejiang University-Quzhou, Quzhou 324000, China +b College of Chemical and Biological Engineering, Zhejiang University, Hangzhou +310027, China +c College of Physical Science and Technology, Bohai University, Jinzhou 121013, +China +d Department of Chemical Engineering, University of Washington, Seattle, WA 98195, +USA + + + +2 + +Contents +1 Principle of the bond polarizability model ............................................................ 1 +2 Training of the bond polarizability model ............................................................. 4 +3 Figures ........................................................................................................................ 7 +References ................................................................................................................... 10 + + + + + +3 + +1 +Principle of the bond polarizability model +This section presents the bond polarizability model (BPM) formalism briefly; detailed +descriptions can be found elsewhere.1 In the BPM, molecular polarizability tensor ( +) +is written as the sum of polarizabilities of all bonds + + +(1) +where + denotes the polarizability of bond in the fixed Cartesian axes. Supposed +that + is the polarizability of bond in its principal axes, + can be rewritten as + + +(2) +Here + is the rotation matrix between the bond’s principal axes and the Cartesian axes. +The bond’s principal axes were constructed as follows: first, the longitudinal axis + is +directed along the vector connecting the bonded two atoms; Second, the two transversal +axes + and + are built orthogonal to + and to each other. + in the bond’s principal +axes reads + + +(3) +where +, +, + are the polarizabilities in the directions of the three principal axes. + +In this work, we employed the zero-order BPM, which is commonly used in the +predictions of polarizabilities.1,2 The diagonal component of + ( +, +) is +expanded in a Taylor series respect to the bond length + and truncated after the second +term + + +(4) + +4 + +Here the superscript 0 denotes the equilibrium state and the superscript denotes the +derivative of polarizability. In addition, a cylindrical bond model was used in the zero- +order BPM, which assumes that + and + are identical.1 Hence, we shall use + to +replace + and + in the following sections. + +All bonds of the same type were described by the same set of parameters. The +parameters + and + are known as the equilibrium parameters, which mainly account +for the variation of polarizability upon a change of the bond’s orientation.1 We used the +polarizabilities of 8 alkanes at equilibrium as the training data set to determine the four +equilibrium parameters +, +, +, + and the equilibrium bond +length parameters +, +. + +The differences between polarizabilities of molecules at equilibrium and molecules +with a stretched C-C or C-H bond were then used to determine the derivative parameters +, +, + and +. The fittings of +, +, + and + were +performed with the use of the singular value decomposition method (SVD).1 + +2 +Training of the bond polarizability model +The equilibrium parameters +, +, + and + in the BPM were +fitted to be -49.571, 25.788, -27.361 and 14.556 +, respectively. However, the +negative values of + and + are not reasonable. The C-C and C-H bonds +should increase the dielectric response in terms of the enhancement of the applied +electric field along the bond’s direction, which indicates that + and + + +5 + +should have positive values.1,2 Moreover, the calculated bond anisotropy values of C-C +and C-H bonds, defined as +, are inconsistent with experimental +results.1,3,4 Zerbi and coworkers4 reported that the C-H bond anisotropy + of +methane was estimated to be 0.305 +. Montero and coworkers3 reported that + +of ethane was about 1.28 + according to the experimental Raman spectra. The failure +of the BPM in predicting the sign of polarizability along the bond’s direction and the +bond anisotropy may be due to the lack of intrinsic relations between + and + in the +BPM model. + +Imposing a restrictive condition for the C-H bond anisotropy in the BPM will remedy +these shortcomings.1 An addition equation defining the C-H bond anisotropy + for the CH4 molecule was introduced in the fitting of the +equilibrium parameters. The new fit values of +, +, + and + +are 1.678, 0.174, 0.777 and 0.484 +, very close to the corresponding values of 1.677, +0.127, 0.779, 0.489 + in the work of Bougeard and coworkers.1 The slight +divergences may be due to the difference of the basis sets and QM packages. The bond +anisotropy + and + were calculated to be 1.504 and 0.293 +, close to the +experimental + of 1.28 + for ethane and + of 0.305 + for methane.3,4 The +consistency between the polarizability tensors predicted by the BPM and calculated by +the DFT method shown in Figure S1a confirms that the trained BPM can predict the +polarizability of alkanes at equilibrium with an accuracy close to the DFT method. The +R2 for the diagonal components is 0.997; for the off-diagonal components, 0.803. + +6 + +The best fit values of the derivative parameters +, +, + and + are 2.881, 0.274, 2.628 and 0.393 +, in accord with the corresponding values +of 2.863, 0.243, 2.743, 0.353 + in the work of Bougeard and coworkers.1 In addition, +the bond length parameters + and + were calculated to be 1.533 and 1.097 +. Figure S1b demonstrates that the derivatives of polarizability associated with a +bond stretch predicted by the BPM are in line with those calculated by the DFT method. +The R2 for the diagonal components of the derivatives of polarizability associated with +a C-C bond stretch is 0.895; for the off-diagonal components, 0.879, while the R2 for +the diagonal components of the derivatives of polarizability associated with a C-H bond +stretch is 0.842; for the off-diagonal components, 0.909. + +3 +Discussion on the large prediction errors of the bond polarizability model +Two reasons may contribute to the high RMSEs of polarizability anisotropy ( +) by +the BPM. First, using the same set of parameters for all C-C bonds is not reasonable. +The anisotropy of C-C bonds is found to be dependent on the local chemical +environment, ranging from 0.6 to 1.4 +.3,5 The situation is also the same for C-H +bonds. Second, the BPM has large errors in predicting the off-diagonal components of +polarizability tensors. This may be originated from the omitting of off-diagonal +parameters in the expression of bond polarizability tensors6 and the omitting of second- +order and higher-order terms in the Taylor series of parameters respect to the bond +length.1,6 However, an attempt to involve more parameters may cause too large +statistical uncertainties.6 + + +7 + + +Figure S1. (a) Correlations between diagonal components of polarizabilities of alkanes +at equilibrium calculated by the DFT method and predicted by the BPM. Correlations +for off-diagonal components are shown in the inset. (b) Correlations between diagonal +components of the derivatives of polarizability associated with a bond stretch calculated +by the DFT method and predicted by the BPM. Correlations for off-diagonal +components are shown in the inset. + + + + + ('n") +αDEI (g'n")10 + C-H pouq 2flefc +C-C pouq 2flefcp9 +p +0.01.1BO +0.2 +40 +b +1'230 +0.0 +2.S +c.S0'0 +-4'0 +2.S +0.51%αl (s'n) +αDEI (g"n) +2'0-5'2 00 5'2 2'0 12 +-4'0-5'0 0'0 5'0 4'010 +SO +30 +-5'2 +40 +eo +10 +08 +0.0 +2.S +0.2 +2.1 +0.01 +2.ST +.S-JS'2 +088 + + +Figure S2. Correlations between isotropic polarizability ( +) of non-equilibrium +structures in the testing data set calculated by the DFT method and number of carbons +( +). The dashed line shows a fitted function of +. + + + +(.U.S) +40'0 +0.00 +bgit +0.0a +DEI +0.01UC +3 +4 +2 +0.01 +0.0S +30'09 + + +Figure S3. Comparisons between parallel and perpendicular Raman spectra of (a) C2H6 +and (b) 2-methylbutane simulated from PACFs by the DFT method, the AlphaML and +the BPM. The latter two Raman spectra are shifted upward. The training data set for +AlphaML includes polarizabilities of 8 alkanes at equilibrium, derivatives of +polarizability associated with a bond stretch and polarizabilities of 40 structures from +MD simulations. + + + +BbW +6 +p +BbW +CsHeDEI +DEI +IMsdqIADEL +IMsnqA +IMsdqlA +BbW +BbWm (cw.) +m (cw.) +0001 +S000 +S200 +3000 +3200 +4000 +0001 +S000 +3000 +3200 +400010 + +References +[1] Smirnov, K. S.; Bougeard, D. Quantum-Chemical Derivation of Electro-Optical +Parameters for Alkanes. Journal of Raman Spectroscopy 2006, 37 (1–3), 100–107. +[2] Chen, Q.; Milner, S. T. Predicting Raman Spectra of Condensed Polymer Phases +from MD Simulations. Macromolecules 2017, 50 (24), 9773–9787. +[3] Martín, J.; Montero, S. Raman Intensities of Ethane and Deuterated Derivatives. +The Journal of Chemical Physics 1984, 80 (10), 4610–4619. +[4] Gussoni, M.; Abbate, S.; Zerbi, G. Raman Intensities: Transferability of +Electrooptical Parameters. Journal of Raman Spectroscopy 1977, 6 (6), 289–298. +[5] Snyder, R. G. A Bond Polarizability Interpretation of the Raman Intensities of +Cyclohexane and Cyclohexane-D12. Journal of Molecular Spectroscopy 1970, 36 (2), +204–221. +[6] Bougeard, D.; Smirnov, K. S. Calculation of Off-Resonance Raman Scattering +Intensities with Parametric Models. Journal of Raman Spectroscopy 2009, 40 (12), +1704–1719. + + diff --git a/OtFRT4oBgHgl3EQfHzf5/content/tmp_files/load_file.txt b/OtFRT4oBgHgl3EQfHzf5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9852b9758ac7b364ae6aefb2935651c6c8279de --- /dev/null +++ b/OtFRT4oBgHgl3EQfHzf5/content/tmp_files/load_file.txt @@ -0,0 +1,1268 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf,len=1267 +page_content='1 A study of simulating Raman spectra for alkanes with a machine learning-based polarizability model Mandi Fang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='b Shi Tang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='a Zheyong Fan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='c Yao Shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='b Nan Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='* Yi He,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='* a Institute of Zhejiang University-Quzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Quzhou 324000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' China b College of Chemical and Biological Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Hangzhou 310027,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' China c College of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Bohai University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Jinzhou 121013,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' China d Department of Chemical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Seattle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' WA 98195,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' USA 2 Abstract Polarizability is closely related to many fundamental characteristics of molecular systems and plays an indispensable role in simulating the Raman spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' However, the calculations of polarizability for large systems still suffers from the limitations of processing ability of the quantum mechanical (QM) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' This work assessed and compared the accuracy of the bond polarizability model (BPM) and a ML-based atomic polarizability model (AlphaML) in predicting polarizability of alkanes and then also investigated the ability of simulating Raman spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' We found that the AlphaML has appreciable advantages over the BPM in learning the polarizability in the training data set and predicting polarizability of molecules that configurational differently from training structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' In addition, the BPM has inherent disadvantages in predicting polarizability anisotropy due to many factors including large uncertainties of estimating bond anisotropy, omitting of off-diagonal parameters in the construction of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' As a result, the BPM has larger errors than the AlphaML in the simulation of anisotropic Raman scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Finally, we demonstrated that both the BPM and AlphaML suffer from transference to alkanes larger than those used in the training data sets, but the problem for the AlphaML can be circumvented by exploring more proper training structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Introduction Polarizability describes the tendency of an atom or molecule to adjust its electron cloud in response to an external electric field,1,2 and therefore plays a vital role in understanding nonlinear optics and the Raman scattering3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' It is also an indispensable ingredient in the development of next-generation polarizable force fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='5,6 Technically, polarizability can be calculated using the quantum mechanical (QM) methods such as the density functional theory (DFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='7,8 However, the computational cost of DFT calculations is proportional to the cube of system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The small-system- size limitations hinder the applications of the DFT method in calculating polarizabilities of systems with more than hundreds of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='9–11 It is possible to circumvent the problem by developing parametric polarizability models, which have a linear scaling of the computational effort with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='12 The bond polarizability model (BPM) is a frequently-used parametric polarizability model, in which the molecular polarizability is treated as the sum of polarizabilities of all chemical bonds in the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='11,13,14 Bougeard and coworkers13 parameterized a BPM to predict polarizabilities for alkanes using molecular polarizabilities calculated by the DFT method as training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The results demonstrate that the trained BPM learn polarizabilities in the data set well and enables transfers to molecules larger than those in the training data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Based on this line of thought, Milner and coworkers14 retrained a BPM for alkanes, and transferred the model to polyethylene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' By combining the BPM with molecular dynamics (MD) simulations, the authors simulated Raman spectra of polyethylene and successfully described the difference between Raman peaks of 4 polyethylene at crystalline and molten phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' In spite of successfully describing the difference in Raman peaks for polyethylene, the BPMs still have large errors when predicting the polarizabilities of alkanes,11,13,14 which seriously questions the quality of the simulated Raman spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The preset functions to map from bonds’ attributes (lengths, orientations etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=') to their polarizabilities may limit the ability of the BPM to describe polarizability under complex chemical environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Machine learning-based polarizability models are considered as an alternative parametric model for predicting molecular polarizability with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Typical packages for this purpose include the AlphaML15, embedded atom neural network (EANN)16 and deep neural network (DNN)17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' In these models, chemical environment of each atom within a cutoff is encoded into mathematical descriptors, and then the descriptors are inputted into a ML model to give a polarizability tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Due to the specially designed descriptors and the powerful fitting capability of ML, these models are able to retain the accuracy of the underlying QM methods but can predict molecular polarizability several orders of magnitude faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='12,15,18 Alkanes are a class of compounds which are extensively studied in the development of the BPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='13,19–21 Two reasons may contribute to the popularity of alkanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Firstly, alkanes have a large quantity of spectroscopic data available which serve as experimental references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Secondly, alkanes and the corresponding polymers (polyolefin) are well suitable to the study on the transferability of the BPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='11,22 Although the machine learning-based polarizability models have achieved great success in predicting 5 polarizabilities of waters and organic compounds15,16,18,23, few researches focus on the prediction of polarizability, transferability of models and simulation of Raman spectra for alkanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The aim of the present work is to assess a ML-based polarizability model in terms of the accuracy of predicting molecular polarizability and simulating Raman spectra for alkanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' We also trained a zero-order BPM for comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Polarizabilities of alkanes containing no more than 5 carbons were used as the training data set and testing data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The performance of the ML-based polarizability model on predicting polarizabilities were evaluated, and then compared with the results produced by the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' In addition, the Raman spectra of ethane and 2-methylbutane were simulated by combining the polarizability models with molecular dynamics (MD) simulations, and the spectra were then compared with those simulated by the combination of the DFT method and MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' We further investigated the transferability of the AlphaML model and the BPM by using larger alkanes containing no more than 11 carbons as a testing data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Simulation Details 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1 DFT calculations Geometries of alkanes were optimized using the B3LYP exchange-correlation functional24 and the 6-31G(d) basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='25 Dispersion corrections were considered with the DFT-D3 method with Becke-Johnson damping (D3BJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='26,27 Based on the optimized structures, molecular polarizabilities were calculated by solving the CPHF equations using the B3LYP functional and the 6-311++G(2d,2p) basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='28 All DFT calculations 6 were carried out using the ORCA-v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0 package29,30 with the RIJCOSX numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='31,32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='2 Preparations of data sets The initial training data set consists of methane, ethane, propane, n-butane, n-pentane and 2-methylbutane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' For n-butane and n-pentane, both the trans and gauche conformations were included, therefore, the initial training data set contains 8 structures in total, which is the same to the training data set for BPM in the work of Bougeard and coworkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='13 An additional training data set containing 95 structures was constructed by stretching the individual C-C and C-H bonds in the data set by ∆푟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='01 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Molecular polarizability of each structure was calculated using the method introduced in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' A testing data set of non-equilibrium molecules was constructed to evaluate the accuracy of the polarizability models in predicting polarizability of molecules that are far away from equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Using structures of 8 alkanes at equilibrium as the initial structures, MD simulations at 300 K were performed to obtain non-equilibrium structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The temperature was controlled using the Berendsen thermostat algorisms with the NVT ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='33 Meanwhile, all C-H bonds were constrained with the SHAKE algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='34 The xTB package35 and the GFN2-xTB36 force field were used for all MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Each simulation has a duration of 5 ps with a time step of 1 fs, and trajectories were saved every 1 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' In this way, 40 structures were collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Examination of transferability was performed for alkanes containing no more than 11 carbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' These structures were firstly generated from SMILES strings in the GDB- 7 11 database37 using the RDKit package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='38 Then, geometry optimizations were performed with the DFT method followed by calculations of molecular polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='3 Principle of the ML-based polarizability model In the present work, we chose the AlphaML15 as a representative of the ML-based polarizability models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The AlphaML is based on a symmetry-adapted Gaussian process regression (SA-GPR) scheme, which is designed for the predictions of tensorial properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='23 We shall introduce the component-wise GPR firstly, which is a simplification of the SA-GPR scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='39 In this scheme, each individual polarizability component 훼\uffff\uffff in the Cartesian frame (푝, 푞 = 푥, 푦, 푧) reads 훼\uffff\uffff(ℬ) = 훼\uffff\uffff\uffff \uffff\uffff\uffff + ∑ 푤\uffff \uffff\uffff푘\uffffℬ�,�풜\uffff\uffff \uffff \uffff\uffff\uffff (1) where 푁 is the number of configurations in the training data set, 훼\uffff\uffff\uffff \uffff\uffff\uffff is the average of the polarizability component 훼\uffff\uffff over the training data set, 푤\uffff \uffff\uffff are the weights, and 푘 is a kernel function that measure the similarity between the target system ℬ and a training structure 풜\uffff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The kernel function commonly used in the GPR is based on a Gaussian similarity 푘\uffffℬ�,�풜\uffff\uffff = exp \uffff−�\uffff퐮�(ℬ)�−�퐮�\uffff풜\uffff\uffff\uffff \uffff \uffff\uffff\uffff \uffff (2) where 휎 is the Gaussian width, and 퐮(⋯ ) is a function that maps the atomic coordinates to a high-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The mapping function 퐮(⋯ ) determines the accuracy and efficiency of the GPR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' It is often constructed using the atomic density representation, which builds a Cartesian reference frame centered on an atom and defines a three-dimensional grid around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' At each grid-point 퐫, the atomic density distribution is calculated in the form 8 휌\uffff(퐫) = ∑ exp \uffff−�\uffff퐫�−�퐫\uffff\uffff \uffff \uffff\uffff\uffff\uffff \uffff \uffff∈\uffff (3) where 푠 identifies an atom type, 퐫\uffff is the coordinate of the central atom, and 훾\uffff is a smearing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 퐮(⋯ ) is given by the set {휌\uffff(퐫), 푠 = 1, ⋯ , 푁\uffff}, where 푁\uffff is the number of atomic species in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The polarizability tensor predicted in this manner is not invariant to rotations in Cartesian space, and therefore the atomic density that uses a Cartesian space representation requires an alignment to a reference structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='39 The AlphaML learns the polarizability using a combination of GPR and SA-GPR scheme23,39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Naturally, the polarizability 훂 is a symmetric rank-2 tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Before fitting, 훂 is decomposed into a scalar component 훼(\uffff) = \uffff훼\uffff\uffff�+�훼\uffff\uffff�+�훼\uffff\uffff\uffff/√3 and a tensorial component 훼(\uffff) = √2 \uffff훼\uffff\uffff�,�훼\uffff\uffff�,�훼\uffff\uffff�,�\uffff\uffff\uffff\uffff\uffff\uffff\uffff\uffff\uffff\uffff\uffff\uffff \uffff√\uffff �,�\uffff\uffff\uffff\uffff\uffff\uffff\uffff \uffff \uffff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='15 The former is fitted using the GPR scheme while the latter is fitted using SA-GPR scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Within the SA-GPR scheme, a tensorial generalization of the smooth overlap of atomic position kernel (λ-SOAP) are employed, which uses the covariant integration of the atomic density in the mapping functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' By using the λ-SOAP for the tensorial component, the AlphaML gets rid of the alignment for the atomic density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Detailed descriptions of the SA-GPR scheme and the λ-SOAP kernels can be found elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='23,39 The training of the AlphaML model is equivalent to the selection of representative reference environments and the determination of the corresponding weights 푤\uffff \uffff\uffff , which is done by minimizing a loss function defining the deviations of predicted polarizabilities from those given in the training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='15,39 For 훼(\uffff), 8 radial functions, 6 angular functions, an environment cutoff of 5 Å and a Gaussian width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='25 Å 9 were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' For 훼(\uffff) , the hyperparameters are the same as those for the scalar component except that a Gaussian width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='35 Å was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='4 Simulations of Raman spectra The isotropic and anisotropic Raman scattering are closely related to the isotropic polarizability 훼\uffff = \uffff훼\uffff\uffff�+�훼\uffff\uffff�+�훼\uffff\uffff\uffff 3 ⁄ and the anisotropic tensor 훂\uffff = 훂 − 훼\uffff퐈 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Neglecting the nuclear quantum effects, the differential cross section of isotropic and anisotropic Raman scattering can be written in terms of the Fourier transform of the polarizability autocorrelation function (PACF)4,11 \uffff \uffff\uffff\uffff \uffff\uffff\uffff\uffff\uffff \uffff\uffff\uffff = \uffff \uffff\uffff ∫ d푡푒\uffff\uffff\uffff\uffff〈훼\uffff�(0)�훼\uffff�(푡)〉 \uffff\uffff \uffff\uffff (4) \uffff \uffff\uffff\uffff \uffff\uffff\uffff\uffff\uffff \uffff\uffff\uffff\uffff\uffff = \uffff \uffff\uffff ∫ d푡푒\uffff\uffff\uffff\uffff〈Tr[훂\uffff(0)훂\uffff(푡)]〉 \uffff\uffff \uffff\uffff (5) where 휔 is the Raman frequency shift, 훺 is the solid angle range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Tr indicates the trace and 〈⋯ 〉 indicates an ensemble average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The parallel and perpendicular spectra that are comparable to the experimental polarized Raman spectra can be obtained by4,11,18 퐼∥(휔) = \uffff \uffff\uffff\uffff \uffff\uffff\uffff\uffff\uffff \uffff\uffff\uffff + \uffff \uffff\uffff \uffff \uffff\uffff\uffff \uffff\uffff\uffff\uffff\uffff \uffff\uffff\uffff\uffff\uffff (6) 퐼\uffff(휔) = \uffff \uffff\uffff \uffff \uffff\uffff\uffff \uffff\uffff\uffff\uffff\uffff \uffff\uffff\uffff\uffff\uffff (7) It is worth mentioning that the simulated spectra only record the line shapes of Raman spectra18, which is independent on the wavelength of the incident light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' To simulate Raman spectra for ethane and 2-methylbutane, we performed a MD simulation with the GFN2-xTB36 force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The simulation has a duration of 25 ps with a time step of 1 fs and trajectories were saved every 2 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Polarizabilities of structures sampled from the MD simulation were calculated by the DFT method, the 10 BPM and the AlphaML model, respectively, and the corresponding PACF along the evolution of trajectories were then obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Finally, we simulated the parallel and perpendicular spectra according to the formulae in Equation 4~7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The Fourier transforms of PACF in Equations 4 and 5 were simplified with the use of the discrete cosine transform (DCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='40 A combination of a sampling interval of 2 fs and a lag time of 10 ps in DCT produces an increment of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='67 cm−1 in frequency, which can capture the highest frequency mode of CH bending with a period of about 23 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Result and discussion In the very beginning, we trained the AlphaML model and the zero-order BPM with the initial training data set containing 8 alkanes at equilibrium and 95 structures with a stretched C-C or C-H bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' We demonstrated the training process and the fitting quality of the AlphaML in the following section, while presenting the training process of the BPM in the supporting material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' In the second step, comparisons were performed between the BPM and the AlphaML model in terms of the accuracy of predicting polarizabilities for molecules that are far away from equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Subsequently, Raman spectra of ethane and 2-methylbutane were simulated by integrating the AlphaML model, the BPM or the DFT method with MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The relationship between the accuracy of polarizability prediction and the quality of simulated Raman spectra were demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Finally, we investigate the transferability of the AlphaML model and the BPM to alkanes larger than those in the training data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1 Training of the AlphaML model We used the training data set containing a total of 103 structures to train the AlphaML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Figure 1a shows explicitly scatter plots of the diagonal and off-diagonal components of polarizabilities calculated by the DFT method and predicted by the AlphaML model for the 8 alkanes at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The AlphaML model yields coefficient of determination (R2) near 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='00 for both the diagonal and off-diagonal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' As presented in the supporting material, the BPM yields R2 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='997 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='803 for the diagonal and off-diagonal components, indicating that the AlphaML possesses a better learning capacity for the off-diagonal components of polarizabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Meanwhile, we also examined the ability of predicting the derivatives of polarizability associated with a bond stretch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The narrow distributions of the derivatives of polarizability around the 푦 = 푥 line manifest that the AlphaML has enough accuracy to describe the tiny changes of polarizabilities associated with bond stretches, as shown in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The R2 for both the diagonal and off-diagonal components of derivatives of polarizability are greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='993, much higher than the corresponding R2 produced by the BPM, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The result echoes the trend that AlphaML has advantage over the BPM in learning the polarizability of molecules in the condition of identical training data sets, which are consistent with what we expected due to the powerful fitting capability of ML12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 12 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' (a) Correlations between diagonal components of polarizabilities of 8 alkanes at equilibrium calculated by the DFT method and predicted by the AlphaML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Correlations for off-diagonal components are shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' (b) Correlations between diagonal components of the derivatives of polarizability associated with a bond stretch calculated by the DFT method and predicted by the AlphaML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Correlations for off-diagonal components are shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The R2 for the diagonal and off-diagonal components of derivatives of polarizability associated with a bond stretch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' R2 (diagonal/off diagonal) Model C C stretch C H stretch BPM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='895/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='879 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='842/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='909 AlphaML 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='996/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='995/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='997 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='S Q00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='S-αDEI (g\'n) αDEI (g\'n") 20-5\'2 0\'0 \'2 2\'0 1\'?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=" -4'0-S'0 0'0 S'0 4'0J0 SO 30 40 eo 10 08 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='Sr 1O08 JS\'2EαDEI (g\'n") αDEI (\'n")10 C-H pouq 2flefcμ 口、 C-C pouq 2flefcμ9 p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1Q10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content="2 40 '0 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='113 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='2 Performance on predicting polarizability of non-equilibrium structures The AlphaML model exhibits an extraordinary ability in learning polarizabilities and derivatives of polarizability from the training data set containing structures at equilibrium and structures with small bond stretches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' A benchmark testing was performed to the AlphaML model and the BPM to evaluate the accuracy in predicting polarizability of structures that are far away from equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Figure 2a shows that both the BPM and the AlphaML model can predict polarizabilities of the non-equilibrium structures in agreement with the DFT results, although the BPM has larger errors in predicting the off-diagonal components than the AlphaML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' This trend is consistent with the fact that the AlphaML model can learn the off-diagonal components of polarizabilities of 8 alkanes at equilibrium better than the BPM as presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' To rule out the effects of rotation operations on molecular polarizabilities, here, we introduced the rotation-invariant isotropic polarizability (훼\uffff) and the rotation-invariant polarizability anisotropy41 (∆훼 = \uffff\uffff\uffff훼\uffff\uffff�−�훼\uffff\uffff\uffff \uffff�+�\uffff훼\uffff\uffff�−�훼\uffff\uffff\uffff \uffff�+�(훼\uffff\uffff�−�훼\uffff\uffff)\uffff�+�6�\uffff훼\uffff\uffff \uffff �+�훼\uffff\uffff \uffff �+�훼\uffff\uffff \uffff \uffff\uffff 2 ⁄ ) to adequately measure the deviations of polarizability predicted by the AlphaML model and the BPM from the corresponding polarizabilities calculated by the DFT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 14 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' (a) Correlations between diagonal components of polarizabilities of non- equilibrium structures calculated by the DFT method and predicted by the AlphaML model and the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Correlations for off-diagonal components are shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' (b) Correlations between isotropic polarizability (훼\uffff) calculated by the DFT method and predicted by the AlphaML model and the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' (c) Correlations between polarizability anisotropy (∆훼) calculated by the DFT method and predicted by the AlphaML model and the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' (d) RMSEs of isotropic polarizability (RMSE\uffff\uffff\uffff) and polarizability anisotropy (RMSE\uffff\uffff\uffff\uffff\uffff) produced by the AlphaML model and the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0S CH 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='5- ableqicr CSHe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='08 (gn) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='s 8H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0 g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content="2D (g') αDEI (g'n') J0'0 50'0 30'0 40'0 20'0 0'0 10'0 80'0 10'0 50'0 30'0 40'0 20'0 e0'0 10'0 80'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content="01 JO'C αDEI (g') ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='CH4BbW BbW JMsdqIA IMsdqIA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='Stoibg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='07 ■ 0.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0815 Both the BPM and the AlphaML can predict 훼\uffff as good as the DFT method, as shown in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The results agree with the trend for the diagonal components of polarizabilities shown in Figure 2a well, as 훼\uffff is the average of the three diagonal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' It is interesting that data in Figure 2b are grouped by their chemical formula of molecules, namely CH4, C2H6, C3H8, C4H10 and C5H12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Moreover, the groups of data distribute around the 푦 = 푥 line almost uniformly, which implies that 훼\uffff may increase linearly with number of carbons in the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' This linear relationship is confirmed in Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' In other words, the additive assumption of polarizabilities is roughly valid for the average of diagonal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Both the BPM and the AlphaML model assume that the molecular polarizabilities are additive,11,15 thus, the high consistency between the predicted 훼\uffff and 훼\uffff calculated by the DFT method is not surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The AlphaML can predict ∆훼 much better than the BPM for non-equilibrium structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Figure 2c clearly shows that the points of ∆훼 predicted by the BPM are far away from the 푦 = 푥 line, indicating rather higher prediction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' This trend is confirmed from high RMSEs of ∆훼 shown in Figure 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' On the contrary, ∆훼 predicted by the AlphaML model agree well with DFT values, which is validated by the low RMSEs of ∆훼 shown in Figure 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The points of ∆훼 predicted by the BPM model are well correlated with those calculated by the DFT method, also having a trend of slight downward offset, as shown in Figure 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The effect of the overall offset on the quality of Raman spectra will be demonstrated in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' By the way, 16 the reasons for the high RMSEs of ∆훼 given by the BPM is deeply discussed in the supporting material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='3 Performance on predicting Raman Spectra In the previous section, we have demonstrated the performance of the AlphaML model on predicting the isotropic polarizability and the polarizability anisotropy for non-equilibrium alkanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The AlphaML model is proven to have better accuracy than the BPM in the condition of identical training data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' In this section, we will evaluate the quality of simulated Raman spectra for ethane and 2-methylbutane and interpret the relationship between the accuracy of polarizability prediction and quality of simulated Raman spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' We calculated the time autocorrelation function of isotropic polarizability ( PACF\uffff\uffff\uffff(푡) = 〈훼\uffff�(0)�훼\uffff�(푡)〉 ) and anisotropic polarizability tensor ( PACF\uffff\uffff\uffff\uffff\uffff(푡) = 〈Tr[훂\uffff(0)훂\uffff(푡)]〉) for ethane using the AlphaML model and the BPM, and compared the time autocorrelation function with the corresponding values calculated by the DFT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Figure 3a shows that the wave shape and amplitude of PACFiso predicted by the AlphaML model and the BPM are close to that calculated by the DFT method, which agrees with the fact that both the AlphaML model and the BPM can predict the 훼\uffff for ethane as good as the DFT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The PACFaniso predicted by the AlphaML model also resembles the PACFaniso calculated by the DFT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' However, the PACFaniso predicted by the BPM has distinctive differences from the PACFaniso calculated by the DFT method in terms of the wave shape and period of change, as shown in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Since the Raman scattering intensity is proportional to the Fourier 17 transform of the PACF,4,11 the obvious change of the wave shape and period may cause the difference in the shifts and the intensity of peaks of the simulated Raman spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Comparison between (a) PACFiso and (b) PACFaniso for ethane calculated by the DFT method and predicted by the AlphaML model and the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The Raman spectra of ethane simulated from PACFs predicted by the AlphaML model are consistent with that simulated from PACFs calculated by the DFT method across a wide range of wavenumbers, as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The Raman spectra simulated from PACFs by the DFT method, the AlphaML model and the BPM can successfully reproduce an array of features reported in experiments21,42, with coincident Raman shifts and approximately close intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' These features include the C-C stretching peak at about 1057 cm-1, CH3 stretching peaks at about 3054 and 3067 cm-1, as shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 0 S 4 e 8 10 0 S 4 8 1O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content="7 (2q) 9miT (q) 9miTDEL DEH J'OE2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0IMsdalA IMsdalA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='7 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='70 U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0 BbW BbW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content="0 0'02." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0 18 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Comparisons between parallel and perpendicular Raman spectra of C2H6 simulated from PACFs by DFT, the AlphaML and the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The latter two Raman spectra were shifted upward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Wavenumber of the Raman Spectra of C2H6 from experiments and simulated from PACFs calculated by DFT (cm-1) skeletal mode description experiment DFT CH3 stretching 2953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='7 3054 CH3 stretching 2968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='7 3067 CH3 deformation 1388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='4 1440 CH3 deformation 1468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1 1502 C-C stretching 994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='8 1057 CH3 rocking 1195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='3 1202 DEI AI: CH3 2{lercuiua L: CH3 2flercuiuaBbWJMsdqIADEI3 1000 00cT 5000 3000 00c8 4000IVV Ⅱ VI IⅡIII: CH3 LoCKIua I : C C 2flercpiuaIMsdqA noitsmioteb sHO :VI BbW I: Ch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' qetoLsiou19 However, both the parallel and perpendicular Raman spectra simulated by the BPM has produced abnormal signal at about 1202 cm-1, inconsistent with the spectra simulated from PACFs by DFT and also inconsistent with the fact that the experimental intensity for the CH rocking peak is extremely weak14,43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' This is due to the large errors in predicting PACFaniso by the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The most significant discrepancy between the Raman spectra simulated from PACFs by the AlphaML and DFT is the relative intensities for the CH3 deformation at about 1440 and 1502 cm-1, as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' We owed the discrepancy to the limited ability of the AlphaML in predicting polarizability of structures that are far away from structures in the training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Although the AlphaML performs much better than the BPM in predicting polarizability anisotropy, as shown in Figure 2c, insufficient samplings44 may still hinder the high-accuracy simulations of Raman spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' An attempt of involving 40 structures from MD simulations into the training data set will produce a right prediction of the relative intensities for the CH3 deformation at about 1440 and 1502 cm-1, as shown in Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Moreover, Figure S3 also demonstrates an excellent consistency between Raman spectra simulated from PACFs by the new trained AlphaML and DFT for 2-methylbutane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='5 Discussions on predicting polarizabilities of larger molecules The additivity assumption for the molecular polarizability endows the AlphaML model and the BPM the ability with the ability of predicting polarizabilities for larger molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' A benchmark testing was performed for alkanes containing no more than 11 carbons to evaluate the extrapolation ability of the AlphaML model and the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' We 20 used the AlphaML model trained with 8 alkanes at equilibrium and additional 95 structures, with no non-equilibrium structures involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' For alkanes containing no more than 5 carbons, which is already included in the training data sets, both the AlphaML and the BPM can predict 훼\uffff and ∆훼 as accurate as the DFT method, as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' When it comes to alkanes containing more than 5 carbons, which is beyond the training data sets, large deviations happen for both the AlphaML model and the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Especially, Figure 5b manifests that the points of ∆훼 predicted by the BPM distributed irregularly and were far away from the y=x line, indicating that the BPM trained with small alkanes has very poor extrapolation ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' On the contrary, the points of ∆훼 predicted by the AlphaML model deviated from the y=x line gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' At the meantime, the deviations produced by the AlphaML model are much lower than those produced by the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' (a) Correlations between 훼\uffff of alkanes containing no more than 11 carbons calculated by the DFT method and predicted by the AlphaML model and the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' (b) Correlations between ∆훼 calculated by the DFT method and predicted by the IMcdalA < opo 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='00 C"HS4VαDEI (g\'n\') αDEI (S\')S 口口 BbW ■口 BbW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content="C°HS0 40'06 300 6 口 8rHg0CH ibg C 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' V10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content="0S C'H/S10." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='02 C4H10H :H0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='01CH40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content="08 40'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='2S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0221 AlphaML model and the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Solid makers stand for alkanes containing no more than 5 carbons, while hollow markers stand for alkanes containing more than 5 carbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The lack of sampling certain local environment is responsible for the large deviations of ∆훼 produced by the AlphaML model for alkanes containing more than 5 carbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' As shown in Figure 6, number of neighbors of the central carbon in n-C5H12 and n- C11H24 within the environment cutoff of 5 Å are 16 and 24, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Therefore, the AlphaML model trained with alkanes containing no more than 5 carbons must be unfamiliar with the local environment of n-C11H24 and thus gives unsatisfied predictions of polarizabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Figure 6 also suggests that a maximum of 9 carbons will be included within environment cutoff of 5 Å for n-alkanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' We inferred that the inclusion of structures containing 9 carbons in the training data set may be beneficial to the transferability of the AlphaML model to larger molecules such as n-C11H24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The consistency between 훼\uffff and ∆훼 calculated by DFT and predicted by the retrained AlphaML model in Figure 7 clearly manifests that the complement of certain local environment with more neighbors will greatly improve the accuracy of predicting 훼\uffff and ∆훼, and thus endow the AlphaML with rational extrapolation ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Having shown the powerful ability of learning and predicting polarizability of the AlphaML model, future work will focus on the accurate prediction of polarizability of large systems such as polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' This work clearly demonstrates the dependence of the accuracy of the AlphaML model on the diversity of the training data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Therefore, to guarantee the successful transference to polymers, sampling those local structures that 22 suffused with atoms is indispensable12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Number of neighbors of the central carbon in n-C5H12 and n-C11H24 within the environment cutoff of 5 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' (a) Correlations between 훼\uffff of alkanes containing no more than 11 carbons calculated by the DFT method and predicted by the retrained AlphaML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' (b) Correlations between ∆훼 calculated by the DFT method and predicted by the retrained AlphaML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Solid makers stand for alkanes containing no more than 9 carbons, while hollow markers stand for alkanes containing more than 9 carbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Conclusion H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content="0240'0 COHSSC°H 3 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='00 HSO加 C Hle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='08 6CH Leq!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='00 C"H/0EUAILoUwGuf cnfoH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0AHO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='2S aHso"0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content="08 40'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0Meappor = Je Meiappo2 = S4VDI (s"n") D (g\'n\')23 In this work, we constructed and compared the zero-order BPM and the ML-based AlphaML model for the prediction of polarizability and simulation of Raman spectra of alkanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' First, the BPM and the AlphaML were trained with polarizability of 8 alkanes at equilibrium together with the derivatives of polarizability associated with a bond stretch, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The accuracy of these two models in the prediction of polarizability for molecules that are far away from equilibrium were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Then the time autocorrelation function of polarizabilities for C2H6 was calculated by these two models and the corresponding Raman spectra were simulated and compared with the Raman spectra by DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Finally, the extrapolation ability of these two models in predicting polarizability of alkanes larger than those in the training data sets were compared, and discussions were made for the AlphaML on the transference to large systems such as polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' We found that the AlphaML has appreciable advantages over the BPM in learning the polarizability and the derivative of polarizability of alkanes using the same training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Both the BPM and AlphaML can appropriately predict the isotropic polarizability for structures that are configurational different from those used in the training data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' However, the BPM has inherent disadvantages in predicting polarizability anisotropy due to many factors including large uncertainties of estimating bond anisotropy, omitting of off-diagonal parameters in the expression of bond polarizability tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' As a result, the BPM has large errors in the simulation of anisotropic Raman scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Finally, we demonstrated that both the BPM and AlphaML suffer from transference to alkanes larger than those used in the training data 24 sets, but the problem for the AlphaML can be circumvented by enhancing samplings properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' AUTHOR INFORMATION Corresponding Author *E-mail: tamas@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' yihezj@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='cn NOTES The authors declare that there is no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' ACKNOWLEDGEMENT This work is supported by the National Key Research and Development Program of China (grant number 2022YFE0106100), and the National Natural Science Foundation of China (grant number 22178299).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Nan Xu would like to thank the financial support provided by the Startup Funds of the Institute of Zhejiang University-Quzhou.' metadata={'source': 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of simulating Raman spectra for alkanes with a machine learning-based polarizability model” Mandi Fang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='b Shi Tang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='a Zheyong Fan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='c Yao Shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='b Nan Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='a,' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' China b College of Chemical and Biological Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Hangzhou 310027,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' China c College of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Bohai University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Jinzhou 121013,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' China d Department of Chemical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Seattle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' WA 98195,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' USA 2 Contents 1 Principle of the bond polarizability model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='. 1 2 Training of the bond polarizability model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 10 3 1 Principle of the bond polarizability model This section presents the bond polarizability model (BPM) formalism briefly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' detailed descriptions can be found elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1 In the BPM, molecular polarizability tensor ( ) is written as the sum of polarizabilities of all bonds (1) where denotes the polarizability of bond in the fixed Cartesian axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Supposed that is the polarizability of bond in its principal axes, can be rewritten as (2) Here is the rotation matrix between the bond’s principal axes and the Cartesian axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The bond’s principal axes were constructed as follows: first, the longitudinal axis is directed along the vector connecting the bonded two atoms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Second, the two transversal axes and are built orthogonal to and to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' in the bond’s principal axes reads (3) where , , are the polarizabilities in the directions of the three principal axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' In this work, we employed the zero-order BPM, which is commonly used in the predictions of polarizabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1,2 The diagonal component of ( , ) is expanded in a Taylor series respect to the bond length and truncated after the second term (4) 4 Here the superscript 0 denotes the equilibrium state and the superscript denotes the derivative of polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' In addition, a cylindrical bond model was used in the zero- order BPM, which assumes that and are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1 Hence, we shall use to replace and in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' All bonds of the same type were described by the same set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The parameters and are known as the equilibrium parameters, which mainly account for the variation of polarizability upon a change of the bond’s orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1 We used the polarizabilities of 8 alkanes at equilibrium as the training data set to determine the four equilibrium parameters , , , and the equilibrium bond length parameters , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The differences between polarizabilities of molecules at equilibrium and molecules with a stretched C-C or C-H bond were then used to determine the derivative parameters , , and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The fittings of , , and were performed with the use of the singular value decomposition method (SVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1 2 Training of the bond polarizability model The equilibrium parameters , , and in the BPM were fitted to be -49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='571, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='788, -27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='361 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='556 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' However, the negative values of and are not reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The C-C and C-H bonds should increase the dielectric response in terms of the enhancement of the applied electric field along the bond’s direction, which indicates that and 5 should have positive values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1,2 Moreover, the calculated bond anisotropy values of C-C and C-H bonds, defined as , are inconsistent with experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1,3,4 Zerbi and coworkers4 reported that the C-H bond anisotropy of methane was estimated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='305 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Montero and coworkers3 reported that of ethane was about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='28 according to the experimental Raman spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The failure of the BPM in predicting the sign of polarizability along the bond’s direction and the bond anisotropy may be due to the lack of intrinsic relations between and in the BPM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Imposing a restrictive condition for the C-H bond anisotropy in the BPM will remedy these shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1 An addition equation defining the C-H bond anisotropy for the CH4 molecule was introduced in the fitting of the equilibrium parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The new fit values of , , and are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='678, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='174, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='777 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='484 , very close to the corresponding values of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='677, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='127, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='779, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='489 in the work of Bougeard and coworkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1 The slight divergences may be due to the difference of the basis sets and QM packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The bond anisotropy and were calculated to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='504 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='293 , close to the experimental of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='28 for ethane and of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='305 for methane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='3,4 The consistency between the polarizability tensors predicted by the BPM and calculated by the DFT method shown in Figure S1a confirms that the trained BPM can predict the polarizability of alkanes at equilibrium with an accuracy close to the DFT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The R2 for the diagonal components is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' for the off-diagonal components, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 6 The best fit values of the derivative parameters , , and are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='881, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='274, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='628 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='393 , in accord with the corresponding values of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='863, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='243, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='743, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='353 in the work of Bougeard and coworkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1 In addition, the bond length parameters and were calculated to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='533 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='097 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Figure S1b demonstrates that the derivatives of polarizability associated with a bond stretch predicted by the BPM are in line with those calculated by the DFT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The R2 for the diagonal components of the derivatives of polarizability associated with a C-C bond stretch is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='895;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' for the off-diagonal components, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='879, while the R2 for the diagonal components of the derivatives of polarizability associated with a C-H bond stretch is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='842;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' for the off-diagonal components, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' 3 Discussion on the large prediction errors of the bond polarizability model Two reasons may contribute to the high RMSEs of polarizability anisotropy ( ) by the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' First, using the same set of parameters for all C-C bonds is not reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The anisotropy of C-C bonds is found to be dependent on the local chemical environment, ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='6 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='3,5 The situation is also the same for C-H bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Second, the BPM has large errors in predicting the off-diagonal components of polarizability tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' This may be originated from the omitting of off-diagonal parameters in the expression of bond polarizability tensors6 and the omitting of second- order and higher-order terms in the Taylor series of parameters respect to the bond length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1,6 However, an attempt to involve more parameters may cause too large statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='6 7 Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' (a) Correlations between diagonal components of polarizabilities of alkanes at equilibrium calculated by the DFT method and predicted by the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Correlations for off-diagonal components are shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' (b) Correlations between diagonal components of the derivatives of polarizability associated with a bond stretch calculated by the DFT method and predicted by the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Correlations for off-diagonal components are shown in the inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' (\'n") αDEI (g\'n")10 C-H pouq 2flefc C-C pouq 2flefcp9 p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='1BO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content="2 40 b 1'230 0." metadata={'source': 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polarizability ( ) of non-equilibrium structures in the testing data set calculated by the DFT method and number of carbons ( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The dashed line shows a fitted function of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content="S) 40'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='00 bgit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='0a DEI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='01UC 3 4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content="0S 30'09 Figure S3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Comparisons between parallel and perpendicular Raman spectra of (a) C2H6 and (b) 2-methylbutane simulated from PACFs by the DFT method, the AlphaML and the BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The latter two Raman spectra are shifted upward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' The training data set for AlphaML includes polarizabilities of 8 alkanes at equilibrium, derivatives of polarizability associated with a bond stretch and polarizabilities of 40 structures from MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' BbW 6 p BbW CsHeDEI DEI IMsdqIADEL IMsnqA IMsdqlA BbW BbWm (cw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=') m (cw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=') 0001 S000 S200 3000 3200 4000 0001 S000 3000 3200 400010 References [1] Smirnov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Calculation of Off-Resonance Raman Scattering Intensities with Parametric Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} +page_content=' Journal of Raman Spectroscopy 2009, 40 (12), 1704–1719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFRT4oBgHgl3EQfHzf5/content/2301.13490v1.pdf'} diff --git a/P9FKT4oBgHgl3EQfhC52/vector_store/index.faiss b/P9FKT4oBgHgl3EQfhC52/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..5c8b274bdf41756e512241345093f0a3c1eb3919 --- /dev/null +++ b/P9FKT4oBgHgl3EQfhC52/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6ee110c89506ef424bd585b68aa363cd0c28371687442cf3a8ca6550f3dc7e7a +size 3342381 diff --git a/P9FKT4oBgHgl3EQfhC52/vector_store/index.pkl 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b/RNFQT4oBgHgl3EQfaTZS/content/tmp_files/2301.13319v1.pdf.txt @@ -0,0 +1,1645 @@ +[Work in progress] Scalable, out-of-the box segmentation of individual +particles from mineral samples acquired with micro CT +Karol Gotkowski∗, Shuvam Gupta, Jose R. A. Godinho, Camila G. S. Tochtrop, Klaus +H. Maier-Hein and Fabian Isensee +aHelmholtz Imaging, German Cancer Research Center, Im Neuenheimer Feld 280, Heidelberg, 69120, Baden-Wuerttemberg, Germany +bDivision of Medical Image Computing, German Cancer Research Center, Im Neuenheimer Feld 280, Heidelberg, 69120, Baden-Wuerttemberg, Germany +cHelmholtz-Zentrum Dresden-Rossendorf, Helmholtz Institute Freiberg for Resource Technology, Chemnitzer Straße +40, Freiberg, 09599, Sachsen, Germany +dPattern Analysis and Learning Group, Department of Radiation Oncology, Heidelberg University Hospital, Im Neuenheimer Feld +672, Heidelberg, 69120, Baden-Wuerttemberg, Germany +A R T I C L E I N F O +Keywords: +individual particle characterization +3D +instance segmentation +deep learning +A B S T R A C T +Minerals are indispensable for a functioning modern society. Yet, their supply is limited causing +a need for optimizing their exploration and extraction both from ores and recyclable materials. +Typically, these processes must be meticulously adapted to the precise properties of the processed +particles, requiring an extensive characterization of their shapes, appearances as well as the overall +material composition. Current approaches perform this analysis based on bulk segmentation and +characterization of particles, and rely on rudimentary postprocessing techniques to separate touching +particles. However, due to their inability to reliably perform this separation as well as the need to +retrain or reconfigure most methods for each new image, these approaches leave untapped potential +to be leveraged. Here, we propose an instance segmentation method that is able to extract individual +particles from large micro CT images taken from mineral samples embedded in an epoxy matrix. +Our approach is based on the powerful nnU-Net framework, introduces a particle size normalization, +makes use of a border-core representation to enable instance segmentation and is trained with a large +dataset containing particles of numerous different materials and minerals. We demonstrate that our +approach can be applied out-of-the box to a large variety of particle types, including materials and +appearances that have not been part of the training set. Thus, no further manual annotations and +retraining are required when applying the method to new mineral samples, enabling substantially +higher scalability of experiments than existing methods. Our code and dataset are made publicly +available. +1. Introduction +Mineral resources are ubiquitous in modern society, yet, +they constitute a finite resource that needs to be acquired +cost-effectively and used responsibly. Our ability to extract +raw materials from the earth’s crust is remarkable, and so are +the recent advances in mineral exploration, ore processing, +reservoir characterization, and the developed technologies to +reuse materials through recycling. Moreover, with the ever- +rising demand for mineral resources [1], these processes +are becoming increasingly automated. Characterization of +mineral particles plays a crucial role in this process by +providing a detailed understanding of the minerals and en- +abling the development of effective automated processes. So +far, bulk characterization of mineral particles is the most +frequently used method for optimizing these processes. Re- +alized through semantic segmentation that segments all par- +ticles without distinguishing between particle instances, it is +either performed through a variation of intensity threshold- +ing Hassan, Airey, Khan and Collop (2012); Becker, Jardine, +∗Corresponding author +karol.gotkowski@dkfz.de (K. Gotkowski); s.gupta@hzdr.de (S. +Gupta); j.godinho@hzdr.de (J.R.A. Godinho); guimar75@hzdr.de (C.G.S. +Tochtrop); klaus.maier-hein@dkfz-heidelberg.de (K.H. Maier-Hein); +f.isensee@dkfz-heidelberg.de (F. Isensee) +ORCID(s): +Miller and Harris (2016); Dominy, Platten, Howard, Elango- +van, Armstrong, Minnitt and Abel (2011); Godinho, Kern, +Renno and Gutzmer (2019) or deep learning approaches +Wang, Li, Xiao, Zhang, Miao and Wang (2021); Filippo, +Gomes, da Costa and Mota (2021); Xiao, Liu, Le, Ji and Sun +(2020); Latif, Bouchard, Maitre, Back and Bédard (2022); +Nie, Zhang and Cao (2022); Liu, Zhang, Liu, Wang and Xia +(2021); Tung, Halim, Wang, Rich, Marjo and Regenauer- +Lieb (2022). The individual particle properties that deter- +mine the behavior of each particle are not accounted for in +this type of analysis, limiting our understanding of the used +minerals and our ability to optimize downstream processes. +A better optimization can be achieved with the characteris- +tics of individual particles Pereira, Frenzel, Khodadadzadeh, +Tolosana-Delgado and Gutzmer (2021), referred to as in- +stance segmentation, in which every particle instance is +segmented and assigned a unique identifier. This has the +advantage that each particle can be further characterized +individually according to its specific shape, size, and particle +histogram[XXX]. Such attempts to characterize individual +particles have been mostly limited to 2D imaging tech- +niques Liu, Zhang, Jing, Wang and Zhao (2020); Baraian, +Kellokumpu, Paaso, Koresaar and Kaartinen (2022); Sun, +Huang, Cheng, Jia, Xiong and Zhang (2022), leading to an +inherent stereological bias that makes the shape and size +quantification of particles unreliable Blannin, Frenzel, Tuşa, +Karol Gotkowski et al.: Preprint submitted to Elsevier +Page 1 of 17 +arXiv:2301.13319v1 [cs.CV] 30 Jan 2023 + +a[Work in progress] Scalable, out-of-the box segmentation of individual particles from mineral samples acquired with micro CT +Birtel, Ivăşcanu, Baker and Gutzmer (2021). Extending in- +dividual particle characterization to 3D is challenging, given +that upwards of ten thousand particles are typically present +in a single image. Even more, the presence of imaging +artifacts as well as highly diverse appearances of particles +and filler materials make this a difficult problem where es- +tablished methods that rely on intensity thresholding Zhou, +Dai, Cheng, Thompson and Leach (2021); Jiang, Shen, Guil- +lard and Einav (2021); Wang, Lin and Miller (2016, 2015); +Guntoro, Ghorbani, Parian, Butcher, Kuva and Rosenkranz +(2021); Godinho, Grilo, Hellmuth and Siddique (2021) may +fail due to their lack of robustness. To solve this problem for +3D instance segmentation more sophisticated deep learning +methods are a necessity Furat, Kirstein, Leißner, Bachmann, +Gutzmer, Peuker and Schmidt (2023); Tang, Da Wang, Niu, +Honeyands, O’Dea, Mostaghimi, Armstrong and Knackstedt +(2023) as they use a multitude of learned features, making +them inherently more robust to these challenges. However, a +current limitation of such approaches is constituted through +touching particles not recognized as separate instances, fal- +sifying any further analysis. Moreover, a common drawback +of all current deep learning approaches used for particle +segmentation is the need to manually segment a subset of +the particles due to the approach’s narrow scope on specific +types of minerals, resulting in a lack of generalization. +To truly pave the way for process optimization based on +individual 3D particle characterization, automated methods +are required with high robustness and generalizability that +can handle diverse sample and particle types out-of-the-box +and thus do not constitute major bottlenecks due to the need +to constantly adapt them or annotate new data. We propose +a fully automated deep learning approach to produce 3D +instance segmentations based on CT images. A large dataset +with a multitude of different materials and minerals is used +for training a nnU-Net in a highly optimized novel training +and inference pipeline ensuring that the inferred instance +segmentations are of excellent quality, robust against arti- +facts, and generalize over a wide variety of materials and +minerals across magnitudes of different particle sizes. Our +framework works out-of-the-box even on completely new +types or combinations of materials and minerals without the +need for additional training data. Moreover, no prior knowl- +edge is required to employ our method. Thus our method +enables a wide range of analyses based on the intrinsic +particle properties measurable based on the inferred instance +segmentations. Our framework and all data are open source +and available under XXX +2. Materials +2.1. Sample preparation +The samples used in our approach consist of dispersed +particles prepared using a standardized procedure (Godinho +et al. 2021a) that uses sugar particles as spacers in the ratio +of 7 g of sugar to 1 g of sample particles. The resin used +is Paladur (Kulzer, Mitsui Chemical Group), a fast-curing +acrylic polymer. The particles of the material together with +the sugar were mixed with methylmethacrylate-copolymer +powder in a mass ratio of 1:1. The methylmethacrylate liquid +resin was added to the solid mixture in a ratio of 3 mL to +10 g. The final paste is left to dry in a tube with a diameter +adequate for the desired voxel size of the scan. Samples +were imaged with a CT scanner (CoreTom from XRE – +Tescan; Ghent, Belgium), while the “XRE – recon” software +(v1.1.0.14, XRE – Tescan, Ghent, Belgium) was used to +reconstruct the 3D images. Scanning conditions differed on a +per-sample basis as to use optimal reconstruction parameters +based on material and mineral compositions. Consequently, +voxel intensities and particle sizes vary between the ob- +tained images. Reconstructed images were processed as 16- +bit images and visualizations were performed using Avizo +software (v9.3.0, Thermo Fisher Scientific, Waltham, MA, +USA) and Dragonfly software (v2021.1, Objects Research +Systems, Montreal, Quebec, Canada). +2.2. Datasets +A diverse set of samples is required in order to train +a model that predicts high-quality instance segmentations +and generalizes well even to materials and minerals not +present in the training distribution. The samples prepared +for our approach cover a wide range of materials, e.g. nat- +ural ores, slags, and recyclable materials like batteries and +crushed electronic devices to fulfill this criterion. For the +development and training of our method, 24 samples were +prepared in total from which we used 41 patches with +either a size of 128x400x400 or 200x400x400 voxels that +had been extracted and annotated following the procedure +outlined in section 3.1. An example of such a prepared and +annotated sample is shown in Figure 1. Model optimization +was performed by running stratified 5-fold cross-validation +on these patches and quantitatively evaluating the result. +For testing the performance of our model, we designed two +test sets reflecting different use cases. The first is an in- +distribution set of in total 8 patches from 8 samples, with +the purpose to evaluate the performance in terms of its +prediction quality on samples that consist of materials and +minerals already seen by the framework during training. The +second is an out-of-distribution set with in total 5 patches +from 5 samples, consisting of materials and minerals entirely +or in their composition unknown to the model to evaluate +the generalization abilities of our method. The datasets are +available online at..... +3. Methodology +Instance segmentation of 3D voxel images is still a +problem mostly solved with classical methods. 3D instance +segmentation models based on deep learning exist [XXX] +but have not been extensively used and evaluated for gen- +eralization and robustness in practice. On the contrary, 3D +semantic segmentation is a well-studied task with a vari- +ety of suitable models proven successful such as U-Nets +and Transformers. Especially nnU-net, a state-of-the-art 3D +semantic segmentation model, has proven its performance, +Karol Gotkowski et al.: Preprint submitted to Elsevier +Page 2 of 17 + +[Work in progress] Scalable, out-of-the box segmentation of individual particles from mineral samples acquired with micro CT +(a) Sample +(b) Sample patch +(c) Patch reference segmentation +Figure 1: Overview of the image Ore1_Zone3_Concentrate and its reference segmentation. +generalization abilities, and robustness on a number of chal- +lenges it has won XXX. However, converting the seman- +tic segmentations predicted by such models into instance +segmentations is challenging as particles tend to have in- +consistent shapes and are often intertwined with each other. +In order to handle such cases, we reformulate the instance +segmentation problem as a semantic segmentation problem +by converting the instances to a border-core representation, +elaborated in 3.2.2, enabling the usage of nnU-Net for 3D +instance segmentation. With this technique at its core, we +propose a new deep learning based method that enables a +fully automated generation of instance segmentations for +individual particle characterization in 3D CT images. The +optimized annotation pipeline developed for this task and +proposed in section 3.1 is used to create training data for +our method due to a lack of openly available reference +instance segmentation data. Subsequently, we use the data +in our training pipeline, as described in section 3.2, to train +a robust model that generalizes well even to unseen types +and compositions of materials and minerals. As a result, +no additional training data or finetuning is required for the +inference process described in section 3.3. +3.1. Reference annotation +3D images of particle dispersions typically contain thou- +sands of individual particles. Annotating entire images, even +with the support of our proposed annotation pipeline, is +too labor-intensive and would yield mostly redundant in- +formation for our method. Thus, utilizing the heterogene- +ity of the images, we take one or several representative +smaller patches from each training sample for annotation. +To enable fast and precise annotations the interactive 3D +viewer Napari Sofroniew, Lambert, Evans, Nunez-Iglesias, +Bokota, Winston, Peña-Castellanos, Yamauchi, Bussonnier, +Pop et al. (2022) is used with multiple plugins to guide +the annotation process. A guide for the annotation process +is available at... For each patch, a semantic segmentation +of the particles is performed through the use of a Random +Forest voxel classifier that is trained with scribble anno- +tations. Random Forest classifiers operate on a broad set +of handcrafted image features such as intensity gradients, +image intensities, and gradient orientations making them +more robust against CT imaging artifacts. The segmentation +process is iteratively repeated with more refined scribbles +until the result is satisfactory. Once completed, remaining +errors are corrected fully manually. An example of this +random forest segmentation is shown in Figure 2. +The semantic segmentation is then converted into an +instance segmentation by assigning all particles a unique +identifier. Particles touching each other are incorrectly given +only a single identifier in this step, which must subsequently +be corrected. For this, a custom particle-splitting tool devel- +oped by us is utilized. A marker is placed on each particle +and a border is drawn between the particles. The particle- +splitting tool then computes the geodesic distance[XXX] of +each voxel in both particles to the defined 2D border and +separates them by applying a watershed algorithm on the +geodesic distances with the particle markers as seeds for the +algorithm. The result is a well-defined 3D border between +the two particles in most cases requiring only a few seconds +of annotation time. The process of this particle separation is +depicted in Figure 3. +3.2. Training pipeline +The training pipeline, depicted in Figure 4, consists first +of a preprocessing stage to normalize the pixel intensities +and homogenize the particle sizes to cope with the size +changes of multiple orders of magnitude (Section 3.2.1). +Next, a border-core representation, shown to perform well +for large-scale cell tracking Isensee, Jaeger, Kohl, Petersen +and Maier-Hein (2021) and thus suitable for identifying +particles in the ten thousands, is used to map the reference +instance segmentations to semantic segmentations (Section +3.2.2). The border-core representations are then used for +training nnU-Net, which has been proven highly successful +on a multitude of diverse datasets (Section 3.2.3). +3.2.1. Preprocessing +Voxel intensities 퐼 of all images are normalized via +z-scoring. To this end, the global mean intensity 휇 and +standard deviation of intensities 휎 are computed over all +Karol Gotkowski et al.: Preprint submitted to Elsevier +Page 3 of 17 + +· ++ +/[Work in progress] Scalable, out-of-the box segmentation of individual particles from mineral samples acquired with micro CT +(a) Patch +(b) Scribbles +(c) Segmentation +Figure 2: Annotation process of creating a semantic segmentation with Random Forest through iterative refinement with scribbles. +(a) A patch is extracted from the image; (b) Scribbles are drawn in an iterative process to create a semantic segmentation through +use of the Random Forest; (c) A semantic segmentation of the particles has successfully been created. +(a) Patch +(b) Scribbles +(c) Segmentation +Figure 3: Annotation process of separating touching particles in order to create an instance segmentation. (a) Two touching +particles are identified; (b) A marker is placed on each particle and a border is defined with our particle-splitting tool; (c) The +two particles are successfully separated and each is assigned a unique identifier. +training images. Then, each voxel intensity is normalized +퐼푛표푟푚 with the following Equation 1. +퐼푛표푟푚 = 퐼 − 휎 +휇 +(1) +Particle sizes typically vary substantially, from XXX +to YYY micrometers. At the same time, CT images are +acquired at varying voxel sizes. Consequently, the size of +particles in voxels varies substantially between images. We +hypothesize that such a broad size distribution hampers the +training of the segmentation model, causing reduced perfor- +mance and reduced robustness. To counteract this problem, +we strive to homogenize the particle size distribution in our +training data. To this end, the original particle size of a +sample is determined by measuring the diameter of a particle +that is about average size within the sample. No sophisticated +methods are required for this as the particle size only needs +to be roughly correct. All training patches are then resized +to a common average patch size. +Figure 4: The training pipeline of our method. +Karol Gotkowski et al.: Preprint submitted to Elsevier +Page 4 of 17 + +Intensity +Particle Size +Border-Core +Training +Sample +Normalization +Normalization +Conversion +(Section 3.2.3) +(Section 3.2.1) +(Section 3.2.1) +(Section 3.2.2)[Work in progress] Scalable, out-of-the box segmentation of individual particles from mineral samples acquired with micro CT +(a) Patch +(b) Instance segmentation +(c) Border-Core representation +Figure 5: The conversion process of a patch (a) from its instance segmentation (b) into its corresponding border-core representation +(c). +3.2.2. Border-core representation +A border-core representation enables any semantic seg- +mentation model to be used for instance segmentation. In +contrast to ad-hoc conversion methods such as watershed +which often fail to correctly separate instances, a border- +core representation is more advantageous due to the intrin- +sic properties of the representation which are subsequently +learned by the model during training. An example of this rep- +resentation is shown in Figure 5. Conversion to the border- +core representation is achieved by eroding every instance +with a fixed width in voxels (see section 4.2) referred to +as border thickness to generate a core, while the eroded +area represents the border of a particle. As a result, the +intrinsic properties of this representation are that every +particle has the same border thickness and that no core of an +instance is connected to a core of another instance. Training +a model with such a representation enables the model to +learn these properties and replicate them during inference. +Such predicted border-core representation can then be easily +converted back into instance segmentations by assigning +each core a unique identifier and dilating the cores back to +the original particle shape as defined by the border. +3.2.3. Model training +We use nnU-Net Isensee et al. (2021) as the semantic +segmentation model in our training and inference pipeline. +nnU-Net is a self-adapting 3D U-Net that creates a finger- +print of relevant properties from a dataset and automat- +ically adapts important training hyperparameters accord- +ingly. Originally developed for 3D biomedical data and be- +coming a state-of-the-art model on multiple medical bench- +mark datasets XXX, nnU-Net has also shown its efficacy in +other domains such as XXX, XXX, and XXX. Its consistent +performance across multiple domains makes it an ideal fit for +the task of mineral particle segmentation in high-resolution +CT images. No modifications have been made to nnU- +Net, except for a touching particle augmentation, which is +detailed in the following. +To a large degree nnU-Net is able to learn on its own the +intrinsic properties of a border-core representation based on +the already existing touching particles in the training data. +However, the number of touching particles in the training +data used in our method is too small for nnU-Net to fully +prevent the miss-prediction of touching particles during in- +ference. This leads to some touching particles being merged +into a single particle. To reduce this error and to further im- +prove the nnU-Net training, we introduce a touching particle +augmentation. This augmentation is applied on every image +patch in a batch during a training iteration with a certain +probability. It selects a random particle within the current +patch (particle P) and a random particle taken from the entire +training dataset (particle D) and copies particle D next to +particle P such that their particle edges touch each other. This +way, nnU-Net learns over time a better understanding of how +to correctly identify touching particles as separate instances +and consequently predict a better border-core representation. +3.3. Inference pipeline +Inference on a new image is performed by first nor- +malizing its voxel intensities (Section 3.2.1). The actual +inference is done in a chunk-patch-based sliding window +approach as the entire sample is too large to store in GPU +memory (Section 3.3.1). For each patch, the particle size +normalization (Section 3.2.1) is performed on-the-fly and +the patch is subsequently passed to the model for predicting +the border-core segmentation (Section 3.2.2). Once a patch +is predicted, its local patch prediction is inserted into its +original position in the global prediction of the sample, +aggregating all patches during the inference process into a +final prediction. This prediction is at last converted into an +instance segmentation and resampled back to its original +particle size, concluding the inference process. +3.3.1. Chunk sampling and aggregation +A sliding window approach is used during inference as +the entire sample is too large to store in GPU memory. In +order to still achieve the best possible quality, it has proven +successful to use a patch overlap when using a sliding win- +dow approach. This means that except for the image edges, +every voxel in the sample is predicted multiple times by the +model, and the resulting predictions for a voxel are averaged. +The model predicts class probabilities for every voxel in a +patch and after the sliding window has reached the end of +Karol Gotkowski et al.: Preprint submitted to Elsevier +Page 5 of 17 + +一[Work in progress] Scalable, out-of-the box segmentation of individual particles from mineral samples acquired with micro CT +Figure 6: The inference pipeline of our method. +the sample, the predicted class probabilities of the sample +are converted into the border-core segmentation. However, +this approach reaches technical limitations for large images +as in our case as the averaged patch predictions of every +patch depend on all surrounding patches. Consequently, +all predicted patches are required to be stored in memory, +resulting in infeasibly large memory consumption of often +more than 100 GB. To solve this issue, we developed a +chunk grid sampling and aggregation strategy as depicted +in Figure 4. The sample is subdivided into multiple chunks. +Each chunk has an overlap with the neighboring chunks of +exactly one patch, ensuring that voxels at the edges of a +chunk are also predicted multiple times through the patch +overlap such that the prediction quality does not decrease at +the chunk edges. A prediction is inferred for each chunk with +the sliding window approach and subsequently saved to disk. +This enables a memory-efficient inference of large images +without compromising segmentation quality and adds only +a minimal inference time overhead. +4. Experimental Setup +4.1. Evaluation metrics +The quality of the inferred instance segmentations is +quantitatively measured with multiple metrics and can be +divided into the categories of segmentation quality metrics +and separation quality metrics. Segmentation quality metrics +used are Dice, a measure for the quality of the foreground- +background segmentation (Equation 2), the Object Level +F1, a measure for the correct identification of particles as +individual instances (Equation 3), and the Instance Dice, a +measure for the quality of the particle instances (Equation +4). Here, true positives, false positives, true negatives and +false negatives are denoted as 푇 푃 , 퐹푃 , 푇 푁 and 퐹푁, +respectively. +퐷푖푐푒 = +2푇 푃 +2푇 푃 + 퐹푃 + 퐹푁 +(2) +푂푏푗푒푐푡퐿푒푣푒푙퐹1 = ... +(3) +퐼푛푠푡푎푛푐푒퐷푖푐푒 = .. +(4) +Separation quality metrics used are the Merger Ratio, a +measure of the ratio of Mergers to the absolute number of +particles or touching particles (Equation 5), and the Splitter +Ratio, a measure of the ratio of Splitters to the absolute +number of particles (Equation 6). A Merger is defined as +two or more touching particles incorrectly predicted as a +Karol Gotkowski et al.: Preprint submitted to Elsevier +Page 6 of 17 + +Border-Core +Sample +Conversion +(Section XXX) +(Section XXX) +Intensity +Particle Size +Segmentation +Normalization +Denormalization +Model +(Section XXX) +(Section XXX) +(Section XXX) +Particle Size +Prediction +Chunk Sampling +Chunk Aggregation +Normalizatior +(Section XXX) +(Section XXX) +(Section XXX) +(Section XXX)[Work in progress] Scalable, out-of-the box segmentation of individual particles from mineral samples acquired with micro CT +Figure 7: The chunk grid sampler used in our inference pipeline. +single particle and a Splitter as a single particle incorrectly +predicted as two or more particles. +푀푒푟푔푒푟푅푎푡푖표 = ... +(5) +푆푝푙푖푡푡푒푟푅푎푡푖표 = ... +(6) +4.2. Training configuration +The training of the nnU-Net is done in PyTorch with +SGD optimizer, a learning rate of 1e-2, a weight decay of +3e-5, a momentum of 0.99 and 1000 epochs of training time. +A target particle size of 60 voxels is used in the particle +normalization stage. In the border-core to instance conver- +sion stage, the small core removal filter uses a minimum +distance of 1 with a threshold of 0.95, and in the final +postprocessing stage, the small particle filter uses a relative +minimum particle size of 0.005. +4.3. Comparison with traditional particle +segmentation +We intend to compare our method against other fre- +quently used methods quantitatively and qualitatively in +section 5. Consequently, we reviewed common segmenta- +tion and postprocessing strategies used to create instance +segmentations in order to derive the standardized instance +segmentation method ThreshWater, which is described in +the following section 4.3.1. +4.3.1. ThreshWater +Individual particle characterization via instance segmen- +tation is mostly performed through thresholding followed by +a postprocessing step designed to label individual particles +and separate the ones that are touching each other. We +facilitate the same method and use it for comparison to +our method. Considering that intensity values differ greatly +between samples, the threshold is manually tuned for each +image. To combat poor performance in low-contrast images, +we erode the resulting noisy particle mask to suppress an +overwhelming number of small false positive particle detec- +tions while retaining the integrity of the true positive larger +particles. To convert this semantic segmentation into an +instance segmentation where the particles are separated and +can be processed individually, we again follow best practices +and make use of a watershed-based Van der Walt, Schön- +berger, Nunez-Iglesias, Boulogne, Warner, Yager, Gouillart +and Yu (2014) splitting procedure. Seed points are generated +by eroding the particle mask. The seeds together with the +masks are then used by the watershed algorithm [cite the +implementation you used] to separate the particles. Both the +amount of erosion for the noise reduction and the amount of +erosion for the core generation have been optimized on the +train dataset. +5. Results +Our model is trained on the 41 annotated training patches +and subsequently used to make predictions on the two test +sets. We perform separate quantitative analyses on the in- +and out-of-distribution test sets using the manually anno- +tated patches. In addition, a thorough qualitative analysis +sheds light on the strengths and weaknesses of our proposed +solution. Throughout our evaluation, we compare results to +ThreshWater, which is based on the frequently used method +of intensity thresholding with subsequent watershed seg- +mentation. +5.1. Quantitative evaluation +Quantitative method performance is measured using the +Dice, Object-level F1, and the Instance Dice. In addition, +Karol Gotkowski et al.: Preprint submitted to Elsevier +Page 7 of 17 + +2 +4 +¥5 +¥７ +９ +14 +3 +4 +¥5 +６ +7 +15 +2 +17 +18 +2 +21 +2 +3 +24 +2 +27 +29 +30 +11 +12 +13 +16 +14 +15 +17 +18 +19 +20 +21 +....... +....... +....... +....... +....... +... +...... +...... +....... +....... +...... +....... +.... +22 +23 +24 +1 +26 +27 +28 +2 +3 +1 +32 +29 +31 +33 +34 +35 +37 +40 +42 +38 +41 +43 +45 +46 +48 +44 +4> +4 +5 +6 +8[Work in progress] Scalable, out-of-the box segmentation of individual particles from mineral samples acquired with micro CT +we perform an in-depth analysis of the method’s ability to +correctly split touching particles using the Merger Ratio and +the Splitter Ratio. For a complete description of the metrics +used, see section 4.1. +5.1.1. In-Distribution evaluation +The in-distribution test set consists of 8 manually an- +notated patches from 8 samples. See section 2.2 for a de- +tailed dataset description. We quantitatively evaluate the +predictions of our proposed method as well as ThreshWater +using the metrics described above. The segmentation quality +results of our method and their comparison to ThreshWater +are shown in Table 1 and Table 2, respectively. On average, +a Dice of 94.84% is achieved over all samples. Even on the +individual samples, the Dice is above 93%, indicating a high +segmentation quality for every sample. Our average Dice is +1.01% better than that of ThreshWater. At this level of pre- +cision, a 1% difference indicates a substantial improvement. +This is especially notable given the fact that ThreshWater +uses a manually tuned threshold for each sample whereas +our method is applied out-of-the-box. Similar to the Dice, +we reach an Object Level F1 score of 95.16% on average +over all samples. This implies a high correct identification +rate of individual particles as instances, which is amongst +the most important properties of an instance segmentation. +By comparison, ThreshWater only achieves an Object Level +F1 score of 82.58%, which is 12.58% less than that of our +method and indicates only a mediocre identification rate of +particles. Our method reaches an Instance Dice of 82.13% +on average while ThreshWater only achieves an Instance +Dice of 69.29% and is thus 12.84% worse compared to +our method. Even though the Instance Dice has similarities +to the Dice, the Dice alone measures only the quality of +the segmentation as a whole. Scenarios such as touching +particles, for which the border is predicted wrong, cannot be +quantified with the Dice alone. Thus a high Instance Dice, +as in our case, also suggests that the borders of individual +particles touching each other are improved by our model. +Following up on the segmentation quality evaluation, we +continue with the discussion on the separation quality evalu- +ation. The results of our method are depicted in Table 3. The +manually annotated patches from the test samples contain +a total of 3020 particles. Of those 3020 particles only 198 +particles are touching, which amounts to 7% of the particles. +This can be attributed to our sample preparation technique, +in which a spacer is used to separate individual particles as +much as possible from each other. Of those 198 touching par- +ticles, only 64 are Mergers (see section 4.1), which amounts +in total to 1.31% of all particles. In terms of Splitters (see +section 4.1), only 11 Splitters appear in all 3020 particles +when using our method, which amounts to 0.58% in total. +When comparing the number of Mergers from our method +with the number of Mergers from ThreshWater in Table 4, +our method produces in total 64 Mergers, while ThreshWater +produces 88. In terms of Splitters, our method produces only +11 Splitters, while ThreshWater produces a considerably +higher number of 329. The reasons for this high number +Name +Dice +Object Level F1 +Instance Dice +Ore1_Zone3_Concentrate +0.9516 +0.9846 +0.8993 +Slag5_3 +0.9463 +0.9831 +0.8955 +TmpName_001 +0.9454 +0.9657 +0.8589 +Ore2_PS850_VS10 +0.9559 +1.0 +0.9320 +Slag6_PS300 +0.9359 +0.9661 +0.8120 +Ore3_Zone1 +0.9614 +0.9548 +0.8424 +Ore3_Zone2 +0.9386 +0.8417 +0.5899 +Slag3_PS300 +0.9518 +0.9167 +0.7401 +Mean +0.9484 +0.9516 +0.8213 +Table 1 +The segmentation quantification results of the in-distribution +set for each sample. +Name +Dice +Object Level F1 +Instance Dice +ThreshWater +0.9383 +0.8258 +0.6929 +Ours +0.9484 +0.9516 +0.8213 +Table 2 +The segmentation quantification results of the in-distribution +set of our method compared to ThreshWater. +of Splitters are investigated in our qualitative evaluation in +section 5.2.4but this quantitative evaluation shows already +the difficulties conventional methods face when facilitated +for the objective of inferring instance segmentations. +5.1.2. Out-Of-Distribution evaluation +The out-of-distribution test set contains samples with +minerals, particle sizes and particle shapes that are not +present in the training dataset. Thus, this evaluation provides +valuable information on how well our method performs on +samples with different characteristics not seen before and +consequently highlights how it can be expected to perform +on new samples. Our out-of-distribution test set contains 5 +annotated patches from 5 different samples. The quantitative +analysis is analogous to the previous section. The segmen- +tation quality results of our method and their comparison to +ThreshWater are shown in Table 5 and Table 7, respectively. +On average our method reaches a Dice of 93.27%, an Object +Level F1 of 89.18% and an Instance Dice of 72.77%. The +Dice and thus the general foreground prediction quality is +on par with our Dice on the in-distribution set and with the +ThreshWater baseline. When looking at the Object Level F1, +we see that it is 5.98% lower than on the in-distribution set. +This dropoff is even more considerable for the Instance Dice +with a reduction of 9.36%. The reduction for both metrics +can be traced to the sample Ore4_PS_Tmp4. In summary, +Ore4_PS_Tmp4 includes a high number of very small parti- +cles ( 4 5 6 8[Work in progress] Scalable, out-of-the box segmentation of individual particles from mineral samples acquired with micro CT we perform an in-depth analysis of the method’s ability to correctly split touching particles using the Merger Ratio and the Splitter Ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' For a complete description of the metrics used, see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' In-Distribution evaluation The in-distribution test set consists of 8 manually an- notated patches from 8 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' See section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='2 for a de- tailed dataset description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' We quantitatively evaluate the predictions of our proposed method as well as ThreshWater using the metrics described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' The segmentation quality results of our method and their comparison to ThreshWater are shown in Table 1 and Table 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' On average, a Dice of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='84% is achieved over all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' Even on the individual samples, the Dice is above 93%, indicating a high segmentation quality for every sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' Our average Dice is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='01% better than that of ThreshWater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' At this level of pre- cision, a 1% difference indicates a substantial improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' This is especially notable given the fact that ThreshWater uses a manually tuned threshold for each sample whereas our method is applied out-of-the-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' Similar to the Dice, we reach an Object Level F1 score of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='16% on average over all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' This implies a high correct identification rate of individual particles as instances, which is amongst the most important properties of an instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' By comparison, ThreshWater only achieves an Object Level F1 score of 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='58%, which is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='58% less than that of our method and indicates only a mediocre identification rate of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' Our method reaches an Instance Dice of 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='13% on average while ThreshWater only achieves an Instance Dice of 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='29% and is thus 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='84% worse compared to our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' Even though the Instance Dice has similarities to the Dice, the Dice alone measures only the quality of the segmentation as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' Scenarios such as touching particles, for which the border is predicted wrong, cannot be quantified with the Dice alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' Thus a high Instance Dice, as in our case, also suggests that the borders of individual particles touching each other are improved by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' Following up on the segmentation quality evaluation, we continue with the discussion on the separation quality evalu- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' The results of our method are depicted in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' The manually annotated patches from the test samples contain a total of 3020 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' Of those 3020 particles only 198 particles are touching, which amounts to 7% of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' This can be attributed to our sample preparation technique, in which a spacer is used to separate individual particles as much as possible from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' Of those 198 touching par- ticles, only 64 are Mergers (see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='1), which amounts in total to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='31% of all particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' In terms of Splitters (see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='1), only 11 Splitters appear in all 3020 particles when using our method, which amounts to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='58% in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' When comparing the number of Mergers from our method with the number of Mergers from ThreshWater in Table 4, our method produces in total 64 Mergers, while ThreshWater produces 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' In terms of Splitters, our method produces only 11 Splitters, while ThreshWater produces a considerably higher number of 329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' The reasons for this high number Name Dice Object Level F1 Instance Dice Ore1_Zone3_Concentrate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9516 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='8993 Slag5_3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9463 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='8955 TmpName_001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9454 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='8589 Ore2_PS850_VS10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9559 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9320 Slag6_PS300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9359 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='8120 Ore3_Zone1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9614 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='8424 Ore3_Zone2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='8417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='5899 Slag3_PS300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9167 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='7401 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9484 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9516 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='8213 Table 1 The segmentation quantification results of the in-distribution set for each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' Name Dice Object Level F1 Instance Dice ThreshWater 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9383 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='8258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='6929 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9484 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='9516 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='8213 Table 2 The segmentation quantification results of the in-distribution set of our method compared to ThreshWater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' of Splitters are investigated in our qualitative evaluation in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='4but this quantitative evaluation shows already the difficulties conventional methods face when facilitated for the objective of inferring instance segmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' Out-Of-Distribution evaluation The out-of-distribution test set contains samples with minerals, particle sizes and particle shapes that are not present in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' Thus, this evaluation provides valuable information on how well our method performs on samples with different characteristics not seen before and consequently highlights how it can be expected to perform on new samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' Our out-of-distribution test set contains 5 annotated patches from 5 different samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' The quantitative analysis is analogous to the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' The segmen- tation quality results of our method and their comparison to ThreshWater are shown in Table 5 and Table 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' On average our method reaches a Dice of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='27%, an Object Level F1 of 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='18% and an Instance Dice of 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='77%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' The Dice and thus the general foreground prediction quality is on par with our Dice on the in-distribution set and with the ThreshWater baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' When looking at the Object Level F1, we see that it is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='98% lower than on the in-distribution set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' This dropoff is even more considerable for the Instance Dice with a reduction of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content='36%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' The reduction for both metrics can be traced to the sample Ore4_PS_Tmp4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFQT4oBgHgl3EQfaTZS/content/2301.13319v1.pdf'} +page_content=' In summary, Ore4_PS_Tmp4 includes a high number of very small parti- cles ( ai+1 [ai < ai+1], where at+1 denotes +a1. We also say that s is oriented if s is cyclic or s is anti-cyclic (see [5, 22, 24]). Given a partial transformation +α ∈ PT n such that Dom(α) = {a1 < · · · < at}, with t ⩾ 0, we say that α is orientation-preserving [orientation- +reversing, oriented] if the sequence of its images (a1α, . . . , atα) is cyclic [anti-cyclic, oriented]. We denote by +POPn the submonoid of PT n of all orientation-preserving transformations and by PORn the submonoid of +PT n of all oriented transformations. +Consider also the inverse submonoids POPIn = POPn ∩ In, of all +∗This work is funded by national funds through the FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.P., under the scope of the +projects UIDB/00297/2020 and UIDP/00297/2020 (NovaMath - Center for Mathematics and Applications). +†This work is funded by national funds through the FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.P., under the scope of the +projects UIDB/00297/2020 and UIDP/00297/2020 (NovaMath - Center for Mathematics and Applications). +1 + +orientation-preserving partial permutations, and PORIn = PORn ∩ In, of all oriented partial permutations, +of PT n. +Notice that, by definition, POIn ⊆ PODIn ⊆ PORIn and POIn ⊆ POPIn ⊆ PORIn. +A monoid presentation is an ordered pair ⟨A | R⟩, where A is a set, often called an alphabet, and R ⊆ A∗×A∗ +is a set of relations of the free monoid A∗ generated by A. A monoid M is said to be defined by a presentation +⟨A | R⟩ if M is isomorphic to A∗/ρR, where ρR denotes the smallest congruence on A∗ containing R. +A presentation for the symmetric group Sn was determined by Moore [25] over a century ago (1897). For +the full transformation monoid Tn, a presentation was given in 1958 by A˘ızenˇstat [1] in terms of a certain type +of two generator presentation for the symmetric group Sn, plus an extra generator and seven more relations. +Presentations for the partial transformation monoid PT n and for the symmetric inverse monoid In were found by +Popova [26] in 1961. In 1962, A˘ızenˇstat [2] and Popova [27] exhibited presentations for the monoids of all order- +preserving transformations and of all order-preserving partial transformations of a finite chain, respectively, and +from the sixties until our days several authors obtained presentations for many classes of monoids. See also +[28], the survey [13] and, for example, [6, 9, 10, 12, 15, 18, 21]. +Let G = (V, E) be a finite simple connected graph. Recall that the (geodesic) distance between two vertices +x and y of G, denoted by dG(x, y), is the length of a shortest path between x and y, i.e. the number of edges +in a shortest path between x and y. We say that a partial transformation α ∈ PT (V ) is a partial isometry or +distance preserving partial transformation of G if dG(xα, yα) = dG(x, y) for all x, y ∈ Dom(α). Let us denote +by DP(G) the submonoid of PT (V ) of all partial isometries of G. Since dG(x, y) = 0 +if and only if +x = y, +for all x, y ∈ V , it follows that DP(G) is an inverse submonoid of I(V ) (see [16]). +Clearly, if G = (V, E) is a complete graph, then DP(G) = I(V ). On the other hand, if Pn is an undirected +path with n vertices then DP(Pn) coincides with the monoid of all partial isometries on Ωn, i.e. the submonoid +DPn = {α ∈ In | |iα − jα| = |i − j|, for all i, j ∈ Dom(α)} of In. The study of partial isometries on Ωn was +initiated by Al-Kharousi et al. [3, 4]. The first of these two papers is dedicated to investigating some combi- +natorial properties of the monoid DPn and of its submonoid ODPn of all order-preserving partial isometries, +in particular, their cardinalities. The second paper presents the study of some of their algebraic properties, +namely Green’s structure and ranks. Presentations for both monoids DPn and ODPn were given by Fernandes +and Quinteiro in [18] and the maximal subsemigroups of ODPn were characterized by Dimitrova in [7]. The +monoid DP(Sn) of all partial isometries of a star graph Sn with n vertices (n ⩾ 1) was considered by Fernandes +and Paulista in [16]. They determined the rank and size of DP(Sn) as well as described its Green’s relations. +A presentation for DP(Sn) was also exhibited in [16]. +Next, for n ⩾ 3, consider the cycle graph Cn = ({1, 2, . . . , n}, {{i, i + 1} | i = 1, 2, . . . , n − 1} ∪ {{1, n}}) +with n vertices. The monoid DP(Cn) of all partial isometries of the cycle graph Cn was studied by Fernandes +and Paulista in [17]. They showed that DP(Cn) is an inverse submonoid of the monoid of all oriented partial +permutations on a chain with n elements and, moreover, that it coincides with the inverse submonoid of In +formed by all restrictions of a dihedral subgroup of order 2n of Sn. This fact motivated the authors of [17] +to call DP(Cn) by dihedral inverse monoid on Ωn. Also in [17], it was determined the cardinal and rank of +DP(Cn) as well as descriptions of its Green’s relations and, furthermore, presentations for DP(Cn). +From now on, as in [8], we denote DP(Cn) by the most appropriate notation DIn. +In this paper, we consider three remarkable submonoids of DIn, namely OPDIn = DIn ∩ POPIn, the +monoid of all orientation-preserving partial isometries of Cn, MDIn = DIn ∩ PODIn, the monoid of all +monotone partial isometries of Cn, and ODIn = DIn ∩ POIn, the monoid of all order-preserving partial +isometries of Cn. Observe that DIn, OPDIn, MDIn and ODIn are all inverse submonoids of the symmetric +inverse monoid In, ODIn ⊆ MDIn and ODIn ⊆ OPDIn. These three monoids were also studied in [8] by +the same authors who characterized their Green’s relations and calculated their cardinals and ranks. Here, +for n ⩾ 4, we aim to determine presentations for the monoids ODIn, MDIn and OPDIn. Notice that, as +ODI3 = POI3, OPDI3 = POPI3 and MDI3 = PODI3, presentations for n = 3 are already known (see +[11, 12, 15]). +Throughout this paper, we take n ⩾ 4. +2 + +For general background on Semigroup Theory and standard notations, we refer to Howie’s book [20]. +We would like to point out that we made considerable use of computational tools, namely GAP [19]. +1 +A note on presentations +In this section, we recall some notions related to the concept of a monoid presentation. +Let A be an alphabet and consider the free monoid A∗ generated by A. The elements of A and of A∗ are +called letters and words, respectively. The empty word is denoted by 1 and we write A+ to express A∗ \ {1}. +For all u ∈ A∗, the power u0 also denotes the empty word. A pair (u, v) of A∗ × A∗ is called a relation of A∗ +and it is usually represented by u = v. A relation u = v of A∗ is said to be a consequence of R if (u, v) ∈ ρR. A +set of representatives W ⊆ A∗ for the congruence ρR is also called a set of forms for the presentation ⟨A | R⟩. +Let X be a generating set of a monoid M and let φ : A −→ M be an injective mapping such that Aφ = X. Let +ϕ : A∗ −→ M be the (surjective) homomorphism of monoids that extends φ to A∗. We say that X satisfies (via +ϕ) a relation u = v of A∗ if uϕ = vϕ. For more details see [23] or [28]. A direct method to find a presentation +for a monoid is described by the following well-known result (e.g. see [28, Proposition 1.2.3]). +Proposition 1.1 Let M be a monoid generated by a set X, let A be an alphabet and let φ : A −→ M be an +injective mapping such that Aφ = X. Let ϕ : A∗ −→ M be the (surjective) homomorphism that extends φ to +A∗ and let R ⊆ A∗ × A∗. Then ⟨A | R⟩ is a presentation for M if and only if the following two conditions are +satisfied: +1. The generating set X of M satisfies (via ϕ) all relations from R; +2. If w1, w2 ∈ A∗ are any two words such that the generating set X of M satisfies (via ϕ) the relation +w1 = w2 then w1 = w2 is a consequence of R. +An usual method to find a presentation for a finite monoid is described by the following result (adapted to +the monoid case from [28, Proposition 3.2.2]). +Proposition 1.2 (Guess and Prove method) Let M be a finite monoid generated by a set X, let A be an +alphabet and let φ : A −→ M be an injective mapping such that Aφ = X. Let ϕ : A∗ −→ M be the (surjective) +homomorphism that extends φ to A∗, let R ⊆ A∗ × A∗ and W ⊆ A∗. Assume that the following conditions are +satisfied: +1. The generating set X of M satisfies (via ϕ) all relations from R; +2. For each word w ∈ X∗, there exists a word w′ ∈ W such that the relation w = w′ is a consequence of R; +3. |W| ⩽ |M|. +Then, M is defined by the presentation ⟨A | R⟩. +Notice that, if W satisfies the above conditions then, in fact, |W| = |M| and W is a set of forms for ⟨A | R⟩. +Given a presentation for a monoid, another method to find a new presentation consists in applying Tietze +transformations. For a monoid presentation ⟨A | R⟩, the four elementary Tietze transformations are: +(T1) Adding a new relation u = v to ⟨A | R⟩, provided that u = v is a consequence of R; +(T2) Deleting a relation u = v from ⟨A | R⟩, provided that u = v is a consequence of R\{u = v}; +(T3) Adding a new generating symbol b and a new relation b = w, where w ∈ A∗; +3 + +(T4) If ⟨A | R⟩ possesses a relation of the form b = w, where b ∈ A, and w ∈ (A\{b})∗, then deleting b from +the list of generating symbols, deleting the relation b = w, and replacing all remaining appearances of b +by w. +The following result is well-known (e.g. see [28]): +Proposition 1.3 Two finite presentations define the same monoid if and only if one can be obtained from the +other by a finite number of elementary Tietze transformations (T1), (T2), (T3) and (T4). +Next, we recall the process, given in [15] (also used in [18]), to obtain a presentation for a finite monoid M +given a presentation for a certain submonoid of M. This method will be applied in Section 4. +Let M be a (finite) monoid, let S be a submonoid of M and y be an element of M such that y2 = 1. Let +us suppose that M is generated by S and y. Let X = {x1, . . . , xk} (k ∈ N) be a generating set of S. Then +Y = X ∪ {y} generates M. Let A = {a1, . . . , ak} be an alphabet, let b be a symbol not belonging to A and +B = A ∪ {b}. Let ϕ : B∗ −→ M be the homomorphism that extends the map ai �−→ xi, for 1 ⩽ i ⩽ k, and +b �−→ y to A∗ and let R ⊆ A∗ × A∗ be such that ⟨A | R⟩ is a presentation for S. Consider a set of forms W for +⟨A | R⟩ and suppose there exist two subsets W1 and W2 of W and a word u0 ∈ A∗ such that W = W1 ∪ W2 +and u0 is a factor of each word in W1. Suppose there exist words v0, v1, . . . , vk ∈ A∗ such that the following +relations over the alphabet B are satisfied (via ϕ) by the generating set Y of M: +( ¯R1) bai = vib, for 1 ⩽ i ⩽ k; +( ¯R2) u0b = v0. +Observe that the relation (over the alphabet B) +( ¯R0) b2 = 1 +is also satisfied (via ϕ) by the generating set Y of M, by hypothesis. +Let ¯R = R ∪ ¯R0 ∪ ¯R1 ∪ ¯R2 and ¯W = W ∪ {wb | w ∈ W2} ⊆ B∗. Then, in [15], Fernandes et al. proved: +Proposition 1.4 ([15, Theorem 2.4]) If W contains the empty word then ¯W is a set of forms for the pre- +sentation ⟨B | ¯R⟩. Moreover, if | ¯W| ⩽ |M| then the monoid M is defined by the presentation ⟨B | ¯R⟩. +2 +On the monoids ODIn, MDIn and OPDIn +Now, we recall some properties of ODIn, MDIn and OPDIn, presented by the authors in [8], that we will +need in the following sections. +Let us consider the following permutations of Ωn of order n and 2, respectively: +g = +�1 +2 +· · · +n − 1 +n +2 +3 +· · · +n +1 +� +and +h = +�1 +2 +· · · +n − 1 +n +n +n − 1 +· · · +2 +1 +� +. +It is clear that g, h ∈ DIn. Moreover, g together with h generate the well-known dihedral group D2n of order +2n (considered as a subgroup of Sn). In fact, we have +D2n = ⟨g, h | gn = 1, h2 = 1, hg = gn−1h⟩ = {1, g, g2, . . . , gn−1, h, hg, hg2, . . . , hgn−1}. +Observe that +gk = +� +1 +2 +· · · +n − k +n − k + 1 +· · · +n +1 + k +2 + k +· · · +n +1 +· · · +k +� +, +i.e. +igk = +� i + k +if 1 ⩽ i ⩽ n − k +i + k − n +if n − k + 1 ⩽ i ⩽ n, +4 + +and +hgk = +�1 +· · · +k +k + 1 +· · · +n +k +· · · +1 +n +· · · +k + 1 +� +, +i.e. +ihgk = +� k − i + 1 +if 1 ⩽ i ⩽ k +n + k − i + 1 +if k + 1 ⩽ i ⩽ n, +for 0 ⩽ k ⩽ n − 1. +Recall that DIn is the submonoid of the monoid PORIn whose elements are precisely all restrictions of +the dihedral group D2n of order 2n. +Let also Cn be the cyclic group of order n generated by g, i.e. +Cn = ⟨g | gn = 1⟩ = {1, g, g2, . . . , gn−1}. +Next, denote by id the identity transformation on Ωn and, for X ⊆ Ωn, by idX the partial identity with +domain X, i.e. idX = id|X. Take +ei = idΩn\{i} = +�1 +· · · +i − 1 +i + 1 +· · · +n +1 +· · · +i − 1 +i + 1 +· · · +n +� +∈ DIn, +for 1 ⩽ i ⩽ n. Clearly, for 1 ⩽ i, j ⩽ n, we have e2 +i = ei and eiej = idΩn\{i,j} = ejei. More generally, for any +X ⊆ Ωn, we get Πi∈Xei = idΩn\X. +Since the elements of DIn are precisely the restrictions of D2n, it is easy to conclude that {g, h, e1, e2, . . . , en} +is a generating set of DIn. Moreover, since gn = 1 and ei = gn−iengi for all i ∈ {1, 2, . . . , n}, it follows that +each set {g, h, ei}, with 1 ⩽ i ⩽ n, also generates DIn (see [8, 17]). +Notice that g ∈ OPDIn, h ∈ MDIn and e1, e2, . . . , en are elements of ODIn, MDIn and OPDIn. +Consider the elements +x = +�1 +2 +· · · +n − 1 +2 +3 +· · · +n +� +and +y = x−1 = +�2 +3 +· · · +n +1 +2 +· · · +n − 1 +� +of ODIn with rank n − 1 and the elements +xi = +�1 +1 + i +1 +n − i + 1 +� +and +yi = x−1 +i += +�1 +n − i + 1 +1 +1 + i +� +, +for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, of ODIn with rank 2. +The following result was proved by the authors in [8]: +Proposition 2.1 ([8, Proposition 4.1 and Theorem 4.3]) For n ⩾ 4, +{x, y, e2, . . . , en−1, x1, x2, . . . , x⌊ n−1 +2 +⌋, y1, y2, . . . , y⌊ n−1 +2 +⌋}, +{h, x, e2, . . . , e⌊ n+1 +2 ⌋, x1, x2, . . . , x⌊ n−1 +2 ⌋, y1, y2, . . . , y⌊ n−1 +2 ⌋} +and +{g, ei, x1, x2, . . . , x⌊ n−1 +2 +⌋}, +with 1 ⩽ i ⩽ n, +are generating sets of minimal size of the monoids ODIn, MDIn and OPDIn, respectively. In particular, the +monoids ODIn, MDIn and OPDIn have ranks n + 2⌊n−1 +2 ⌋, 2 + 3⌊n−1 +2 ⌋ and 2 + ⌊n−1 +2 ⌋, respectively. +Observe that n + 2⌊n−1 +2 ⌋ = 2n − 3+(−1)n +2 +. +We end this section by recalling that also in [8, Theorem 2.1] the authors showed the following equality: +|ODIn| = 3 · 2n + (n + 1)n(n − 1) +6 +− 1 + (−1)n +8 +n2 − 2n − 2. +(1) +5 + +3 +Presentations for ODIn +In this section, we first determine a presentation for ODIn on 2n + 1−(−1)n +2 +generators and, secondly, by using +Tietze transformations, we deduce another presentation for ODIn on 2n − 3+(−1)n +2 +generators. +Consider the alphabet A = {x, y, e1, . . . , en, x1, . . . , x⌊ n−1 +2 +⌋, y1, . . . , y⌊ n−1 +2 ⌋} and the set R formed by the +following monoid relations: +(R1) e2 +i = ei, for 1 ⩽ i ⩽ n; +(R2) xy = en and yx = e1; +(R3) xe1 = x and e1y = y; +(R4) eiej = ejei, for 1 ⩽ i < j ⩽ n; +(R5) eix = xei+1, for 1 ⩽ i ⩽ n − 1; +(R6) xiyi = e2 · · · eiei+2 · · · en, yixi = e2 · · · en−ien−i+2 · · · en, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋; +(R7) xiej = xi, ejyi = yi, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, 2 ⩽ j ⩽ n and j ̸= n − i + 1; +(R8) ejxi = xi, yiej = yi, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, 2 ⩽ j ⩽ n and j ̸= i + 1; +(R9) e1xi = xie1 = xn−2ien−2i+1 · · · en−ien−i+2 · · · en, e1yi = yie1 = yn−2ie1 · · · eiei+2 · · · e2i, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋; +(R10) xien−i+1 = ei+1xi = yiei+1 = en−i+1yi = e2 · · · en, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋; +(R11) xe2 · · · en = e1 · · · en. +Observe that |R| = 1 +2(5n2 − (1 + 2(−1)n)n + (−1)n+1 + 5). +Throughout Section 3, we represent the congruence ρR of A∗ by ≈. +We aim to show that the monoid ODIn is defined by the presentation ⟨A | R⟩. To this end, our strategy is +to use Proposition 1.2 together with the known presentation of a submonoid of ODIn determined by Fernandes +in [14]. Therefore, we begin to recall this presentation as well as some other auxiliary results that will also be +useful to us here. +Let us denote by CIn the cyclic inverse monoid on Ωn, i.e. the inverse submonoid of the symmetric inverse +monoid on Ωn consisting of all restrictions of the cyclic group Cn, and by OCIn the submonoid of CIn formed by +all order-preserving elements of CIn. Clearly, OCIn is also a submonoid of ODIn. Moreover, {x, y, e1, . . . , en} +is a generating set of OCIn and |OCIn| = 3 · 2n − 2n − 2 [14, Theorem 1.4]. Furthermore, if U is the set of +relations R1 to R5 together with relation R11 over the alphabet C = {x, y, e1, . . . , en}, we have: +Proposition 3.1 ([14, Theorem 2.16]) The monoid OCIn is defined by the presentation ⟨C | U⟩ on n + 2 +generators and 1 +2(n2 + 3n + 8) relations. +The following lemma was crucial to prove the above result. Also here, it will be very useful. +Lemma 3.2 ([14, Lemma 2.15]) Let u ∈ C∗. Then, there exist z ∈ {x, y}, v ∈ {e1, . . . , en}∗ and 0 ⩽ r ⩽ +n − 1 such that u = zrv is a consequence of U. +Next, recall that the relations +xjei = xj = en−i+1xj and eiyj = yj = yjen−i+1, for 1 ⩽ i ⩽ j ⩽ n, +(2) +are consequences of U (see [14, Lemma 2.11]). +In order to find a set of words W satisfying condition 2 of Proposition 1.2 for ⟨A | R⟩, we now present a +series of lemmas. +The following lemma is easy to deduce from (2) and R10: +6 + +Lemma 3.3 Let 1 ⩽ i ⩽ ⌊n−1 +2 ⌋. The following relations are consequences of R: +1. yrxi = yre2 · · · en, for r ⩾ n − i; +2. yryi = yre2 · · · en, for r ⩾ i; +3. xixs = e2 · · · enxs, for s ⩾ i; +4. yixs = e2 · · · enxs, for s ⩾ n − i. +Proof. We prove the first relation. For the remaining three ones we can argue in a similar way. +By (2), we have yn−i ≈ yn−iei+1 and so, by R10, we get yn−ixi ≈ yn−iei+1xi ≈ yn−ie2 · · · en, whence +yrxi = yr−n+iyn−ixi ≈ yr−n+iyn−ie2 · · · en = yre2 · · · en, as required. +Lemma 3.4 The following relations are consequences of R: +1. xixj = yiyj = e2 · · · en, for 1 ⩽ i, j ⩽ ⌊n−1 +2 ⌋; +2. xiyj = yjxi = e2 · · · en, for 1 ⩽ i, j ⩽ ⌊n−1 +2 ⌋ and i ̸= j. +Proof. Let 1 ⩽ i, j ⩽ ⌊n−1 +2 ⌋. +Then j + 1 ̸= n − i + 1. In fact, if j + 1 = n − i + 1 then +1 ⩽ j ⩽ ⌊n−1 +2 ⌋ =⇒ 1 ⩽ n − i ⩽ ⌊n−1 +2 ⌋ =⇒ ⌊n−1 +2 ⌋ < n − ⌊n−1 +2 ⌋ ⩽ i, +which is a contradiction. Notice that j + 1 ̸= n − i + 1 is equivalent to i + 1 ̸= n − j + 1. Hence, by R7, we have +xi ≈ xiej+1 and ei+1yj ≈ yj. Thus, by R1, R4 and R10, we get +xixj ≈ xiej+1xj ≈ xie2 · · · en ≈ xien−i+1e2 · · · en−ien−i+2 · · · en ≈ e2 · · · ene2 · · · en−ien−i+2 · · · en ≈ e2 · · · en +and +yiyj ≈ yiei+1yj ≈ e2 · · · enyj ≈ e2 · · · en−jen−j+2 · · · enen−j+1yj ≈ e2 · · · en−jen−j+2 · · · ene2 · · · en ≈ e2 · · · en +(observe that i, j < n implies n − i + 1, n − j + 1 > 1), which proves property 1. +Now, in order to prove property 2, suppose also that i ̸= j. Then n − j + 1 ̸= n − i + 1 and i + 1 ̸= j + 1, +whence xi ≈ xien−j+1, by R7, and yjei+1 ≈ yj, by R8. Thus, by R1, R4, R10 and the above calculations, we +obtain +xiyj ≈ xien−j+1yj ≈ xie2 · · · en ≈ e2 · · · en +and +yjxi ≈ yjei+1xi ≈ yje2 · · · en ≈ yjej+1e2 · · · ejej+2 · · · en ≈ e2 · · · ene2 · · · ejej+2 · · · en ≈ e2 · · · en, +as required. +Lemma 3.5 Let u ∈ C∗ and 1 ⩽ i ⩽ ⌊n−1 +2 ⌋. Then: +1. There exists u′ ∈ C∗ such that uxi ≈ u′ or there exists 0 ⩽ r ⩽ n − i − 1 such that uxi ≈ yrxi; +2. There exists u′ ∈ C∗ such that uyi ≈ u′ or there exists 0 ⩽ r ⩽ i − 1 such that uyi ≈ yryi. +7 + +Proof. We prove property 1. The argument for proving property 2 is analogous. +Let z ∈ {x, y}, v ∈ {e1, . . . , en}∗ and 0 ⩽ r ⩽ n − 1 be such that u ≈ zrv, by Lemma 3.2. +Since +v ∈ {e1, . . . , en}∗, by R1, R4 and R8, we have vxi ≈ v′xi, for some v′ ∈ {1, e1, ei+1, e1ei+1}. +If v′ = et +1ei+1, for some t ∈ {0, 1}, then uxi ≈ zrv′xi = zret +1ei+1xi ≈ zret +1e2 · · · en ∈ C∗, by using R10. +If v′ = e1 then uxi ≈ zrv′xi = zre1xi ≈ zrxn−2ien−2i+1 · · · en−ien−i+2 · · · en ∈ C∗, by using R9. +Now, suppose that v′ = 1. Then uxi ≈ zrxi. If r = 0 then there is nothing more to prove. So, suppose also +that r > 0. +If z = x then uxi ≈ xrxi ≈ xre1xi ≈ xrxn−2ien−2i+1 · · · en−ien−i+2 · · · en ∈ C∗, by R3 and R9. +If z = y then uxi ≈ yrxi. If r ⩽ n − i − 1 there is nothing more to prove. On the other hand, if r ⩾ n − i +then, by Lemma 3.3, we get uxi ≈ yrxi ≈ yre2 · · · en ∈ C∗. +Thus, the proof of property 1 is complete, as required. +Let ϕ : A∗ −→ ODIn be the homomorphism of monoids that extends the mapping A −→ ODIn defined by +x �−→ x, +y �−→ y, +ei �−→ ei, for 1 ⩽ i ⩽ n, +xj �−→ xj and yj �−→ yj, for 1 ⩽ j ⩽ ⌊n−1 +2 ⌋. +Notice that we are using the same symbols for the letters of the alphabet A and for the generators of ODIn, +which simplifies notation and, within the context, will not cause ambiguity. +The following subsets of A∗, in a sense, were motivated by Lemma 3.3: +W1 = {yrxixs | 0 ⩽ r, s ⩽ n − 1, 1 ⩽ i ⩽ ⌊n−1 +2 ⌋ and s + 1 ⩽ i ⩽ n − r − 1} +and +W2 = {yryixs | 0 ⩽ r, s ⩽ n − 1, 1 ⩽ i ⩽ ⌊n−1 +2 ⌋ and r + 1 ⩽ i ⩽ n − s − 1}. +Observe that |W1 ∪ W2| = 1 +6(n + 1)n(n − 1) − 1 +8(1 + (−1)n)n2, i.e. the number of order-preserving restrictions +of {hgk | 0 ⩽ k ⩽ n − 1} with rank (greater than or) equal to 2, except those that are also restrictions of +{gk | 0 ⩽ k ⩽ n − 1}. In fact, (W1 ∪ W2)ϕ is precisely this set of transformations of ODIn with rank 2. +Furthermore, we have: +Lemma 3.6 Let w ∈ A∗. Then, there exists w′ ∈ C∗ ∪ W1 ∪ W2 such that w ≈ w′. +Proof. We proceed by induction on |w|. +If |w| = 1 then w ∈ C ∪ W1 ∪ W2 and so there is nothing to prove. +As induction hypothesis, assume that the lemma is valid for all words w ∈ A∗ such that |w| = k ⩾ 1. +Let w ∈ A∗ be such that |w| = k + 1. +If w ∈ C∗ then there is nothing to prove. Hence, suppose that w ∈ A∗ \ C∗. Let u ∈ A∗ and a ∈ A be such +that w = ua. We will consider several cases. +case 1. u ∈ C∗. +Therefore a ∈ A\C. So, by Lemma 3.5, there exists u′ ∈ C∗ such that ua ≈ u′ or there exists 0 ⩽ r ⩽ n−i−1 +such that ua ≈ yrxi, with a = xi for some 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, or there exists 0 ⩽ r ⩽ i − 1 such that ua ≈ yryi, +with a = yi for some 1 ⩽ i ⩽ ⌊n−1 +2 ⌋. Hence, in any of these three situations, we obtain w′ ∈ C∗ ∪ W1 ∪ W2 such +that w = ua ≈ w′. +case 2. u ∈ A∗ \ C∗. +Since |u| = k then, by the induction hypothesis, there exists u′ ∈ C∗ ∪ W1 ∪ W2 such that u′ ≈ u. +case 2.1. u′ ∈ C∗. +If a ∈ C then w = ua ≈ u′a ∈ C∗. On the other hand, if a ∈ A \ C then, as in case 1, there exists +w′ ∈ C∗ ∪ W1 ∪ W2 such that u′a ≈ w′ and so such that w = ua ≈ u′a ≈ w′. +case 2.2. u′ ∈ W1, i.e. u′ = yrxixs, for some 0 ⩽ r, s ⩽ n − 1, 1 ⩽ i ⩽ ⌊n−1 +2 ⌋ and s + 1 ⩽ i ⩽ n − r − 1. +Then, we have w = ua ≈ u′a = yrxixsa. +8 + +case 2.2.1. a = e1. +If s > 0 then, by R3, xse1 ≈ xs and so w ≈ yrxixse1 ≈ yrxixs ∈ W1. On the other hand, if s = 0 then, by +R9, we have w ≈ yrxie1 ≈ yrxn−2ien−2i+1 · · · en−ien−i+2 · · · en ∈ C∗. +case 2.2.2. a = ej, with 2 ⩽ j ⩽ n. +If j ⩽ s then, by (2), w ≈ yrxixsej ≈ yrxixs ∈ W1. +Now, suppose that s < j. Then, by R5, we get xsej ≈ ej−sxs and so w ≈ yrxixsej ≈ yrxiej−sxs. If +j − s ̸= 1 and j − s ̸= n − i + 1 then, by R7, w ≈ yrxiej−sxs ≈ yrxixs ∈ W1. +If j − s = 1 then, by +R9, w ≈ yrxie1xs ≈ yrxn−2ien−2i+1 · · · en−ien−i+2 · · · enxs ∈ C∗. Finally, if j − s = n − i + 1 then, by R10, +w ≈ yrxien−i+1xs ≈ yre2 · · · enxs ∈ C∗. +case 2.2.3. a = x. +If s + 1 < i then w ≈ yrxixs+1 ∈ W1. On the other hand, if s + 1 ⩾ i (in fact s + 1 = i) then, by Lemma +3.3, we get w ≈ yrxixs+1 ≈ yre2 · · · enxs+1 ∈ C∗. +case 2.2.4. a = y. +If s = 0 then, by R3 and R9, w ≈ yrxiy ≈ yrxie1y ≈ yrxn−2ien−2i+1 · · · en−ien−i+2 · · · eny ∈ C∗. On the +other hand, if s > 0 then, by R2, R5 (noticing that s − 1 < n) and R7 (noticing that n − s + 1 > n − i + 1), we +have w ≈ yrxixsy ≈ yrxixs−1en ≈ yrxien−s+1xs−1 ≈ yrxixs−1 ∈ W1. +case 2.2.5. a = xj, with 1 ⩽ j ⩽ ⌊n−1 +2 ⌋. +If s = 0 then, by Lemma 3.4, we have w ≈ yrxixj ≈ yre2 · · · en ∈ C∗. +So, suppose that s > 0. Then, by R3 and R9, we get +w ≈ yrxixsxj ≈ yrxixse1xj ≈ yrxixsxn−2jen−2j+1 · · · en−jen−j+2 · · · en. +(3) +If s + n − 2j ⩾ i then from (3), by Lemma 3.3, we get w ≈ yre2 · · · enxs+n−2jen−2j+1 · · · en−jen−j+2 · · · en ∈ C∗. +On the other hand, suppose that s + n − 2j + 1 ⩽ i. If s ⩾ j then n − s + 1 ⩽ s + n − 2j + 1 ⩽ i, whence +n − i + 1 ⩽ s ⩽ i − 1 and so n + 2 ⩽ 2i ⩽ 2⌊n−1 +2 ⌋ ⩽ n − 1, a contradiction. +Therefore s < j and so +n − 2j + 1 ⩽ s + n − 2j + 1 ⩽ n − j. Hence, by (3), R4, R5 and R9, we obtain +w +≈ +yrxixs+n−2jes+n−2j+1en−2j+1 · · · es+n−2jes+n−2j+2 · · · en−jen−j+2 · · · en +≈ +yrxie1xs+n−2jen−2j+1 · · · es+n−2jes+n−2j+2 · · · en−jen−j+2 · · · en +≈ +yrxn−2ien−2i+1 · · · en−ien−i+2 · · · enxs+n−2jen−2j+1 · · · es+n−2jes+n−2j+2 · · · en−jen−j+2 · · · en ∈ C∗. +case 2.2.6. a = yj, with 1 ⩽ j ⩽ ⌊n−1 +2 ⌋. +If s = 0 then, by R6 and Lemma 3.4, respectively, we have +w ≈ yrxiyj ≈ +� yre2 · · · eiei+2 · · · en ∈ C∗ +if i = j +yre2 · · · en ∈ C∗ +otherwise. +Suppose that s > 0. Then, by R3 and R9, we get +w ≈ yrxixsyj ≈ yrxixse1yj ≈ yrxixsyn−2je1 · · · ejej+2 · · · e2j. +(4) +Next, by R2, R5 and R7 (noticing that s + 1 ⩽ i implies n − s + 1 > n − i + 1), we have +xixsyn−2j ≈ xixs−1enyn−2j−1 ≈ xien−s+1xs−1yn−2j−1 ≈ xixs−1yn−2j−1 +and, repeating this process as long as possible, we obtain +xixsyn−2j ≈ + + + +xi +if s = n − 2j +xixs−n+2j +if s > n − 2j +xiyn−2j−s +if s < n − 2j. +(5) +From (4) and (5), we have: if s = n − 2j, by R9, +w ≈ yrxie1e2 · · · ejej+2 · · · e2j ≈ yrxn−2ien−2i+1 · · · en−ien−i+2 · · · ene2 · · · ejej+2 · · · e2j ∈ C∗; +9 + +and, if s < n − 2j, by R3 and R9, +w +≈ +yrxiyn−2j−se1 · · · ejej+2 · · · e2j +≈ +yrxie1yn−2j−se1 · · · ejej+2 · · · e2j +≈ +yrxn−2ien−2i+1 · · · en−ien−i+2 · · · enyn−2j−se1 · · · ejej+2 · · · e2j ∈ C∗. +Now, observe that s + 1 ⩽ i implies n − s − 1 ⩾ n − i ⩾ n − ⌊n−1 +2 ⌋ = ⌊n +2 ⌋ + 1 > ⌊n−1 +2 ⌋ ⩾ j, i.e. j < n − s − 1 +and so j > s − n + 2j + 1. Therefore, if s > n − 2j, from (4) and (5), we have +w +≈ +yrxixs−n+2je1 · · · ejej+2 · · · e2j +≈ +yrxixs−n+2jes−n+2j+1e1 · · · es−n+2jes−n+2j+2 · · · ejej+2 · · · e2j +≈ +yrxie1xs−n+2je1 · · · es−n+2jes−n+2j+2 · · · ejej+2 · · · e2j +≈ +yrxn−2ien−2i+1 · · · en−ien−i+2 · · · enxs−n+2je1 · · · es−n+2jes−n+2j+2 · · · ejej+2 · · · e2j ∈ C∗, +by R4, R5 and R9, thus completing the proof of the lemma for the case 2.2. +case 2.3. u′ ∈ W2, i.e. u′ = yryixs, for some 0 ⩽ r, s ⩽ n − 1, 1 ⩽ i ⩽ ⌊n−1 +2 ⌋ and r + 1 ⩽ i ⩽ n − s − 1. +Then, we have w = ua ≈ u′a = yryixsa. +If a ∈ C then proceeding analogously to the cases 2.2.1-2.2.4, we can find w′ ∈ C∗ ∪ W2 such that w ≈ w′. +Thus, it remains to study the case a ∈ A \ C, which we divide in two cases. +First, let us consider a = xj, for some 1 ⩽ j ⩽ ⌊n−1 +2 ⌋. +If s = 0 then, by R6 and Lemma 3.4, respectively, we have +w ≈ yryixj ≈ +� yre2 · · · en−ien−i+2 · · · en ∈ C∗ +if i = j +yre2 · · · en ∈ C∗ +otherwise. +Now, suppose that s > 0. Then, by R3 and R9, we get +w ≈ yryixsxj ≈ yryixse1xj ≈ yryixsxn−2jen−2j+1 · · · en−jen−j+2 · · · en. +(6) +If s + n − 2j ⩾ n − i then +w ≈ yryixs+n−2jen−2j+1 · · · en−jen−j+2 · · · en ≈ yre2 · · · enxs+n−2jen−2j+1 · · · en−jen−j+2 · · · en ∈ C∗, +by (6) and Lemma 3.3. +So, suppose that s + n − 2j ⩽ n − i − 1, i.e. i ⩽ n − (s + n − 2j) − 1. +If s + n − 2j = n − j then, by (6), (2) and R5, we have w ≈ yryixs+n−2jen−j+2 · · · en ≈ yryie2 · · · ejxs+n−2j. +In addition, if j ⩽ i, by R8, we obtain w ≈ yryie2 · · · ejxs+n−2j ≈ yryixs+n−2j ∈ W2. Otherwise, by R4, R8 and +R10, it follows that w ≈ yryie2 · · · eiei+2 · · · ejei+1xs+n−2j ≈ yryiei+1xs+n−2j ≈ yre2 · · · enxs+n−2j ∈ C∗. +On the other hand, suppose that s + n − 2j ̸= n − j. Since s + n − 2j + 1 > n − 2j + 1, then s + n − 2j + 1 ∈ +L = {n − 2j + 1, . . . , n − j, n − j + 2, . . . , n} and so, by (6), R4, R5 and R9, we have +w +≈ +yryixs+n−2jen−2j+1 · · · en−jen−j+2 · · · en +≈ +yryixs+n−2jes+n−2j+1Πt∈L\{s+n−2j+1}et +≈ +yryie1xs+n−2jΠt∈L\{s+n−2j+1}et +≈ +yryn−2ie1 · · · eiei+2 · · · e2ixs+n−2jΠt∈L\{s+n−2j+1}et ∈ C∗, +which completes the study of this case. +Finally, let us move on to our last case by considering a = yj, for some 1 ⩽ j ⩽ ⌊n−1 +2 ⌋. +If s = 0 then, by Lemma 3.4, we have w ≈ yryiyj ≈ yre2 · · · en ∈ C∗. +So, suppose that s > 0. Then, by R3 and R9, we get +w ≈ yryixsyj ≈ yryixse1yj ≈ yryixsyn−2je1 · · · ejej+2 · · · e2j. +(7) +10 + +Next, by R2, R5 and R8 (noticing that i ⩽ n − s − 1 implies i + 1 < n − s + 1), we have +yixsyn−2j ≈ yixs−1enyn−2j−1 ≈ yien−s+1xs−1yn−2j−1 ≈ yixs−1yn−2j−1. +By repeating this process as long as possible, we obtain +yixsyn−2j ≈ + + + +yi +if s = n − 2j +yixs−n+2j +if s > n − 2j +yiyn−2j−s +if s < n − 2j. +(8) +If s = n − 2j, by (7), (8) and R9, we have +w ≈ yryie1e2 · · · ejej+2 · · · e2j ≈ yryn−2ie1 · · · eiei+2 · · · e2ie2 · · · ejej+2 · · · e2j ∈ C∗. +On the other hand, if s < n − 2j, by (7), (8), R3 and R9, we get +w +≈ +yryiyn−2j−se1 · · · ejej+2 · · · e2j +≈ +yryie1yn−2j−se1 · · · ejej+2 · · · e2j +≈ +yryn−2ie1 · · · eiei+2 · · · e2iyn−2j−se1 · · · ejej+2 · · · e2j ∈ C∗. +Now, suppose that s > n − 2j. +In addition, suppose first that s − n + 2j = j. Then, by (7), (8), (2) and R5, we obtain +w ≈ yryixje1 · · · ejej+2 · · · e2j ≈ yryixjej+2 · · · e2j ≈ yryie2 · · · ejxj. +Since i ⩽ n − s − 1 < n − s = j, then w ≈ yryiei+1xj ≈ yre2 · · · enxj ∈ C∗, by R8, R4 and R10. +Secondly, suppose that s − n + 2j ̸= j. Since s − n < 0, then s − n + 2j + 1 ⩽ 2j and so s − n + 2j + 1 ∈ +K = {1, . . . , j, j + 2, . . . 2j}. Hence, by (7), (8), R4, R5 and R9, we get +w +≈ +yryixs−n+2je1 · · · ejej+2 · · · e2j +≈ +yryixs−n+2jes−n+2j+1Πt∈K\{s−n+2j+1}et +≈ +yryie1xs−n+2jΠt∈K\{s−n+2j+1}et +≈ +yryn−2ie1 · · · eiei+2 · · · e2ixs−n+2jΠt∈K\{s−n+2j+1}et ∈ C∗. +Therefore, we have exhausted all possible cases, completing the proof of the lemma. +Now, let us choose a set of forms W0 for the presentation ⟨C | U⟩. Then, for each w ∈ C∗ there exists (a +unique) w′ ∈ W0 such that w′ρUw. Moreover, as the monoid OCIn is defined by the presentation ⟨C | U⟩, by +Proposition 3.1, we have |W0| = |OCIn| = 3 · 2n − 2n − 2. +Let W = W0 ∪ W1 ∪ W2. Then, by (1), +|W| = |W0| + |W1 ∪ W2| = 3 · 2n − 2n − 2 + 1 +6(n + 1)n(n − 1) − 1 +8(1 + (−1)n)n2 = |ODIn| +and, by Lemma 3.6, for each word w ∈ A∗, there exists w′ ∈ W such that w ≈ w′. +On the other hand, it is a routine matter to check: +Lemma 3.7 The generating set {x, y, e1, e2, . . . , en, x1, . . . , x⌊ n−1 +2 +⌋, y1, . . . , y⌊ n−1 +2 ⌋} of ODIn satisfies (via ϕ) all +relations from R. +Therefore, the conditions of Proposition 1.2 are satisfied and so we have: +Theorem 3.8 The monoid ODIn is defined by the presentation ⟨A | R⟩ on 2n + 1−(−1)n +2 +generators and +1 +2(5n2 − (1 + 2(−1)n)n + (−1)n+1 + 5) relations. +11 + +Next, by using Tietze transformations and applying Proposition 1.3, we deduce from the above presentation +for ODIn a new presentation on a minimal size set of generators of ODIn given in Proposition 2.1. +Let us consider the alphabet +B = {x, y, e2, . . . , en−1, x1, x2, . . . , x⌊ n−1 +2 ⌋, y1, y2, . . . , y⌊ n−1 +2 ⌋} = A \ {e1, en}. +Basically, we first apply T4 with each of the relations R2 and then, of the resulting relations, we eliminate the +trivial ones and some deduced from others. +This procedure was applied in [14] to the set U of relations (R1 to R5 together with relation R11) on the +alphabet C = {x, y, e1, . . . , en}, having resulted in the following set of 1 +2(n2 + 3n) monoid relations (which we +can consider on the alphabet B): +(V1) e2 +i = ei, for 2 ⩽ i ⩽ n − 1; +(V2) xyx = x and yxy = y; +(V3) yx2y = xy2x; +(V4) eiej = ejei, for 2 ⩽ i < j ⩽ n − 1; +(V5) xyei = eixy and yxei = eiyx, for 2 ⩽ i ⩽ n − 1; +(V6) xei+1 = eix, for 2 ⩽ i ⩽ n − 2; +(V7) x2y = en−1x and yx2 = xe2; +(V8) yxe2 · · · en−1xy = xe2 · · · en−1xy. +Performing the same procedure to relations R6 to R10 on the alphabet A, we may routinely obtain the +following 2n2 − (2 + (−1)n)n − 1 +2(3 + (−1)n) monoid relations on the alphabet B: +(V9) xiyi = e2 · · · eiei+2 · · · en−1xy, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋; +y1x1 = e2 · · · en−1; yixi = e2 · · · en−ien−i+2 · · · en−1xy, for 2 ⩽ i ⩽ ⌊n−1 +2 ⌋; +(V10) xiej = xi and ejyi = yi, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, 2 ⩽ j ⩽ n − 1 and j ̸= n − i + 1; +(V11) ejxi = xi and yiej = yi, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, 2 ⩽ j ⩽ n − 1 and j ̸= i + 1; +(V12) xixy = xyxi = xi and xyyi = yixy = yi, for 2 ⩽ i ⩽ ⌊n−1 +2 ⌋; xyx1 = x1 and y1xy = y1; +(V13) yxx1 = x1yx = xn−2en−1; yxxi = xiyx = xn−2ien−2i+1 · · · en−ien−i+2 · · · en−1xy, for 2 ⩽ i ⩽ ⌊n−1 +2 ⌋; +yxyi = yiyx = yn−2i+1xe2 · · · eiei+2 · · · e2i, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋; +(V14) x1xy = e2x1 = y1e2 = xyy1 = e2 · · · en−1xy; xien−i+1 = ei+1xi = yiei+1 = en−i+1yi = e2 · · · en−1xy, for +2 ⩽ i ⩽ ⌊n−1 +2 ⌋. +Thus, defining V as the set of monoid relations on the alphabet B consisting of relations V1 to V14, we have: +Theorem 3.9 The monoid ODIn is defined by the presentation ⟨B | V ⟩ on 2n − 3+(−1)n +2 +generators and +1 +2(5n2 − (1 + 2(−1)n)n + (−1)n+1 − 3) relations. +12 + +4 +Presentations for MDIn +We begin this section by determining a presentation for MDIn on 2n− 1+(−1)n +2 +generators. For this purpose, we +will apply Proposition 1.4. Next, by using Tietze transformations, we deduce another presentation for MDIn +on 2 + 3⌊n−1 +2 ⌋ generators. +Let us consider the alphabet ¯B = B ∪ {h} = {h, x, y, e2, . . . , en−1, x1, . . . , x⌊ n−1 +2 +⌋, y1, . . . , y⌊ n−1 +2 +⌋} and let +¯ϕ : ¯B∗ −→ MDIn be the homomorphism of monoids that extends the mapping ¯B −→ MDIn defined by +h �−→ h, +x �−→ x, +y �−→ y, +ei �−→ ei, for 2 ⩽ i ⩽ n − 1, +xj �−→ xj and yj �−→ yj, for 1 ⩽ j ⩽ ⌊n−1 +2 ⌋. +Notice that ¯B ¯ϕ is a generating set of MDIn with 2n − 1+(−1)n +2 +elements. +Next, observe that ∅ = e1e2 · · · en−1en, +�1 +1 +� += e2 · · · en−1en and +�i +j +� += +�i +1 +��1 +1 +��1 +j +� += ei+1 · · · enyi−1e2 · · · en−1enxj−1ej+1 · · · en, +for 1 ⩽ i, j ⩽ n. Therefore, let u0 = e2 · · · en−1xy ∈ B∗ and let W ′ +1 be the subset of B∗ formed by the following +1 + n2 words: +1. yxu0; +2. ei+1 · · · en−1xyiu0xj−1ej+1 · · · en−1xy, for 1 ⩽ i, j ⩽ n − 1; +3. ei+1 · · · en−1xyiu0xn−1, for 1 ⩽ i ⩽ n − 1; +4. yn−1u0xj−1ej+1 · · · en−1xy, for 1 ⩽ j ⩽ n − 1; and +5. yn−1u0xn−1. +Then ¯ϕ is a bijection from W ′ +1 onto {∅} ∪ { +�i +j +� +| 1 ⩽ i, j ⩽ n} and so we can choose a set of forms W ′ for the +presentation ⟨B | V ⟩ of ODIn such that W ′ contains the empty word and W ′ +1 ⊂ W ′. Let W ′ +2 = W ′ \ W ′ +1 and +consider the subset ¯W = W ′ ∪ {wh | w ∈ W ′ +2} of ¯B∗. +Notice that | ¯W| = |W ′| + |W ′ +2| = |W ′| + |W ′| − |W ′ +1| = 2|ODIn| − n2 − 1 = |MDIn|. +Let 1 ⩽ i ⩽ ⌊n−1 +2 ⌋. Then +hxih = +�n − i +n +i +n +� += yn−i−1xixi−1 +and +hyih = +� +i +n +n − i +n +� += yi−1yixn−i−1. +On the other hand, we also have +hxh = y, +hyh = x, +heih = en−i+1, for 1 ⩽ i ⩽ n, +and +e2 · · · enh = +�1 +n +� += xn−1. +Therefore, the monoid relations (on the alphabet ¯B) +h2 = 1, +hx = yh, +hy = xh, +hei = en−i+1h, for 2 ⩽ i ⩽ n − 1, +hxi = yn−i−1xixi−1h and hyi = yi−1yixn−i−1h, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, +and +e2 · · · en−1xyh = xn−1 +are all satisfied (via ¯ϕ) by the generating set {h, x, y, e2, . . . , en−1, x1, . . . , x⌊ n−1 +2 +⌋, y1, . . . , y⌊ n−1 +2 ⌋} of MDIn. +Now, let ¯V be the set of monoid relations V (relations V1 to V14 considered on the alphabet ¯B) together +with the following n + ⌊n+1 +2 ⌋ + 1−(−1)n +2 +monoid relations on the alphabet ¯B: +( ¯V0) h2 = 1; +13 + +( ¯V1) hx = yh; hei = en−i+1h, for 2 ⩽ i ⩽ ⌊n+1 +2 ⌋; +hxi = yn−i−1xixi−1h and hyi = yi−1yixn−i−1h, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋; +( ¯V2) e2 · · · en−1xyh = xn−1. +Since the relation hy = xh is a consequence of h2 = 1 and hx = yh and the relations hei = en−i+1h, with +2 ⩽ i ⩽ n − 1, are consequences of h2 = 1 and hei = en−i+1h, with 2 ⩽ i ⩽ ⌊n+1 +2 ⌋, by Proposition 1.4, we +conclude that: +Theorem 4.1 The monoid MDIn is defined by the presentation ⟨ ¯B | ¯V ⟩ on 2n − 1+(−1)n +2 +generators and +1 +2(5n2 + (2 − 2(−1)n)n − 3+5(−1)n +2 +) relations. +Next, like in Section 3, by using Tietze transformations and applying Proposition 1.3, we deduce from the +presentation ⟨ ¯B | ¯V ⟩ of MDIn a new presentation on a minimal size set of generators of MDIn provided by +Proposition 2.1. So, consider the alphabet +¯B′ = {h, x, e2, . . . , e⌊ n+1 +2 ⌋, x1, x2, . . . , x⌊ n−1 +2 +⌋, y1, y2, . . . , y⌊ n−1 +2 +⌋}. +Hence, since y = hxh and ei = hen−i+1h, for ⌊n+1 +2 ⌋ + 1 ⩽ i ⩽ n − 1 (as transformations), we can apply T1 by +adding the relations y = hxh and ei = hen−i+1h, for ⌊n+1 +2 ⌋ + 1 ⩽ i ⩽ n − 1. Next, we apply T4 with each of +these relations and then, of the resulting relations, we eliminate the trivial ones and some deduced from others. +Performing this procedure to ¯V , we may routinely obtain the following set ¯V ′ of 2n2 + 7−(−1)n +4 +n − 2(−1)n − 1 +monoid relations on the alphabet ¯B′: +(V ′ +1) e2 +i = ei, for 2 ⩽ i ⩽ ⌊n+1 +2 ⌋; +(V ′ +2) (xh)2x = x; +(V ′ +3) xhx2hxh = hxhx2hx; +(V ′ +4) eiej = ejei, for 2 ⩽ i < j ⩽ ⌊n+1 +2 ⌋; eihejh = hejhei, for 2 ⩽ i ⩽ ⌊n+1 +2 ⌋ and 2 ⩽ j ⩽ ⌊n +2 ⌋; +(V ′ +5) (xh)2ei = ei(xh)2 and (hx)2ei = ei(hx)2, for 2 ⩽ i ⩽ ⌊n+1 +2 ⌋; +(V ′ +6) xei+1 = eix, for 2 ⩽ i ⩽ ⌊n−1 +2 ⌋; xhe⌊ n +2 ⌋h = e⌊ n+1 +2 ⌋x; xheih = hei+1hx, for 2 ⩽ i ⩽ ⌊n−2 +2 ⌋; +(V ′ +7) x(xh)2 = he2hx and (hx)2x = xe2; +(V ′ +8) (hx)2e2 · · · e⌊ n+1 +2 ⌋he2 · · · e⌊ n +2 ⌋(hx)2 = xe2 · · · e⌊ n+1 +2 ⌋he2 · · · e⌊ n +2 ⌋(hx)2; +(V ′ +9) xiyi = e2 · · · eiei+2 · · · e⌊ n+1 +2 ⌋he2 · · · e⌊ n +2 ⌋h(xh)2, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋; y1x1 = e2 · · · e⌊ n+1 +2 ⌋he2 · · · e⌊ n +2 ⌋h; +yixi = e2 · · · e⌊ n+1 +2 ⌋he2 · · · ei−1ei+1 · · · e⌊ n +2 ⌋h(xh)2, for 2 ⩽ i ⩽ ⌊n−1 +2 ⌋; +(V ′ +10) xiej = xi and ejyi = yi, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋ and 2 ⩽ j ⩽ ⌊n+1 +2 ⌋; +xihejh = xi and hejhyi = yi, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, 2 ⩽ j ⩽ ⌊n +2 ⌋ and j ̸= i; +(V ′ +11) ejxi = xi and yiej = yi, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, 2 ⩽ j ⩽ ⌊n+1 +2 ⌋ and j ̸= i + 1; +hejhxi = xi and yihejh = yi, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋ and 2 ⩽ j ⩽ ⌊n +2 ⌋; +(V ′ +12) xi(xh)2 = (xh)2xi = xi and (xh)2yi = yi(xh)2 = yi, for 2 ⩽ i ⩽ ⌊n−1 +2 ⌋; (xh)2x1 = x1 and y1(xh)2 = y1; +(V ′ +13) (hx)2x1 = x1(hx)2 = xn−2he2h; +(hx)2xi = xi(hx)2 = xn−2ien−2i+1 · · · e⌊ n+1 +2 ⌋he2 · · · ei−1ei+1 · · · e⌊ n +2 ⌋h(xh)2, for 2 ⩽ i ⩽ ⌊n−1 +2 ⌋; +(hx)2yi = yi(hx)2 = hxn−2i+1hxe2 · · · eiei+2 · · · e⌊ n+1 +2 ⌋hen−2i+1 · · · e⌊ n +2 ⌋h, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋; +14 + +(V ′ +14) x1(xh)2 = e2x1 = y1e2 = (xh)2y1 = e2 · · · e⌊ n+1 +2 ⌋he2 · · · e⌊ n +2 ⌋h(xh)2; +xiheih = ei+1xi = yiei+1 = heihyi = e2 · · · e⌊ n+1 +2 ⌋he2 · · · e⌊ n +2 ⌋h(xh)2, for 2 ⩽ i ⩽ ⌊n−1 +2 ⌋; +( ¯V ′ +0) h2 = 1; +( ¯V ′ +1) he⌊ n+1 +2 ⌋ = e⌊ n+1 +2 ⌋h, if n is odd; +hxi = hxn−i−1hxixi−1h and hyi = hxi−1hyixn−i−1h, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋; +( ¯V ′ +2) e2 · · · e⌊ n+1 +2 ⌋he2 · · · e⌊ n +2 ⌋(hx)2 = xn−1. +Thus, we have: +Theorem 4.2 The monoid MDIn is defined by the presentation ⟨ ¯B′ | ¯V ′⟩ on 2 + 3⌊n−1 +2 ⌋ generators and +2n2 + 7−(−1)n +4 +n − 2(−1)n − 1 relations. +5 +Presentations for OPDIn +As in both previous sections, we first determine a presentation for OPDIn on an extended set of generators, +namely, with n+⌊n−1 +2 ⌋+1 generators, and then, through Tietze transformations, we deduce another presentation +for OPDIn on a minimum size set of generators, i.e. on 2 + ⌊n−1 +2 ⌋ generators. +Consider the alphabet D = {g, e1, . . . , en, x1, . . . , x⌊ n−1 +2 ⌋} and let ψ : D∗ −→ OPDIn be the homomorphism +of monoids that extends the mapping D −→ OPDIn defined by +g �−→ g, +ei �−→ ei, for 1 ⩽ i ⩽ n, +xj �−→ xj, for 1 ⩽ j ⩽ ⌊n−1 +2 ⌋. +Let Q be the set formed by the following 1 +2(3n2 + (1 − (−1)n)n + 3 − 1+(−1)n +2 +) monoid relations: +(Q1) gn = 1; +(Q2) e2 +i = ei, for 1 ⩽ i ⩽ n; +(Q3) eiej = ejei, for 1 ⩽ i < j ⩽ n; +(Q4) ge1 = eng and gei+1 = eig, for 1 ⩽ i ⩽ n − 1; +(Q5) ge1 · · · en = e1 · · · en; +(Q6) e1xi = xie1 = gn−2ie1 · · · en−ien−i+2 · · · en, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋; +(Q7) xiej = xi, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, 2 ⩽ j ⩽ n and j ̸= n − i + 1; +(Q8) ejxi = xi, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, 2 ⩽ j ⩽ n and j ̸= i + 1; +(Q9) xien−i+1 = ei+1xi = e2 · · · en, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋; +(Q10) (xigi)2 = e2 · · · eiei+2 · · · en, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋. +Our aim is to show that the monoid OPDIn is defined by the presentation ⟨D | Q⟩. As in Section 3, we +will make use of results of [14], this time in view to applying Proposition 1.1. +We begin by noticing that it is a routine matter to check: +Lemma 5.1 The set of generators {g, e1, e2, . . . , en, x1, . . . , x⌊ n−1 +2 ⌋} of OPDIn satisfies (via ψ) all relations +from Q. +15 + +Observe that, as a consequence of the previous lemma, if u, v ∈ D∗ are such that the relation u = v is a +consequence of Q, then uψ = vψ. +Now, let us recall that the cyclic inverse monoid CIn is generated by {g, e1} (see [14]) and, even more so, +by {g, e1, . . . , en}. +Let us consider the alphabet D0 = {g, e1, . . . , en} and denote by Q0 the subset of Q consisting of relations +Q1 to Q5. Then, we have: +Proposition 5.2 ([14, Theorem 2.6]) The monoid CIn is defined by the presentation ⟨D0 | Q0⟩ on n + 1 +generators and 1 +2(n2 + 3n + 4) relations. +To prove this result, the author used the property given by the following lemma, which we will also use here: +Lemma 5.3 ([14, Lemma 2.4]) Let u ∈ D∗ +0. Then, there exist 0 ⩽ m ⩽ n − 1, 1 ⩽ i1 < · · · < ik ⩽ n and +0 ⩽ k ⩽ n such that the relation u = gmei1 · · · eik is a consequence of relations Q1 to Q4. +Observe that, it is easy to show that, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, the relations +eigm = +� gmei+m +if 0 ⩽ m ⩽ n − i +gmei+m−n +if n − i + 1 ⩽ m ⩽ n − 1 +and +gmei = +� ei−mgm +if 0 ⩽ m ⩽ i − 1 +en+i−mgm +if i ⩽ m ⩽ n − 1 +(9) +are consequences of Q4. +Combining (9) with Lemma 5.3, we immediately obtain the symmetric result of the latter one: +Lemma 5.4 Let u ∈ D∗ +0. Then, there exist 0 ⩽ m ⩽ n − 1, 1 ⩽ i1 < · · · < ik ⩽ n and 0 ⩽ k ⩽ n such that the +relation u = ei1 · · · eikgm is a consequence of relations Q1 to Q4. +From now on, we denote the congruence ρQ of D∗ again by ≈. +By Lemma 3.4, we can conclude: +Lemma 5.5 For all 1 ⩽ i, j ⩽ ⌊n−1 +2 ⌋, xixj ≈ e2 · · · en. +Next, we prove a series of lemmas. +Lemma 5.6 Let u ∈ D∗ +0 and let 1 ⩽ i, j ⩽ ⌊n−1 +2 ⌋. Then, there exists v ∈ D∗ +0 ∪xiD∗ +0 ∪D∗ +0xj such that xiuxj ≈ v. +Proof. By Lemma 5.3, there exist 0 ⩽ m ⩽ n − 1, 1 ⩽ i1 < · · · < ik ⩽ n and 0 ⩽ k ⩽ n such that +u ≈ gmei1 · · · eik. +If i1 = 1 (with k > 0) then +xiuxj ≈ xigmei2 · · · eike1xj ≈ xigmei2 · · · eikgn−2je1 · · · en−jen−j+2 · · · en ∈ xiD∗ +0, +by Q3 and Q6. +If j + 1 ∈ {i1, . . . , ik} (with k > 0) then, being 1 ⩽ ℓ ⩽ k such that j + 1 = iℓ, we have +xiuxj ≈ xigmei1 · · · eiℓ−1eiℓ+1 · · · eikej+1xj ≈ xigmei1 · · · eiℓ−1eiℓ+1 · · · eike2 · · · en ∈ xiD∗ +0, +by Q3 and Q9. +Now, suppose that either k = 0 or i1 > 1 and j+1 ̸∈ {i1, . . . , ik} (with k > 0). Then, by Q8, ei1 · · · eikxj ≈ xj +and so xiuxj ≈ xigmxj. +If m = 0 then, by Lemma 5.5, xiuxj ≈ xixj ≈ e2 · · · en ∈ D∗ +0. So, suppose that m > 0. +If m ̸= i then n − i + 1 ̸= n − m + 1 ⩾ 2 and so +xiuxj ≈ xien−m+1gmxj ≈ xigme1xj ≈ xigmgn−2je1 · · · en−jen−j+2 · · · en ∈ xiD∗ +0, +16 + +by Q7, (9) and Q6. +If m ̸= j then j + 1 ̸= m + 1 ⩾ 2 and so +xiuxj ≈ xigmem+1xj ≈ xie1gmxj ≈ gn−2ie1 · · · en−ien−i+2 · · · engmxj ∈ D∗ +0xj, +by Q8, (9) and Q6. +Finally, if m = i = j then +xiuxj ≈ xigixi ≈ xigixigign−i = (xigi)2gn−i ≈ e2 · · · eiei+2 · · · engn−i ∈ D∗ +0, +by Q1 and Q10, as required. +Lemma 5.7 Let w ∈ D∗. Then, there exists w′ ∈ D∗ +0 ∪ D∗ +0x1D∗ +0 ∪ · · · ∪ D∗ +0x⌊ n−1 +2 ⌋D∗ +0 such that w ≈ w′. +Proof. We proceed by induction on the number of occurrences of the letters x1, . . . , x⌊ n−1 +2 +⌋ in a word w ∈ D∗. +If w ∈ D∗ has no occurrences of the letters x1, . . . , x⌊ n−1 +2 ⌋ then w ∈ D∗ +0 and so there is nothing to prove. +Hence, for k ⩾ 1, suppose that the lemma is valid for all words in D∗ with k − 1 occurrences of the letters +x1, . . . , x⌊ n−1 +2 +⌋. +Let w be a word of D∗ with k occurrences of the letters x1, . . . , x⌊ n−1 +2 +⌋. Then w = u0xi1u1 · · · xik−1uik−1xikuk, +for some u0, u1, . . . , uk ∈ D∗ +0 and 1 ⩽ i1, . . . , ik ⩽ ⌊n−1 +2 ⌋. Hence, by induction hypothesis, there exists u′ ∈ +D∗ +0 ∪ D∗ +0x1D∗ +0 ∪ · · · ∪ D∗ +0x⌊ n−1 +2 +⌋D∗ +0 such that u0xi1u1 · · · xik−1uik−1 ≈ u′ and so w ≈ u′xikuk. +If u′ ∈ D∗ +0 then the proof is finished. +So, suppose that u′ = u′ +0xiu′ +1, for some u′ +0, u′ +1 ∈ D∗ +0 and some +1 ⩽ i ⩽ ⌊n−1 +2 ⌋. Thus, by Lemma 5.6, there exists v ∈ D∗ +0 ∪ xiD∗ +0 ∪ D∗ +0xik such that xiu′ +1xik ≈ v, whence +w ≈ u′ +0xiu′ +1xikuk ≈ u′ +0vuk ∈ D∗ +0 ∪ D∗ +0xiD∗ +0 ∪ D∗ +0xikD∗ +0, as required. +Lemma 5.8 Let w ∈ D∗. Then, there exists u ∈ D∗ +0 such that w ≈ u or there exist 1 ⩽ i ⩽ ⌊n−1 +2 ⌋ and +0 ⩽ r, s ⩽ n − 1 such that w ≈ grxigs. +Proof. First, by Lemma 5.7, take w′ ∈ D∗ +0 ∪ D∗ +0x1D∗ +0 ∪ · · · ∪ D∗ +0x⌊ n−1 +2 ⌋D∗ +0 such that w ≈ w′. If w′ ∈ D∗ +0 then +the proof is finished. So, suppose that w′ = uxiv, for some u, v ∈ D∗ +0 and some 1 ⩽ i ⩽ ⌊n−1 +2 ⌋. Then, by +Lemmas 5.3 and 5.4, u ≈ grei1 · · · eik and v ≈ ej1 · · · ejℓgs, for some 0 ⩽ r, s ⩽ n − 1, 1 ⩽ i1 < · · · < ik ⩽ n, +1 ⩽ j1 < · · · < jℓ ⩽ n and 0 ⩽ k, ℓ ⩽ n. Hence w ≈ grei1 · · · eikxiej1 · · · ejℓgs. +If i1 = 1 (with k > 0) then, by Q3 and Q6, +w ≈ grei2 · · · eike1xiej1 · · · ejℓgs ≈ grei2 · · · eikgn−2ie1 · · · en−ien−i+2 · · · enej1 · · · ejℓgs ∈ D∗ +0. +If i + 1 ∈ {i1, . . . , ik} (with k > 0) then, being 1 ⩽ p ⩽ k such that i + 1 = ip, we have +w ≈ grei1 · · · eip−1eip+1 · · · eikei+1xiej1 · · · ejℓgs ≈ grei1 · · · eip−1eip+1 · · · eike2 · · · enej1 · · · ejℓgs ∈ D∗ +0, +by Q3 and Q9. +If j1 = 1 (with ℓ > 0) then, by Q6, +w ≈ grei1 · · · eikxie1ej2 · · · ejℓgs ≈ grei1 · · · eikgn−2ie1 · · · en−ien−i+2 · · · enej2 · · · ejℓgs ∈ D∗ +0. +If n − i + 1 ∈ {j1, . . . , jℓ} (with ℓ > 0) then, being 1 ⩽ q ⩽ ℓ such that n − i + 1 = jq, we have +w ≈ grei1 · · · eikxien−i+1ej1 · · · ejq−1ejq+1 · · · ejℓgs ≈ grei1 · · · eike2 · · · enej1 · · · ejq−1ejq+1 · · · ejℓgs ∈ D∗ +0, +by Q3 and Q9. +Finally, suppose that none of the four previous cases occurs. Then, by Q7 and Q8, ei1 · · · eikxiej1 · · · ejℓ ≈ xi +and so w ≈ grxigs, as required. +17 + +For 1 ⩽ i ⩽ ⌊n−1 +2 ⌋ and 0 ⩽ r, s ⩽ n − 1, let us consider the transformation grxigs. It is a routine matter to +check that +grxigs = + + + + + + + + + + + + + + + +� +1 +1 + i +1 + s +a +� +if r = 0 +�n − r + 1 +n − r + i + 1 +1 + s +a +� +if r > 0 and 1 ⩽ i ⩽ r − 1 +�i − r + 1 +n − r + 1 +a +1 + s +� +if r > 0 and r ⩽ i ⩽ ⌊n−1 +2 ⌋, +with +a = +� s − i + 1 +if 1 ⩽ i ⩽ s +n + s − i + 1 +if s + 1 ⩽ i ⩽ ⌊n−1 +2 ⌋. +Hence, it is easy to show that +xi = grxigs if and only if r = s = 0. +(10) +Now, recall that {g, e1} generates CIn and that {g, e1, x1, x2, . . . , x⌊ n−1 +2 ⌋} is a minimal generating set of +OPDIn. Therefore, noticing also that gn = 1, we have +grxigs ̸∈ CIn, for all 1 ⩽ i ⩽ ⌊n−1 +2 ⌋ and r, s ⩾ 0, +(11) +and +xj = grxigs, with 1 ⩽ i, j ⩽ ⌊n−1 +2 ⌋ and r, s ⩾ 0, implies i = j. +(12) +We are now in a position to prove our first objective of this section. +Theorem 5.9 The monoid OPDIn is defined by the presentation ⟨D | Q⟩ on n + ⌊n−1 +2 ⌋ + 1 generators and +1 +2(3n2 + (1 − (−1)n)n + 3 − 1+(−1)n +2 +) relations. +Proof. Given Lemma 5.1, by Proposition 1.1, it remains to prove that w1 ≈ w2 for all words w1, w2 ∈ D∗ such +that w1ψ = w2ψ. So, let w1, w2 ∈ D∗ be such that w1ψ = w2ψ. +By Lemma 5.8, for k ∈ {1, 2}, there exists uk ∈ D∗ +0 such that wk ≈ uk (and then wkψ = ukψ ∈ CIn) or there +exist 1 ⩽ ik ⩽ ⌊n−1 +2 ⌋ and 0 ⩽ rk, sk ⩽ n−1 such that wk ≈ grkxikgsk (and then, by (11), wkψ = (grkxikgsk)ψ = +grkxikgsk ̸∈ CIn). Since w1ψ = w2ψ, we can only have: (case 1) w1 ≈ u1 and w2 ≈ u2, for some u1, u2 ∈ D∗ +0; +or (case 2) w1 ≈ gr1xi1gs1 and w2 ≈ gr2xi2gs2, for some 1 ⩽ i1, i2 ⩽ ⌊n−1 +2 ⌋ and 0 ⩽ r1, s1, r2, s2 ⩽ n − 1. +If case 1 occurs, then u1ψ = w1ψ = w2ψ = u2ψ and so u1ρQ0u2, since ⟨D0 | Q0⟩ is a presentation of CIn, +by Proposition 5.2, which implies that u1 ≈ u2 and thus w1 ≈ w2. +Now, suppose we have case 2. Then gr1xi1gs1 = w1ψ = w2ψ = gr2xi2gs2 and so xi1 = gr2−r1+r3xi2gs2−s1+s3, +where +r3 = +� 0 +if r1 ⩽ r2 +n +if r1 > r2 +and +s3 = +� 0 +if s1 ⩽ s2 +n +if s1 > s2, +whence i1 = i2, by (12). Thus, it follows by (10) that r2 − r1 + r3 = 0 = s2 − s1 + s3. If r3 = n then n = r1 − r2, +which is a contradiction, since 0 ⩽ r1, r2 ⩽ n − 1. Therefore r3 = 0 and, analogously, s3 = 0, whence r1 = r2 +and s1 = s2. Thus w1 ≈ gr1xi1gs1 = gr2xi2gs2 ≈ w2, as required. +Next, by using Tietze transformations and applying Proposition 1.3, we deduce from the previous presen- +tation for OPDIn a new one on the minimal set of generators {g, e1, x1, x2, . . . , x⌊ n−1 +2 ⌋} of OPDIn. +By noticing that ei = gn−i+1e1gi−1 for 2 ⩽ i ⩽ n (as transformations), we proceed as follows: first, by +applying T1, we add the relations ei = gn−i+1e1gi−1, for 2 ⩽ i ⩽ n; secondly, we apply T4 with each of the +relations ei = gn−i+1e1gi−1 with 2 ⩽ i ⩽ n; finally, by using the relation Q1, we simplify the new relations +obtained, eliminating the trivial ones or those that are deduced from others. By performing this procedure for +each of the sets of relations Q1 to Q10, it is a routine matter to check that we can obtain the following set Q′ +of 1 +2(3n2 − (3 + (−1)n)n + 5 − 1+(−1)n +2 +) monoid relations on the alphabet D′ = {g, e1, x1, x2, . . . , x⌊ n−1 +2 +⌋}: +18 + +(Q′ +1) gn = 1; +(Q′ +2) e2 +1 = e1; +(Q′ +3) e1gn−j+ie1gn−i+j = gn−j+ie1gn−i+je1, for 1 ⩽ i < j ⩽ n; +(Q′ +5) g(e1gn−1)n = (e1gn−1)n; +(Q′ +6) e1xi = xie1 = gn−2i(e1gn−1)n−i−1e1(e1gn−1)i−1, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋; +(Q′ +7) xign−j+1e1gj−1 = xi, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, 2 ⩽ j ⩽ n and j ̸= n − i + 1; +(Q′ +8) gn−j+1e1gj−1xi = xi, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋, 2 ⩽ j ⩽ n and j ̸= i + 1; +(Q′ +9) xigie1gn−i = gn−ie1gixi = gn−1(e1gn−1)n−1, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋; +(Q′ +10) (xigi)2 = gn−1(e1gn−1)i−2e1gn−2(e1gn−1)n−i−1, for 1 ⩽ i ⩽ ⌊n−1 +2 ⌋. +Thus, we have: +Theorem 5.10 The monoid OPDIn is defined by the presentation ⟨D′ | Q′⟩ on 2 + ⌊n−1 +2 ⌋ generators and +1 +2(3n2 − (3 + (−1)n)n + 5 − 1+(−1)n +2 +) relations. +References +[1] A.Ya. A˘ızenˇstat, Defining relations of finite symmetric semigroups, Mat. Sb. N. S. 45 (1958), 261–280 +(Russian). +[2] A.Ya. A˘ızenˇstat, The defining relations of the endomorphism semigroup of a finite linearly ordered set, +Sibirsk. Mat. 3 (1962), 161–169 (Russian). +[3] F. Al-Kharousi, R. Kehinde and A. Umar, Combinatorial results for certain semigroups of partial isometries +of a finite chain, Australas. J. Combin. 58 (2014), 365–375. +[4] F. Al-Kharousi, R. Kehinde and A. Umar, On the semigroup of partial isometries of a finite chain, Commun. +Algebra 44 (2016), 639–647. +[5] P.M. Catarino and P.M. Higgins, The monoid of orientation-preserving mappings on a chain, Semigroup +Forum 58 (1999), 190–206. +[6] S. Cical`o, V.H. Fernandes and C. Schneider, Partial transformation monoids preserving a uniform partition, +Semigroup Forum 90 (2015), 532–544. +[7] I. Dimitrova, The Maximal Subsemigroups of the Semigroup of all Partial Order-preserving Isometries, +Proceedings of the 5-th International Scientific Conference FMNS-2013, Vol. 1 (2013), 95–101. +[8] I. Dimitrova, V.H. Fernandes, J. Koppitz and T.M. Quinteiro, On three remarkable submonoids of the di- +hedral inverse monoid on a finite set, arXiv:2301.01519 (2023), https://doi.org/10.48550/arXiv.2301.01519. +[9] J. East, Generators and relations for partition monoids and algebras, J. Algebra 339 (2011), 1–26. +[10] Y.-Y. Feng, A. Al-Aadhami, I. Dolinka, J. East and V. Gould, Presentations for singular wreath products, +J. Pure Appl. Algebra 223 (2019), 5106–5146. +[11] V.H. Fernandes, The monoid of all injective orientation preserving partial transformations on a finite chain, +Commun. Algebra 28 (2000), 3401–3426. +19 + +[12] V.H. Fernandes, The monoid of all injective order preserving partial transformations on a finite chain, +Semigroup Forum 62 (2001), 178-204. +[13] V.H. Fernandes, Presentations for some monoids of partial transformations on a finite chain: a survey, +Semigroups, Algorithms, Automata and Languages, eds. Gracinda M. S. Gomes & Jean-´Eric Pin & Pedro +V. Silva, World Scientific (2002), 363–378. +[14] V.H. Fernandes, On the cyclic inverse monoid on a finite set, arXiv:2211.02155 (2022), +https://doi.org/10.48550/arXiv.2211.02155. +[15] V.H. Fernandes, G.M.S. Gomes and M.M. Jesus, Presentations for some monoids of injective partial trans- +formations on a finite chain, Southeast Asian Bull. Math. 28 (2004), 903–918. +[16] V.H. Fernandes and T. Paulista, On the monoid of partial isometries of a finite star graph, Commun. +Algebra (DOI 10.1080/00927872.2022.2121404). Online (2022). +[17] V.H. Fernandes and T. Paulista, On the monoid of partial isometries of a cycle graph, arXiv:2205.02196v2 +(2022), https://doi.org/10.48550/arXiv.2205.02196. +[18] V.H. Fernandes and T.M. Quinteiro, Presentations for monoids of finite partial isometries, Semigroup +Forum 93 (2016), 97–110. +[19] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.11.1; 2021. +(https://www.gap-system.org) +[20] J.M. Howie, Fundamentals of Semigroup Theory, Oxford, Oxford University Press, 1995. +[21] J.M. Howie and N. Ruˇskuc, Constructions and presentations for monoids, Commun. Algebra 22 (1994), +6209–6224. +[22] P.M. Higgins and A. Vernitski, Orientation-preserving and orientation-reversing mappings: a new descrip- +tion, Semigroup Forum 104 (2022), 509–514. +[23] G. Lallement, Semigroups and Combinatorial Applications, John Wiley & Sons, New York, 1979. +[24] D. McAlister, Semigroups generated by a group and an idempotent, Commun. Algebra 26 (1998), 515–547. +[25] E.H. Moore, Concerning the abstract groups of order k! and 1 +2k! holohedrically isomorphic with the sym- +metric and the alternating substitution groups on k letters, Proc. London Math. Soc. 28 (1897), 357–366. +[26] L.M. Popova, The defining relations of certain semigroups of partial transformations of a finite set, +Leningrad. Gos. Ped. Inst. Uˇcen. Zap. 218 (1961), 191–212 (Russian). +[27] L.M. Popova, Defining relations of a semigroup of partial endomorphisms of a finite linearly ordered set, +Leningrad. Gos. Ped. Inst. Uˇcen. Zap. 238 (1962), 78–88 (Russian). +[28] N. Ruˇskuc, Semigroup Presentations, Ph.D. Thesis, University of St-Andrews, 1995. +Ilinka Dimitrova, Department of Mathematics, Faculty of Mathematics and Natural Science, South-West University ”Neofit +Rilski”, 2700 Blagoevgrad, Bulgaria; e-mail: ilinka dimitrova@swu.bg. +V´ıtor H. Fernandes, Center for Mathematics and Applications (NovaMath) and Department of Mathematics, Faculdade de +Ciˆencias e Tecnologia, Universidade Nova de Lisboa, Monte da Caparica, 2829-516 Caparica, Portugal; e-mail: vhf@fct.unl.pt. +J¨org Koppitz, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria; e-mail: +koppitz@math.bas.bg. +Teresa M. Quinteiro, Instituto Superior de Engenharia de Lisboa, 1950-062 Lisboa, Portugal. Also: Center for Mathematics +and Applications (NovaMath), Faculdade de Ciˆencias e Tecnologia, Universidade Nova de Lisboa, Monte da Caparica, 2829-516 +Caparica, Portugal; e-mail: tmelo@adm.isel.pt. +20 + diff --git a/RdE0T4oBgHgl3EQfkgGS/content/tmp_files/load_file.txt b/RdE0T4oBgHgl3EQfkgGS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..88a05425fdc390947f96d736f359326347af7dee --- /dev/null +++ b/RdE0T4oBgHgl3EQfkgGS/content/tmp_files/load_file.txt @@ -0,0 +1,1042 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf,len=1041 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='02474v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='RA] 6 Jan 2023 Presentations for three remarkable submonoids of the dihedral inverse monoid on a finite set I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Dimitrova, V´ıtor H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Fernandes∗ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Koppitz and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Quinteiro† January 9, 2023 Abstract In this paper we consider the submonoids OPDIn, MDIn and ODIn of the dihedral inverse monoid DIn of all orientation-preserving, monotone and order-preserving transformations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Our goal is to exhibit presentations for each of these three monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 2020 Mathematics subject classification: 20M20, 20M05, 05C12, 05C25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Keywords: dihedral inverse monoid, transformations, orientation, monotonicity, presentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Introduction Let Ω be a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Denote by PT (Ω) the monoid (under composition) of all partial transformations on Ω, by T (Ω) the submonoid of PT (Ω) of all full transformations on Ω, by I(Ω) the symmetric inverse monoid on Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' the inverse submonoid of PT (Ω) of all partial permutations on Ω, and by S(Ω) the symmetric group on Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' the subgroup of PT (Ω) of all permutations on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If Ω is a finite set with n elements (n ∈ N), say Ω = Ωn = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , n}, as usual, we denote PT (Ω), T (Ω), I(Ω) and S(Ω) simply by PT n, Tn, In and Sn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Recall that the rank of a partial transformation α ∈ PT n is the size of Im(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' On the other hand, the rank of a monoid M is the minimum size of a generating set of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Next, suppose that Ωn is a chain, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Ωn = {1 < 2 < · · · < n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' An element α ∈ PT n is called order-preserving [order-reversing] if x ⩽ y implies xα ⩽ yα [xα ⩾ yα], for all x, y ∈ Dom(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' A partial transformation is said to be monotone if it is order-preserving or order-reversing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We denote by POn the submonoid of PT n of all order-preserving transformations and by PODn the submonoid of PT n of all monotone transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let also POIn = POn ∩ In, the monoid of all order-preserving partial permutations of Ωn, and PODIn = PODn ∩ In, the monoid of all monotone partial permutations of Ωn, which are inverse submonoids of PT n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let s = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , at) be a sequence of t (t ⩾ 0) elements from the chain Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We say that s is cyclic [anti- cyclic] if there exists no more than one index i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , t} such that ai > ai+1 [ai < ai+1], where at+1 denotes a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We also say that s is oriented if s is cyclic or s is anti-cyclic (see [5, 22, 24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Given a partial transformation α ∈ PT n such that Dom(α) = {a1 < · · · < at}, with t ⩾ 0, we say that α is orientation-preserving [orientation- reversing, oriented] if the sequence of its images (a1α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , atα) is cyclic [anti-cyclic, oriented].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We denote by POPn the submonoid of PT n of all orientation-preserving transformations and by PORn the submonoid of PT n of all oriented transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Consider also the inverse submonoids POPIn = POPn ∩ In, of all ∗This work is funded by national funds through the FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=', under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (NovaMath - Center for Mathematics and Applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' †This work is funded by national funds through the FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=', under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (NovaMath - Center for Mathematics and Applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 1 orientation-preserving partial permutations, and PORIn = PORn ∩ In, of all oriented partial permutations, of PT n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Notice that, by definition, POIn ⊆ PODIn ⊆ PORIn and POIn ⊆ POPIn ⊆ PORIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' A monoid presentation is an ordered pair ⟨A | R⟩, where A is a set, often called an alphabet, and R ⊆ A∗×A∗ is a set of relations of the free monoid A∗ generated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' A monoid M is said to be defined by a presentation ⟨A | R⟩ if M is isomorphic to A∗/ρR, where ρR denotes the smallest congruence on A∗ containing R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' A presentation for the symmetric group Sn was determined by Moore [25] over a century ago (1897).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' For the full transformation monoid Tn, a presentation was given in 1958 by A˘ızenˇstat [1] in terms of a certain type of two generator presentation for the symmetric group Sn, plus an extra generator and seven more relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Presentations for the partial transformation monoid PT n and for the symmetric inverse monoid In were found by Popova [26] in 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' In 1962, A˘ızenˇstat [2] and Popova [27] exhibited presentations for the monoids of all order- preserving transformations and of all order-preserving partial transformations of a finite chain, respectively, and from the sixties until our days several authors obtained presentations for many classes of monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' See also [28], the survey [13] and, for example, [6, 9, 10, 12, 15, 18, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let G = (V, E) be a finite simple connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Recall that the (geodesic) distance between two vertices x and y of G, denoted by dG(x, y), is the length of a shortest path between x and y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' the number of edges in a shortest path between x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We say that a partial transformation α ∈ PT (V ) is a partial isometry or distance preserving partial transformation of G if dG(xα, yα) = dG(x, y) for all x, y ∈ Dom(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let us denote by DP(G) the submonoid of PT (V ) of all partial isometries of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Since dG(x, y) = 0 if and only if x = y, for all x, y ∈ V , it follows that DP(G) is an inverse submonoid of I(V ) (see [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Clearly, if G = (V, E) is a complete graph, then DP(G) = I(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' On the other hand, if Pn is an undirected path with n vertices then DP(Pn) coincides with the monoid of all partial isometries on Ωn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' the submonoid DPn = {α ∈ In | |iα − jα| = |i − j|, for all i, j ∈ Dom(α)} of In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The study of partial isometries on Ωn was initiated by Al-Kharousi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The first of these two papers is dedicated to investigating some combi- natorial properties of the monoid DPn and of its submonoid ODPn of all order-preserving partial isometries, in particular, their cardinalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The second paper presents the study of some of their algebraic properties, namely Green’s structure and ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Presentations for both monoids DPn and ODPn were given by Fernandes and Quinteiro in [18] and the maximal subsemigroups of ODPn were characterized by Dimitrova in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The monoid DP(Sn) of all partial isometries of a star graph Sn with n vertices (n ⩾ 1) was considered by Fernandes and Paulista in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' They determined the rank and size of DP(Sn) as well as described its Green’s relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' A presentation for DP(Sn) was also exhibited in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Next, for n ⩾ 3, consider the cycle graph Cn = ({1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , n}, {{i, i + 1} | i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , n − 1} ∪ {{1, n}}) with n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The monoid DP(Cn) of all partial isometries of the cycle graph Cn was studied by Fernandes and Paulista in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' They showed that DP(Cn) is an inverse submonoid of the monoid of all oriented partial permutations on a chain with n elements and, moreover, that it coincides with the inverse submonoid of In formed by all restrictions of a dihedral subgroup of order 2n of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' This fact motivated the authors of [17] to call DP(Cn) by dihedral inverse monoid on Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Also in [17], it was determined the cardinal and rank of DP(Cn) as well as descriptions of its Green’s relations and, furthermore, presentations for DP(Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' From now on, as in [8], we denote DP(Cn) by the most appropriate notation DIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' In this paper, we consider three remarkable submonoids of DIn, namely OPDIn = DIn ∩ POPIn, the monoid of all orientation-preserving partial isometries of Cn, MDIn = DIn ∩ PODIn, the monoid of all monotone partial isometries of Cn, and ODIn = DIn ∩ POIn, the monoid of all order-preserving partial isometries of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Observe that DIn, OPDIn, MDIn and ODIn are all inverse submonoids of the symmetric inverse monoid In, ODIn ⊆ MDIn and ODIn ⊆ OPDIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' These three monoids were also studied in [8] by the same authors who characterized their Green’s relations and calculated their cardinals and ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Here, for n ⩾ 4, we aim to determine presentations for the monoids ODIn, MDIn and OPDIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Notice that, as ODI3 = POI3, OPDI3 = POPI3 and MDI3 = PODI3, presentations for n = 3 are already known (see [11, 12, 15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Throughout this paper, we take n ⩾ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 2 For general background on Semigroup Theory and standard notations, we refer to Howie’s book [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We would like to point out that we made considerable use of computational tools, namely GAP [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 1 A note on presentations In this section, we recall some notions related to the concept of a monoid presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let A be an alphabet and consider the free monoid A∗ generated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The elements of A and of A∗ are called letters and words, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The empty word is denoted by 1 and we write A+ to express A∗ \\ {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' For all u ∈ A∗, the power u0 also denotes the empty word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' A pair (u, v) of A∗ × A∗ is called a relation of A∗ and it is usually represented by u = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' A relation u = v of A∗ is said to be a consequence of R if (u, v) ∈ ρR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' A set of representatives W ⊆ A∗ for the congruence ρR is also called a set of forms for the presentation ⟨A | R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let X be a generating set of a monoid M and let φ : A −→ M be an injective mapping such that Aφ = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let ϕ : A∗ −→ M be the (surjective) homomorphism of monoids that extends φ to A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We say that X satisfies (via ϕ) a relation u = v of A∗ if uϕ = vϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' For more details see [23] or [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' A direct method to find a presentation for a monoid is described by the following well-known result (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' see [28, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1 Let M be a monoid generated by a set X, let A be an alphabet and let φ : A −→ M be an injective mapping such that Aφ = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let ϕ : A∗ −→ M be the (surjective) homomorphism that extends φ to A∗ and let R ⊆ A∗ × A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then ⟨A | R⟩ is a presentation for M if and only if the following two conditions are satisfied: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The generating set X of M satisfies (via ϕ) all relations from R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If w1, w2 ∈ A∗ are any two words such that the generating set X of M satisfies (via ϕ) the relation w1 = w2 then w1 = w2 is a consequence of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' An usual method to find a presentation for a finite monoid is described by the following result (adapted to the monoid case from [28, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2 (Guess and Prove method) Let M be a finite monoid generated by a set X, let A be an alphabet and let φ : A −→ M be an injective mapping such that Aφ = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let ϕ : A∗ −→ M be the (surjective) homomorphism that extends φ to A∗, let R ⊆ A∗ × A∗ and W ⊆ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Assume that the following conditions are satisfied: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The generating set X of M satisfies (via ϕ) all relations from R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' For each word w ∈ X∗, there exists a word w′ ∈ W such that the relation w = w′ is a consequence of R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' |W| ⩽ |M|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, M is defined by the presentation ⟨A | R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Notice that, if W satisfies the above conditions then, in fact, |W| = |M| and W is a set of forms for ⟨A | R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Given a presentation for a monoid, another method to find a new presentation consists in applying Tietze transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' For a monoid presentation ⟨A | R⟩, the four elementary Tietze transformations are: (T1) Adding a new relation u = v to ⟨A | R⟩, provided that u = v is a consequence of R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (T2) Deleting a relation u = v from ⟨A | R⟩, provided that u = v is a consequence of R\\{u = v};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (T3) Adding a new generating symbol b and a new relation b = w, where w ∈ A∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 3 (T4) If ⟨A | R⟩ possesses a relation of the form b = w, where b ∈ A, and w ∈ (A\\{b})∗, then deleting b from the list of generating symbols, deleting the relation b = w, and replacing all remaining appearances of b by w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The following result is well-known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' see [28]): Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3 Two finite presentations define the same monoid if and only if one can be obtained from the other by a finite number of elementary Tietze transformations (T1), (T2), (T3) and (T4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Next, we recall the process, given in [15] (also used in [18]), to obtain a presentation for a finite monoid M given a presentation for a certain submonoid of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' This method will be applied in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let M be a (finite) monoid, let S be a submonoid of M and y be an element of M such that y2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let us suppose that M is generated by S and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , xk} (k ∈ N) be a generating set of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then Y = X ∪ {y} generates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let A = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , ak} be an alphabet, let b be a symbol not belonging to A and B = A ∪ {b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let ϕ : B∗ −→ M be the homomorphism that extends the map ai �−→ xi, for 1 ⩽ i ⩽ k, and b �−→ y to A∗ and let R ⊆ A∗ × A∗ be such that ⟨A | R⟩ is a presentation for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Consider a set of forms W for ⟨A | R⟩ and suppose there exist two subsets W1 and W2 of W and a word u0 ∈ A∗ such that W = W1 ∪ W2 and u0 is a factor of each word in W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Suppose there exist words v0, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , vk ∈ A∗ such that the following relations over the alphabet B are satisfied (via ϕ) by the generating set Y of M: ( ¯R1) bai = vib, for 1 ⩽ i ⩽ k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' ( ¯R2) u0b = v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Observe that the relation (over the alphabet B) ( ¯R0) b2 = 1 is also satisfied (via ϕ) by the generating set Y of M, by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let ¯R = R ∪ ¯R0 ∪ ¯R1 ∪ ¯R2 and ¯W = W ∪ {wb | w ∈ W2} ⊆ B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, in [15], Fernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' proved: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4 ([15, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4]) If W contains the empty word then ¯W is a set of forms for the pre- sentation ⟨B | ¯R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Moreover, if | ¯W| ⩽ |M| then the monoid M is defined by the presentation ⟨B | ¯R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 2 On the monoids ODIn, MDIn and OPDIn Now, we recall some properties of ODIn, MDIn and OPDIn, presented by the authors in [8], that we will need in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let us consider the following permutations of Ωn of order n and 2, respectively: g = �1 2 · · n − 1 n 2 3 · · n 1 � and h = �1 2 · · n − 1 n n n − 1 · · 2 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' It is clear that g, h ∈ DIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Moreover, g together with h generate the well-known dihedral group D2n of order 2n (considered as a subgroup of Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' In fact, we have D2n = ⟨g, h | gn = 1, h2 = 1, hg = gn−1h⟩ = {1, g, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , gn−1, h, hg, hg2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , hgn−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Observe that gk = � 1 2 · · n − k n − k + 1 · · n 1 + k 2 + k · · n 1 · · k � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' igk = � i + k if 1 ⩽ i ⩽ n − k i + k − n if n − k + 1 ⩽ i ⩽ n, 4 and hgk = �1 · · k k + 1 · · n k · · 1 n · · k + 1 � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' ihgk = � k − i + 1 if 1 ⩽ i ⩽ k n + k − i + 1 if k + 1 ⩽ i ⩽ n, for 0 ⩽ k ⩽ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Recall that DIn is the submonoid of the monoid PORIn whose elements are precisely all restrictions of the dihedral group D2n of order 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let also Cn be the cyclic group of order n generated by g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Cn = ⟨g | gn = 1⟩ = {1, g, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , gn−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Next, denote by id the identity transformation on Ωn and, for X ⊆ Ωn, by idX the partial identity with domain X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' idX = id|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Take ei = idΩn\\{i} = �1 · · i − 1 i + 1 · · n 1 · · i − 1 i + 1 · · n � ∈ DIn, for 1 ⩽ i ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Clearly, for 1 ⩽ i, j ⩽ n, we have e2 i = ei and eiej = idΩn\\{i,j} = ejei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' More generally, for any X ⊆ Ωn, we get Πi∈Xei = idΩn\\X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Since the elements of DIn are precisely the restrictions of D2n, it is easy to conclude that {g, h, e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en} is a generating set of DIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Moreover, since gn = 1 and ei = gn−iengi for all i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , n}, it follows that each set {g, h, ei}, with 1 ⩽ i ⩽ n, also generates DIn (see [8, 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Notice that g ∈ OPDIn, h ∈ MDIn and e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en are elements of ODIn, MDIn and OPDIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Consider the elements x = �1 2 · · n − 1 2 3 · · n � and y = x−1 = �2 3 · · n 1 2 · · n − 1 � of ODIn with rank n − 1 and the elements xi = �1 1 + i 1 n − i + 1 � and yi = x−1 i = �1 n − i + 1 1 1 + i � , for 1 ⩽ i ⩽ ⌊n−1 2 ⌋, of ODIn with rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The following result was proved by the authors in [8]: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1 ([8, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3]) For n ⩾ 4, {x, y, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en−1, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , y⌊ n−1 2 ⌋}, {h, x, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , e⌊ n+1 2 ⌋, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , y⌊ n−1 2 ⌋} and {g, ei, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋}, with 1 ⩽ i ⩽ n, are generating sets of minimal size of the monoids ODIn, MDIn and OPDIn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' In particular, the monoids ODIn, MDIn and OPDIn have ranks n + 2⌊n−1 2 ⌋, 2 + 3⌊n−1 2 ⌋ and 2 + ⌊n−1 2 ⌋, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Observe that n + 2⌊n−1 2 ⌋ = 2n − 3+(−1)n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We end this section by recalling that also in [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1] the authors showed the following equality: |ODIn| = 3 · 2n + (n + 1)n(n − 1) 6 − 1 + (−1)n 8 n2 − 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (1) 5 3 Presentations for ODIn In this section, we first determine a presentation for ODIn on 2n + 1−(−1)n 2 generators and, secondly, by using Tietze transformations, we deduce another presentation for ODIn on 2n − 3+(−1)n 2 generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Consider the alphabet A = {x, y, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , y⌊ n−1 2 ⌋} and the set R formed by the following monoid relations: (R1) e2 i = ei, for 1 ⩽ i ⩽ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (R2) xy = en and yx = e1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (R3) xe1 = x and e1y = y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (R4) eiej = ejei, for 1 ⩽ i < j ⩽ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (R5) eix = xei+1, for 1 ⩽ i ⩽ n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (R6) xiyi = e2 · · · eiei+2 · · · en, yixi = e2 · · · en−ien−i+2 · · · en, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (R7) xiej = xi, ejyi = yi, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋, 2 ⩽ j ⩽ n and j ̸= n − i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (R8) ejxi = xi, yiej = yi, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋, 2 ⩽ j ⩽ n and j ̸= i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (R9) e1xi = xie1 = xn−2ien−2i+1 · · · en−ien−i+2 · · · en, e1yi = yie1 = yn−2ie1 · · · eiei+2 · · · e2i, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (R10) xien−i+1 = ei+1xi = yiei+1 = en−i+1yi = e2 · · · en, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (R11) xe2 · · · en = e1 · · · en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Observe that |R| = 1 2(5n2 − (1 + 2(−1)n)n + (−1)n+1 + 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Throughout Section 3, we represent the congruence ρR of A∗ by ≈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We aim to show that the monoid ODIn is defined by the presentation ⟨A | R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' To this end, our strategy is to use Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2 together with the known presentation of a submonoid of ODIn determined by Fernandes in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Therefore, we begin to recall this presentation as well as some other auxiliary results that will also be useful to us here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let us denote by CIn the cyclic inverse monoid on Ωn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' the inverse submonoid of the symmetric inverse monoid on Ωn consisting of all restrictions of the cyclic group Cn, and by OCIn the submonoid of CIn formed by all order-preserving elements of CIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Clearly, OCIn is also a submonoid of ODIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Moreover, {x, y, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en} is a generating set of OCIn and |OCIn| = 3 · 2n − 2n − 2 [14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Furthermore, if U is the set of relations R1 to R5 together with relation R11 over the alphabet C = {x, y, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en}, we have: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1 ([14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='16]) The monoid OCIn is defined by the presentation ⟨C | U⟩ on n + 2 generators and 1 2(n2 + 3n + 8) relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The following lemma was crucial to prove the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Also here, it will be very useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2 ([14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='15]) Let u ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, there exist z ∈ {x, y}, v ∈ {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en}∗ and 0 ⩽ r ⩽ n − 1 such that u = zrv is a consequence of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Next, recall that the relations xjei = xj = en−i+1xj and eiyj = yj = yjen−i+1, for 1 ⩽ i ⩽ j ⩽ n, (2) are consequences of U (see [14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' In order to find a set of words W satisfying condition 2 of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2 for ⟨A | R⟩, we now present a series of lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The following lemma is easy to deduce from (2) and R10: 6 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3 Let 1 ⩽ i ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The following relations are consequences of R: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' yrxi = yre2 · · · en, for r ⩾ n − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' yryi = yre2 · · · en, for r ⩾ i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' xixs = e2 · · · enxs, for s ⩾ i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' yixs = e2 · · · enxs, for s ⩾ n − i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We prove the first relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' For the remaining three ones we can argue in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' By (2), we have yn−i ≈ yn−iei+1 and so, by R10, we get yn−ixi ≈ yn−iei+1xi ≈ yn−ie2 · · · en, whence yrxi = yr−n+iyn−ixi ≈ yr−n+iyn−ie2 · · · en = yre2 · · · en, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4 The following relations are consequences of R: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' xixj = yiyj = e2 · · · en, for 1 ⩽ i, j ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' xiyj = yjxi = e2 · · · en, for 1 ⩽ i, j ⩽ ⌊n−1 2 ⌋ and i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let 1 ⩽ i, j ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then j + 1 ̸= n − i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' In fact, if j + 1 = n − i + 1 then 1 ⩽ j ⩽ ⌊n−1 2 ⌋ =⇒ 1 ⩽ n − i ⩽ ⌊n−1 2 ⌋ =⇒ ⌊n−1 2 ⌋ < n − ⌊n−1 2 ⌋ ⩽ i, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Notice that j + 1 ̸= n − i + 1 is equivalent to i + 1 ̸= n − j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Hence, by R7, we have xi ≈ xiej+1 and ei+1yj ≈ yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Thus, by R1, R4 and R10, we get xixj ≈ xiej+1xj ≈ xie2 · · · en ≈ xien−i+1e2 · · · en−ien−i+2 · · · en ≈ e2 · · · ene2 · · · en−ien−i+2 · · · en ≈ e2 · · · en and yiyj ≈ yiei+1yj ≈ e2 · · · enyj ≈ e2 · · · en−jen−j+2 · · · enen−j+1yj ≈ e2 · · · en−jen−j+2 · · · ene2 · · · en ≈ e2 · · · en (observe that i, j < n implies n − i + 1, n − j + 1 > 1), which proves property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Now, in order to prove property 2, suppose also that i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then n − j + 1 ̸= n − i + 1 and i + 1 ̸= j + 1, whence xi ≈ xien−j+1, by R7, and yjei+1 ≈ yj, by R8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Thus, by R1, R4, R10 and the above calculations, we obtain xiyj ≈ xien−j+1yj ≈ xie2 · · · en ≈ e2 · · · en and yjxi ≈ yjei+1xi ≈ yje2 · · · en ≈ yjej+1e2 · · · ejej+2 · · · en ≈ e2 · · · ene2 · · · ejej+2 · · · en ≈ e2 · · · en, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='5 Let u ∈ C∗ and 1 ⩽ i ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' There exists u′ ∈ C∗ such that uxi ≈ u′ or there exists 0 ⩽ r ⩽ n − i − 1 such that uxi ≈ yrxi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' There exists u′ ∈ C∗ such that uyi ≈ u′ or there exists 0 ⩽ r ⩽ i − 1 such that uyi ≈ yryi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We prove property 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The argument for proving property 2 is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let z ∈ {x, y}, v ∈ {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en}∗ and 0 ⩽ r ⩽ n − 1 be such that u ≈ zrv, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Since v ∈ {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en}∗, by R1, R4 and R8, we have vxi ≈ v′xi, for some v′ ∈ {1, e1, ei+1, e1ei+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If v′ = et 1ei+1, for some t ∈ {0, 1}, then uxi ≈ zrv′xi = zret 1ei+1xi ≈ zret 1e2 · · · en ∈ C∗, by using R10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If v′ = e1 then uxi ≈ zrv′xi = zre1xi ≈ zrxn−2ien−2i+1 · · · en−ien−i+2 · · · en ∈ C∗, by using R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Now, suppose that v′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then uxi ≈ zrxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If r = 0 then there is nothing more to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' So, suppose also that r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If z = x then uxi ≈ xrxi ≈ xre1xi ≈ xrxn−2ien−2i+1 · · · en−ien−i+2 · · · en ∈ C∗, by R3 and R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If z = y then uxi ≈ yrxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If r ⩽ n − i − 1 there is nothing more to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' On the other hand, if r ⩾ n − i then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3, we get uxi ≈ yrxi ≈ yre2 · · · en ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Thus, the proof of property 1 is complete, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let ϕ : A∗ −→ ODIn be the homomorphism of monoids that extends the mapping A −→ ODIn defined by x �−→ x, y �−→ y, ei �−→ ei, for 1 ⩽ i ⩽ n, xj �−→ xj and yj �−→ yj, for 1 ⩽ j ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Notice that we are using the same symbols for the letters of the alphabet A and for the generators of ODIn, which simplifies notation and, within the context, will not cause ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' The following subsets of A∗, in a sense, were motivated by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3: W1 = {yrxixs | 0 ⩽ r, s ⩽ n − 1, 1 ⩽ i ⩽ ⌊n−1 2 ⌋ and s + 1 ⩽ i ⩽ n − r − 1} and W2 = {yryixs | 0 ⩽ r, s ⩽ n − 1, 1 ⩽ i ⩽ ⌊n−1 2 ⌋ and r + 1 ⩽ i ⩽ n − s − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Observe that |W1 ∪ W2| = 1 6(n + 1)n(n − 1) − 1 8(1 + (−1)n)n2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' the number of order-preserving restrictions of {hgk | 0 ⩽ k ⩽ n − 1} with rank (greater than or) equal to 2, except those that are also restrictions of {gk | 0 ⩽ k ⩽ n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' In fact, (W1 ∪ W2)ϕ is precisely this set of transformations of ODIn with rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Furthermore, we have: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='6 Let w ∈ A∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, there exists w′ ∈ C∗ ∪ W1 ∪ W2 such that w ≈ w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We proceed by induction on |w|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If |w| = 1 then w ∈ C ∪ W1 ∪ W2 and so there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' As induction hypothesis, assume that the lemma is valid for all words w ∈ A∗ such that |w| = k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let w ∈ A∗ be such that |w| = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If w ∈ C∗ then there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Hence, suppose that w ∈ A∗ \\ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let u ∈ A∗ and a ∈ A be such that w = ua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We will consider several cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' u ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Therefore a ∈ A\\C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' So, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='5, there exists u′ ∈ C∗ such that ua ≈ u′ or there exists 0 ⩽ r ⩽ n−i−1 such that ua ≈ yrxi, with a = xi for some 1 ⩽ i ⩽ ⌊n−1 2 ⌋, or there exists 0 ⩽ r ⩽ i − 1 such that ua ≈ yryi, with a = yi for some 1 ⩽ i ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Hence, in any of these three situations, we obtain w′ ∈ C∗ ∪ W1 ∪ W2 such that w = ua ≈ w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' u ∈ A∗ \\ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Since |u| = k then, by the induction hypothesis, there exists u′ ∈ C∗ ∪ W1 ∪ W2 such that u′ ≈ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' u′ ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If a ∈ C then w = ua ≈ u′a ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' On the other hand, if a ∈ A \\ C then, as in case 1, there exists w′ ∈ C∗ ∪ W1 ∪ W2 such that u′a ≈ w′ and so such that w = ua ≈ u′a ≈ w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' u′ ∈ W1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' u′ = yrxixs, for some 0 ⩽ r, s ⩽ n − 1, 1 ⩽ i ⩽ ⌊n−1 2 ⌋ and s + 1 ⩽ i ⩽ n − r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, we have w = ua ≈ u′a = yrxixsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 8 case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' a = e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If s > 0 then, by R3, xse1 ≈ xs and so w ≈ yrxixse1 ≈ yrxixs ∈ W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' On the other hand, if s = 0 then, by R9, we have w ≈ yrxie1 ≈ yrxn−2ien−2i+1 · · · en−ien−i+2 · · · en ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' a = ej, with 2 ⩽ j ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If j ⩽ s then, by (2), w ≈ yrxixsej ≈ yrxixs ∈ W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Now, suppose that s < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, by R5, we get xsej ≈ ej−sxs and so w ≈ yrxixsej ≈ yrxiej−sxs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If j − s ̸= 1 and j − s ̸= n − i + 1 then, by R7, w ≈ yrxiej−sxs ≈ yrxixs ∈ W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If j − s = 1 then, by R9, w ≈ yrxie1xs ≈ yrxn−2ien−2i+1 · · · en−ien−i+2 · · · enxs ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Finally, if j − s = n − i + 1 then, by R10, w ≈ yrxien−i+1xs ≈ yre2 · · · enxs ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' a = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If s + 1 < i then w ≈ yrxixs+1 ∈ W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' On the other hand, if s + 1 ⩾ i (in fact s + 1 = i) then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3, we get w ≈ yrxixs+1 ≈ yre2 · · · enxs+1 ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' a = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If s = 0 then, by R3 and R9, w ≈ yrxiy ≈ yrxie1y ≈ yrxn−2ien−2i+1 · · · en−ien−i+2 · · · eny ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' On the other hand, if s > 0 then, by R2, R5 (noticing that s − 1 < n) and R7 (noticing that n − s + 1 > n − i + 1), we have w ≈ yrxixsy ≈ yrxixs−1en ≈ yrxien−s+1xs−1 ≈ yrxixs−1 ∈ W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' a = xj, with 1 ⩽ j ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If s = 0 then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4, we have w ≈ yrxixj ≈ yre2 · · · en ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' So, suppose that s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, by R3 and R9, we get w ≈ yrxixsxj ≈ yrxixse1xj ≈ yrxixsxn−2jen−2j+1 · · · en−jen−j+2 · · · en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (3) If s + n − 2j ⩾ i then from (3), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3, we get w ≈ yre2 · · · enxs+n−2jen−2j+1 · · · en−jen−j+2 · · · en ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' On the other hand, suppose that s + n − 2j + 1 ⩽ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If s ⩾ j then n − s + 1 ⩽ s + n − 2j + 1 ⩽ i, whence n − i + 1 ⩽ s ⩽ i − 1 and so n + 2 ⩽ 2i ⩽ 2⌊n−1 2 ⌋ ⩽ n − 1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Therefore s < j and so n − 2j + 1 ⩽ s + n − 2j + 1 ⩽ n − j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Hence, by (3), R4, R5 and R9, we obtain w ≈ yrxixs+n−2jes+n−2j+1en−2j+1 · · · es+n−2jes+n−2j+2 · · · en−jen−j+2 · · · en ≈ yrxie1xs+n−2jen−2j+1 · · · es+n−2jes+n−2j+2 · · · en−jen−j+2 · · · en ≈ yrxn−2ien−2i+1 · · · en−ien−i+2 · · · enxs+n−2jen−2j+1 · · · es+n−2jes+n−2j+2 · · · en−jen−j+2 · · · en ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' a = yj, with 1 ⩽ j ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If s = 0 then, by R6 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4, respectively, we have w ≈ yrxiyj ≈ � yre2 · · · eiei+2 · · · en ∈ C∗ if i = j yre2 · · · en ∈ C∗ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Suppose that s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, by R3 and R9, we get w ≈ yrxixsyj ≈ yrxixse1yj ≈ yrxixsyn−2je1 · · · ejej+2 · · · e2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (4) Next, by R2, R5 and R7 (noticing that s + 1 ⩽ i implies n − s + 1 > n − i + 1), we have xixsyn−2j ≈ xixs−1enyn−2j−1 ≈ xien−s+1xs−1yn−2j−1 ≈ xixs−1yn−2j−1 and, repeating this process as long as possible, we obtain xixsyn−2j ≈ \uf8f1 \uf8f2 \uf8f3 xi if s = n − 2j xixs−n+2j if s > n − 2j xiyn−2j−s if s < n − 2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (5) From (4) and (5), we have: if s = n − 2j, by R9, w ≈ yrxie1e2 · · · ejej+2 · · · e2j ≈ yrxn−2ien−2i+1 · · · en−ien−i+2 · · · ene2 · · · ejej+2 · · · e2j ∈ C∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 9 and, if s < n − 2j, by R3 and R9, w ≈ yrxiyn−2j−se1 · · · ejej+2 · · · e2j ≈ yrxie1yn−2j−se1 · · · ejej+2 · · · e2j ≈ yrxn−2ien−2i+1 · · · en−ien−i+2 · · · enyn−2j−se1 · · · ejej+2 · · · e2j ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Now, observe that s + 1 ⩽ i implies n − s − 1 ⩾ n − i ⩾ n − ⌊n−1 2 ⌋ = ⌊n 2 ⌋ + 1 > ⌊n−1 2 ⌋ ⩾ j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' j < n − s − 1 and so j > s − n + 2j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Therefore, if s > n − 2j, from (4) and (5), we have w ≈ yrxixs−n+2je1 · · · ejej+2 · · · e2j ≈ yrxixs−n+2jes−n+2j+1e1 · · · es−n+2jes−n+2j+2 · · · ejej+2 · · · e2j ≈ yrxie1xs−n+2je1 · · · es−n+2jes−n+2j+2 · · · ejej+2 · · · e2j ≈ yrxn−2ien−2i+1 · · · en−ien−i+2 · · · enxs−n+2je1 · · · es−n+2jes−n+2j+2 · · · ejej+2 · · · e2j ∈ C∗, by R4, R5 and R9, thus completing the proof of the lemma for the case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' u′ ∈ W2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' u′ = yryixs, for some 0 ⩽ r, s ⩽ n − 1, 1 ⩽ i ⩽ ⌊n−1 2 ⌋ and r + 1 ⩽ i ⩽ n − s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, we have w = ua ≈ u′a = yryixsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If a ∈ C then proceeding analogously to the cases 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4, we can find w′ ∈ C∗ ∪ W2 such that w ≈ w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Thus, it remains to study the case a ∈ A \\ C, which we divide in two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' First, let us consider a = xj, for some 1 ⩽ j ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If s = 0 then, by R6 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4, respectively, we have w ≈ yryixj ≈ � yre2 · · · en−ien−i+2 · · · en ∈ C∗ if i = j yre2 · · · en ∈ C∗ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Now, suppose that s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, by R3 and R9, we get w ≈ yryixsxj ≈ yryixse1xj ≈ yryixsxn−2jen−2j+1 · · · en−jen−j+2 · · · en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (6) If s + n − 2j ⩾ n − i then w ≈ yryixs+n−2jen−2j+1 · · · en−jen−j+2 · · · en ≈ yre2 · · · enxs+n−2jen−2j+1 · · · en−jen−j+2 · · · en ∈ C∗, by (6) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' So, suppose that s + n − 2j ⩽ n − i − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' i ⩽ n − (s + n − 2j) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If s + n − 2j = n − j then, by (6), (2) and R5, we have w ≈ yryixs+n−2jen−j+2 · · · en ≈ yryie2 · · · ejxs+n−2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' In addition, if j ⩽ i, by R8, we obtain w ≈ yryie2 · · · ejxs+n−2j ≈ yryixs+n−2j ∈ W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Otherwise, by R4, R8 and R10, it follows that w ≈ yryie2 · · · eiei+2 · · · ejei+1xs+n−2j ≈ yryiei+1xs+n−2j ≈ yre2 · · · enxs+n−2j ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' On the other hand, suppose that s + n − 2j ̸= n − j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Since s + n − 2j + 1 > n − 2j + 1, then s + n − 2j + 1 ∈ L = {n − 2j + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , n − j, n − j + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , n} and so, by (6), R4, R5 and R9, we have w ≈ yryixs+n−2jen−2j+1 · · · en−jen−j+2 · · · en ≈ yryixs+n−2jes+n−2j+1Πt∈L\\{s+n−2j+1}et ≈ yryie1xs+n−2jΠt∈L\\{s+n−2j+1}et ≈ yryn−2ie1 · · · eiei+2 · · · e2ixs+n−2jΠt∈L\\{s+n−2j+1}et ∈ C∗, which completes the study of this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Finally, let us move on to our last case by considering a = yj, for some 1 ⩽ j ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If s = 0 then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4, we have w ≈ yryiyj ≈ yre2 · · · en ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' So, suppose that s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, by R3 and R9, we get w ≈ yryixsyj ≈ yryixse1yj ≈ yryixsyn−2je1 · · · ejej+2 · · · e2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (7) 10 Next, by R2, R5 and R8 (noticing that i ⩽ n − s − 1 implies i + 1 < n − s + 1), we have yixsyn−2j ≈ yixs−1enyn−2j−1 ≈ yien−s+1xs−1yn−2j−1 ≈ yixs−1yn−2j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' By repeating this process as long as possible, we obtain yixsyn−2j ≈ \uf8f1 \uf8f2 \uf8f3 yi if s = n − 2j yixs−n+2j if s > n − 2j yiyn−2j−s if s < n − 2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (8) If s = n − 2j, by (7), (8) and R9, we have w ≈ yryie1e2 · · · ejej+2 · · · e2j ≈ yryn−2ie1 · · · eiei+2 · · · e2ie2 · · · ejej+2 · · · e2j ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' On the other hand, if s < n − 2j, by (7), (8), R3 and R9, we get w ≈ yryiyn−2j−se1 · · · ejej+2 · · · e2j ≈ yryie1yn−2j−se1 · · · ejej+2 · · · e2j ≈ yryn−2ie1 · · · eiei+2 · · · e2iyn−2j−se1 · · · ejej+2 · · · e2j ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Now, suppose that s > n − 2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' In addition, suppose first that s − n + 2j = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, by (7), (8), (2) and R5, we obtain w ≈ yryixje1 · · · ejej+2 · · · e2j ≈ yryixjej+2 · · · e2j ≈ yryie2 · · · ejxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Since i ⩽ n − s − 1 < n − s = j, then w ≈ yryiei+1xj ≈ yre2 · · · enxj ∈ C∗, by R8, R4 and R10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Secondly, suppose that s − n + 2j ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Since s − n < 0, then s − n + 2j + 1 ⩽ 2j and so s − n + 2j + 1 ∈ K = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , j, j + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 2j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Hence, by (7), (8), R4, R5 and R9, we get w ≈ yryixs−n+2je1 · · · ejej+2 · · · e2j ≈ yryixs−n+2jes−n+2j+1Πt∈K\\{s−n+2j+1}et ≈ yryie1xs−n+2jΠt∈K\\{s−n+2j+1}et ≈ yryn−2ie1 · · · eiei+2 · · · e2ixs−n+2jΠt∈K\\{s−n+2j+1}et ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Therefore, we have exhausted all possible cases, completing the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Now, let us choose a set of forms W0 for the presentation ⟨C | U⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, for each w ∈ C∗ there exists (a unique) w′ ∈ W0 such that w′ρUw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Moreover, as the monoid OCIn is defined by the presentation ⟨C | U⟩, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1, we have |W0| = |OCIn| = 3 · 2n − 2n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let W = W0 ∪ W1 ∪ W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, by (1), |W| = |W0| + |W1 ∪ W2| = 3 · 2n − 2n − 2 + 1 6(n + 1)n(n − 1) − 1 8(1 + (−1)n)n2 = |ODIn| and, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='6, for each word w ∈ A∗, there exists w′ ∈ W such that w ≈ w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' On the other hand, it is a routine matter to check: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='7 The generating set {x, y, e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , y⌊ n−1 2 ⌋} of ODIn satisfies (via ϕ) all relations from R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Therefore, the conditions of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2 are satisfied and so we have: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='8 The monoid ODIn is defined by the presentation ⟨A | R⟩ on 2n + 1−(−1)n 2 generators and 1 2(5n2 − (1 + 2(−1)n)n + (−1)n+1 + 5) relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 11 Next, by using Tietze transformations and applying Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3, we deduce from the above presentation for ODIn a new presentation on a minimal size set of generators of ODIn given in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let us consider the alphabet B = {x, y, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en−1, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , y⌊ n−1 2 ⌋} = A \\ {e1, en}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Basically, we first apply T4 with each of the relations R2 and then, of the resulting relations, we eliminate the trivial ones and some deduced from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' This procedure was applied in [14] to the set U of relations (R1 to R5 together with relation R11) on the alphabet C = {x, y, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en}, having resulted in the following set of 1 2(n2 + 3n) monoid relations (which we can consider on the alphabet B): (V1) e2 i = ei, for 2 ⩽ i ⩽ n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V2) xyx = x and yxy = y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V3) yx2y = xy2x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V4) eiej = ejei, for 2 ⩽ i < j ⩽ n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V5) xyei = eixy and yxei = eiyx, for 2 ⩽ i ⩽ n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V6) xei+1 = eix, for 2 ⩽ i ⩽ n − 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V7) x2y = en−1x and yx2 = xe2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V8) yxe2 · · · en−1xy = xe2 · · · en−1xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Performing the same procedure to relations R6 to R10 on the alphabet A, we may routinely obtain the following 2n2 − (2 + (−1)n)n − 1 2(3 + (−1)n) monoid relations on the alphabet B: (V9) xiyi = e2 · · · eiei+2 · · · en−1xy, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' y1x1 = e2 · · · en−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' yixi = e2 · · · en−ien−i+2 · · · en−1xy, for 2 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V10) xiej = xi and ejyi = yi, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋, 2 ⩽ j ⩽ n − 1 and j ̸= n − i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V11) ejxi = xi and yiej = yi, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋, 2 ⩽ j ⩽ n − 1 and j ̸= i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V12) xixy = xyxi = xi and xyyi = yixy = yi, for 2 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' xyx1 = x1 and y1xy = y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V13) yxx1 = x1yx = xn−2en−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' yxxi = xiyx = xn−2ien−2i+1 · · · en−ien−i+2 · · · en−1xy, for 2 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' yxyi = yiyx = yn−2i+1xe2 · · · eiei+2 · · · e2i, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V14) x1xy = e2x1 = y1e2 = xyy1 = e2 · · · en−1xy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' xien−i+1 = ei+1xi = yiei+1 = en−i+1yi = e2 · · · en−1xy, for 2 ⩽ i ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Thus, defining V as the set of monoid relations on the alphabet B consisting of relations V1 to V14, we have: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='9 The monoid ODIn is defined by the presentation ⟨B | V ⟩ on 2n − 3+(−1)n 2 generators and 1 2(5n2 − (1 + 2(−1)n)n + (−1)n+1 − 3) relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 12 4 Presentations for MDIn We begin this section by determining a presentation for MDIn on 2n− 1+(−1)n 2 generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' For this purpose, we will apply Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Next, by using Tietze transformations, we deduce another presentation for MDIn on 2 + 3⌊n−1 2 ⌋ generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let us consider the alphabet ¯B = B ∪ {h} = {h, x, y, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en−1, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , y⌊ n−1 2 ⌋} and let ¯ϕ : ¯B∗ −→ MDIn be the homomorphism of monoids that extends the mapping ¯B −→ MDIn defined by h �−→ h, x �−→ x, y �−→ y, ei �−→ ei, for 2 ⩽ i ⩽ n − 1, xj �−→ xj and yj �−→ yj, for 1 ⩽ j ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Notice that ¯B ¯ϕ is a generating set of MDIn with 2n − 1+(−1)n 2 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Next, observe that ∅ = e1e2 · · · en−1en, �1 1 � = e2 · · · en−1en and �i j � = �i 1 ��1 1 ��1 j � = ei+1 · · · enyi−1e2 · · · en−1enxj−1ej+1 · · · en, for 1 ⩽ i, j ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Therefore, let u0 = e2 · · · en−1xy ∈ B∗ and let W ′ 1 be the subset of B∗ formed by the following 1 + n2 words: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' yxu0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' ei+1 · · · en−1xyiu0xj−1ej+1 · · · en−1xy, for 1 ⩽ i, j ⩽ n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' ei+1 · · · en−1xyiu0xn−1, for 1 ⩽ i ⩽ n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' yn−1u0xj−1ej+1 · · · en−1xy, for 1 ⩽ j ⩽ n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' yn−1u0xn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then ¯ϕ is a bijection from W ′ 1 onto {∅} ∪ { �i j � | 1 ⩽ i, j ⩽ n} and so we can choose a set of forms W ′ for the presentation ⟨B | V ⟩ of ODIn such that W ′ contains the empty word and W ′ 1 ⊂ W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let W ′ 2 = W ′ \\ W ′ 1 and consider the subset ¯W = W ′ ∪ {wh | w ∈ W ′ 2} of ¯B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Notice that | ¯W| = |W ′| + |W ′ 2| = |W ′| + |W ′| − |W ′ 1| = 2|ODIn| − n2 − 1 = |MDIn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let 1 ⩽ i ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then hxih = �n − i n i n � = yn−i−1xixi−1 and hyih = � i n n − i n � = yi−1yixn−i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' On the other hand, we also have hxh = y, hyh = x, heih = en−i+1, for 1 ⩽ i ⩽ n, and e2 · · · enh = �1 n � = xn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Therefore, the monoid relations (on the alphabet ¯B) h2 = 1, hx = yh, hy = xh, hei = en−i+1h, for 2 ⩽ i ⩽ n − 1, hxi = yn−i−1xixi−1h and hyi = yi−1yixn−i−1h, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋, and e2 · · · en−1xyh = xn−1 are all satisfied (via ¯ϕ) by the generating set {h, x, y, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en−1, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , y⌊ n−1 2 ⌋} of MDIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Now, let ¯V be the set of monoid relations V (relations V1 to V14 considered on the alphabet ¯B) together with the following n + ⌊n+1 2 ⌋ + 1−(−1)n 2 monoid relations on the alphabet ¯B: ( ¯V0) h2 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 13 ( ¯V1) hx = yh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' hei = en−i+1h, for 2 ⩽ i ⩽ ⌊n+1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' hxi = yn−i−1xixi−1h and hyi = yi−1yixn−i−1h, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' ( ¯V2) e2 · · · en−1xyh = xn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Since the relation hy = xh is a consequence of h2 = 1 and hx = yh and the relations hei = en−i+1h, with 2 ⩽ i ⩽ n − 1, are consequences of h2 = 1 and hei = en−i+1h, with 2 ⩽ i ⩽ ⌊n+1 2 ⌋, by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4, we conclude that: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1 The monoid MDIn is defined by the presentation ⟨ ¯B | ¯V ⟩ on 2n − 1+(−1)n 2 generators and 1 2(5n2 + (2 − 2(−1)n)n − 3+5(−1)n 2 ) relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Next, like in Section 3, by using Tietze transformations and applying Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3, we deduce from the presentation ⟨ ¯B | ¯V ⟩ of MDIn a new presentation on a minimal size set of generators of MDIn provided by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' So, consider the alphabet ¯B′ = {h, x, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , e⌊ n+1 2 ⌋, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , y⌊ n−1 2 ⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Hence, since y = hxh and ei = hen−i+1h, for ⌊n+1 2 ⌋ + 1 ⩽ i ⩽ n − 1 (as transformations), we can apply T1 by adding the relations y = hxh and ei = hen−i+1h, for ⌊n+1 2 ⌋ + 1 ⩽ i ⩽ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Next, we apply T4 with each of these relations and then, of the resulting relations, we eliminate the trivial ones and some deduced from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Performing this procedure to ¯V , we may routinely obtain the following set ¯V ′ of 2n2 + 7−(−1)n 4 n − 2(−1)n − 1 monoid relations on the alphabet ¯B′: (V ′ 1) e2 i = ei, for 2 ⩽ i ⩽ ⌊n+1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V ′ 2) (xh)2x = x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V ′ 3) xhx2hxh = hxhx2hx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V ′ 4) eiej = ejei, for 2 ⩽ i < j ⩽ ⌊n+1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' eihejh = hejhei, for 2 ⩽ i ⩽ ⌊n+1 2 ⌋ and 2 ⩽ j ⩽ ⌊n 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V ′ 5) (xh)2ei = ei(xh)2 and (hx)2ei = ei(hx)2, for 2 ⩽ i ⩽ ⌊n+1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V ′ 6) xei+1 = eix, for 2 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' xhe⌊ n 2 ⌋h = e⌊ n+1 2 ⌋x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' xheih = hei+1hx, for 2 ⩽ i ⩽ ⌊n−2 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V ′ 7) x(xh)2 = he2hx and (hx)2x = xe2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V ′ 8) (hx)2e2 · · · e⌊ n+1 2 ⌋he2 · · · e⌊ n 2 ⌋(hx)2 = xe2 · · · e⌊ n+1 2 ⌋he2 · · · e⌊ n 2 ⌋(hx)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V ′ 9) xiyi = e2 · · · eiei+2 · · · e⌊ n+1 2 ⌋he2 · · · e⌊ n 2 ⌋h(xh)2, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' y1x1 = e2 · · · e⌊ n+1 2 ⌋he2 · · · e⌊ n 2 ⌋h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' yixi = e2 · · · e⌊ n+1 2 ⌋he2 · · · ei−1ei+1 · · · e⌊ n 2 ⌋h(xh)2, for 2 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V ′ 10) xiej = xi and ejyi = yi, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋ and 2 ⩽ j ⩽ ⌊n+1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' xihejh = xi and hejhyi = yi, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋, 2 ⩽ j ⩽ ⌊n 2 ⌋ and j ̸= i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V ′ 11) ejxi = xi and yiej = yi, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋, 2 ⩽ j ⩽ ⌊n+1 2 ⌋ and j ̸= i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' hejhxi = xi and yihejh = yi, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋ and 2 ⩽ j ⩽ ⌊n 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V ′ 12) xi(xh)2 = (xh)2xi = xi and (xh)2yi = yi(xh)2 = yi, for 2 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (xh)2x1 = x1 and y1(xh)2 = y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (V ′ 13) (hx)2x1 = x1(hx)2 = xn−2he2h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (hx)2xi = xi(hx)2 = xn−2ien−2i+1 · · · e⌊ n+1 2 ⌋he2 · · · ei−1ei+1 · · · e⌊ n 2 ⌋h(xh)2, for 2 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (hx)2yi = yi(hx)2 = hxn−2i+1hxe2 · · · eiei+2 · · · e⌊ n+1 2 ⌋hen−2i+1 · · · e⌊ n 2 ⌋h, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 14 (V ′ 14) x1(xh)2 = e2x1 = y1e2 = (xh)2y1 = e2 · · · e⌊ n+1 2 ⌋he2 · · · e⌊ n 2 ⌋h(xh)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' xiheih = ei+1xi = yiei+1 = heihyi = e2 · · · e⌊ n+1 2 ⌋he2 · · · e⌊ n 2 ⌋h(xh)2, for 2 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' ( ¯V ′ 0) h2 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' ( ¯V ′ 1) he⌊ n+1 2 ⌋ = e⌊ n+1 2 ⌋h, if n is odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' hxi = hxn−i−1hxixi−1h and hyi = hxi−1hyixn−i−1h, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' ( ¯V ′ 2) e2 · · · e⌊ n+1 2 ⌋he2 · · · e⌊ n 2 ⌋(hx)2 = xn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Thus, we have: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2 The monoid MDIn is defined by the presentation ⟨ ¯B′ | ¯V ′⟩ on 2 + 3⌊n−1 2 ⌋ generators and 2n2 + 7−(−1)n 4 n − 2(−1)n − 1 relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 5 Presentations for OPDIn As in both previous sections, we first determine a presentation for OPDIn on an extended set of generators, namely, with n+⌊n−1 2 ⌋+1 generators, and then, through Tietze transformations, we deduce another presentation for OPDIn on a minimum size set of generators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' on 2 + ⌊n−1 2 ⌋ generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Consider the alphabet D = {g, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋} and let ψ : D∗ −→ OPDIn be the homomorphism of monoids that extends the mapping D −→ OPDIn defined by g �−→ g, ei �−→ ei, for 1 ⩽ i ⩽ n, xj �−→ xj, for 1 ⩽ j ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let Q be the set formed by the following 1 2(3n2 + (1 − (−1)n)n + 3 − 1+(−1)n 2 ) monoid relations: (Q1) gn = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q2) e2 i = ei, for 1 ⩽ i ⩽ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q3) eiej = ejei, for 1 ⩽ i < j ⩽ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q4) ge1 = eng and gei+1 = eig, for 1 ⩽ i ⩽ n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q5) ge1 · · · en = e1 · · · en;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q6) e1xi = xie1 = gn−2ie1 · · · en−ien−i+2 · · · en, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q7) xiej = xi, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋, 2 ⩽ j ⩽ n and j ̸= n − i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q8) ejxi = xi, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋, 2 ⩽ j ⩽ n and j ̸= i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q9) xien−i+1 = ei+1xi = e2 · · · en, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q10) (xigi)2 = e2 · · · eiei+2 · · · en, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Our aim is to show that the monoid OPDIn is defined by the presentation ⟨D | Q⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' As in Section 3, we will make use of results of [14], this time in view to applying Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We begin by noticing that it is a routine matter to check: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1 The set of generators {g, e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋} of OPDIn satisfies (via ψ) all relations from Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 15 Observe that, as a consequence of the previous lemma, if u, v ∈ D∗ are such that the relation u = v is a consequence of Q, then uψ = vψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Now, let us recall that the cyclic inverse monoid CIn is generated by {g, e1} (see [14]) and, even more so, by {g, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let us consider the alphabet D0 = {g, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , en} and denote by Q0 the subset of Q consisting of relations Q1 to Q5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, we have: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2 ([14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='6]) The monoid CIn is defined by the presentation ⟨D0 | Q0⟩ on n + 1 generators and 1 2(n2 + 3n + 4) relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' To prove this result, the author used the property given by the following lemma, which we will also use here: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3 ([14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4]) Let u ∈ D∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, there exist 0 ⩽ m ⩽ n − 1, 1 ⩽ i1 < · · · < ik ⩽ n and 0 ⩽ k ⩽ n such that the relation u = gmei1 · · · eik is a consequence of relations Q1 to Q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Observe that, it is easy to show that, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋, the relations eigm = � gmei+m if 0 ⩽ m ⩽ n − i gmei+m−n if n − i + 1 ⩽ m ⩽ n − 1 and gmei = � ei−mgm if 0 ⩽ m ⩽ i − 1 en+i−mgm if i ⩽ m ⩽ n − 1 (9) are consequences of Q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Combining (9) with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3, we immediately obtain the symmetric result of the latter one: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4 Let u ∈ D∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, there exist 0 ⩽ m ⩽ n − 1, 1 ⩽ i1 < · · · < ik ⩽ n and 0 ⩽ k ⩽ n such that the relation u = ei1 · · · eikgm is a consequence of relations Q1 to Q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' From now on, we denote the congruence ρQ of D∗ again by ≈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4, we can conclude: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='5 For all 1 ⩽ i, j ⩽ ⌊n−1 2 ⌋, xixj ≈ e2 · · · en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Next, we prove a series of lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='6 Let u ∈ D∗ 0 and let 1 ⩽ i, j ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, there exists v ∈ D∗ 0 ∪xiD∗ 0 ∪D∗ 0xj such that xiuxj ≈ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3, there exist 0 ⩽ m ⩽ n − 1, 1 ⩽ i1 < · · · < ik ⩽ n and 0 ⩽ k ⩽ n such that u ≈ gmei1 · · · eik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If i1 = 1 (with k > 0) then xiuxj ≈ xigmei2 · · · eike1xj ≈ xigmei2 · · · eikgn−2je1 · · · en−jen−j+2 · · · en ∈ xiD∗ 0, by Q3 and Q6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If j + 1 ∈ {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , ik} (with k > 0) then, being 1 ⩽ ℓ ⩽ k such that j + 1 = iℓ, we have xiuxj ≈ xigmei1 · · · eiℓ−1eiℓ+1 · · · eikej+1xj ≈ xigmei1 · · · eiℓ−1eiℓ+1 · · · eike2 · · · en ∈ xiD∗ 0, by Q3 and Q9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Now, suppose that either k = 0 or i1 > 1 and j+1 ̸∈ {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , ik} (with k > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, by Q8, ei1 · · · eikxj ≈ xj and so xiuxj ≈ xigmxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If m = 0 then, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='5, xiuxj ≈ xixj ≈ e2 · · · en ∈ D∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' So, suppose that m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If m ̸= i then n − i + 1 ̸= n − m + 1 ⩾ 2 and so xiuxj ≈ xien−m+1gmxj ≈ xigme1xj ≈ xigmgn−2je1 · · · en−jen−j+2 · · · en ∈ xiD∗ 0, 16 by Q7, (9) and Q6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If m ̸= j then j + 1 ̸= m + 1 ⩾ 2 and so xiuxj ≈ xigmem+1xj ≈ xie1gmxj ≈ gn−2ie1 · · · en−ien−i+2 · · · engmxj ∈ D∗ 0xj, by Q8, (9) and Q6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Finally, if m = i = j then xiuxj ≈ xigixi ≈ xigixigign−i = (xigi)2gn−i ≈ e2 · · · eiei+2 · · · engn−i ∈ D∗ 0, by Q1 and Q10, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='7 Let w ∈ D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, there exists w′ ∈ D∗ 0 ∪ D∗ 0x1D∗ 0 ∪ · · · ∪ D∗ 0x⌊ n−1 2 ⌋D∗ 0 such that w ≈ w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' We proceed by induction on the number of occurrences of the letters x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋ in a word w ∈ D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If w ∈ D∗ has no occurrences of the letters x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋ then w ∈ D∗ 0 and so there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Hence, for k ⩾ 1, suppose that the lemma is valid for all words in D∗ with k − 1 occurrences of the letters x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Let w be a word of D∗ with k occurrences of the letters x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then w = u0xi1u1 · · · xik−1uik−1xikuk, for some u0, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , uk ∈ D∗ 0 and 1 ⩽ i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , ik ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Hence, by induction hypothesis, there exists u′ ∈ D∗ 0 ∪ D∗ 0x1D∗ 0 ∪ · · · ∪ D∗ 0x⌊ n−1 2 ⌋D∗ 0 such that u0xi1u1 · · · xik−1uik−1 ≈ u′ and so w ≈ u′xikuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If u′ ∈ D∗ 0 then the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' So, suppose that u′ = u′ 0xiu′ 1, for some u′ 0, u′ 1 ∈ D∗ 0 and some 1 ⩽ i ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Thus, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='6, there exists v ∈ D∗ 0 ∪ xiD∗ 0 ∪ D∗ 0xik such that xiu′ 1xik ≈ v, whence w ≈ u′ 0xiu′ 1xikuk ≈ u′ 0vuk ∈ D∗ 0 ∪ D∗ 0xiD∗ 0 ∪ D∗ 0xikD∗ 0, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='8 Let w ∈ D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, there exists u ∈ D∗ 0 such that w ≈ u or there exist 1 ⩽ i ⩽ ⌊n−1 2 ⌋ and 0 ⩽ r, s ⩽ n − 1 such that w ≈ grxigs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' First, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='7, take w′ ∈ D∗ 0 ∪ D∗ 0x1D∗ 0 ∪ · · · ∪ D∗ 0x⌊ n−1 2 ⌋D∗ 0 such that w ≈ w′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If w′ ∈ D∗ 0 then the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' So, suppose that w′ = uxiv, for some u, v ∈ D∗ 0 and some 1 ⩽ i ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, by Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='4, u ≈ grei1 · · · eik and v ≈ ej1 · · · ejℓgs, for some 0 ⩽ r, s ⩽ n − 1, 1 ⩽ i1 < · · · < ik ⩽ n, 1 ⩽ j1 < · · · < jℓ ⩽ n and 0 ⩽ k, ℓ ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Hence w ≈ grei1 · · · eikxiej1 · · · ejℓgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If i1 = 1 (with k > 0) then, by Q3 and Q6, w ≈ grei2 · · · eike1xiej1 · · · ejℓgs ≈ grei2 · · · eikgn−2ie1 · · · en−ien−i+2 · · · enej1 · · · ejℓgs ∈ D∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If i + 1 ∈ {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , ik} (with k > 0) then, being 1 ⩽ p ⩽ k such that i + 1 = ip, we have w ≈ grei1 · · · eip−1eip+1 · · · eikei+1xiej1 · · · ejℓgs ≈ grei1 · · · eip−1eip+1 · · · eike2 · · · enej1 · · · ejℓgs ∈ D∗ 0, by Q3 and Q9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If j1 = 1 (with ℓ > 0) then, by Q6, w ≈ grei1 · · · eikxie1ej2 · · · ejℓgs ≈ grei1 · · · eikgn−2ie1 · · · en−ien−i+2 · · · enej2 · · · ejℓgs ∈ D∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If n − i + 1 ∈ {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , jℓ} (with ℓ > 0) then, being 1 ⩽ q ⩽ ℓ such that n − i + 1 = jq, we have w ≈ grei1 · · · eikxien−i+1ej1 · · · ejq−1ejq+1 · · · ejℓgs ≈ grei1 · · · eike2 · · · enej1 · · · ejq−1ejq+1 · · · ejℓgs ∈ D∗ 0, by Q3 and Q9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Finally, suppose that none of the four previous cases occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then, by Q7 and Q8, ei1 · · · eikxiej1 · · · ejℓ ≈ xi and so w ≈ grxigs, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 17 For 1 ⩽ i ⩽ ⌊n−1 2 ⌋ and 0 ⩽ r, s ⩽ n − 1, let us consider the transformation grxigs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' It is a routine matter to check that grxigs = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 � 1 1 + i 1 + s a � if r = 0 �n − r + 1 n − r + i + 1 1 + s a � if r > 0 and 1 ⩽ i ⩽ r − 1 �i − r + 1 n − r + 1 a 1 + s � if r > 0 and r ⩽ i ⩽ ⌊n−1 2 ⌋, with a = � s − i + 1 if 1 ⩽ i ⩽ s n + s − i + 1 if s + 1 ⩽ i ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Hence, it is easy to show that xi = grxigs if and only if r = s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (10) Now, recall that {g, e1} generates CIn and that {g, e1, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋} is a minimal generating set of OPDIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Therefore, noticing also that gn = 1, we have grxigs ̸∈ CIn, for all 1 ⩽ i ⩽ ⌊n−1 2 ⌋ and r, s ⩾ 0, (11) and xj = grxigs, with 1 ⩽ i, j ⩽ ⌊n−1 2 ⌋ and r, s ⩾ 0, implies i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (12) We are now in a position to prove our first objective of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='9 The monoid OPDIn is defined by the presentation ⟨D | Q⟩ on n + ⌊n−1 2 ⌋ + 1 generators and 1 2(3n2 + (1 − (−1)n)n + 3 − 1+(−1)n 2 ) relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Given Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1, by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='1, it remains to prove that w1 ≈ w2 for all words w1, w2 ∈ D∗ such that w1ψ = w2ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' So, let w1, w2 ∈ D∗ be such that w1ψ = w2ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='8, for k ∈ {1, 2}, there exists uk ∈ D∗ 0 such that wk ≈ uk (and then wkψ = ukψ ∈ CIn) or there exist 1 ⩽ ik ⩽ ⌊n−1 2 ⌋ and 0 ⩽ rk, sk ⩽ n−1 such that wk ≈ grkxikgsk (and then, by (11), wkψ = (grkxikgsk)ψ = grkxikgsk ̸∈ CIn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Since w1ψ = w2ψ, we can only have: (case 1) w1 ≈ u1 and w2 ≈ u2, for some u1, u2 ∈ D∗ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' or (case 2) w1 ≈ gr1xi1gs1 and w2 ≈ gr2xi2gs2, for some 1 ⩽ i1, i2 ⩽ ⌊n−1 2 ⌋ and 0 ⩽ r1, s1, r2, s2 ⩽ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If case 1 occurs, then u1ψ = w1ψ = w2ψ = u2ψ and so u1ρQ0u2, since ⟨D0 | Q0⟩ is a presentation of CIn, by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='2, which implies that u1 ≈ u2 and thus w1 ≈ w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Now, suppose we have case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Then gr1xi1gs1 = w1ψ = w2ψ = gr2xi2gs2 and so xi1 = gr2−r1+r3xi2gs2−s1+s3, where r3 = � 0 if r1 ⩽ r2 n if r1 > r2 and s3 = � 0 if s1 ⩽ s2 n if s1 > s2, whence i1 = i2, by (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Thus, it follows by (10) that r2 − r1 + r3 = 0 = s2 − s1 + s3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' If r3 = n then n = r1 − r2, which is a contradiction, since 0 ⩽ r1, r2 ⩽ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Therefore r3 = 0 and, analogously, s3 = 0, whence r1 = r2 and s1 = s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Thus w1 ≈ gr1xi1gs1 = gr2xi2gs2 ≈ w2, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Next, by using Tietze transformations and applying Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='3, we deduce from the previous presen- tation for OPDIn a new one on the minimal set of generators {g, e1, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋} of OPDIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' By noticing that ei = gn−i+1e1gi−1 for 2 ⩽ i ⩽ n (as transformations), we proceed as follows: first, by applying T1, we add the relations ei = gn−i+1e1gi−1, for 2 ⩽ i ⩽ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' secondly, we apply T4 with each of the relations ei = gn−i+1e1gi−1 with 2 ⩽ i ⩽ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' finally, by using the relation Q1, we simplify the new relations obtained, eliminating the trivial ones or those that are deduced from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' By performing this procedure for each of the sets of relations Q1 to Q10, it is a routine matter to check that we can obtain the following set Q′ of 1 2(3n2 − (3 + (−1)n)n + 5 − 1+(−1)n 2 ) monoid relations on the alphabet D′ = {g, e1, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' , x⌊ n−1 2 ⌋}: 18 (Q′ 1) gn = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q′ 2) e2 1 = e1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q′ 3) e1gn−j+ie1gn−i+j = gn−j+ie1gn−i+je1, for 1 ⩽ i < j ⩽ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q′ 5) g(e1gn−1)n = (e1gn−1)n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q′ 6) e1xi = xie1 = gn−2i(e1gn−1)n−i−1e1(e1gn−1)i−1, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q′ 7) xign−j+1e1gj−1 = xi, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋, 2 ⩽ j ⩽ n and j ̸= n − i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q′ 8) gn−j+1e1gj−1xi = xi, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋, 2 ⩽ j ⩽ n and j ̸= i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q′ 9) xigie1gn−i = gn−ie1gixi = gn−1(e1gn−1)n−1, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' (Q′ 10) (xigi)2 = gn−1(e1gn−1)i−2e1gn−2(e1gn−1)n−i−1, for 1 ⩽ i ⩽ ⌊n−1 2 ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Thus, we have: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='10 The monoid OPDIn is defined by the presentation ⟨D′ | Q′⟩ on 2 + ⌊n−1 2 ⌋ generators and 1 2(3n2 − (3 + (−1)n)n + 5 − 1+(−1)n 2 ) relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='Ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' A˘ızenˇstat, Defining relations of finite symmetric semigroups, Mat.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Zap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 218 (1961), 191–212 (Russian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' [27] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Popova, Defining relations of a semigroup of partial endomorphisms of a finite linearly ordered set, Leningrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Gos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Ped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Uˇcen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Zap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 238 (1962), 78–88 (Russian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' [28] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Ruˇskuc, Semigroup Presentations, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Thesis, University of St-Andrews, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Ilinka Dimitrova, Department of Mathematics, Faculty of Mathematics and Natural Science, South-West University ”Neofit Rilski”, 2700 Blagoevgrad, Bulgaria;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' e-mail: ilinka dimitrova@swu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='bg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' V´ıtor H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Fernandes, Center for Mathematics and Applications (NovaMath) and Department of Mathematics, Faculdade de Ciˆencias e Tecnologia, Universidade Nova de Lisboa, Monte da Caparica, 2829-516 Caparica, Portugal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' e-mail: vhf@fct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' J¨org Koppitz, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' e-mail: koppitz@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='bas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='bg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Teresa M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Quinteiro, Instituto Superior de Engenharia de Lisboa, 1950-062 Lisboa, Portugal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' Also: Center for Mathematics and Applications (NovaMath), Faculdade de Ciˆencias e Tecnologia, Universidade Nova de Lisboa, Monte da Caparica, 2829-516 Caparica, Portugal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' e-mail: tmelo@adm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='isel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content='pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} +page_content=' 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfkgGS/content/2301.02474v1.pdf'} diff --git a/RtE4T4oBgHgl3EQf_g5X/vector_store/index.faiss b/RtE4T4oBgHgl3EQf_g5X/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..30feff45a64419f000d725bd978c10b8de74a859 --- /dev/null +++ b/RtE4T4oBgHgl3EQf_g5X/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:119468160bd4c63366ef548fb95f346438f102774b2d8ea1c92ae932cab51485 +size 3473453 diff --git a/RtE4T4oBgHgl3EQf_g5X/vector_store/index.pkl b/RtE4T4oBgHgl3EQf_g5X/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..604362357e5b5005a213052a0ee3336518581b48 --- /dev/null +++ b/RtE4T4oBgHgl3EQf_g5X/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4c3dc011a97631eb99e4d15ca8f85f1fa7050687e9cb7b02aa1a774f4a5bd912 +size 112300 diff --git a/UNE0T4oBgHgl3EQfVADN/content/tmp_files/2301.02259v1.pdf.txt b/UNE0T4oBgHgl3EQfVADN/content/tmp_files/2301.02259v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..920463a40f355868f08e0111b53ff7d8908edd1b --- /dev/null +++ b/UNE0T4oBgHgl3EQfVADN/content/tmp_files/2301.02259v1.pdf.txt @@ -0,0 +1,2612 @@ +Prepared for submission to JHEP +Ensemble averaging in JT gravity from +entanglement in Matrix Quantum Mechanics +Gabriele Di Ubaldo,1 Giuseppe Policastro2 +1Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique, 91191, Gif-sur-Yvette, +France +2Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne +Université, Université de Paris, F-75005 Paris, France +E-mail: gabriele.diubaldo@ipht.fr, giuseppe.policastro@ens.fr +Abstract: We consider the generalization of a matrix integral with arbitrary spectral +curve ρ0(E) to a 0+1D theory of matrix quantum mechanics (MQM). Using recent tech- +niques for 1D quantum systems at large-N, we formulate a hydrodynamical effective theory +for the eigenvalues. The result is a simple 2D free boson BCFT on a curved background, +describing the quantum fluctuations of the eigenvalues around ρ0(E), which is now the +large-N limit of the quantum expectation value of the eigenvalue density operator ˆρ(E). +The average over the ensemble of random matrices becomes a quantum expectation value. +Equal-time density correlations reproduce the results (including non-perturbative correc- +tions) of random matrix theory. This suggests an interpretation of JT gravity as dual to a +one-time-point reduction of MQM. +As an application, we compute the Rényi entropy associated to a bipartition of the eigenval- +ues. We match a previous result by Hartnoll and Mazenc for the c = 1 matrix model dual +to two-dimensional string theory and extend it to arbitrary ρ0(E). The hydrodynamical +theory provides a clear picture of the emergence of spacetime in two dimensional string +theory. The entropy is naturally finite and displays a large amount of short range entan- +glement, proportional to the microcanonical entropy. We also compute the reduced density +matrix for a subset of n < N eigenvalues. +arXiv:2301.02259v1 [hep-th] 5 Jan 2023 + +Contents +1 +Introduction +2 +1.1 +Motivation +2 +1.2 +Overview and results +5 +2 +Quantum hydrodynamics of random matrix eigenvalues +6 +2.1 +Eigenvalues as fermions +6 +2.2 +Effective hydrodynamics of the eigenvalue density ρ(E) +9 +2.3 +2D CFT for the quantum fluctuations of ρ(E) +12 +3 +Spectral correlations and entanglement +15 +3.1 +Spectral correlations +16 +3.1.1 +Non perturbative corrections to density of eigenvalues ⟨ρ(E)⟩ +17 +3.1.2 +The ramp and plateau in ⟨ρ(E1)ρ(E2)⟩ +18 +3.2 +Entanglement entropy +20 +3.2.1 +Emergence of spacetime in 2D string theory +24 +3.3 +Reduced density matrix for n < N eigenvalues +26 +4 +Open questions and future work +28 +– 1 – + +1 +Introduction +Random Matrix Models have been studied for a long time, as they provide a powerful +computational tool and a great source of insight with many applications in different fields, +from nuclear physics to condensed matter to high-energy physics.1 +One of the most intriguing applications arises from the connection to quantum gravity. +It was first observed by ’t Hooft [17] that a theory with matrix degrees of freedom can be +interpreted as a theory of random surfaces in the large-N limit and thus, in many cases, it +can be connected to some 2d quantum gravity/string theory. The perturbative expansion +in Feynman diagrams can be reorganized as a topological expansion in the genus of the +surface, with 1/N playing the role of the expansion parameter (string coupling). This idea +has found its most concrete realization so far in the AdS/CFT correspondence [18–20]. In +its most basic and well-understood instance, this correspondence relates a gravity theory +on 5D Anti-de Sitter space to a SU(N) gauge theory on the 4D boundary. Despite the +fact that the correspondence has a very precise formulation and has been tested to great +accuracy, its perhaps most striking conceptual aspect, namely the emergence of spacetime +from the matrix degrees of freedom, is still poorly understood. The correspondence gives in +principle a complete definition of quantum gravity in AdS, since the boundary theory is well +defined non-perturbatively (e.g. by the CFT axioms); however, because it is a weak/strong +coupling duality, it is still difficult to use it in order to find detailed answers to fundamental +questions, such as the information loss paradox, and the statistical interpretation of the +Bekenstein-Hawking entropy in terms of black hole microstates. +1.1 +Motivation +Driven by the desire to understand these questions in a simplified setting, there have been +many recent developments in low dimensional holography. +The SYK model [21, 22] is +composed of a large number of fermions interacting with disordered couplings. In the large- +N limit the low-energy sector of the model is described holographically by 2-dimensional +Jackiw-Teitelboim (JT) gravity, or equivalently by a 1D Schwarzian theory [23–25]. The +SYK model and JT gravity were shown to exhibit quantum chaos as universally described +by random matrix theory [26, 27]. The paper [28] showed a much stronger connection to +random matrices: the partition function of JT gravity on a surface of arbitrary genus and +number of boundaries agrees with the perturbative expansion of a certain matrix integral, +thus solving the theory to all orders in the genus expansion. The matrix integral is in- +terpreted as an average over an ensemble of Hamiltonians and the matrix eigenvalues as +the energy levels dual to gravitational microstates. It was also noted that JT gravity can +be seen as the p → ∞ limit of (2, p) minimal strings which were long known to be dual +to matrix models [10, 13, 29–31]. The study of non-perturbative effects in these models +can then help us understand the detailed structure of gravitational microstates. As a con- +sequence much effort has been devoted to this pursuit (see [32] for a review). However +1There are many reviews on the subject. For general aspects see, in rough order of complexity [1–6]. For +applications to low dimensional gravity and string theory see [7–14]. For applications to chaotic systems +see [15, 16]. +– 2 – + +many interconnected questions still remain. The matrix integral does not provide a unique +non-perturbative completion of JT gravity [33]. The bulk theory does not seem to have, +at first sight, a well defined dual quantum mechanical system, but rather an ensemble of +them. The presence of connected geometries and the consequent lack of factorization pose +a deep puzzle about the nature of the gravitational path integral [34]. A more explicit +understanding of the emergence of spacetime from the dual degrees of freedom remains to +be attained [35]. +In this work we discuss some of these issues by considering a generalization of the kind of +matrix integral dual to JT-gravity, given by a 0 + 1D theory of matrix quantum mechanics +[36]: +S = +� +dt tr +�1 +2 +˙H2 + V (H) +� +. +(1.1) +with an arbitrary potential V (H). The classical average over the matrix ensemble becomes +a quantum path integral: +� +dHe−NtrV (H) → +� +DH(t)e−S. +(1.2) +We can think of the original matrix integral as a matrix quantum mechanics with +one-time-point (as discussed in [37] for SYK) meaning that we look at a single instant of +time where the dynamics are frozen. We will make this statement precise and show that +we can reproduce matrix integral results from equal time correlations in matrix quantum +mechanics. In particular, we have a quantum density operator ˆρ(E) whose expectation +value is the ensemble-averaged density of eigenvalues: +ρ(E) = ⟨ˆρ(E)⟩ +(1.3) +and similarly for higher correlations. The spectral curve ρ0(E), defined as the large N limit +of ρ(E), can be chosen to be that of any specific matrix integral. This offers a possible +interpretation of JT gravity as being dual to a one-time-point matrix quantum mechanics +with the appropriate spectral curve ρJT +0 (E) [28]. +Matrix quantum mechanics is richer than a a matrix integral, first and foremost because +it is a quantum mechanical theory. +The eigenvalues are described by a wavefunction +ψN(E1, . . . , EN) instead of a classical probability distribution ρN(E1, . . . , EN) as in random +matrix theory. Since the matrix eigenvalues describe the microstates {Ei} of JT gravity, +we might think of matrix quantum mechanics as describing their associated wavefunctions +|Ei⟩. It is then natural to consider the entanglement between eigenvalues. The average +over the ensemble of random matrices becomes a quantum expectation value in Hilbert +space. Thus, the statistical fluctuations due to ensemble averaging over Hamiltonians may +now be interpreted as quantum fluctuations of a single quantum mechanical system. These +observations point to Matrix Quantum Mechanics as an interesting generalization of the +matrix integral dual to JT gravity. +In two spacetime dimensions there is another instance of holographic duality: the duality +between two-dimensional string theory and the c = 1 matrix model [38], a theory of matrix +quantum mechanics with a specific potential. This duality precedes AdS/CFT and has +– 3 – + +been extensively checked both perturbatively and non-perturbatively. 2 Two-dimensional +string theory and JT gravity form part of the same family of theories. A minimal string +consists of a Liouville CFT with cL > 25 and a minimal model with cM < 1 coupled by +anomaly cancellation. In the limit p → ∞ of the (2, p) minimal string, which corresponds +to JT gravity, we have that cM → −∞. Instead, two dimensional string theory consists of +a cL = 25 Liouville theory and a cM = 1 free boson. Thus JT gravity and two-dimensional +string theory lie at opposite ends of the same spectrum of worldsheet theories given by +Liouville theory coupled respectively to cM = −∞ and cM = 1.3 Despite knowing they are +related, the relation between the two dualities has yet to be understood explicitly (See [50– +53] for related work). From the matrix model point of view, JT gravity is dual to a matrix +integral over a single matrix while, by discretizing time, matrix quantum mechanics can be +thought of as the continuum limit of a chain of q matrices [54]. Thus JT is dual to a single +matrix while two-dimensional string theory is dual to an infinite number of matrices, one +for each instant of time. Understanding better the relationship between the two dualities +could help elucidate various aspects of JT gravity. For example, in 2D string theory the +dual is a single quantum mechanical system and no averaging is involved. Spacetime can +be thought of as emergent from the continuum of eigenvalues at large N and locality can be +probed by the entanglement between the eigenvalues [55–57], as we will demonstrate. The +worldsheet description present in minimal and 2D string theories allows for a detailed study +of non-perturbative effects [41, 42, 44, 58–65] and their matching to the matrix model. Un- +derstanding the relation between JT gravity and two-dimensional string theory at the level +of the dual matrix models motivates a new consideration of matrix quantum mechanics. +Finally, the duality between the c = 1 matrix model and two-dimensional string theory is +a perfect playground to study the emergence of spacetime from matrix degrees of freedom +in gauge theories since, at large N, the eigenvalues form a continuum that is directly re- +lated to the dual spacetime. The relation between spacetime and eigenvalue-space can be +tested in various ways, e.g. using local observables, scattering of the excitations, or using +entanglement entropy, as was done in [55]. The motivation of this last paper was to apply +to the c = 1 matrix model the insight, gained in AdS/CFT with the Ryu-Takayanagi for- +mula, of the essential role that entanglement plays in the emergence of spacetime [66, 67]. +In this paper we will give a different and more comprehensive perspective on the eigen- +value/spacetime relation by explicitly constructing the geometry of eigenvalue-space that +corresponds to the spacetime geometry in a natural way and studying its entanglement +properties. The entanglement between eigenvalues is an example of entanglement in target +space. Characterizing the entanglement of target space degrees of freedom is essential to +understand spacetime in string theory and holography, and recently there has been a grow- +ing interest in the subject, see [68–75] . +2See [8–11] for reviews. See [39–44] for extensive recent work on matching scattering amplitudes. See +[45–49] for recent related work on black holes +3For a review of the Liouville approach, see [14] +– 4 – + +1.2 +Overview and results +We start sec. 2 by recalling some basic facts about Matrix Quantum Mechanics. Eigenvalue +repulsion enforces fermionic statistics for the eigenvalues which can be mapped to a system +of fermions in an external potential. We introduce a second-quantized fermionic field Ψ(E) +which gives the eigenvalue density operator ˆρ(E) = Ψ†(E)Ψ(E). The density of eigenvalues +is the expectation value ρ(E) = ⟨ˆρ(E)⟩ which is, at leading order in the large N limit, +equal to the spectral curve ρ(E) ≈ ρ0(E). We then proceed in sec. 2.2 to illustrate the +construction of an effective hydrodynamical theory for the eigenvalues valid for arbitrary +ρ0(E) . The construction follows from recent developments in the study of 1D many body +quantum systems in external potentials [76–78]. It can be seen as a generalization of the +collective field theory approach [12] to arbitrary potentials. In sec. 2.3 we discuss quantum +fluctuations of the eigenvalues in the effective theory. One can show that the quantum +hydrodynamical fluctuations of the eigenvalues are described by a 2D free boson CFT on +a curved background determined by ρ0(E) with boundaries at the edge of the spectrum +where ρ0(E∗) = 0. +In section 3 we proceed to use the 2D CFT to study the different properties of the +eigenvalues. We start by computing spectral correlations in sec. 3.1 which are now given +by correlation functions of the density operator: ⟨ˆρ(E)⟩ and ⟨ρ(E1)ρ(E2)⟩. These are given +by correlation functions of vertex operators in the CFT. We reproduce the leading non- +perturbative corrections to the density of states ρ(E) and to the level-correlation ρ(E1, E2) +as described in sec. 5 of [28] by considering equal-time correlations. In other words, we +reproduce the oscillations of ρ(E) around the semiclassical density ρ0(E) and the terms in +ρ(E1, E2) corresponding to the ramp and plateau in the spectral form factor (i.e. the sine +kernel). In matrix quantum mechanics these spectral correlations arise due to quantum +fluctuations of a single quantum mechanical system, as opposed to statistical fluctuations +due to ensemble averaging. +This matching provides evidence to support the idea that +a matrix integral and consequently JT gravity might be interpreted as a one-time-point +matrix quantum mechanics with the same spectral curve ρ0(E). +In sec. 3.2 we consider the entanglement between the eigenvalues. We compute the +Rényi entropies for a bipartition (0, E)∪(E, ER), where ER is the right edge of the eigenvalue +density, finding some interesting features. For non double-scaled matrix models, where the +density has a right edge ER, we see that the entanglement entropy follows a “Page curve” +(as a function of the lenght of the interval) as required by unitarity and comes down +instead of growing indefinitely. This feature is lost in double-scaled models where ER → ∞ +indicating that indeed we are missing states from the spectrum. The entanglement entropy +is naturally finite due to the mean spacing between the eigenvalues +1 +ρ0(E) ∼ e−S0 acting +as a UV cutoff.4 In two-dimensional string theory we can interpret the finiteness of the +entropy as due to gs stringy effects regulating the divergence as first noted in [55]. We notice +that the entanglement entropy Sent(E) present a leading contribution proportional to the +4While finishing this paper, the work [79] appeared which discusses the finiteness of the entanglement +entropy in matrix quantum mechanics. Their methods are different and the discussion is complementary. +In particular, they discuss a vanishing potential V = 0 while we treat arbitrary potentials. +– 5 – + +microcanical entropy in the window E ± dE such that Sent(E) ∝ S0(E) = log(ρ0(E)), +indicating a large amount of short range entanglement between eigenvalues close to the +boundary. We also compute the entanglement entropy for an interval bipartition (E1, E2), +extending the results of [55] for the c = 1 matrix model to arbitrary spectral curves ρ0(E). +We provide constructive evidence for the proposed map between the eigenvalue-space and +the emergent spacetime in two-dimensional string theory[55, 57] and the identification of +the spacetime geometry with the geometry of the Fermi surface. +In sec. 3.3 we compute the one eigenvalue reduced density matrix obtained by tracing +out N − 1 eigenvalues, corresponding to the fermion one-body density matrix g(E, E′) = +⟨Ψ†(E)Ψ(E′)⟩. We also write the general expression for the n eigenvalue density matrix. +We conclude in sec. 4 with a discussion of open questions and possible future work. +2 +Quantum hydrodynamics of random matrix eigenvalues +We study the quantum mechanics of a random N × N hermitian matrix H(t) in a generic +potential V (H) with the following action: +S = N +� +dt tr +�1 +2 +˙H2 + V (H) +� +. +(2.1) +The eigenvalues (E1 . . . EN) of H no longer obey a classical probability distribution +as in Random Matrix Theory. Instead they are now described by a quantum mechanical +wavefunction ψN(E). We will now briefly summarize some well known facts about ma- +trix quantum mechanics (MQM) and derive the Schrodinger equation for the N-eigenvalue +wavefunction ψN(E). More details can be found in the above mentioned reviews [2, 7–11]. +2.1 +Eigenvalues as fermions +To study the eigenvalues we diagonalize the matrix H: +H = Ω†E Ω +(2.2) +where Ω ∈ SU(N) and E = diag(E1, . . . , EN). This change of variables has a non-trivial +jacobian which modifies the path integral measure DH(t): +� +DH = +� +DΩ +� +i +DEi∆2(E), +(2.3) +where ∆(E) = � +i βBKT [80–83]. +Consider now the Schrodinger equation for the singlet sector HχN(E) = ϵχN(E). The +wavefunctions χN(E) are clearly symmetric functions of the eigenvalues. By defining a +completely anti-symmetric wavefunction ψN(E) = ∆(E)χN(E), the Schrodinger equation +now reads: +N +� +i=1 +HiψN(E) = +N +� +i +ϵiψN(E), +Hi = −1 +2 +d2 +dE2 +i ++ V (Ei). +(2.7) +The Hamiltonian acting on ψN(E) is now a sum of single-particle Hamiltonians. The wave- +function ψN(E) is completely antisymmetric by construction due to the antisymmetry of +the Vandermonde and vanishes whenever Ei = Ej. We see that the eigenvalue repulsion +of random matrices enforces the Pauli exclusion principle. The eigenvalues Ei are then +equivalent to a system of N fermions each in an external potential V (E), interacting only +through the exclusion principle/eigenvalue repulsion. +The many-body ground state wavefunction ψN(E) can be obtained by first solving for +– 7 – + +the single particle wavefunctions ψϵ(E) and building the Slater determinant ψN(E) = +1 +√ +N!detij(ψϵi(Ej)) which involves a single fermion in each energy level up to the Fermi en- +ergy ϵF , the energy of the last (N-th) fermion. We will not do this as it involves solving the +Schrodinger equation for a specific choice of potential with the resulting Slater wavefunc- +tions ψN being complicated expressions for large N. We will instead describe the system +in second quantization by introducing a fermionic field Ψ(E) [55, 84] with the following +Hamiltonian H : +H = N +� +dEΨ†(E) +� +− 1 +2N2 +d2 +dE2 + V (E) +� +Ψ(E). +(2.8) +The fermionic field Ψ(E) can be expressed as a mode expansion with creation/annihilation +operators aϵ, a† +ϵ weighted by the single particle wavefunctions ψϵ(E): +Ψ(t, E) = +� +dϵe−iϵtaϵψϵ(E). +(2.9) +The fermions fill the potential V (E) up to the Fermi energy ϵF . +We can control how +the potential is filled by introducing a chemical potential µ = NϵF in the Hamiltonian +H − µΨ†Ψ. The system forms a Fermi surface |µ⟩ on which the operators aϵ, a† +ϵ act as +follows: +aϵ |µ⟩ = 0 +ϵ > µ, +a† +ϵ |µ⟩ = 0 +ϵ < µ. +(2.10) +The presence of a Fermi sea corresponds to having a finite density of eigenvalues ρ0(E) +in RMT. In what follows we will employ recent techniques from condensed matter [85] +describing the quantum fluctuations of the Fermi surface by a 2D effective hydrodynamical +theory . +Correspondingly, one can develop a quantum hydrodynamical theory for the eigenvalue +density ρ(E), describing the fluctuations around a semiclassical background ρ0(E) given by +the RMT spectral curve. Quantum fluctuations on top of the Fermi surface involving the +creation/annihilation of a single eigenvalue will produce non-perturbative effects in +1 +ρ0(E) ∼ +e−S0 (similarly as described in sec. 5 of [28]). The effective theory will be a simple free boson +CFT on a curved background with a boundary. This simple description allows us to study +many interesting features of Matrix Quantum Mechanics. We can access non-perturbative +physics like the oscillations in the density of states ρ(E) and the plateau in the two level +correlation ρ(E1, E2) which are a consequence of the underlying discreetness of the spectrum +of H. We can also compute observables that do not have a clear classical counterpart such +as the reduced density matrix obtained by tracing out k-out-of-N eigenvalues and the +spectrum of Renyi entropies for arbitrary bipartition (E1, E2). +Incorporating the chemical potential, we arrive at the following Hamiltonian, which is the +starting point for the rest of discussion: +H = +� +dEΨ†(t, E) +� +−1 +2 +d2 +dE2 + (V (E) − µ) +� +Ψ(t, E). +(2.11) +– 8 – + +2.2 +Effective hydrodynamics of the eigenvalue density ρ(E) +We now give a self-contained review of some recent developments in the study of 1D many- +body quantum systems in external potentials via hydrodynamics [76, 86–88]. The hydro- +dynamics approach to 1D quantum systems was introduced a few years ago in [77, 78] and +has been rapidly developing ever since, see the recent lectures [89] for a review. We will +only introduce the necessary tools for our purposes. +Conformal Field Theory in 2D is a well-proven technique in addressing 1D critical quantum +systems [90]. It is commonly used to describe low energy excitations around a fixed energy +scale (such as the Fermi energy ϵF ) and so it is not a priori possible to apply it to inho- +mogeneous systems, where there is a varying energy scale due to an external potential or +out-of-equilibrium dynamics. On the other hand, hydrodynamics is useful to describe inho- +mogeneous systems at mesoscopic scales, large enough to contain a macroscopic number of +degrees of freedom but smaller than the characteristic scale of the inhomogeneities. In [76], +they obtained a 2D CFT describing inhomogeneous 1D quantum systems using hydrody- +namics. The CFT lives on a non-trivial background metric encoding the inhomogeneities +of the original system. +Let us start by considering a many-body quantum system composed of N particles with a +finite density ρ(x) in the large N limit in an interval x ∈ (xL, xR). This means that the +quantum density operator ˆρ(x) = Ψ†(x)Ψ(x) acquires a VEV ρ(x) ≡ ⟨Ψ†(x)Ψ(x)⟩. The +VEV introduces a length scale in the system corresponding to the local average spacing +between particles d(x) = +1 +ρ(x). In Random Matrix Theory there is a finite density of eigen- +values ρ(E) due to eigenvalue repulsion, which is analogous to the non-zero VEV of the +quantum density operator ˆρ. We will make this correspondence precise in MQM: since the +eigenvalues are fermions we have x = E and we have that ρ(E) = ⟨ˆρ(E)⟩. The mean level +spacing is then equal to d = +1 +ρ(E) ∼ e−S0.5 The key assumption to develop hydrodynamics +for inhomogeneous systems is the separation of scales, meaning there exists an intermediate +mesoscopic scale ℓ such that: +d ≪ ℓ ≪ +ρ(x) +∂xρ(x), +(2.12) +where +ρ +∂ρ is the characteristic length scale of the inhomogeneities. +The scale ℓ is then +small enough such that the system is quasi-homogeneous and large enough to contain a +thermodynamically large number of particles. These scales provide both UV and IR cutoffs +in the effective theory defined at energy scales Λ such that: +∂xρ(x) +ρ(x) += ΛIR ≪ Λ ≪ ΛUV = 1 +d = ρ(x), +(2.13) +Having understood the characteristic scales and the regime of validity of the effective theory, +we will now focus on a specific system: the Lieb-Liniger gas of interacting bosons in an +5This is not the first instance where the spectral density ρ(E) is identified with a VEV, in [91] +ρ(E) is identified as the order parameter responsible for Causal Symmetry Breaking in the universal late +time behaviour of chaotic systems. +– 9 – + +external potential. It is defined by the following Hamiltonian: +H = +� +dx +� +Φ† +�¯h2∂2 +x +2m + V (x) +� +Φ + g +2Φ†2Φ2 +� +, +(2.14) +where Φ(x) is a bosonic field [Φ(x), Φ(x′)] = δ(x − x′). This system can be solved exactly +via Bethe-Ansatz in the homogeneous V = 0 case [92]. In the limit of hard-core bosons +g → ∞ it is equivalent to a system of free fermions in the potential V (x) and thus describes +the eigenvalues of MQM. This limit is often referred to as the Tonks-Girardeau gas in +the literature. The mapping between hard-core bosons and free fermions is made via a +Jordan-Wigner string: +Ψ†(x) = eiπ +� +y