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filter=lfs diff=lfs merge=lfs -text +NtFLT4oBgHgl3EQfOS8u/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +8tE4T4oBgHgl3EQf3A1I/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text diff --git a/0NAzT4oBgHgl3EQfRPtW/content/tmp_files/2301.01212v1.pdf.txt b/0NAzT4oBgHgl3EQfRPtW/content/tmp_files/2301.01212v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..564fa8e09cd07dc633910975f30dcf9ba04c0d76 --- /dev/null +++ b/0NAzT4oBgHgl3EQfRPtW/content/tmp_files/2301.01212v1.pdf.txt @@ -0,0 +1,596 @@ +arXiv:2301.01212v1 [q-fin.RM] 31 Dec 2022 +Assessment of creditworthiness models privacy-preserving +training with synthetic data +Ricardo Mu˜noz-Cancino1, Cristi´an Bravo2, Sebasti´an A. R´ıos3, and Manuel Gra˜na4 +1,3Business Intelligence Research Center (CEINE), Industrial Engineering Department, University of +Chile, Beauchef 851, Santiago 8370456, Chile +2Department of Statistical and Actuarial Sciences, The University of Western Ontario,1151 Richmond +Street, London, Ontario, N6A 5B7, Canada. +4Computational Intelligence Group, University of Basque Country, 20018 San Sebasti´an, Spain. +Abstract +Credit scoring models are the primary instrument used by financial institutions to manage +credit risk. The scarcity of research on behavioral scoring is due to the difficult data access. +Financial institutions have to maintain the privacy and security of borrowers’ information +refrain them from collaborating in research initiatives. In this work, we present a methodology +that allows us to evaluate the performance of models trained with synthetic data when they +are applied to real-world data. Our results show that synthetic data quality is increasingly +poor when the number of attributes increases. However, creditworthiness assessment models +trained with synthetic data show a reduction of 3% of AUC and 6% of KS when compared with +models trained with real data. These results have a significant impact since they encourage +credit risk investigation from synthetic data, making it possible to maintain borrowers’ privacy +and to address problems that until now have been hampered by the availability of information. +Keywords: credit scoring; synthetic data; generative adversarial networks; variational au- +toencoders +1 +Introduction +For decades financial institutions have used mathematical models to determine borrowers’ credit- +worthiness and consequently manage credit risk. The main objective of these models is to char- +acterize each borrower with the probability of not complying with their contractual obligations +The Basel Committee on Banking Supervision (2000), avoiding to give loans to applicants that +will not be able to pay them back. Despite all the years of research on credit scoring, there is still +little done on behavioral scoring models, which are the credit scoring models used on those clients +who have already been granted a loan, because it requires large volumes of data and a relevant +∗NOTICE: This is a preprint of a published work. Changes resulting from the publishing process, such as editing, corrections, +structural formatting, and other quality control mechanisms may not be reflected in this version of the document. Please cite this work +as follows: Mu˜noz-Cancino, R., Bravo, C., R´ıos, S.A., Gra˜na, M. (2022). Assessment of Creditworthiness Models Privacy-Preserving +Training with Synthetic Data. In: , et al. Hybrid Artificial Intelligent Systems. HAIS 2022. Lecture Notes in Computer Science(), vol +13469. Springer, Cham. https://doi.org/10.1007/978-3-031-15471-3 32 +∗E-mail +addresses: +rimunoz@uchile.cl +(Ricardo +Mu˜noz-Cancino), +cbravoro@uwo.ca +(Cristi´an +Bravo), +srios@dii.uchile.cl (Sebasti´an A. R´ıos), manuel.grana@ehu.es (Manuel Gra˜na) +1 + +historical depth Goh and Lee (2019); Kennedy et al. (2013). In addition, financial institutions are +often reluctant to collaborate in this type of investigation due to concerns about data security and +personal privacy. Until now, the use of synthetic data in credit scoring is mainly restricted to bal- +ancing the minority class in classification problems using the traditional SMOTE Gici´c and Subasi +(2019), variational autoencoders Wan et al. (2017), and lately generative adversarial networks +Fiore et al. (2019); Lei et al. (2020); Ngwenduna and Mbuvha (2021). In these studies, synthetic +records of the minority class are generated, and the original data set is augmented. In this paper, +we present a framework that allows us to train a model on synthetic data and then apply it to +real-world data. We also analyze if the model copes with data drift by applying both models to +real-world data representing the same problem but obtaining the dataset one year later. The main +findings of our work are: +• It is possible to train a model on synthetic data that achieves good performance in real +situations. +• As the number of features increases, the synthesized data quality gets worse. +• There is a performance cost for working in a privacy-preserving environment. +This cost +corresponds to a loss of predictive power of approximately 3% if measured in AUC and 6% +in KS. +2 +Related Work +2.1 +Credit Scoring +Credit scoring aims to manage credit risk, defined as the potential for a borrower to default on +established contractual obligations The Basel Committee on Banking Supervision (2000). These +models intensively use borrower data, demographic information, payment behavior, and even al- +ternative data sources such as social networks Mu˜noz-Cancino et al. (2021); ´Oskarsd´ottir et al. +(2019), psychometrics Djeundje et al. (2021), and geolocation Simumba et al. (2021). +2.2 +Generative models for synthetic data generation +Generative models are a subset of machine learning models whose main objective is to learn +the real-data distribution and then to generate consistent samples from the learned distribution. +Working with synthetic data allows addressing problems where real-data is expensive to obtain, +where a large dataset is needed to train a model, or where the real-data is sensitive or cannot +be shared Torres (2018). For years, statistical methods were the most used ones to estimate the +real-world data joint distribution. In this group, Gaussian Mixture Models are the most utilized +for this task when there are fewer continuous variables. At the same time, Bayesian Networks are +commonly used for discrete variables. The main problem of these methods is dealing with datasets +containing numerical and categorical variables. They also present problems when the continuous +variables have more than one mode and the categorical variables present small categories Xu et al. +(2020). During the last years, deep learning models have gained popularity to generate synthetic +data due to their performance and because they allow us to deal with the problems mentioned +above. The generative adversarial networks and the variational autoencoders stand out within +these models. +2 + +2.2.1 +Generative Adversarial Networks +Generative adversarial networks are a deep learning framework based on a game theory scenario +where a generator network G(·) must compete with a discriminator network D(·). The generator +network produces samples of synthetic data that attempt to emulate real data. In contrast, the +discriminator network aims to differentiate between real examples from the training dataset and +synthetic samples obtained from the generator Goodfellow et al. (2016). +Its most basic form, +vanilla GAN, G(·) maps a vector z from a multivariate Gaussian distribution N (0, I) to a vector +ˆx in the data domain X. While D(·) outputs a probability that indicates whether ˆx is a real +training samples or a fake sample drawn from G(·) Xu et al. (2020). +The generator G(·) and +the discriminator D(·) are alternatively optimized to train a GAN. Vanilla GANs have two main +problems, representing unbalanced categorical features and expressing numerical features having +multiple modes. To solve this, Xu et al. (2019) Xu et al. (2019) present a conditional generator +(CTGAN) that samples records from each category according to the log-frequency; this way, the +generator can explore all discrete values. Moreover, the multimodal distributions are handled using +kernel density estimation to assess the number of modes in each numerical feature. +2.2.2 +Variational autoencoders +Autoencoders (AE) are an unsupervised machine learning method that enables two main objec- +tives: low-dimensional representation and synthetic data generation. +Variational Autoencoder +Kingma and Welling (2013) interpret the latent space produced by the encoder as a probability +distribution modeling the training samples as independent random variables, assuming the poste- +rior distribution defined by the encoder qθ(z|x) and generative distribution pφ(x|z) defined by the +decoder. To accomplish that the encoder produces two vectors as output, one of means and the +other of standard deviations, which are the parameters to be optimized in the model. Xu et al. +(2019) Xu et al. (2019) present TVAE, a variational autoencoder adaption for tabular data, using +the same pre-processing as in CTGAN and the evidence lower bound (ELBO) loss. +3 +Methodology and Experimental Design +3.1 +Dataset +In this work, we use a dataset provided by a financial institution already used for research on credit +scoring Mu˜noz-Cancino et al. (2021,2). This dataset includes each borrower financial information +and social interactions features over two periods: January 2018 and January 2019; each dataset +contains 500,000 individuals. Each borrower is labeled based on their payment behavior in the +following 12-month observation period. Each borrower in the 2018 dataset is labeled as a defaulter +if it was more than 90 days past due between February 2018 and January 2019 and is labeled as a +non-defaulter if it was not more than 90 days past due. Borrowers from the Jan-2019 dataset are +similarly tagged. This dataset contains three feature subsets: XF in corresponds to the borrower’s +financial information, XDegree corresponds to the number of connections the borrower has in the +social interaction network, and XSocInt are the features extracted from the social interactions. +3 + +3.2 +Synthetic data generation +A step to privacy-preserving credit scoring model building is to generate a synthetic dataset that +mimics real-world behavior. In order to accomplish this, we compare the performance of two state- +of-the-art synthetic data generators, CTGAN and TVAE, defined in Sect. 2. The first experiment +(S01) only compares these methods using borrowers’ features XF in. The objective of this stage +is to find a method to generate synthetic data from real data, and it is not part of this study +to find the best way to generate them. Despite not generating an exhaustive search for the best +hyper-parameters, we will test two different architectures (Arch) for each synthesizer. Arch A is +the default configuration for both methods. In the case of CTGAN, Arch B set up the generator +with two linear residual layers and the discriminator with two linear layers, both of size (64, 64). +In the case of TVAE Arch B, set hidden layers of (64, 64) for both the encoder and the decoder. +Then, in experiment S02, we train a new synthesizer using the best architecture from S01. This +experiment uses the borrowers’ features XF in and exclusively one feature from the network data, +the node degree XDegree. We only include node degree because its feature enables us to reconstruct +an entire network using the random graphs generators. Finally, in experiment S03, the borrowers +and social interaction features (XF in + XDegree + XSocInt) are used to train a synthesizer. This +experiment corresponds to the traditional approach to generating synthetic data from a dataset +using social interaction features. +3.3 +Borrower’s creditworthiness assessment +The objective of this stage is to have a framework that allows us to estimate the borrower’s cred- +itworthiness from a feature set. +This modeling framework is based on previous investigations +Mu˜noz-Cancino et al. (2021,2). This stage begins by discarding attributes with low or null predic- +tive power and selecting uncorrelated attributes. The correlation-based selection method begins by +selecting the attribute with the highest predictive power. It then discards the possible selections +if the correlation exceeds a threshold ρ. This step is repeated until no attributes are left to select. +To ensure the model generalization capability, we work under a K-fold cross-validation scheme; +in this way, the feature selection and the model training use K-1 folds, and the evaluation is car- +ried out with the remaining fold. Additionally, we use two holdout datasets, one generated with +information from the same year as the training dataset but not contained. The second contains +information from one year later. Both the results of the validation fold and the holdout dataset +are stored to use a t-test later to compare different models (Flach, 2012, Ch. 12). +3.4 +Evaluation Metrics +In this section, we describe a set of metrics that will help us to evaluate the performance of the +synthetic data generators and the classification models used for creditworthiness assessment. The +area under the curve (AUC) is a performance measure used to evaluate classification models +Bradley (1997). The AUC is an overall measure of performance that can be interpreted as the +average of the true positive rate for all possible values of the false positive rate. A higher AUC +indicates a higher overall performance of the classification model Park Seong Ho (2004). Another +classification performance measure is the F-measure. This metric is calculated as the harmonic +mean between precision and recall. It is beneficial for dichotomous outputs and when there is +no preference between maximizing the model’s precision or recall Hripcsak and Rothschild (2005). +Kolmogorov-Smirnov (KS) statistic measures the distance separating two cumulative distributions +4 + +Hodges (1958). The KS statistic ranges between 0 and 1 and is defined as D = maxx |F1(x)−F2(x)|, +where F1 and F2 are two cumulative distributions. In the case of creditworthiness assessment, +we are interested in the difference between the cumulative distributions of defaulters and non- +defaulters, and a higher D indicates a higher discriminatory power. +However, in the case of +synthetic data generation, we are interested in the real data distribution and the synthetic data +distribution being as similar as possible; in this way, a lower D indicates a better synthetic data +generation. In order for all the acceptance criteria to be the same, we define the KSTest as 1−D; +in this way, a higher KSTest indicates a better synthetic data generator. In the synthetic data +generation problem, the KS is only valid to measure the performance for continuous features; to +handle categorical features, we will use the chi-square test (CS). The CS is a famous test to assess +the independence of two events McHugh (2013). We will call CSTest to the resulting p-value for +this test. Therefore a small value indicates we can reject the null hypothesis that synthetic data +has the real data distribution. In the synthetic data generation problem, we want to maximize the +CSTest. +3.5 +Experimental setup +The parameters of the univariate selection are set at KSmin = 0.01 and AUCmin = 0.53, i.e., we +discard feature with a univarite performance lower than KSmin or AUCmin. In the multivariate +selection process, we set ρ = 0.7 in the process to avoid high correlated features Akoglu (2018). The +N-Fold Cross-Validation stage is carried out considering N = 10, and in each iteration, the results +of regularized logistic regression and gradient boosting Friedman (2001) models are displayed. +4 +Results and Discussion +In this section, we present the results of our methodology. We start with the implementation +details. Then, we compare the synthesizers, and finally, we analyze the creditworthiness assessment +performance of the models trained using synthetic data. +4.1 +Implementation Details +In this work, we used the Python implementations of Networkx v2.6.3 Hagberg et al. (2008) and +Synthetic Data Vault (SDV) v5.0.0 Patki et al. (2016) for networks statistics and synthetic data +generation, respectively. To conduct the experiments, we used a laptop with 8 CPU cores Intel i7 +and 32 GB of RAM. +4.2 +Synthetic Data Generation Performance +The first objective is to analyze the performance of the methods to generate synthetic data pre- +sented above, CTGAN and TVAE. Table 1 shows the results obtained. The features Synthesizer +training features corresponds to the training feature set, while Arq indicates the network architec- +ture defined in Sect. 3.2. The experiment S01 consisted in comparing both synthesizer using two +different architectures. It is observed that a reduction in the number of layers reduces the execution +times considerably in both cases, being TVAE, the one that presented the fastest execution times. +KSTest show us the performance to synthesize continuous features, where TVAE achieves better +performance than CTGAN. The difference between TVAE architectures is almost negligible when +5 + +evaluate continuous features performance. The performance to synthesize categorical features is +measured using CSTest. In this case, TVAE obtained higher performance again, the differences +between architectures is slightly higher to architecture A. Another popular approach to measuring +the synthesizer performance is training a classifier to distinguish between real and synthetic data. +The column Logistic Detection in Table 1 shows the result after training a logistic regression model; +the value displayed corresponds to the complementary F-measure. In this way, values closer to 1 +indicate that the classifier cannot distinguish between real and synthetic data, and values closer +to 0 mean the classifier efficiently detects synthetic data. It can be seen that TVAE achieve the +best performance, but this performance decreases as we include more features to the synthesizer. +Table 1: Synthetic data generators performance +Experiment +Synthesizer training features +Synthesizer +Arch +Exec Time (m) +CSTest +KSTest +Logistic Detection +S01 +XF in +CTGAN +A +410 +0.836 +0.864 +0.697 +B +260 +0.861 +0.846 +0.749 +TVAE +A +230 +0.962 +0.868 +0.803 +B +130 +0.952 +0.861 +0.756 +S02 +XF in + XDegree +TVAE +B +140 +0.935 +0.836 +0.644 +S03 +XF in + XDegree + XSocInt +TVAE +A +400 +0.924 +0.809 +0.539 +S03 +XF in + XDegree + XSocInt +TVAE +B +320 +0.907 +0.825 +0.542 +S03 +XF in + XDegree + XSocInt +TVAE +B +465 +0.930 +0.819 +0.513 +4.3 +Creditworthiness assessment performance on real data +This section establishes a comparison line for the performance of the models trained with synthetic +data. In order to establish this comparison, we first trained classifiers using real-world data and +tested their performance using the holdout datasets previously defined. Table 2 shows the results +of training models according to the methodology described in 3.3. The performance is measured +using AUC and KS on the two holdout datasets; the 10-folds mean and its standard deviation are +shown for each statistic. For each feature set, we trained two classifiers, logistic regression and +gradient boosting. The results show that gradient boosting obtains better results compared to +logistic regression. More details of this comparison are shown in Table 3, where we quantify the +higher predictive power of gradient boosting. +Table 2: Creditworthiness assessment performance for models trained on real data +Classifier training +features +Classifier +Holdout 2018 +Holdout 2019 +AUC +KS +AUC +KS +XF in +GB +0.88 ± 0.001 +0.59 ± 0.002 +0.82 ± 0.001 +0.50 ± 0.002 +XF in +LR +0.87 ± 0.001 +0.58 ± 0.001 +0.82 ± 0.001 +0.50 ± 0.002 +XF in + XDegree + XSocInt +GB +0.88 ± 0.001 +0.59 ± 0.002 +0.82 ± 0.001 +0.50 ± 0.002 +XF in + XDegree + XSocInt +LR +0.87 ± 0.001 +0.58 ± 0.002 +0.83 ± 0.001 +0.50 ± 0.002 +XDegree + XSocInt +GB +0.61 ± 0.002 +0.17 ± 0.002 +0.62 ± 0.001 +0.18 ± 0.002 +XDegree + XSocInt +LR +0.60 ± 0.001 +0.17 ± 0.002 +0.61 ± 0.001 +0.18 ± 0.002 +Based on the results presented above, we will select gradient boosting for the comparisons +against the models trained on synthetic data that we will present in the next section. +4.4 +Creditworthiness assessment performance on synthetic data +This section aims to know how the performance of a creditworthiness assessment model (the +classifier) behaves when trained on synthetic data and applied to real-world data. Table 4 shows +6 + +Table 3: Gradient boosting and logistic regression comparison on real data (holdout 2018) +Classifier training features +AUC diff (%) +KS diff (%) +AUC diff p-value +KS diff p-value +XF in +0.70% +1.65% +0.000 +0.000 +XF in + XDegree + XSocInt +0.84% +1.91% +0.000 +0.000 +XDegree + XSocInt +1.65% +2.36% +0.000 +0.000 +the performance indicators on real-world data. Considering all synthesizers are trained with at +least the feature set XF in, the results of training the classifier with XF in are also displayed for all +synthesizers. It is observed that regardless of the synthesizer, training the classifier incorporating +at least feature set XF in produces similar performances in 2018 except in S02. However, when we +analyze how much the model degrades, the model trained with synthetic XF in from synthesizer +S01 is the one that suffers a minor discrimination power loss. It can be explained in part that a +better synthesizer manages to capture better the proper relationship between the borrower features +and the default. +Table 4: Creditworthiness assessment performance on real data for model trained on synthetic data +Synthesizer +Experiment +Classifier training +features +holdout 2018 +holdout 2019 +AUC +KS +AUC +KS +S01 +XF in +0.85 ± 0.003 +0.53 ± 0.002 +0.82 ± 0.002 +0.48 ± 0.002 +S02 +XF in +0.82 ± 0.001 +0.51 ± 0.001 +0.80 ± 0.001 +0.46 ± 0.002 +S03 +XF in +0.85 ± 0.002 +0.55 ± 0.002 +0.80 ± 0.002 +0.46 ± 0.002 +S03 +XF in + XDegree + XSocInt +0.85 ± 0.002 +0.56 ± 0.003 +0.80 ± 0.002 +0.47 ± 0.003 +S03 +XDegree + XSocInt +0.60 ± 0.002 +0.16 ± 0.002 +0.61 ± 0.003 +0.18 ± 0.002 +The comparison of performance obtained by the models trained with synthetic data against +the models trained on real-world data is presented in Table 5. We can understand this comparison +as the cost of using synthetic data, and it corresponds to the loss of predictive power to preserve +the borrower’s privacy. We can observe that in the best cases, this decrease in predictive power is +approximately 3% and 6% when we measure the performance in AUC and KS, respectively. +Table 5: Comparison between models trained using synthetic data and models trained on real data. ∗∗ +denotes when the difference is statistically significant using 0.05 as the p-value threshold, while ∗ uses 0.1. +Synthesizer +Experiment +Classifier training +features +holdout 2018 +holdout 2019 +AUC diff +KS diff +AUC diff +KS diff +S01 +XF in +-3.59%∗∗ +-10.09%∗∗ +-0.86%∗∗ +-3.92%∗∗ +S02 +XF in +-6.24%∗∗ +-13.24%∗∗ +-3.32%∗∗ +-6.48%∗∗ +S03 +XF in +-2.81%∗∗ +-6.01%∗∗ +-3.21%∗∗ +-6.70%∗∗ +S03 +XF in + XDegree + XSocInt +-3.12%∗∗ +-5.68%∗∗ +-2.54%∗∗ +-4.73%∗∗ +S03 +XDegree + XSocInt +-1.85%∗∗ +-4.31%∗∗ +-0.69%∗∗ +1.10%∗ +5 +Conclusions +This work aimed to use synthetic data to train creditworthiness assessment models. We used a +massive dataset of 1 million individuals and trained state-of-the-art synthesizer methods to obtain +synthetic data and achieve this goal. Then, we presented a training framework that allows us to +analyze trained models with synthetic data and observe their performance on real-world data. In +addition, we observed their performance one year after being trained to see how susceptible they +7 + +are to data drift. Our results show that lower quality synthetic data is obtained as we increase the +number of attributes in the synthesizer. Despite this, it is possible to use these data to train models +that obtain good results in real-world scenarios, with only a reduction in the predictive power of +approximately 3% and 6% when we measure the performance in AUC and KS, respectively. These +findings are of great relevance since they allow us to train accurate creditworthiness models. At the +same time, we keep borrowers’ privacy and encourage financial institutions to strengthen ties with +academia and foster collaboration and research in credit scoring without the privacy and security +restrictions. +6 +Future Work +Our future work will delve into how to synthesize social interactions’ information in the form of +graphs and not as added attributes to the training dataset since, as we show, this deteriorates the +quality of the synthetic data. +Acknowledgements +This work would not have been accomplished without the financial support of CONICYT-PFCHA +/ DOCTORADO BECAS CHILE / 2019-21190345. The second author acknowledges the support +of the Natural Sciences and Engineering Research Council of Canada (NSERC) [Discovery Grant +RGPIN-2020-07114]. This research was undertaken, in part, thanks to funding from the Canada +Research Chairs program. The last author thanks the support of MICIN UNDER project PID2020- +116346GB-I00. +References +Akoglu, H. (2018). 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CoRR, abs/1907.00503. +9 + diff --git a/0NAzT4oBgHgl3EQfRPtW/content/tmp_files/load_file.txt b/0NAzT4oBgHgl3EQfRPtW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c4c04984a4ea07ebae83e13c844fac8f34b543f9 --- /dev/null +++ b/0NAzT4oBgHgl3EQfRPtW/content/tmp_files/load_file.txt @@ -0,0 +1,561 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf,len=560 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='01212v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='RM] 31 Dec 2022 Assessment of creditworthiness models privacy-preserving training with synthetic data Ricardo Mu˜noz-Cancino1, Cristi´an Bravo2, Sebasti´an A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' R´ıos3, and Manuel Gra˜na4 1,3Business Intelligence Research Center (CEINE), Industrial Engineering Department, University of Chile, Beauchef 851, Santiago 8370456, Chile 2Department of Statistical and Actuarial Sciences, The University of Western Ontario,1151 Richmond Street, London, Ontario, N6A 5B7, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 4Computational Intelligence Group, University of Basque Country, 20018 San Sebasti´an, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Abstract Credit scoring models are the primary instrument used by financial institutions to manage credit risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The scarcity of research on behavioral scoring is due to the difficult data access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Financial institutions have to maintain the privacy and security of borrowers’ information refrain them from collaborating in research initiatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In this work, we present a methodology that allows us to evaluate the performance of models trained with synthetic data when they are applied to real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Our results show that synthetic data quality is increasingly poor when the number of attributes increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' However, creditworthiness assessment models trained with synthetic data show a reduction of 3% of AUC and 6% of KS when compared with models trained with real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' These results have a significant impact since they encourage credit risk investigation from synthetic data, making it possible to maintain borrowers’ privacy and to address problems that until now have been hampered by the availability of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Keywords: credit scoring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' synthetic data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' generative adversarial networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' variational au- toencoders 1 Introduction For decades financial institutions have used mathematical models to determine borrowers’ credit- worthiness and consequently manage credit risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The main objective of these models is to char- acterize each borrower with the probability of not complying with their contractual obligations The Basel Committee on Banking Supervision (2000), avoiding to give loans to applicants that will not be able to pay them back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Despite all the years of research on credit scoring, there is still little done on behavioral scoring models, which are the credit scoring models used on those clients who have already been granted a loan, because it requires large volumes of data and a relevant ∗NOTICE: This is a preprint of a published work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Changes resulting from the publishing process, such as editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this version of the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Please cite this work as follows: Mu˜noz-Cancino, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=', Bravo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=', R´ıos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=', Gra˜na, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Assessment of Creditworthiness Models Privacy-Preserving Training with Synthetic Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In: , et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Hybrid Artificial Intelligent Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' HAIS 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Lecture Notes in Computer Science(), vol 13469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Springer, Cham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='1007/978-3-031-15471-3 32 ∗E-mail addresses: rimunoz@uchile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='cl (Ricardo Mu˜noz-Cancino), cbravoro@uwo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='ca (Cristi´an Bravo), srios@dii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='uchile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='cl (Sebasti´an A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' R´ıos), manuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='grana@ehu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='es (Manuel Gra˜na) 1 historical depth Goh and Lee (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Kennedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In addition, financial institutions are often reluctant to collaborate in this type of investigation due to concerns about data security and personal privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Until now, the use of synthetic data in credit scoring is mainly restricted to bal- ancing the minority class in classification problems using the traditional SMOTE Gici´c and Subasi (2019), variational autoencoders Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2017), and lately generative adversarial networks Fiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Ngwenduna and Mbuvha (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In these studies, synthetic records of the minority class are generated, and the original data set is augmented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In this paper, we present a framework that allows us to train a model on synthetic data and then apply it to real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' We also analyze if the model copes with data drift by applying both models to real-world data representing the same problem but obtaining the dataset one year later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The main findings of our work are: It is possible to train a model on synthetic data that achieves good performance in real situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' As the number of features increases, the synthesized data quality gets worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' There is a performance cost for working in a privacy-preserving environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' This cost corresponds to a loss of predictive power of approximately 3% if measured in AUC and 6% in KS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 2 Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='1 Credit Scoring Credit scoring aims to manage credit risk, defined as the potential for a borrower to default on established contractual obligations The Basel Committee on Banking Supervision (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' These models intensively use borrower data, demographic information, payment behavior, and even al- ternative data sources such as social networks Mu˜noz-Cancino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' ´Oskarsd´ottir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2019), psychometrics Djeundje et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2021), and geolocation Simumba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='2 Generative models for synthetic data generation Generative models are a subset of machine learning models whose main objective is to learn the real-data distribution and then to generate consistent samples from the learned distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Working with synthetic data allows addressing problems where real-data is expensive to obtain, where a large dataset is needed to train a model, or where the real-data is sensitive or cannot be shared Torres (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' For years, statistical methods were the most used ones to estimate the real-world data joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In this group, Gaussian Mixture Models are the most utilized for this task when there are fewer continuous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' At the same time, Bayesian Networks are commonly used for discrete variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The main problem of these methods is dealing with datasets containing numerical and categorical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' They also present problems when the continuous variables have more than one mode and the categorical variables present small categories Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' During the last years, deep learning models have gained popularity to generate synthetic data due to their performance and because they allow us to deal with the problems mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The generative adversarial networks and the variational autoencoders stand out within these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='1 Generative Adversarial Networks Generative adversarial networks are a deep learning framework based on a game theory scenario where a generator network G(·) must compete with a discriminator network D(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The generator network produces samples of synthetic data that attempt to emulate real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In contrast, the discriminator network aims to differentiate between real examples from the training dataset and synthetic samples obtained from the generator Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Its most basic form, vanilla GAN, G(·) maps a vector z from a multivariate Gaussian distribution N (0, I) to a vector ˆx in the data domain X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' While D(·) outputs a probability that indicates whether ˆx is a real training samples or a fake sample drawn from G(·) Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The generator G(·) and the discriminator D(·) are alternatively optimized to train a GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Vanilla GANs have two main problems, representing unbalanced categorical features and expressing numerical features having multiple modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' To solve this, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2019) Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2019) present a conditional generator (CTGAN) that samples records from each category according to the log-frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' this way, the generator can explore all discrete values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Moreover, the multimodal distributions are handled using kernel density estimation to assess the number of modes in each numerical feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='2 Variational autoencoders Autoencoders (AE) are an unsupervised machine learning method that enables two main objec- tives: low-dimensional representation and synthetic data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Variational Autoencoder Kingma and Welling (2013) interpret the latent space produced by the encoder as a probability distribution modeling the training samples as independent random variables, assuming the poste- rior distribution defined by the encoder qθ(z|x) and generative distribution pφ(x|z) defined by the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' To accomplish that the encoder produces two vectors as output, one of means and the other of standard deviations, which are the parameters to be optimized in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2019) Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2019) present TVAE, a variational autoencoder adaption for tabular data, using the same pre-processing as in CTGAN and the evidence lower bound (ELBO) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 3 Methodology and Experimental Design 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='1 Dataset In this work, we use a dataset provided by a financial institution already used for research on credit scoring Mu˜noz-Cancino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2021,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' This dataset includes each borrower financial information and social interactions features over two periods: January 2018 and January 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' each dataset contains 500,000 individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Each borrower is labeled based on their payment behavior in the following 12-month observation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Each borrower in the 2018 dataset is labeled as a defaulter if it was more than 90 days past due between February 2018 and January 2019 and is labeled as a non-defaulter if it was not more than 90 days past due.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Borrowers from the Jan-2019 dataset are similarly tagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' This dataset contains three feature subsets: XF in corresponds to the borrower’s financial information, XDegree corresponds to the number of connections the borrower has in the social interaction network, and XSocInt are the features extracted from the social interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='2 Synthetic data generation A step to privacy-preserving credit scoring model building is to generate a synthetic dataset that mimics real-world behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In order to accomplish this, we compare the performance of two state- of-the-art synthetic data generators, CTGAN and TVAE, defined in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The first experiment (S01) only compares these methods using borrowers’ features XF in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The objective of this stage is to find a method to generate synthetic data from real data, and it is not part of this study to find the best way to generate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Despite not generating an exhaustive search for the best hyper-parameters, we will test two different architectures (Arch) for each synthesizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Arch A is the default configuration for both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In the case of CTGAN, Arch B set up the generator with two linear residual layers and the discriminator with two linear layers, both of size (64, 64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In the case of TVAE Arch B, set hidden layers of (64, 64) for both the encoder and the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Then, in experiment S02, we train a new synthesizer using the best architecture from S01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' This experiment uses the borrowers’ features XF in and exclusively one feature from the network data, the node degree XDegree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' We only include node degree because its feature enables us to reconstruct an entire network using the random graphs generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Finally, in experiment S03, the borrowers and social interaction features (XF in + XDegree + XSocInt) are used to train a synthesizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' This experiment corresponds to the traditional approach to generating synthetic data from a dataset using social interaction features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='3 Borrower’s creditworthiness assessment The objective of this stage is to have a framework that allows us to estimate the borrower’s cred- itworthiness from a feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' This modeling framework is based on previous investigations Mu˜noz-Cancino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2021,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' This stage begins by discarding attributes with low or null predic- tive power and selecting uncorrelated attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The correlation-based selection method begins by selecting the attribute with the highest predictive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' It then discards the possible selections if the correlation exceeds a threshold ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' This step is repeated until no attributes are left to select.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' To ensure the model generalization capability, we work under a K-fold cross-validation scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' in this way, the feature selection and the model training use K-1 folds, and the evaluation is car- ried out with the remaining fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Additionally, we use two holdout datasets, one generated with information from the same year as the training dataset but not contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The second contains information from one year later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Both the results of the validation fold and the holdout dataset are stored to use a t-test later to compare different models (Flach, 2012, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='4 Evaluation Metrics In this section, we describe a set of metrics that will help us to evaluate the performance of the synthetic data generators and the classification models used for creditworthiness assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The area under the curve (AUC) is a performance measure used to evaluate classification models Bradley (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The AUC is an overall measure of performance that can be interpreted as the average of the true positive rate for all possible values of the false positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' A higher AUC indicates a higher overall performance of the classification model Park Seong Ho (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Another classification performance measure is the F-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' This metric is calculated as the harmonic mean between precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' It is beneficial for dichotomous outputs and when there is no preference between maximizing the model’s precision or recall Hripcsak and Rothschild (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Kolmogorov-Smirnov (KS) statistic measures the distance separating two cumulative distributions 4 Hodges (1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The KS statistic ranges between 0 and 1 and is defined as D = maxx |F1(x)−F2(x)|, where F1 and F2 are two cumulative distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In the case of creditworthiness assessment, we are interested in the difference between the cumulative distributions of defaulters and non- defaulters, and a higher D indicates a higher discriminatory power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' However, in the case of synthetic data generation, we are interested in the real data distribution and the synthetic data distribution being as similar as possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' in this way, a lower D indicates a better synthetic data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In order for all the acceptance criteria to be the same, we define the KSTest as 1−D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' in this way, a higher KSTest indicates a better synthetic data generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In the synthetic data generation problem, the KS is only valid to measure the performance for continuous features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' to handle categorical features, we will use the chi-square test (CS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The CS is a famous test to assess the independence of two events McHugh (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' We will call CSTest to the resulting p-value for this test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Therefore a small value indicates we can reject the null hypothesis that synthetic data has the real data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In the synthetic data generation problem, we want to maximize the CSTest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='5 Experimental setup The parameters of the univariate selection are set at KSmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='01 and AUCmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='53, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=', we discard feature with a univarite performance lower than KSmin or AUCmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In the multivariate selection process, we set ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='7 in the process to avoid high correlated features Akoglu (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The N-Fold Cross-Validation stage is carried out considering N = 10, and in each iteration, the results of regularized logistic regression and gradient boosting Friedman (2001) models are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 4 Results and Discussion In this section, we present the results of our methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' We start with the implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Then, we compare the synthesizers, and finally, we analyze the creditworthiness assessment performance of the models trained using synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='1 Implementation Details In this work, we used the Python implementations of Networkx v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='3 Hagberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2008) and Synthetic Data Vault (SDV) v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='0 Patki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' (2016) for networks statistics and synthetic data generation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' To conduct the experiments, we used a laptop with 8 CPU cores Intel i7 and 32 GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='2 Synthetic Data Generation Performance The first objective is to analyze the performance of the methods to generate synthetic data pre- sented above, CTGAN and TVAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Table 1 shows the results obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The features Synthesizer training features corresponds to the training feature set, while Arq indicates the network architec- ture defined in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The experiment S01 consisted in comparing both synthesizer using two different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' It is observed that a reduction in the number of layers reduces the execution times considerably in both cases, being TVAE, the one that presented the fastest execution times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' KSTest show us the performance to synthesize continuous features, where TVAE achieves better performance than CTGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The difference between TVAE architectures is almost negligible when 5 evaluate continuous features performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The performance to synthesize categorical features is measured using CSTest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In this case, TVAE obtained higher performance again, the differences between architectures is slightly higher to architecture A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Another popular approach to measuring the synthesizer performance is training a classifier to distinguish between real and synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The column Logistic Detection in Table 1 shows the result after training a logistic regression model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' the value displayed corresponds to the complementary F-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In this way, values closer to 1 indicate that the classifier cannot distinguish between real and synthetic data, and values closer to 0 mean the classifier efficiently detects synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' It can be seen that TVAE achieve the best performance, but this performance decreases as we include more features to the synthesizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Table 1: Synthetic data generators performance Experiment Synthesizer training features Synthesizer Arch Exec Time (m) CSTest KSTest Logistic Detection S01 XF in CTGAN A 410 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='836 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='697 B 260 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='749 TVAE A 230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='868 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='803 B 130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='952 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='756 S02 XF in + XDegree TVAE B 140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='836 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='644 S03 XF in + XDegree + XSocInt TVAE A 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='809 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='539 S03 XF in + XDegree + XSocInt TVAE B 320 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='542 S03 XF in + XDegree + XSocInt TVAE B 465 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='930 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='819 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='513 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='3 Creditworthiness assessment performance on real data This section establishes a comparison line for the performance of the models trained with synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In order to establish this comparison, we first trained classifiers using real-world data and tested their performance using the holdout datasets previously defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Table 2 shows the results of training models according to the methodology described in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The performance is measured using AUC and KS on the two holdout datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' the 10-folds mean and its standard deviation are shown for each statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' For each feature set, we trained two classifiers, logistic regression and gradient boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The results show that gradient boosting obtains better results compared to logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' More details of this comparison are shown in Table 3, where we quantify the higher predictive power of gradient boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Table 2: Creditworthiness assessment performance for models trained on real data Classifier training features Classifier Holdout 2018 Holdout 2019 AUC KS AUC KS XF in GB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 XF in LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 XF in + XDegree + XSocInt GB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 XF in + XDegree + XSocInt LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 XDegree + XSocInt GB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 XDegree + XSocInt LR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 Based on the results presented above, we will select gradient boosting for the comparisons against the models trained on synthetic data that we will present in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='4 Creditworthiness assessment performance on synthetic data This section aims to know how the performance of a creditworthiness assessment model (the classifier) behaves when trained on synthetic data and applied to real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Table 4 shows 6 Table 3: Gradient boosting and logistic regression comparison on real data (holdout 2018) Classifier training features AUC diff (%) KS diff (%) AUC diff p-value KS diff p-value XF in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='70% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='65% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='000 XF in + XDegree + XSocInt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='84% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='91% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='000 XDegree + XSocInt 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='65% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='36% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='000 the performance indicators on real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Considering all synthesizers are trained with at least the feature set XF in, the results of training the classifier with XF in are also displayed for all synthesizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' It is observed that regardless of the synthesizer, training the classifier incorporating at least feature set XF in produces similar performances in 2018 except in S02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' However, when we analyze how much the model degrades, the model trained with synthetic XF in from synthesizer S01 is the one that suffers a minor discrimination power loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' It can be explained in part that a better synthesizer manages to capture better the proper relationship between the borrower features and the default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Table 4: Creditworthiness assessment performance on real data for model trained on synthetic data Synthesizer Experiment Classifier training features holdout 2018 holdout 2019 AUC KS AUC KS S01 XF in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 S02 XF in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 S03 XF in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 S03 XF in + XDegree + XSocInt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='003 S03 XDegree + XSocInt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='002 The comparison of performance obtained by the models trained with synthetic data against the models trained on real-world data is presented in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' We can understand this comparison as the cost of using synthetic data, and it corresponds to the loss of predictive power to preserve the borrower’s privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' We can observe that in the best cases, this decrease in predictive power is approximately 3% and 6% when we measure the performance in AUC and KS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Table 5: Comparison between models trained using synthetic data and models trained on real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' ∗∗ denotes when the difference is statistically significant using 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='05 as the p-value threshold, while ∗ uses 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Synthesizer Experiment Classifier training features holdout 2018 holdout 2019 AUC diff KS diff AUC diff KS diff S01 XF in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='59%∗∗ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='09%∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='86%∗∗ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='92%∗∗ S02 XF in 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='24%∗∗ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='24%∗∗ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='32%∗∗ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='48%∗∗ S03 XF in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='81%∗∗ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='01%∗∗ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='21%∗∗ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='70%∗∗ S03 XF in + XDegree + XSocInt 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='12%∗∗ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='68%∗∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='54%∗∗ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='73%∗∗ S03 XDegree + XSocInt 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='85%∗∗ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='31%∗∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='69%∗∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content='10%∗ 5 Conclusions This work aimed to use synthetic data to train creditworthiness assessment models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' We used a massive dataset of 1 million individuals and trained state-of-the-art synthesizer methods to obtain synthetic data and achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Then, we presented a training framework that allows us to analyze trained models with synthetic data and observe their performance on real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' In addition, we observed their performance one year after being trained to see how susceptible they 7 are to data drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Our results show that lower quality synthetic data is obtained as we increase the number of attributes in the synthesizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Despite this, it is possible to use these data to train models that obtain good results in real-world scenarios, with only a reduction in the predictive power of approximately 3% and 6% when we measure the performance in AUC and KS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' These findings are of great relevance since they allow us to train accurate creditworthiness models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' At the same time, we keep borrowers’ privacy and encourage financial institutions to strengthen ties with academia and foster collaboration and research in credit scoring without the privacy and security restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' 6 Future Work Our future work will delve into how to synthesize social interactions’ information in the form of graphs and not as added attributes to the training dataset since, as we show, this deteriorates the quality of the synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' Acknowledgements This work would not have been accomplished without the financial support of CONICYT-PFCHA / DOCTORADO BECAS CHILE / 2019-21190345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NAzT4oBgHgl3EQfRPtW/content/2301.01212v1.pdf'} +page_content=' The second author acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) [Discovery Grant RGPIN-2020-07114].' metadata={'source': 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b/19E2T4oBgHgl3EQf5QiW/content/tmp_files/2301.04189v1.pdf.txt @@ -0,0 +1,753 @@ +Spectral edge-to-edge topological state transfer in diamond photonic lattices +Gabriel C´aceres-Aravena†,1, 2 Basti´an Real†,1, 2 Diego Guzm´an-Silva,1, 2 Paloma Vildoso,1, 2 +Ignacio Salinas,1, 2 Alberto Amo,3 Tomoki Ozawa,4 and Rodrigo A. Vicencio1, 2, ∗ +1Departamento de F´ısica, Facultad de Ciencias F´ısicas y Matem´aticas, Universidad de Chile, Chile +2Millenium Institute for Research in Optics - MIRO, Chile +3Univ. Lille, CNRS, UMR 8523—PhLAM—Physique des Lasers Atomes et Mol´ecules, F-59000 Lille, France +4Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, Sendai 980-8577, Japan +(Dated: January 12, 2023) +Transfer of information between topological edge states is a robust way of spatially manipulating +quantum states while preserving their coherence in lattice environments. This method is particularly +efficient when the edge modes are kept within the topological gap of the lattice during the transfer. +In this work we show experimentally the transfer of photonic modes between topological edge states +located at opposite ends of a dimerized one-dimensional photonic lattice. We use a diamond lattice +of coupled waveguides and show that the transfer is insensitive both to the presence of a high density +of states in the form of a flat band at an energy close to that of the edge states, and to the presence +of disorder in the hoppings. We explore dynamics in the waveguide lattice using wavelength-scan +method, where different input wavelength translates into different effective waveguide length. These +results open the way to the implementation of more efficient protocols based on the active driving +of the hoppings. +Topological edge states are a remarkable resource to +engineer photonic systems with isolated modes protected +from the presence of disorder. In two-dimensional lat- +tices, they can be used to fabricate topological edge mode +lasers with distributed gain and quantized orbital mo- +mentum [1, 2], to transfer single photons around cor- +ners in elaborated photonic circuits [3, 4], and to de- +sign topological frequency combs with enhanced effi- +ciency [5, 6]. +One dimensional systems such as the +Su-Schrieffer-Heeger (SSH) lattice are particularly in- +teresting because topological edge and interface modes +are hosted deep into the topological gap of the lattice. +This gap protection has been shown to be beneficial +to preserve the quantum state of photons in boundary +modes [7, 8]. Interestingly, the presence of topological +edge modes on both sides of one-dimensional lattices can +be used to transfer a state from one edge of the lattice +to the other with high fidelity with the advantage of be- +ing protected from certain types of disorder due to the +topological nature of the system. Such edge state trans- +fer is a promising route to store and manipulate photonic +quantum states in on-chip lattice environments. +Most topological edge transfer protocols rely on the +adiabatic evolution of the lattice such that an edge mode +is driven into quasi-bulk modes and again into an edge +mode at the other side [9–17]. +While these protocols +present an optimized transfer rate and fidelity, they are +limited by the adiabaticity condition that requires the +adiabatic passage to be slow enough to avoid the Zen- +ner coupling of the edge state information into the bulk +modes [18, 19]. Furthermore, the presence of disorder in +the lattice would enhance this coupling. A variation of +these protocols include counter-adiabatic driving meth- +ods [20]. Recently, a different route has been proposed +based on the coherent coupling of edge modes within the +gap [19, 21, 22]. The great advantage of this approach +is that edge modes are kept well into the topological gap +throughout the protocol, ensuring a high fidelity in re- +duced times. The simplest version of the coherent state +transfer of topological edge states is via passive evanes- +cent coupling of the exponential tails of edge modes at op- +posite sides of the finite size lattice. In this case, coherent +transfer between edge modes takes place at a frequency +determined by the tail overlap, which can be controlled +via the size of the gap. Observation of such coherent os- +cillations was reported in a short SSH lattice for Rydberg +atoms [23]. +In this work, we demonstrate coherent edge-to-edge +transfer of light in a dimerized diamond lattice of coupled +waveguides employing a spectral tomographic technique. +More importantly, we show experimentally that the fi- +delity of the transfer is robust to a number of perturba- +tions in the system. First, we show that orthogonality of +eigenmodes in our undriven protocol preserves the trans- +fer even in the presence of a high density of states in the +form of a flat band at energies close to that of the edge +states. Second, we demonstrate that the transfer mech- +anism is robust to the presence of lattice defects thanks +to the underlying chiral symmetry of the system. The +experimental proof of principle we report in this work +can be significantly sped up by applying a number of +driven techniques based on the modulating of the hop- +pings in time and the use of concatenated topological +lattices [19, 22, 24]. Such techniques are readily imple- +mentable in lattices of coupled waveguides. +To demonstrate the topological edge transfer we use +a diamond lattice of coupled waveguides with different +intracell (t1) and intercell (t2) hoppings, as sketched in +Fig. 1(a). The lattice has three sites per unit cell, denoted +as A, B and C sites. Considering a tight-binding coupled- +mode approach, the evolution of the optical field at every +arXiv:2301.04189v1 [physics.optics] 10 Jan 2023 + +2 +d2 +AB5HicdZDLTgIxFIb +P4A3xhrp04gmrsgM4G1H +4sYlRrkMCGdcgYaOpe0 +HRMy4Q10ZdSdT+QL+DZ2 +ABM1+q+nv9vcv7jxYIr +bdsfVm5peWV1Lb9e2Njc +2t4p7u61VJRIhk0WiUh2P +KpQ8BCbmuBnVgiDTyBb +W98lfnte5SKR+GdnsToB +nQYcp8zqs3odtCv9Islu +3yZqUrmcJaBXNs5Q4ZX +umEizU6Bfe4OIJQGmg +mqVNexY+2mVGrOBE4LvU +RhTNmYDrFrMKQBKjedrT +olx34kiR4hmb2/Z1MaKDU +JPJMJqB6p3142/MvrJtq +/cFMexonGkJmI8fxEB2 +RrDEZcIlMi4kByiQ3WxI +2opIybe5SMPW/OpL/oVUp +O9Vy5aZWqh8tDpGHAziE +E3DgHOpwDQ1oAoMhPMIL +vFq+9WA9Wc/zaM5a/NmH +H7LePgFuhotw +d1 +AB +5HicdZDLTgIxFIbP4A3xhr +p04gmrsgM4G1H4sYlRrkM +CGdcgYaOpe0HRMy4Q10ZdS +dT+QL+DZ2ABM1+q+nv9vc +v7jxYIrbdsfVm5peWV1Lb9 +e2Njc2t4p7u61VJRIhk0WiU +h2PKpQ8BCbmuBnVgiDTyB +bW98lfnte5SKR+GdnsToBn +QYcp8zqs3odtB3+sWSXb7M +VCVzOMvArjm2c0qcsj1TCRZ +q9IvUHEkgBDzQRVquvYs +XZTKjVnAqeFXqIwpmxMh9g +1GNIAlZvOVp2SYz+SRI+Qz +N7fsykNlJoEnskEVI/Uby8b +/uV1E+1fuCkP40RjyEzEeH +4iI5I1pgMuESmxcQAZKb +LQkbUmZNncpmPpfHcn/0Kq +UnWq5clMr1Y8Wh8jDARzC +ThwDnW4hgY0gcEQHuEFXi3 +ferCerOd5NGct/uzD1lvn +20Ii28= +dv +AB5HicdZ +DLTgIxFIbP4A3xhrp04gmriYzgLcdiRuXGAVJYEI6 +5Qw0dC5pOySE8Aa6MurOJ/IFfBs7gIka/Vdfz/83O +f/xE8GVdpwPK7e0vLK6l8vbGxube8Ud/eaKk4lw +aLRSxbPlUoeIQNzbXAViKRhr7Ae394lfn3I5SKx9G +dHifohbQf8YAzqs3otcdYslx7MVCFzOMvAqbqO +e0pc25mpBAvVu8X3Ti9maYiRZoIq1XadRHsTKjVnAq +eFTqowoWxI+9g2GNEQlTeZrTolx0EsiR4gmb2/Zyc +0VGoc+iYTUj1Qv71s+JfXTnVw4U14lKQaI2YixgtS +QXRMsakxyUyLcYGKJPcbEnYgErKtLlLwdT/6kj+h +2bZdit2+aZaqh0tDpGHAziE3DhHGpwDXVoAIM+PMI +LvFqB9WA9Wc/zaM5a/NmH7LePgHT/ou0 +(d) +(c) +fs +glass wafer +(a) +(b) +(e1) (e2) ++ ++ +Figure 1. (a) Sketch of a dimerized diamond lattice, with A, +B and C the sites of the unit cell. Thick (thin) line denotes a +strong (weak) hopping, and t1 (t2) indicates the intra(inter)- +cell coupling constant. Top (bottom) panel schematizes the +trivial (topological) case t1 > t2 (t1 < t2). +(b) Spectrum +as a function of δ for a finite (lines) and an infinite (shaded +area) lattice. +The vertical line denotes δ = 1. +The color +indicates the IPR for all the states. Inset: amplitude profiles +of edge states at δ = 0.4. (c) Sketch of the fs laser writing +technique. (d) Microscope image of a diamond lattice with +{d1, d2} = {35, 25} µm (δ = 0.37) and dv = 32 µm. Output +images for the lattice in (d) and for the excitation at (e1) a +B left edge site and (e2) an in-phase A-C right edge sites. +Yellow ellipses indicate the excited sites. +site of the n-th unit cell is written as: +−i∂zAn = t1Bn + t2Bn+1 , +−i∂zBn = t1(An + Cn) + t2(An−1 + Cn−1) , +(1) +−i∂zCn = t1Bn + t2Bn+1 . +Here An, Bn and Cn are the amplitudes of the opti- +cal field at the n-th unit cell. +z describes the coordi- +nate along the waveguides and the dynamical variable. +Moreover, the hopping strengths among nearest-neighbor +(NN) sites can be varied experimentally by adjusting the +lattice distances [25]. We then define the control param- +eter δ ≡ t1/t2 to characterize the different regimes. We +assume an infinite system and impose a Bloch-like ansatz +in Eq. (1), obtaining the following bands: +βz(kx) = 0, ±t2 +� +2 [δ2 + 2δ cos(kxa) + 1] , +(2) +where βz is the propagation constant (energy), a is +the lattice constant and kx the quasimomentum. +The +spectrum is composed of two dispersive and one flat +band (FB) [see shaded areas and horizontal light-blue +line at βz = 0 in Fig. 1(b), respectively]. +The gap +in between both dispersive bands has a size equal to +2 +√ +2t2|δ − 1|. For δ = 1, this gap closes and the three +bands touch each other at the edges of the Brillouin +zone [26]. +The diamond lattice possesses the smallest +experimentally reported FB states [26, 27], with an in- +verse participation ratio (IPR) [28] of 1/2 [represented +as light-blue color in Fig. 1(b)]. +Specifically, in the +bases of Wannier functions in the A, B and C sites, +the FB eigenvector is given by: |vF B⟩ = {1, 0, −1}/ +√ +2, +and the ones corresponding to the dispersive bands +are |v±⟩ = {eiφ(kx), ± +√ +2, eiφ(kx)}/ +√ +2, where φ(kx) = +arctan(− sin(kxa)/[δ + cos(kxa)]). +Even though this lattice has three sites per unit +cell, it exhibits similar topological features to the SSH +model [29], when varying the parameter δ [30]. Indeed, +a quantized Zak phase of a value 0 or π can be found +when δ > 1 (t1 > t2) or δ < 1 (t1 < t2), respectively. +In this case, the nontrivial phase is protected by inver- +sion symmetry between An and Cn and by chiral sym- +metry [30, 31]. Thus, we expect the appearance of two +edge states at zero propagation constant on a lattice with +open boundaries when δ < 1. To corroborate this, we +compute the spectrum as a function of δ for dimerized +diamond lattices of 9 unit cells [see full lines in Fig. 1(b)]. +It can be clearly seen that two states at zero frequency +(lighter blue) transform into two dispersive states (darker +blue) around δ = 1. When increasing δ, the degeneracy +between them is removed at around δ = 0.7 (splitting +∆βe +z ∼ 0.06) [32] due to the finite size of the lattice. The +flat band remains unchanged at βz = 0, for any value +of δ. +The IPR (denoted by color) shows very clearly +the transition from localized edge states (IPR = 1 or +1/2, light blue) into extended propagating modes (IPR +∼ 1/N, black). +Figs. 1(b)-insets show the two edge states for δ = 0.4. +They exhibit exponentially localized amplitudes at both +edges. On the left edge, these states present a null ampli- +tude at A and C sites, whereas the states have a null am- +plitude at B sites at the right edge. Moreover, one edge +state is antisymmetric (bottom inset) and the other one is +symmetric (top inset), with respect to the opposite edge. +They decay exponentially into the bulk as (−δ)|n−ne| for +a semi-infinite system, exhibiting a phase shift of π at +consecutive B or A,C sites, depending on the specific +edge (ne). Notice in Fig. 1(b) that the edge states are +degenerate for δ ≲ 0.7; consequently, the sum of these +states gives a state fully localized at the left edge with +amplitude on B sites only and, conversely, the subtrac- +tion of them gives a state fully localized at the right edge +with amplitude on A and C sublattices. For δ ≳ 0.7, the +degeneracy of the edge states is lifted, and their frequency +deviates from 0 to ±βe +z. Therefore, the excitation of sites +at the edges is expected to induce an oscillatory pattern +in between both surfaces with a frequency βe +z [33], with +a long-distance state transfer occurring on a dynamical + +(a) +>1 +Trivial +A +Bs<1 +Topological +A +B +t1 +C4 +-1 +1 +2 +βz +0 +t2 +-2 +-4 +0 +0.5 +1 +1.5 +23 +scale ztransfer = π/βe +z [32]. +We fabricate several dimerized diamond photonic lat- +tices, of 9 unit cells each, by using a femtosecond (fs) +laser writing technique [25, 32, 34], as it is sketched in +Fig. 1(c). +For a first set of experiments, the diamond +geometry is defined by distances d1, d2 and dv = 32 µm, +as described in Fig. 1(d). For these values, the diagonal +(NN) distance was swept in the interval {25.6, 43.1} µm, +as d1 and d2 were varied in the interval {20, 40} µm +in steps of 1 µm. +The hopping coefficients (which de- +cay exponentially on waveguide separation [25]) range in +the interval ∼ {0.03, 0.21} cm−1 at a wavelength of 640 +nm [32]. Fig. 1(d) shows an output facet of a lattice with +d1 = 35 and d2 = 25 µm, with t1 = 0.05 and t2 = 0.14 +cm−1 (δ = 0.37). We first test the quality of the lat- +tices by exciting them at different input positions using +a 640 nm laser beam (see Ref. [32] for a complete charac- +terization). For example, topological edge states can be +efficiently excited by injecting light directly at the lattice +boundaries [28, 32, 35, 36]. Fig. 1(e1) shows the output +profile after a B-edge site excitation, with a clear expo- +nential decaying profile from the edge into the bulk. The +excitation at the right boundary requires a more com- +plicated input condition with two in-phase beams. The +result of this is shown in Fig. 1(e2), with an output profile +formed by A and C sites mostly. +We propose a novel technique to characterize the lat- +tice dynamics. Instead of measuring the output profiles +at different z values (which also implies the fabrication of +a larger number of lattices), we implement a wavelength- +scan method: The dynamics of a wavepacket injected in +the input facet of a lattice is revealed when varying the +input wavelength coming from a Supercontinuum (SC) +laser source. In general, the lattice dynamics depends on +the excitation wavelength λ: the longer the wavelength +the wider the mode profile and the larger the coupling +constants [25, 32]. In this way, by tuning the input wave- +length, we can study the same lattice at different effective +lengths. +We first consider a diamond lattice with d1 = d2 = +30 µm. We excite a B site at the central 5-th unit cell and +scan the input wavelength in the interval 600 − 760 nm, +with a step of 10 nm. Fig. 2(a) shows the output inten- +sity for three selected λ’s [32]. Fig. 2(b) shows the second +moment (width), defined as m2 ≡ � +n(n−nc)2Pn, versus +wavelength, with Pn ≡ |An|2 +|Bn|2 +|Cn|2 the unit cell +power and nc ≡ � +n nPn the center of mass. We observe +a growing diffraction pattern [27], with a width that in- +creases almost linearly with the input wavelength [a lin- +ear fit is included in Fig. 2(b)]. m2 ∼ z corresponds to a +diffusive regime [37], as expected for discrete diffraction; +therefore, a λ increment produces an effectively larger +propagation distance z or a larger coupling constant t1,2. +A dimerized diamond lattice has two hoppings which +simultaneously change while λ is modified. Since we ob- +serve a linear dependence of coupling constants on wave- +length, we can assume δ as a constant, as a first approx- +(c) +600 nm +680 nm +650 nm +700 nm +(a) +(b) +Figure 2. (a) Output intensity profiles for a B-site central +excitation, for a lattice with d1 = d2 = 30 µm, at the indicated +wavelength. +(b) m2 versus λ. +(c) md versus d1 (bottom) +and δ (top). Insets in (c) show the profile at 700 nm for the +indicated case. The bar shows the standard deviation. Yellow +ellipses indicate the excited sites. +imation. We use the wavelength-scan method to exper- +imentally determine nc for all the output profiles, after +exciting a B site at the central (5-th) unit cell of 17 dimer- +ized lattices, having different values of δ. For each lattice, +we average nc over λ and obtain the averaged beam dis- +placement md, from which the topological invariant can +be inferred [38, 39]. A topologically trivial lattice has a +md = 0, as an indication of a zero Zak phase. A topolog- +ically non-trivial system will shift this value to md ∼ 0.5, +corresponding to a π Zak phase [38]. Our collected results +are shown in Fig. 2(c). We observe that for d1 < 30 µm +(δ > 1) the lattice is topologically trivial and the prop- +agation shows a md around zero. For 30 ⩽ d1 ⩽ 34 µm +(0.4 < δ ⩽ 1), a transition region without a well defined +topological phase is observed. For d1 ⩾ 35 µm (δ ⩽ 0.4), +the lattices express a clear averaged beam displacement +around 0.5, implying a nontrivial Zak phase. Therefore, +the wavelength-scan method gives us valuable informa- +tion about the dynamics on a specific lattice, and it be- +comes a key method to determine its topological phase +on a finite size configuration. +The number of unit cells in the lattice affects the edge + +4 +¥3 +2 +0.5 +0.3 +0.1 +0.8 +0.6 +0.4 +0.2 +0.0 +25 +30 +35 +40 +di [um]10 +8 +6 +2 +4 +2 +0 +600 +650 +700 +750 +Wavelength ^ [nm]4 +(d) +650 nm +680 nm +710 nm +740 nm +(a) +650 nm +670 nm +690 nm +710 nm +730 nm +660 nm +680 nm +700 nm +720 nm +740 nm +640 nm +(c) +(b) +Figure 3. (a) Output profiles of a non-trivial diamond pho- +tonic lattice at different λ’s, after a B-edge excitation (see +yellow ellipse). (b) Fidelity versus wavelength for topological +(black), trivial (gray), topological + defect (red), and trivial ++ defect (orange) lattices. (c) Microscope image of a topolog- +ical lattice plus two coupling defects (see dashed rectangle). +(d) Same than (a) for a topological lattice with a coupling +defect. +state properties: +the fewer unit cells, the shorter the +range of δ in which the edge states keep degenerate in +βz [32]. +When the degeneracy is lifted, the two edge +modes hybridize. Therefore, an input on one edge will +excite coherently both modes and result in periodic os- +cillations of the amplitude at the two edges. +Then, a +transfer of light from one edge to the other becomes pos- +sible [23, 33]. +To experimentally demonstrate this we +fabricate a topological lattice with 9 unit cells and dis- +tances d1 = 18, d2 = 14, and dv = 14 µm (t1 = 0.30 and +t2 = 0.42 cm−1 at 640 nm, and δ = 0.71). The trivial +lattice (δ = 1.40) is obtained by inverting these distances +to d1 = 14 and d2 = 18 µm. We decreased the distances +to increase the coupling coefficients and favor a faster +transport in between the edges, while staying at the non- +degenerate situation. Again, we use a SC laser source in +the range 610 − 740 nm and sweep the input wavelength +in steps of 10 nm. We excite the system by injecting light +at the B left edge waveguide, as shown in Fig. 3(a). For +λ ≲ 670 nm, the intensity profiles are well localized at the +left edge, with most of the light intensity at the B sublat- +tices, with a profile resembling the edge state [Fig. 1(e1)]. +The edge states splitting manifests for λ ≈ 680 nm, where +we start observing a smooth population of the opposite +edge, with a weak excitation of the lattice center (a weak +background radiation is always observed because of the +excitation of dispersive modes [32]). The connection in +between both edge patterns [see Figs. 1(b)-insets and (e)], +with a B-site exponential decaying profile at the left sur- +face and an A,C exponential profile at the right edge, +becomes evident for ∼ 710 nm. The spectral state trans- +fer phenomenon starts occurring at λ ≳ 720 nm: the +light injected at one edge is mostly transferred into the +opposite edge [32]. This shows a very interesting trans- +port mechanism which does not require that the light +explores the whole lattice; in this case, the light is sud- +denly transferred from one edge into the other without +interacting with the lattice bulk. +We define the fidelity F for an edge-to-edge light +transfer by measuring the normalized transferred in- +tensity at the opposite lattice edge: F ≡ (|Aedge|2 + +|Cedge|2)/ � +n Pn. If all the light reaches the two right- +most sites F = 1, and F = 0 in the fully opposite +case. We show our results in Fig. 3(b), where we plot +the fidelity F versus λ, for topological and trivial lat- +tices. We observe how the topological (black) and the +trivial (gray) cases have a similar dynamical scale; i.e., +both processes occur approximately at the same speed. +However, the fidelity at the A, C surface is larger for the +topological lattice (∼ 61%). The trivial lattice presents +a standard discrete diffraction pattern [27], with the en- +ergy exploring the whole lattice while it moves from one +edge into the other [32], as the wavelength increases [sim- +ilar to Fig. 2(a)]. Therefore, once the light arrives at the +A, C right edge it is reflected back due to the absence of +the edge states. The fidelity in this case decreases to a +∼ 40%. +A remarkable feature of the state transfer between +topological edge states is the resilience to certain types of +disorder. Although the fabrication process can produce +random on-site or inter-site defects, we fabricate a cou- +ple of lattices with a symmetric coupling defect, as the +one shown in Fig. 3(c). We design a different distance in +between the fourth and the fifth cells and inside the fifth +cell [see dashed rectangle in Fig. 3(c)]. Specifically, we +set this distance to 23 µm, implying a coupling defect of +0.18 cm−1. Fig. 3(d) shows a set of output images at the +indicated values of λ, for the topological lattice with a +defect. We notice that this defect produces some reflec- +tion and trapping of energy at short wavelengths, con- +sequently, not all the energy is edge-to-edge transferred. +Despite this, a significant amount of the light excites the +topological right-edge state composed of A and C sites. +The fidelity is ∼ 26% for the topological case, whereas it +drops to ∼ 15% for the trivial lattice. +These numbers show that a trivial lattice undergoes +a stronger back reflection caused by the defect, because +the light explores the whole lattice and interacts strongly +with it. On the other hand, in the topological case, the +light does not travel across the lattice and excite effi- +ciently the edge state without the need of arriving at the +boundary by standard transportation. The fidelity is not +perfect in none of the topological cases because a single +B-site input always excites part of the dispersive spec- + +0.6 +0.5 +0.4 +F 0.3 +0.2 +0.1 +0.0 +620 +640 +660 +680 +700 +720 +740 +Wavelength ^ [nm]888888885 +trum, in which the modes extend over the entire lattice. +Nonetheless, the strong difference between the topologi- +cal and trivial cases is the key of success for a topological +state transfer process, which occurs due to the excitation +of exponentially localized edge states which live at both +edges simultaneously and deep in the gap of the lattice +spectrum. This could be a proof of concept for a long +distance sensor, which detects away from the interaction +region. +The wavelength-scan method proposed in this work of- +fers a tool for investigating the dynamics in lattices of +coupled waveguides. Using this method, we evidenced +the nontrivial topology of dimerized diamond lattices by +measuring the averaged beam displacement and, addi- +tionally, we demonstrated an edge-to-edge transfer of +light via the excitation of the topological edge states. +This transfer is partially robust to defects across the +lattice bulk and, hence, it has the potential for a pre- +cise wavelength filter, as well as an efficient information +transport mechanism between two distant ports or by +concatenating several topological lattices in the quantum +domain [12, 14]. +Acknowledgments. This work was supported in part +by Millennium Science Initiative Program ICN17−012, +and FONDECYT Grant 1191205. +A.A. acknowledges +the support of European Research Council grant Emer- +genTopo (865151), the French government through the +Programme Investissement d’Avenir (I-SITE ULNE / +ANR-16-IDEX-0004 ULNE) managed by the Agence Na- +tionale de la Recherche, the Labex CEMPI (ANR-11- +LABX-0007) and the CPER Wavetech. T.O. acknowl- +edges the support from JSPS KAKENHI Grant No. +JP20H01845, JST PRESTO Grant No. JPMJPR19L2, +and JST CREST Grant No.JPMJCR19T1. +†Both authors contributed equally. +∗ rvicencio@uchile.cl +[1] B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, +and B. Kant´e, Nonreciprocal lasing in topological cavities +of arbitrary geometries, Science 358, 636 (2017). +[2] M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, +M. Segev, D. N. Christodoulides, and M. Khajavikhan, +Topological insulator laser: Experiments, Science 359, +aar4005 (2018). +[3] S. 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Szameit, Experimental observa- +tion of superdiffusive transport in random dimer lattices, +New Journal of Physics 15, 013045 (2013). +[38] S. Longhi, Probing one-dimensional topological phases +in waveguide lattices with broken chiral symmetry, Opt. +Lett. 43, 4639 (2018). +[39] Z.-Q. Jiao, S. Longhi, X.-W. Wang, J. Gao, W.-H. Zhou, +Y. Wang, Y.-X. Fu, L. Wang, R.-J. Ren, L.-F. Qiao, +and X.-M. Jin, Experimentally detecting quantized zak +phases without chiral symmetry in photonic lattices, +Phys. Rev. Lett. 127, 147401 (2021). + diff --git a/19E2T4oBgHgl3EQf5QiW/content/tmp_files/load_file.txt b/19E2T4oBgHgl3EQf5QiW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..31227fd65ff4bd562202412fa61f2bd664304e28 --- /dev/null +++ b/19E2T4oBgHgl3EQf5QiW/content/tmp_files/load_file.txt @@ -0,0 +1,665 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf,len=664 +page_content='Spectral edge-to-edge topological state transfer in diamond photonic lattices Gabriel C´aceres-Aravena†,1, 2 Basti´an Real†,1, 2 Diego Guzm´an-Silva,1, 2 Paloma Vildoso,1, 2 Ignacio Salinas,1, 2 Alberto Amo,3 Tomoki Ozawa,4 and Rodrigo A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' Vicencio1, 2, ∗ 1Departamento de F´ısica, Facultad de Ciencias F´ısicas y Matem´aticas, Universidad de Chile, Chile 2Millenium Institute for Research in Optics - MIRO, Chile 3Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' Lille, CNRS, UMR 8523—PhLAM—Physique des Lasers Atomes et Mol´ecules, F-59000 Lille, France 4Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, Sendai 980-8577, Japan (Dated: January 12, 2023) Transfer of information between topological edge states is a robust way of spatially manipulating quantum states while preserving their coherence in lattice environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' This method is particularly efficient when the edge modes are kept within the topological gap of the lattice during the transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' In this work we show experimentally the transfer of photonic modes between topological edge states located at opposite ends of a dimerized one-dimensional photonic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' We use a diamond lattice of coupled waveguides and show that the transfer is insensitive both to the presence of a high density of states in the form of a flat band at an energy close to that of the edge states, and to the presence of disorder in the hoppings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' We explore dynamics in the waveguide lattice using wavelength-scan method, where different input wavelength translates into different effective waveguide length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' These results open the way to the implementation of more efficient protocols based on the active driving of the hoppings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' Topological edge states are a remarkable resource to engineer photonic systems with isolated modes protected from the presence of disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' In two-dimensional lat- tices, they can be used to fabricate topological edge mode lasers with distributed gain and quantized orbital mo- mentum [1, 2], to transfer single photons around cor- ners in elaborated photonic circuits [3, 4], and to de- sign topological frequency combs with enhanced effi- ciency [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' One dimensional systems such as the Su-Schrieffer-Heeger (SSH) lattice are particularly in- teresting because topological edge and interface modes are hosted deep into the topological gap of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' This gap protection has been shown to be beneficial to preserve the quantum state of photons in boundary modes [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' Interestingly, the presence of topological edge modes on both sides of one-dimensional lattices can be used to transfer a state from one edge of the lattice to the other with high fidelity with the advantage of be- ing protected from certain types of disorder due to the topological nature of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' Such edge state trans- fer is a promising route to store and manipulate photonic quantum states in on-chip lattice environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' Most topological edge transfer protocols rely on the adiabatic evolution of the lattice such that an edge mode is driven into quasi-bulk modes and again into an edge mode at the other side [9–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' While these protocols present an optimized transfer rate and fidelity, they are limited by the adiabaticity condition that requires the adiabatic passage to be slow enough to avoid the Zen- ner coupling of the edge state information into the bulk modes [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' Furthermore, the presence of disorder in the lattice would enhance this coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' A variation of these protocols include counter-adiabatic driving meth- ods [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' Recently, a different route has been proposed based on the coherent coupling of edge modes within the gap [19, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' The great advantage of this approach is that edge modes are kept well into the topological gap throughout the protocol, ensuring a high fidelity in re- duced times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' The simplest version of the coherent state transfer of topological edge states is via passive evanes- cent coupling of the exponential tails of edge modes at op- posite sides of the finite size lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' In this case, coherent transfer between edge modes takes place at a frequency determined by the tail overlap, which can be controlled via the size of the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' Observation of such coherent os- cillations was reported in a short SSH lattice for Rydberg atoms [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' In this work, we demonstrate coherent edge-to-edge transfer of light in a dimerized diamond lattice of coupled waveguides employing a spectral tomographic technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' More importantly, we show experimentally that the fi- delity of the transfer is robust to a number of perturba- tions in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' First, we show that orthogonality of eigenmodes in our undriven protocol preserves the trans- fer even in the presence of a high density of states in the form of a flat band at energies close to that of the edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' Second, we demonstrate that the transfer mech- anism is robust to the presence of lattice defects thanks to the underlying chiral symmetry of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' The experimental proof of principle we report in this work can be significantly sped up by applying a number of driven techniques based on the modulating of the hop- pings in time and the use of concatenated topological lattices [19, 22, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' Such techniques are readily imple- mentable in lattices of coupled waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' To demonstrate the topological edge transfer we use a diamond lattice of coupled waveguides with different intracell (t1) and intercell (t2) hoppings, as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' The lattice has three sites per unit cell, denoted as A, B and C sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' Considering a tight-binding coupled- mode approach, the evolution of the optical field at every arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content='04189v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content='optics] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content='10 Jan 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content='d2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQf5QiW/content/2301.04189v1.pdf'} +page_content=' 1 THz), and less +than the thermal energy (kBT/h ≈ 6 GHz), which ex- +cludes the possibility of avoided crossings associated with +coherent QPS. +Our demonstration of the ability to control the tun- +nelling of single flux-quanta represents important progress +towards applications of QPS devices in quantum infor- +mation processing that have been proposed elsewhere. +The low loss in our device suggests the potential for high +tunnelling rates in QPS devices without a significant in- +crease in T1-type decoherence. In addition to this, there is +potential for a QPS digital logic processing device, based +on the deterministic transfer of single flux-quanta [15– +18]. Utilising quantum tunnelling of flux in such a device +should enable significant reduction in the heat dissipa- +tion associated with each gate [34], a reduction that will +be necessary for the scaling up of systems beyond the +1,000-qubit level. +FIG. 1. (a) Flux-dependent energy spectrum of a continuous +superconducting loop, with blue dashed line highlighting the +ground state. A single flux-quantum may tunnel into the loop +at the degeneracy point — highlighted in red. N is the winding +number and Φ0 the flux quantum. (b) Optical (main image; +blue contrast) and SEM (inset; grey contrast) images of the +NbN nanowire-embedded loop located at the short-circuited +termination of the CPW resonator. The nanowires in this +device are 25-nm wide and 200-nm long. The lower and upper +leads to the loop are connected to the CPW centre conductor +and superconducting ground plane respectively. +II. +FABRICATION AND EXPERIMENTAL +DETAILS +We fabricated nanowire-embedded resonators from 10- +nm-thick films of superconducting NbN. The NbN was +deposited on c-axis oriented sapphire substrates by dc +magnetron sputtering of a 99.99% pure niobium target in +a 1:1 Ar:N2 atmosphere at a pressure of 5 × 10−3 mbar +and a power of 150 W. The resulting film was measured to +have critical temperature Tc = 8.55 K and sheet resistance +Rsq = 1.2 kΩ/sq. + +(a) +N=0 +N=1 +N=2 +Energy +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +Applied Flux (Φo) +(b) +sapphire +NbN +μm +100um3 +FIG. 2. (a) Upper panel: Single-tone spectroscopy of nanowire-embedded resonator measured at T = 305 mK and ⟨n⟩ ≈ 50. +|S21| is plotted as a function of frequency and applied magnetic field. The top axis shows the applied magnetic flux Φapp seen by +the nanowire loop, which is inferred from the periodicity of the resonator tuning. Dashed white lines show resonant frequencies +corresponding to the calculated energy states of the loop. Lower panel: Magnetic-field dependence of the measured intrinsic +quality factor. (b) Calculated energy states of the loop and values extracted from the measured resonant frequency. +Quarter-wavelength resonators were patterned by +electron-beam lithography (EBL) into a 300-nm layer +of PMMA resist. Multiple resonators on each chip are +capacitively coupled to a common feedline and are pat- +terned with narrow loops galvanically coupled at the +short-circuited end. At this stage, the loops contain ‘pre- +cursor’ nanowires designed with a width of 400 nm. The +pattern was transferred into the NbN film by reactive ion +etching (RIE) using a 2:5 volume ratio of CHF3 and SF6 +at 100 W and 100 mbar. +Nanowires are then patterned into the loops using a +neon focused-ion-beam, whereby a beam of Ne ions is +accelerated from an atomically-defined tip onto the sample +surface with spot size down to 2 nm and sufficient energy +to sputter the metal film [35, 36]. The precursor wires +were milled to a width of 25 nm (see Fig. 1(c)) using a 15- +keV Ne beam. A Ne dose of 0.5 nC/µm−2 was sufficient +to clear the 10-nm film. +The sample was wire-bonded to a copper printed-circuit- +board (PCB) and enclosed within an ECCOSORB-lined +brass sample box. This was cooled to a temperature of +T ∼ 300 mK using a 3He refrigerator. Measurement of the +rf response of the device was made using a Vector Network +Analyser (VNA) via an input line with 60 dB of atten- +uation to reduce thermal noise from room temperature. +Signals in the output line were amplified with a high- +electron-mobility-transistor (HEMT) amplifier. Global +magnetic field was applied perpendicular to the plane of +the loop using a superconducting solenoid and a precision +current source. The lines supplying current to the coil +were filtered at room temperature with an upper cut-off +frequency of 9.2 kHz. +III. +RESULTS AND DISCUSSION +A. Flux Dependence of Resonant Frequency +In this paper, we present results on a single NbN +nanowire-embedded CPW resonator (see Supplemental +Information for comparison of multiple devices). We mea- +sured the forward transmission (S21) through the on-chip +feedline, where the λ/4 resonators appear as a notch-type +resonance. The upper panel of Fig. 2(a) shows the main +result of this work: under an applied magnetic field, the +resonance tuning shows discontinuous changes of gradi- +ent at periodic values of the applied field. As we will +demonstrate in the remainder of this paper, these dis- +continuities occur when two stable states with winding +number differing by one become degenerate and are due +to single-flux-quantum tunnelling mediated by quantum +phase-slips in the nanowire [37]. +The magnetic-field periodicity is 153 µT, which cor- +responds to a single flux-quantum in our loop assum- +ing a flux-focusing factor F = 1.7 [38]. +The data in +Fig. 2(a) corresponds to a single direction of magnetic +field sweep, but sweeps in the opposite direction were +found to give the same result. We also observe a non- +periodic, parabolic decrease of the resonant frequency as +the magnitude of the applied field is increased. This is the +expected [39, 40] kinetic-inductance tuning of the NbN +resonator, and can be parametrised by a phenomenologi- +cal field-scale B⋆ = 8 mT. + +(a) +Dapp(@o) +(b) +-3 +-2 +-1 +0 +2 +3 +S21| (dB) +45.0 +500 +3.384 +3.383 +45.2 +400 +Frequency (GHz) +3.382 +45.4 +3.381 +300 +45.6 +3.380 +45.8 +E (GHz) +3.379 +200 +3.378 +46.0 +100 +3.377 +46.2 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +10×-01 +N=-1 N=0 +N=1 +N=2 +calculated +-100 +extracted from measured vo +3 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +4 +-2 +0 +Applied Field (mT) +Φapp(Φo)4 +Figure 2(a) also shows periodic variation of Qi of the +resonance as a function of applied field with the same +field period as the resonant frequency. Quality factors +were obtained by an analytical fit [41] to the equation +|S21(ν)| = aeiαe−i2πντ +� +1 − +� +Ql +|Qc| +� +eiφ +1 + 2iQl(ν/ν0 − 1) +� +, +(1) +where ν is the probe frequency and ν0 is the resonance +frequency. +Qc and Ql are the coupling and loaded +quality factors respectively, and obey the relationship +1/Ql = 1/Qi + 1/Qc. +φ accounts for the effect of +impedance mismatches in the circuit, the scale factor +a represents the change in amplitude due to any attenua- +tion/amplification in the measurement chain, α describes +any initial phase offset of the signal, and τ accounts +for frequency-dependent cable delay. We find that Qi +decreases approximately quadratically from 4 × 103 at +δΦ ≡ (Φapp −NΦ0)/Φ0 = 0 to 3.4×103 at δΦ = 1/2. We +attribute this small change to non-equilibrium quasiparti- +cles excited by the induced screening current providing an +extra loss mechanism [42]. We observe no sharp decrease +in Qi at δΦ = 1/2, which suggests that the heat dissipated +by the quantum phase-slip itself is not large enough to +cause extra losses in the resonator. We note that the +intrinsic quality factor exceeds any currently reported in +the literature for QPS devices, and discuss this further in +the Supplemental Information. +The periodic tuning of the resonance is well fitted by +a model of an inductive superconducting loop remaining +in its ground state (see Fig. 2(b)), where the system is +allowed to move between adjacent parabolas by trans- +ferring a single flux-quantum through the nanowire at +Φapp = (N + 1/2)Φ0. +The loop is made up of a wide section and a narrow +section (as shown in Fig. 1(b)), and so can be modelled +as two nonlinear kinetic inductances in series. The flux- +dependent kinetic inductance of the loop Lk(Φ) is there- +fore +Lk(I) = Lk,1(0) +� +1 + +� +Is +I⋆,1 +�2� ++ Lk,2(0) +� +1 + +� +Is +I⋆,2 +�2� +, +(2) +where I⋆,1 and I⋆,2 are known to be of the order of the +critical current in the wide and narrow section respectively +[43]. Since the screening current is Is = Φ/Lk, we can +insert this into Eq. 2 and solve for Lk(Φ). +The input impedance Zin of a λ/4 CPW resonator +terminated by an inductive load, as a function of frequency +and load impedance, is +Zin(ν, ZL) = Z0 +ZL + iZ0 tan +� 2πνl +c +� +Z0 + iZL tan +� 2πνl +c +�, +(3) +where Z0 is the characteristic impedance of the resonator, +ZL(Φ) = i2πνLk(Φ) is the impedance of the inductive +load, c is the speed of light in the resonator, and l is its +FIG. 3. |S21| measured at Φapp = 0 and Φapp = Φ0/2 along +with fit (black line) to a linear resonance model. This shows +that at Φapp = Φ0/2 the response is linear, so the current in +the nanowire is well below the critical current. +length. At resonance, Im{Zin} → ∞, so given Lk(Φ) one +can numerically calculate ν0(Φ), or given ν0(Φ) one can +numerically calculate Lk(Φ). +One can also calculate the flux-dependent free energy +of the loop E(Φ) from Lk(Φ), using the relation +L−1 +k += d2E +dΦ2 . +(4) +To obtain the free energy, we simply numerically integrate +the inverse of the inductance twice with respect to flux. +We calculated Lk(Φ) for our device from Eq. 2 using +only independently determined parameters. A critical +current density of Jc = 4.4 × 105 Acm−2 was obtained +from a dc measurement of a track etched into the NbN +film, and a sheet kinetic inductance of Lsq = 0.34 nH/sq +was inferred from the zero-field ν0 of the resonator. The +geometry of the loop was measured by SEM and this was +used to calculate Lk(0) and I⋆ (we set I⋆ = Ic). We then +calculated the white dashed lines in Fig. 2(a) using Eq. 3, +and the solid black lines in Fig. 2(b) from Eq. 4. The blue +points in Fig. 2(b) were extracted from the measured ν0 +using Eqs. 3 and 4. +B. Mechanisms for Flux Quantum Transfer +The periodic tuning of the resonator and the associated +fit to the calculated energy states of the loop constitute +strong evidence that the loop remains in its ground state +for all values of Φapp, and this is made possible by a +single flux-quantum transferring into or out of the loop at +δΦ = 1/2. It is important to establish the mechanism by +which the flux is able to enter the loop, so we now turn our +attention to the transitions between flux states. Across +multiple devices on multiple chips, we found an onset + +-45.0 +6Φ= 0 +-45.2 +45.4 +(dB +45.6 +S21 +45.8 +-46.0 +6Φ = 1/2 +46.2 +3.378 +3.380 +3.382 +3.384 +Frequency(GHz)5 +of flux-periodic tuning in devices containing nanowires +with w ≲ 35 nm. This dependence of the behaviour on +nanowire width suggests that the flux tunnelling occurs +in the nanowires, and not in the wider part of the loop. +We can now examine some possible physical processes +that could occur in the nanowires and see how the data +fits with them. +-Does the nanowire current exceed its critical current +Inw +c ? +Niobium-nitride resonators commonly exhibit a +nonlinear S21 response when they are driven with a high +microwave power [44] as a result of the current-induced +nonlinear kinetic inductance. As we see in Eq. 2, the +kinetic inductance is quadratically dependent on (I/Ic)2, +and so the nonlinearity must be dominated by the part +of the conductor with the lowest Ic. This is confirmed by +measurements of our circuits, where we find that nanowire- +embedded resonators show a much higher degree of non- +linearity in their S21 response than bare resonators (see +Supplemental Information). The quadratic nature of the +nonlinearity suggests that a strongly nonlinear response is +a consequence of the magnitude of the rf current Ires in the +resonator reaching a significant fraction of the nanowire +critical current Inw +c . Our NbN resonator readout method +therefore gives us an indirect readout of whether the +current in the nanowire is close to Inw +c . +Figure 3 shows the S21 response of the nanowire- +embedded resonator at δΦ = 0 and δΦ = 1/2, both +measured in the low-power limit with an estimated res- +onator photon population of ⟨n⟩ ≈ 50. In both cases, the +response is linear and well fitted by Eq. 1. We calculate, +using the relation Is = dE(Φ)/dΦ that the maximum +induced screening current in the nanowires Is(δΦ = 1/2) +is 120 nA, an order of magnitude less than Inw +c . Crucially, +the lack of nonlinearity of the resonance at δΦ = 1/2, +along with the fact that Qi remains a significant fraction +of its zero-field value, means that the nanowires are not +being driven close to their critical current by the applied +flux. By a similar argument, we know that the nanowires +are not being driven through Tc by a local heating process, +as this would also result in nonlinearity of the resonance +at δΦ = 1/2 due to dissipation in the highly resistive +normal metal. +-Is the nanowire a constriction Josephson-junction? +‘Dayem-bridge’ Josephson-junction SQUIDs are com- +monly embedded within CPW resonators [45, 46] to +provide a flux-tunable resonant frequency. +However, +when the SQUID loop has a large inductance, one ob- +serves hysteretic tuning, characterised by the parameter +βL = 2LIc/Φ0. When βL ≳ 1, the SQUID behaviour be- +comes hysteretic with applied flux and the resonator will +exhibit discontinuous jumps in the resonant frequency, +as observed in [47, 48]. Our device does not undergo +any discontinuous jumps, and the tuning over a single +flux-quantum is symmetric, so βL < 1. Given this and the +known loop inductance, we can set an upper bound on the +critical current of any Josephson junction of IJJ +c +< 100 nA. +This bound is 10× smaller than the expected transport +critical current of our nanowires and also less than Is(δΦ0). +FIG. 4. Normalised tunnelling flux ∆φ+ +N (defined in the main +text), as a function of the winding number N. +The solid +line corresponds to half a flux quantum, ∆φ+ +N = 0.5. +In- +set: Magnetic-field dependence of resonant frequency of the +nanowire-embedded resonator up to an applied field of 2.4 +mT. The black dotted lines mark transitions between winding +number states (φ+ +N), and the red dotted lines mark the energy +minimum of each winding number state (φmin,N). +Therefore, it is unrealistic to conclude that the flux-tuning +we observe is a consequence of a dc SQUID formed of +Dayem-bridge Josephson junctions. We also note that +the closeness of the fit shown in Fig. 2 suggests there is +no contribution to the flux-dependent inductance from a +Josephson junction, which would add a 1/ cos Φ term to +Eq. 2. +-Is this a thermal or quantum process? Figure 4 shows +that tunnelling always occurs at degeneracy. Following +Petkovi´c et al. [30], ∆φ+ +N is defined as ∆φ+ +N = φ+ +N−φmin,N, +where φ+ +N is the normalised flux φ = Φ/Φ0 at which tun- +nelling from state N to state N + 1 occurs, and φmin,N is +the flux that minimises the loop free-energy for a particu- +lar winding number N. Our data shows the periodicity, +defined in this way, to be half a flux quantum for all values +of the winding number except N = 1 (we attribute the +latter exception to enhanced flux-focusing at low magnetic +field). This is in contrast to [30], where a characteristic +dependence of ∆φ+ +N on N is shown to be a defining fea- +ture of thermally-activated phase-slips. By following the +method of [30], we calculate that ∆φ+ +N ≈ 300 would be +required in order for the energy barrier to phase slips +in our nanowire to be tuned to ≲ kBT. +Correspond- +ingly, we can estimate (see Supplemental Information) +that ΓQPS = 35 MHz for our nanowire, and we calculate +a temperature-dependent ΓTAPS that is below 1 Hz for +T < 1.5 K. Therefore, at our measurement temperature, +quantum tunnelling of flux is overwhelmingly more likely +than a thermal transition. For comparison, the inverse +experimental timescale is 1/τE ≈ 0.2 Hz since, for each + +0.5 +0.4 +3.380 +Resonant Frequency (GHz) +3.378 +0.3 ++z +3.374 +0.2 +3.372 +3.370 +0.1 +3.368 +3.366 +0.0 +0.0 +0.5 +1.0 +1.5 +2.0 +Applied Field (mT) +0 +2 +4 +6 +8 +10 +12 +14 +16 +Winding NumberN6 +setpoint of the magnetic field, it takes the VNA approxi- +mately 5 s to collect S21 data across the resonance. Be- +cause kBT/h > ΓQPS ≫ 1/τE — and to our knowledge +this is the first reported study in this regime — when +our device bias is swept through the degeneracy point, +a single quantum phase-slip always occurs before we are +able to observe the system in a higher-energy metastable +state. +IV. +CONCLUSIONS +We have used a Ne FIB to fabricate NbN nanowires +with widths down to 25 nm embedded within CPW res- +onators. +We observe periodic modulation of resonant +frequency and intrinsic quality factor, which is consis- +tent with quantum tunnelling of individual flux quanta +mediated by quantum phase-slip, occurring when states +of different winding number become degenerate. This +behaviour has been observed in resonators with intrinsic +quality factor, Qi, up to 2.7 × 104 at 300 mK, which to +our knowledge is the highest quality factor measured in +quantum phase-slip experiments — note that the losses +here are significantly lower than suggested by comparable +reports [10]. We estimate that the QPS rate is of the +order 10–100 MHz, which means that the tunnelling of +a single flux-quantum is effectively deterministic on the +timescale of microseconds. This result shows the suitabil- +ity of the Ne FIB process for fabricating QPS devices. We +also suggest that an incoherent QPS device with a high +QPS rate such as ours could be promising for classical +digital logic processing applications, where the quantum +nature of the flux tunnelling implies a reduction in heat +dissipation compared with current state-of-the-art devices, +opening up a route to resolving an important roadblock +to the upscaling of qubit control electronics. +ACKNOWLEDGMENTS +The authors thank O. W. Kennedy for useful dis- +cussions. The authors gratefully acknowledge funding +from the United Kingdom Engineering and Physical +Sciences Research Council, Grant Nos. EP/L015242/1, +EP/J017329/1, and EP/T001062/1. +[1] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, +R. Barends, S. 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Appl. 11, 014006 (2019). + diff --git a/4tE3T4oBgHgl3EQfQQmX/content/tmp_files/load_file.txt b/4tE3T4oBgHgl3EQfQQmX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2a37e898383bea7e119c1c14f40ff3a43bdd9728 --- /dev/null +++ b/4tE3T4oBgHgl3EQfQQmX/content/tmp_files/load_file.txt @@ -0,0 +1,898 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf,len=897 +page_content='Controllable tunnelling of single flux-quanta mediated by quantum phase-slip in disordered superconducting loops J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Potter,∗ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Fenton, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Warburton London Centre for Nanotechnology, UCL, 17-19 Gordon Street, London WC1H 0AH, United Kingdom (Dated: January 12, 2023) Quantum phase-slip (QPS) is the exact dual to the well-known Josephson effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Although there are numerous proposals for applications of QPS devices, experimental work to develop these remains in the relatively early stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Significant barriers to exploiting QPS nanowires for useful technologies still exist, such as establishing robust nanowire fabrication methods that allow coupling to low-loss circuits, and demonstrating control over the QPS process with an experimenter-controlled external bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Here we report experiments which show that both of these barriers have been overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We present measurements at 300 mK of NbN coplanar waveguide (CPW) resonators embedded with nanowires fabricated using a neon focused ion-beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The internal quality factor exceeds 2 × 104 — significantly higher than previously reported in comparable experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The resonator frequency tunes periodically with an applied magnetic field, revealing tunnelling of the order parameter that always occurs at half-integer values of the applied flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' In contrast to previous studies of single QPS, the order-parameter tunnelling is shown to be adiabatic, demonstrating improved control over energy dissipation in nanowire QPS circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Our results highlight a promising pathway towards realising low-loss nanowire-based QPS devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' INTRODUCTION Quantum circuits based on superconducting materials are currently the leading candidate for the implementa- tion of a scalable quantum computer, already beginning to tackle relevant computation and simulation problems [1, 2] and recently demonstrating ‘quantum advantage’ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The Josephson junction is near ubiquitous in these circuits, providing the necessary nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' A quantum phase-slip nanowire is the flux-charge dual to the Joseph- son junction [4], and in theory every Josephson junction- based circuit has a QPS dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' As well as being proposed as a qubit with favourable properties over traditional Josephson-based technology [5, 6], the QPS nanowire’s dual property of a nonlinear quantum capacitance enables potential applications such as novel qubit-qubit couplers [7], parametric amplification for qubit readout, and a primary quantum current standard [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' As yet, the huge potential of QPS nanowires remains to be fully exploited, and two key reasons for this stand out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Firstly, consistent and reliable fabrication of materials and nanowires with the requisite properties remains challenging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' secondly, full control over individual QPS events in a nanowire has not yet been demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The QPS phenomenon is most pronounced in quasi- one-dimensional superconducting nanowires, by which we mean that the cross-sectional dimensions of the nanowire are comparable to the coherence length ξ [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' In these nanowires, quantum (or indeed thermal) fluctuations can lead to complete suppression of the superconducting order parameter over the cross-section of the wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' This in turn leads to a sudden change of 2π in the phase difference ∗ jamie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='potter@npl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='uk Present address: National Physical Laboratory, Teddington, TW11 0LW, United Kingdom between the two ends of the wire, accompanied by the transfer of a quantised amount of magnetic flux equal to the magnetic flux quantum Φ0 = h/2e through the wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' This tunnelling of a flux quantum can be coherent [10–12] or incoherent [13], depending on the relative scales of the phase-slip energy ES and the inductive energy EL of the nanowire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' When ES/EL < 1, the magnetic flux quantum number is well-defined, and incoherent transfer of individual flux-quanta is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' This is in direct analogy to small Josephson junctions with large charging energy, where tunnelling of single Cooper pairs can be observed [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' It is the incoherent QPS regime that is the focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Incoherent QPS occur probabilis- tically, with a frequency characterised by the phase-slip rate ΓS ≡ ES/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' However if one waits for a time τ that is much longer than 1/ΓS, then flux-quantum tunnelling will occur with extremely high likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' A device dis- playing deterministic transfer of quantised amounts of magnetic flux may find useful applications in tasks such as energy-efficient classical digital logic processing [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Development of such a device is a key enabling technol- ogy for control of superconducting quantum processors at technologically-useful scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Historically, phase slips were studied in externally con- nected, current-biased nanowires, where the collective effect of many phase slips manifests as a resistance below Tc [20–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' However, in order to isolate a single phase- slip, it is necessary to incorporate the nanowire into a flux-biased superconducting loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The flux-dependent energy states of a continuous superconducting loop are described by a set of parabolas (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 1(a)), where each parabola corresponds to a unique value N of the phase winding number, or equivalently the number of flux quanta in the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' A single phase-slip corresponds to a transition between neighbouring parabolas, and if no lower-energy state is available at a particular external flux bias, then phase slips are forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' When sweeping the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='04411v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='supr-con] 11 Jan 2023 2 externally applied flux, tunnelling of a single flux-quantum becomes allowed at Φapp = (N + 1/2)Φ0 (known as the degeneracy point, highlighted in red in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 1(a)) [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Tunnelling will occur when the system passes the degen- eracy point if the rate of phase slips is much greater than the rate at which the flux is swept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' However, if the flux is swept slowly with respect to ΓS, tunnelling will cause the system to enter a metastable state and it will then un- dergo irreversible relaxation to the ground state at some later time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' A number of recent experiments [28–31] have demonstrated relaxation via QPS from a higher-energy metastable state, but controlled single-flux-quantum tun- nelling when passing through the degeneracy point has not previously been demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' In this paper, we demonstrate for the first time single- flux-quantum tunnelling occurring at the degeneracy point in a continuous superconducting loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The flux quanta tunnel through NbN nanowires embedded in the loop and this is read out via coupling to a high-quality coplanar waveguide (CPW) resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' An important innovation in our fabrication technique is that the nanowires were fabri- cated by neon focused-ion-beam (FIB) milling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The FIB process enables the repeatable fabrication of nanowires with width w ≈ 25 nm ensuring a large phase-slip rate, while introducing minimal losses to the host resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' FIB milling has previously been shown to be compati- ble with low-dissipation superconducting circuits [32, 33];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' however, this is the first report of the use of FIB to fabri- cate a device in which quantum phase-slip behaviour has been measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Our results show flux-periodic tuning of the resonant frequency ν0 while maintaining a high intrin- sic quality factor Qi at all values of applied flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We show that a single quantum phase-slip always occurs when ad- jacent winding-number states become degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' This is ensured by a phase-slip rate — we estimate ΓS = 35 MHz — which is large in comparison to 1/τE, where τE is the experimental timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' However, the phase-slip rate is less than the inductive energy (EL/h > 1 THz), and less than the thermal energy (kBT/h ≈ 6 GHz), which ex- cludes the possibility of avoided crossings associated with coherent QPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Our demonstration of the ability to control the tun- nelling of single flux-quanta represents important progress towards applications of QPS devices in quantum infor- mation processing that have been proposed elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The low loss in our device suggests the potential for high tunnelling rates in QPS devices without a significant in- crease in T1-type decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' In addition to this, there is potential for a QPS digital logic processing device, based on the deterministic transfer of single flux-quanta [15– 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Utilising quantum tunnelling of flux in such a device should enable significant reduction in the heat dissipa- tion associated with each gate [34], a reduction that will be necessary for the scaling up of systems beyond the 1,000-qubit level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' (a) Flux-dependent energy spectrum of a continuous superconducting loop, with blue dashed line highlighting the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' A single flux-quantum may tunnel into the loop at the degeneracy point — highlighted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' N is the winding number and Φ0 the flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' (b) Optical (main image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' blue contrast) and SEM (inset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' grey contrast) images of the NbN nanowire-embedded loop located at the short-circuited termination of the CPW resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The nanowires in this device are 25-nm wide and 200-nm long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The lower and upper leads to the loop are connected to the CPW centre conductor and superconducting ground plane respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' FABRICATION AND EXPERIMENTAL DETAILS We fabricated nanowire-embedded resonators from 10- nm-thick films of superconducting NbN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The NbN was deposited on c-axis oriented sapphire substrates by dc magnetron sputtering of a 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='99% pure niobium target in a 1:1 Ar:N2 atmosphere at a pressure of 5 × 10−3 mbar and a power of 150 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The resulting film was measured to have critical temperature Tc = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='55 K and sheet resistance Rsq = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='2 kΩ/sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' (a) N=0 N=1 N=2 Energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='5 Applied Flux (Φo) (b) sapphire NbN μm 100um3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' (a) Upper panel: Single-tone spectroscopy of nanowire-embedded resonator measured at T = 305 mK and ⟨n⟩ ≈ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' |S21| is plotted as a function of frequency and applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The top axis shows the applied magnetic flux Φapp seen by the nanowire loop, which is inferred from the periodicity of the resonator tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Dashed white lines show resonant frequencies corresponding to the calculated energy states of the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Lower panel: Magnetic-field dependence of the measured intrinsic quality factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' (b) Calculated energy states of the loop and values extracted from the measured resonant frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Quarter-wavelength resonators were patterned by electron-beam lithography (EBL) into a 300-nm layer of PMMA resist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Multiple resonators on each chip are capacitively coupled to a common feedline and are pat- terned with narrow loops galvanically coupled at the short-circuited end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' At this stage, the loops contain ‘pre- cursor’ nanowires designed with a width of 400 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The pattern was transferred into the NbN film by reactive ion etching (RIE) using a 2:5 volume ratio of CHF3 and SF6 at 100 W and 100 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Nanowires are then patterned into the loops using a neon focused-ion-beam, whereby a beam of Ne ions is accelerated from an atomically-defined tip onto the sample surface with spot size down to 2 nm and sufficient energy to sputter the metal film [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The precursor wires were milled to a width of 25 nm (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 1(c)) using a 15- keV Ne beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' A Ne dose of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='5 nC/µm−2 was sufficient to clear the 10-nm film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The sample was wire-bonded to a copper printed-circuit- board (PCB) and enclosed within an ECCOSORB-lined brass sample box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' This was cooled to a temperature of T ∼ 300 mK using a 3He refrigerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Measurement of the rf response of the device was made using a Vector Network Analyser (VNA) via an input line with 60 dB of atten- uation to reduce thermal noise from room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Signals in the output line were amplified with a high- electron-mobility-transistor (HEMT) amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Global magnetic field was applied perpendicular to the plane of the loop using a superconducting solenoid and a precision current source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The lines supplying current to the coil were filtered at room temperature with an upper cut-off frequency of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='2 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Flux Dependence of Resonant Frequency In this paper, we present results on a single NbN nanowire-embedded CPW resonator (see Supplemental Information for comparison of multiple devices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We mea- sured the forward transmission (S21) through the on-chip feedline, where the λ/4 resonators appear as a notch-type resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 2(a) shows the main result of this work: under an applied magnetic field, the resonance tuning shows discontinuous changes of gradi- ent at periodic values of the applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' As we will demonstrate in the remainder of this paper, these dis- continuities occur when two stable states with winding number differing by one become degenerate and are due to single-flux-quantum tunnelling mediated by quantum phase-slips in the nanowire [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The magnetic-field periodicity is 153 µT, which cor- responds to a single flux-quantum in our loop assum- ing a flux-focusing factor F = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='7 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 2(a) corresponds to a single direction of magnetic field sweep, but sweeps in the opposite direction were found to give the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We also observe a non- periodic, parabolic decrease of the resonant frequency as the magnitude of the applied field is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' This is the expected [39, 40] kinetic-inductance tuning of the NbN resonator, and can be parametrised by a phenomenologi- cal field-scale B⋆ = 8 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' (a) Dapp(@o) (b) 3 2 1 0 2 3 S21| (dB) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='0 500 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='384 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='383 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='2 400 Frequency (GHz) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='382 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='381 300 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='380 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='8 E (GHz) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='379 200 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='378 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='0 100 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='377 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='6 10×-01 N=-1 N=0 N=1 N=2 calculated 100 extracted from measured vo 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='6 4 2 0 Applied Field (mT) Φapp(Φo)4 Figure 2(a) also shows periodic variation of Qi of the resonance as a function of applied field with the same field period as the resonant frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Quality factors were obtained by an analytical fit [41] to the equation |S21(ν)| = aeiαe−i2πντ � 1 − � Ql |Qc| � eiφ 1 + 2iQl(ν/ν0 − 1) � , (1) where ν is the probe frequency and ν0 is the resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Qc and Ql are the coupling and loaded quality factors respectively, and obey the relationship 1/Ql = 1/Qi + 1/Qc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' φ accounts for the effect of impedance mismatches in the circuit, the scale factor a represents the change in amplitude due to any attenua- tion/amplification in the measurement chain, α describes any initial phase offset of the signal, and τ accounts for frequency-dependent cable delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We find that Qi decreases approximately quadratically from 4 × 103 at δΦ ≡ (Φapp −NΦ0)/Φ0 = 0 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='4×103 at δΦ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We attribute this small change to non-equilibrium quasiparti- cles excited by the induced screening current providing an extra loss mechanism [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We observe no sharp decrease in Qi at δΦ = 1/2, which suggests that the heat dissipated by the quantum phase-slip itself is not large enough to cause extra losses in the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We note that the intrinsic quality factor exceeds any currently reported in the literature for QPS devices, and discuss this further in the Supplemental Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The periodic tuning of the resonance is well fitted by a model of an inductive superconducting loop remaining in its ground state (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 2(b)), where the system is allowed to move between adjacent parabolas by trans- ferring a single flux-quantum through the nanowire at Φapp = (N + 1/2)Φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The loop is made up of a wide section and a narrow section (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 1(b)), and so can be modelled as two nonlinear kinetic inductances in series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The flux- dependent kinetic inductance of the loop Lk(Φ) is there- fore Lk(I) = Lk,1(0) � 1 + � Is I⋆,1 �2� + Lk,2(0) � 1 + � Is I⋆,2 �2� , (2) where I⋆,1 and I⋆,2 are known to be of the order of the critical current in the wide and narrow section respectively [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Since the screening current is Is = Φ/Lk, we can insert this into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 2 and solve for Lk(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The input impedance Zin of a λ/4 CPW resonator terminated by an inductive load, as a function of frequency and load impedance, is Zin(ν, ZL) = Z0 ZL + iZ0 tan � 2πνl c � Z0 + iZL tan � 2πνl c �, (3) where Z0 is the characteristic impedance of the resonator, ZL(Φ) = i2πνLk(Φ) is the impedance of the inductive load, c is the speed of light in the resonator, and l is its FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' |S21| measured at Φapp = 0 and Φapp = Φ0/2 along with fit (black line) to a linear resonance model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' This shows that at Φapp = Φ0/2 the response is linear, so the current in the nanowire is well below the critical current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' At resonance, Im{Zin} → ∞, so given Lk(Φ) one can numerically calculate ν0(Φ), or given ν0(Φ) one can numerically calculate Lk(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' One can also calculate the flux-dependent free energy of the loop E(Φ) from Lk(Φ), using the relation L−1 k = d2E dΦ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' (4) To obtain the free energy, we simply numerically integrate the inverse of the inductance twice with respect to flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We calculated Lk(Φ) for our device from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 2 using only independently determined parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' A critical current density of Jc = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='4 × 105 Acm−2 was obtained from a dc measurement of a track etched into the NbN film, and a sheet kinetic inductance of Lsq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='34 nH/sq was inferred from the zero-field ν0 of the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The geometry of the loop was measured by SEM and this was used to calculate Lk(0) and I⋆ (we set I⋆ = Ic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We then calculated the white dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 2(a) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 3, and the solid black lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 2(b) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The blue points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 2(b) were extracted from the measured ν0 using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Mechanisms for Flux Quantum Transfer The periodic tuning of the resonator and the associated fit to the calculated energy states of the loop constitute strong evidence that the loop remains in its ground state for all values of Φapp, and this is made possible by a single flux-quantum transferring into or out of the loop at δΦ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' It is important to establish the mechanism by which the flux is able to enter the loop, so we now turn our attention to the transitions between flux states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Across multiple devices on multiple chips, we found an onset 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='0 6Φ= 0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='4 (dB 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='6 S21 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='8 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='0 6Φ = 1/2 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='378 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='380 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='382 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='384 Frequency(GHz)5 of flux-periodic tuning in devices containing nanowires with w ≲ 35 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' This dependence of the behaviour on nanowire width suggests that the flux tunnelling occurs in the nanowires, and not in the wider part of the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We can now examine some possible physical processes that could occur in the nanowires and see how the data fits with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Does the nanowire current exceed its critical current Inw c ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Niobium-nitride resonators commonly exhibit a nonlinear S21 response when they are driven with a high microwave power [44] as a result of the current-induced nonlinear kinetic inductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' As we see in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 2, the kinetic inductance is quadratically dependent on (I/Ic)2, and so the nonlinearity must be dominated by the part of the conductor with the lowest Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' This is confirmed by measurements of our circuits, where we find that nanowire- embedded resonators show a much higher degree of non- linearity in their S21 response than bare resonators (see Supplemental Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The quadratic nature of the nonlinearity suggests that a strongly nonlinear response is a consequence of the magnitude of the rf current Ires in the resonator reaching a significant fraction of the nanowire critical current Inw c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Our NbN resonator readout method therefore gives us an indirect readout of whether the current in the nanowire is close to Inw c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Figure 3 shows the S21 response of the nanowire- embedded resonator at δΦ = 0 and δΦ = 1/2, both measured in the low-power limit with an estimated res- onator photon population of ⟨n⟩ ≈ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' In both cases, the response is linear and well fitted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We calculate, using the relation Is = dE(Φ)/dΦ that the maximum induced screening current in the nanowires Is(δΦ = 1/2) is 120 nA, an order of magnitude less than Inw c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Crucially, the lack of nonlinearity of the resonance at δΦ = 1/2, along with the fact that Qi remains a significant fraction of its zero-field value, means that the nanowires are not being driven close to their critical current by the applied flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' By a similar argument, we know that the nanowires are not being driven through Tc by a local heating process, as this would also result in nonlinearity of the resonance at δΦ = 1/2 due to dissipation in the highly resistive normal metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Is the nanowire a constriction Josephson-junction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' ‘Dayem-bridge’ Josephson-junction SQUIDs are com- monly embedded within CPW resonators [45, 46] to provide a flux-tunable resonant frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' However, when the SQUID loop has a large inductance, one ob- serves hysteretic tuning, characterised by the parameter βL = 2LIc/Φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' When βL ≳ 1, the SQUID behaviour be- comes hysteretic with applied flux and the resonator will exhibit discontinuous jumps in the resonant frequency, as observed in [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Our device does not undergo any discontinuous jumps, and the tuning over a single flux-quantum is symmetric, so βL < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Given this and the known loop inductance, we can set an upper bound on the critical current of any Josephson junction of IJJ c < 100 nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' This bound is 10× smaller than the expected transport critical current of our nanowires and also less than Is(δΦ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Normalised tunnelling flux ∆φ+ N (defined in the main text), as a function of the winding number N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The solid line corresponds to half a flux quantum, ∆φ+ N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' In- set: Magnetic-field dependence of resonant frequency of the nanowire-embedded resonator up to an applied field of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='4 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The black dotted lines mark transitions between winding number states (φ+ N), and the red dotted lines mark the energy minimum of each winding number state (φmin,N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Therefore, it is unrealistic to conclude that the flux-tuning we observe is a consequence of a dc SQUID formed of Dayem-bridge Josephson junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We also note that the closeness of the fit shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 2 suggests there is no contribution to the flux-dependent inductance from a Josephson junction, which would add a 1/ cos Φ term to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Is this a thermal or quantum process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Figure 4 shows that tunnelling always occurs at degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Following Petkovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' [30], ∆φ+ N is defined as ∆φ+ N = φ+ N−φmin,N, where φ+ N is the normalised flux φ = Φ/Φ0 at which tun- nelling from state N to state N + 1 occurs, and φmin,N is the flux that minimises the loop free-energy for a particu- lar winding number N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Our data shows the periodicity, defined in this way, to be half a flux quantum for all values of the winding number except N = 1 (we attribute the latter exception to enhanced flux-focusing at low magnetic field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' This is in contrast to [30], where a characteristic dependence of ∆φ+ N on N is shown to be a defining fea- ture of thermally-activated phase-slips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' By following the method of [30], we calculate that ∆φ+ N ≈ 300 would be required in order for the energy barrier to phase slips in our nanowire to be tuned to ≲ kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Correspond- ingly, we can estimate (see Supplemental Information) that ΓQPS = 35 MHz for our nanowire, and we calculate a temperature-dependent ΓTAPS that is below 1 Hz for T < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Therefore, at our measurement temperature, quantum tunnelling of flux is overwhelmingly more likely than a thermal transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' For comparison, the inverse experimental timescale is 1/τE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='2 Hz since, for each 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='380 Resonant Frequency (GHz) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='3 +z 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='374 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='372 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='370 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='368 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='366 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='0 Applied Field (mT) 0 2 4 6 8 10 12 14 16 Winding NumberN6 setpoint of the magnetic field, it takes the VNA approxi- mately 5 s to collect S21 data across the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Be- cause kBT/h > ΓQPS ≫ 1/τE — and to our knowledge this is the first reported study in this regime — when our device bias is swept through the degeneracy point, a single quantum phase-slip always occurs before we are able to observe the system in a higher-energy metastable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' CONCLUSIONS We have used a Ne FIB to fabricate NbN nanowires with widths down to 25 nm embedded within CPW res- onators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We observe periodic modulation of resonant frequency and intrinsic quality factor, which is consis- tent with quantum tunnelling of individual flux quanta mediated by quantum phase-slip, occurring when states of different winding number become degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' This behaviour has been observed in resonators with intrinsic quality factor, Qi, up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content='7 × 104 at 300 mK, which to our knowledge is the highest quality factor measured in quantum phase-slip experiments — note that the losses here are significantly lower than suggested by comparable reports [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We estimate that the QPS rate is of the order 10–100 MHz, which means that the tunnelling of a single flux-quantum is effectively deterministic on the timescale of microseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' This result shows the suitabil- ity of the Ne FIB process for fabricating QPS devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' We also suggest that an incoherent QPS device with a high QPS rate such as ours could be promising for classical digital logic processing applications, where the quantum nature of the flux tunnelling implies a reduction in heat dissipation compared with current state-of-the-art devices, opening up a route to resolving an important roadblock to the upscaling of qubit control electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Kennedy for useful dis- cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' The authors gratefully acknowledge funding from the United Kingdom Engineering and Physical Sciences Research Council, Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' EP/L015242/1, EP/J017329/1, and EP/T001062/1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Burkett, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Bushnell, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Chiaro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Collins, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Giustina, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Graff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Habegger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Har- rigan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Ho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Hong, T.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Jiang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Jones, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Kafri, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Kechedzhi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Kelly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Kim, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Klimov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Ko- rotkov, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Kostritsa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Landhuis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Laptev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Lind- mark, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Lucero, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Martin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Martinis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Mc- Clean, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' McEwen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Megrant, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Mi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Mohseni, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Mruczkiewicz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Mutus, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} +page_content=' Naaman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE3T4oBgHgl3EQfQQmX/content/2301.04411v1.pdf'} 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J. Bevins1,2,∗, Stefan Heimersheim3, Irene Abril-Cabezas3, Anastasia Fialkov2,3, +Eloy de Lera Acedo1,2, William Handley1,2, Saurabh Singh4, and Rennan Barkana5,6 +1 Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge, CB3 0HE, UK +2Kavli Institute for Cosmology, Madingley Road, Cambridge CB3 0HA, UK +3 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK +4Raman Research Institute, C V Raman Avenue, Sadashivanagar, Bangalore 560080, India +5School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, 69978, Israel +6Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540, USA and +∗ htjb2@cam.ac.uk +(Dated: January 10, 2023) +Observations of the first billion years of cosmic history are currently limited. We demonstrate +the synergy between observations of the sky-averaged 21-cm signal from neutral hydrogen and +interferometric measurements of the corresponding spatial fluctuations. By jointly analysing data +from SARAS3 (redshift z ≈ 15−25) and limits from HERA (z ≈ 8 and 10), we produce the tightest +constraints to date on the astrophysics of galaxies 200 million years after the Big Bang. We disfavour +at 95% confidence scenarios in which power spectra are ≥ 126 mK2 at z = 25 and the sky-averaged +signals are ≤ −277 mK. +PROBING THE INFANT UNIVERSE +The infant Universe, corresponding to cosmic time between 100 and 700 million years after the Big Bang, remains +largely unexplored. This epoch in cosmic history covers the birth of primordial stars, formation of the very first black +holes and assembly of the earliest galaxies. The nature of the first bright objects to form and the exact timing of these +events are yet to be constrained by observations. Theoretical studies and numerical simulations suggest that first +stars form between z ∼ 20 − 60, i.e. around 35 − 200 million years after the Big Bang[1–3], and, thus, are currently +out of reach of the modern telescopes. +The newly launched James Webb Space Telescope (JWST) [4–6], with an increased sensitivity over previous in- +struments such as the Hubble Space Telescope (HST), is advancing the observational frontier [7] with the prospect of +infrared observations of the brightest early galaxies out to z ≈ 20 [4]. A complementary view of the infant Universe +will be offered by radio telescopes such as the upcoming Square Kilometre Array (SKA) [8–10] which will probe +the astrophysical processes at early times by measuring the redshifted 21-cm line of neutral hydrogen gas located +in-between the first star forming regions. This signal, sensitive to the processes of star formation, cosmic heating and +reionziation, can inform us about the state of the Universe between 100 and 1000 million years after the Big Bang +[11–13]) and provide an insight into the properties of the first sources of light [8–10]. This joined effort across the broad +wavelength range will lift the veil on the period between the formation of the Cosmic Microwave Background (CMB), +when the Universe was approximately 300, 000 years old, and the end of the Epoch of Reionization (EoR). +The field of 21-cm cosmology is rapidly evolving with the very first, yet unconfirmed, detection of the sky-averaged +(or global) 21-cm signal made with the EDGES Low Band antenna and reported in 2018 [14]. This tentative detection +is much deeper than what is predicted by conventional theoretical modelling [15–19] and, thus, is hard to interpret, +and the cosmological nature itself of the EDGES absorption feature has been disputed by a number of works [20–24] +which suggest the existence of unaccounted for systematics in the data. However, if this signal is of cosmological +origin, the redshift range of the detected absorption feature implies an onset of efficient star formation at z ∼ 22 +followed (with some delay) by a strong heating of the neutral gas. To explain the anomalously strong absorption, +exotic mechanisms need to be invoked such as cooling of neutral gas via interactions with charged cold dark matter +[25–32] or production of strong radio background in addition to the CMB [33–36], e.g. by radio-luminous galaxies +such as considered here. +Although the EDGES signal is the only detection of the high-redshift 21-cm signal reported to date, advances have +been made by other existing radio telescopes with upper limits reported by both the interferometers such as PAPER +[37], MWA [38, 39], LOFAR [40, 41], AARTFAAC [42] and HERA [43–45] on the 21-cm power spectrum, which +quantifies the variation in the 21-cm brightness field as a function of angular scale and time, and on the global 21-cm +signal by SARAS [46–49], EDGES High Band [50–52] and LEDA [53]. These upper limits are becoming increasingly +constraining and have already being used to put limits on the properties of high-redshift luminous objects [44–49, 54]. +The ongoing and upcoming experiments including SARAS [55], MIST [56], REACH [57], LOFAR [58], NenuFAR [59], +arXiv:2301.03298v1 [astro-ph.CO] 9 Jan 2023 + +2 +HERA [60], the SKA and future proposed space based missions such as DAPPER and FARSIDE [61] aim to further +improve our understanding of the infant Universe. +Existing upper limits on the 21-cm signal provide the first very weak constraints on the astrophysical sources at a +broad range of redshifts. In this paper, we take the current tightest publicly available upper limits to date from the +HERA interferometer on the magnitude of the 21-cm power spectrum at redshifts z ≈ 8 and ≈ 10 which provides a +window to the EoR when ultraviolet photons emitted by first massive galaxies efficiently ionize the neutral hydrogen +gas [44], and the SARAS3 radiometer [55] that probes the sky-averaged 21-cm emission at higher redshifts between +z ≈ 15 − 25 when the first stars and X-ray emitting objects are expected to have formed in small galaxies during +the Cosmic Dawn. We combine these two data sets for the first time to improve constraints on the properties of the +first galaxies and the state of the neutral gas. We develop the methods to perform the joint analysis using a machine +learning enhanced Bayesian workflow and pave the way for future discoveries. We find that when considered together, +these two experiments provide a better leverage on theoretical scenarios that bridge across the wide redshift range +compared to the constraints from each individual experiment. In synergy, the two experiments leave only 64.9+0.3 +−0.1% +of the explored broad theoretical parameter space to be consistent with the joint data set in comparison to 92.3+0.3 +−0.1% +for SARAS3 and 79.0+0.5 +−0.2% for HERA alone. We use the joint analysis to limit star formation efficiency, minimum +halo mass for star formation, X-ray luminosity of early emitters and the radio luminosity of early galaxies. The joint +analysis disfavours at 68% confidence a combination of galaxies with X-ray emission that is ≲ 33 and radio emission +that is ≳ 32 times as efficient as present day galaxies. +In Methodology we review the data and the specifics of the experiments incorporated in our analysis. This is followed +by a discussion about the synergies between the power spectrum and sky-averaged 21-cm experiments in the same +section. We present the implications of our work for the astrophysical constraints and the validity of the EDGES +absorption feature as a sky-averaged 21-cm signal in Results. This is followed by a summary in Conclusion. Additional +information about the methodology and additional results can be found in Supplementary materials. +METHODOLOGY +The SARAS3 radiometer experiment took 15 hours of data, integrated in the frequency range 55 − 85 MHz, from a +lake in Southern India. After corrections were made for the receiver noise temperature, radio frequency interference +and emission from the water beneath the antenna, the data was appropriately scaled by the total efficiency of the +system to produce a measurement of the average sky temperature which is expected to include the Galactic and +extra-galactic foregrounds as well as the 21-cm signal. SARAS3 reported a null detection of the EDGES absorption +feature with a confidence of 95.3% [55] and more recently the data has been used to place constraints on the properties +of the first galaxies which we discuss in more detail below [49]. +SARAS2, the previous iteration of the instrumentation recorded data at higher frequencies (lower redshifts) of +110 − 200 MHz and was found to contain a non-smooth systematic structure possibly caused by emission from the +ground that the antenna was placed on. The data has been used to derive constraints on galaxies in the infant +Universe initially by fitting the foreground and systematic structure together with high order polynomials [46, 47] and +subsequently by modelling the two components separately [48]. In Supplementary materials we combine constraints +from the latter with constraints from SARAS3 and HERA. +To date, interferometric experiments have only observed upper limits of the cosmological 21-cm power spectrum, +which still allow for a large range of realistic astrophysical scenarios. The tightest constraints are derived from the +data from the HERA interferometer [43], followed by MWA in the redshift range z = 6.5−8.7 [38] and LOFAR, which +provide the tightest upper limits at redshifts z ∼ 9.1 [40] and z ∼ 9.3 − 10.6 [41]. The HERA telescope is a radio +interferometer located in the Karoo Desert of South Africa [60]. Already, this first public data release delivered the +strongest constraints on the 21-cm power spectrum to date. The publicly available HERA data[62] from the analysis +of Internal Data Release 2 that we use in this paper is based on 18 nights of observations, with 39 antennas operating +at science quality level. For our analysis, we employ the publicly available spherically averaged power spectra derived +from this data [43], in the wavenumber range k = 0.128 hMpc−1 to 0.960 hMpc−1 and from the two bands focusing +on redshifts z = 7.9 and 10.3. In Supplementary materials we consider the implications of including the upper limits +on the power spectrum from MWA and LOFAR in our joint analysis. +In 21-cm cosmology, data analysis efforts are increasingly employing Bayes theorem +P(θ|D, M) = L(θ)π(θ) +Z +, +(1) +to derive constraints on the astrophysical scenarios of the early Universe. Here the likelihood, L(θ), represents the + +3 +probability that we observe the data, D, from SARAS3 or HERA, given a particular model, M. The prior, π(θ), +represents our assumed knowledge before we consider any data and the evidence, Z, is a normalisation constant. The +posterior, P(θ|D, M), tells us which parts of the parameter space, θ, given the data and chosen model, are more +probable than others. The evaluation of Bayes theorem is usually performed with Nested Sampling or Markov Chain +Monte Carlo (MCMC) algorithms (see Supplementary materials). In many 21-cm experiments, θ is composed of +parameters that describe instrumental effects, θI, foregrounds, θfg, and the astrophysical processes that influence the +21-cm signal, θ21. We typically refer to the set of θI and θfg as the nuisance parameters. Since we are only interested +in the 21-cm signal, we work with the marginal or nuisance-free posteriors and nuisance-free likelihoods, L(θ21), which +can be estimated using normalising flows and the marginal Bayesian statistics code margarine [63, 64]. We give an +overview of this in Supplementary materials however we note that it allows for efficient combination of the HERA and +SARAS3 constraints. We detail the astrophysical processes included in the modelling, and the definition of θ21, next. +In order to realistically model the range of time covered by the Cosmic Dawn and Epoch of Reionization, we need +a consistent modelling of the cosmological and astrophysical processes from redshift 60, when star formation might +have began, all the way to redshift 5 at the end of the EoR. To that end we employ semi-numerical simulations +[18, 65–68] that have previously been used in the HERA [44], SARAS2 [48] and SARAS3 [49] analyses as well +as similar analysis of the LOFAR [54] and EDGES High-Band [52] data. We model the three-dimensional 21-cm +field as a function of cosmic time during the infant Universe taking into account important astrophysical process +including the Wouthuysen-Field (WF) effect [68–70], heating of the intergalactic medium by X-ray [67], Ly-α [18] +and CMB photons [35], multiple scattering of Ly-α, relative velocity between dark matter and gas [65], feedback of +Lyman-Werner radiation on star formation [66], and radio emission from galaxies [36]. The key parameters in the +astrophysical model are: the star formation efficiency, f∗, which quantifies the percentage of the baryonic mass in +the star forming halos that is converted into stars; the minimum virial circular velocity, Vc, which is proportional +to the cube root of the halo mass, M; the X-ray production efficiency, fX, which is directly proportional to the +X-ray luminosity per star formation rate, LX/SFR measured in erg s−1 M−1 +⊙ yr, between 0.2 and 95 keV; the CMB +optical depth, τ; finally, we model the contribution of high redshift radio-luminous galaxies to the radio background +by specifying a radio production efficiency, fradio, which is proportional to the radio luminosity per star formation +rate, Lr, measured in W Hz−1 M−1 +⊙ +yr at 150 MHz, and normalized such that it has a value of one for the present +day population of radio galaxies [36]. The X-ray spectral energy density is modelled based on a population of X-ray +binaries as in [71]. In our Bayesian analysis, θ21, therefore, comprises the set of parameters {f∗, Vc, fX, τ, fradio} or +equivalently {f∗, M, LX/SFR, τ, Lr/SFR}. +We explore wide prior ranges on all the parameters in an attempt to let the data inform us about the high-redshift +astrophysical processes. Specifically, the model for radio-luminous galaxies that we employ here is not conventionally +considered in 21-cm cosmology where typically the CMB is assumed to be the only source of radio background photons. +Here we expect that early galaxies will contribute to the radio background, thus increasing the amplitude of both the +sky-averaged 21-cm signal [72] and the power spectrum [35, 36]. +Both the sky-averaged 21-cm signal and the power spectrum rely on the same underlying physics, and constraints +from experiments targeting the different probes can be effectively combined to improve our knowledge of the infant +Universe. The semi-numerical simulations take of order a few hours to produce a signal per parameter set, which +is impractical for nested sampling or MCMC runs. We therefore train neural networks on outputs of the detailed +simulations. Typically, these networks take of order a few tens of milliseconds to evaluate meaning they are much +more well suited for computationally intensive fitting algorithms. The specific emulators used in this analysis and +more details regarding the signal modelling can be found in Supplementary materials. +RESULTS +Although the constraints presented here are weak, through the novelty of combining the previously reported HERA +and SARAS3 constraints we produce the tightest constraints to date on the properties of the infant Universe as +detailed below. This is the first time a joint analysis between a global signal data and interferometric limits has been +attempted. +To visualize the importance of combining the constraining power of HERA and SARAS3 we show, in the top panel +of Fig. 1, constraints on the ratio of the spin temperature of neutral hydrogen and background radiation temperature. +The background radiation temperature is the sum of the CMB temperature and radio background from galaxies. +The ratio determines the maximum absorption of the sky-averaged 21-cm signal, the smaller the ratio the larger the +signal can be. The SARAS3 limits on the 21-cm signal (grey markers) correspond to lower limits on Ts/Tr, with the +corresponding 1 and 2 σ contours shown as lines and extrapolated out of the SARAS3 band. Our simulations provide + +4 +FIG. 1. Key results from the joint analysis. Top Panel: The ratio of the spin temperature of neutral hydrogen, Ts, and +the radio background temperature, Tr, as a function of redshift for the joint HERA and SARAS3 analysis in green. We show the +HERA and SARAS3 68% and 95% confidence constraints in blue and grey respectively as triangles at the relevant redshifts and +solid and dashed lines. As a guideline, we show the ratio for Tr = TCMB and assuming adiabatically cooled gas in an expanding +Universe in the absence of any heating but with saturated coupling between Ts and the gas kinetic temperature (dashed black +line). Bottom Left: The 2D PDF from the joint analysis on the minimum virial circular velocity, Vc, in combination with the +star formation efficiency, f∗, marginalising over fradio, fX and τ. The solid black line shows the 68% contour, approximated by +the pink dashed line, and the black dashed line shows the 95% contour. The joint analysis disfavours low values of Vc and high +f∗ corresponding to efficient star formation. Bottom right: The constraint on the X-ray and radio luminosities from the joint +analysis marginalising over Vc, f∗ and τ. The joint analysis disfavours at 68% confidence low X-ray efficiencies in combination +with high radio production efficiencies. +a natural link between the power spectra and the global quantities, e.g. Ts/Tr, meaning that we can use the limits on +the power spectrum from HERA to derive an equivalent constraint on Ts/Tr. These constraints are shown in Fig. 1 +(blue markers and lines). The joint constraint, as shown by the green contours, provides the strongest constraints on +the ratio, and in particular at redshifts z = 15 − 20, gives better constraints than either experiment alone. Further, +the combination of the two experimental data sets improves the constraints at intermediate redshifts over a pure +extrapolation of each one of the sets of constraints. As a guideline, the dashed black line in the figure shows the +ratio for Tr = TCMB, i.e. in the absence of radio emission from galaxies, and assuming adiabatically cooled gas in +an expanding Universe in the absence of any astrophysical heating sources but with saturated coupling between the +21-cm spin temperature and the gas kinetic temperature. This limit is often used in the literature to give context to +the constraints (e.g. [44, 45]). However, we note that for the models tested here the gas does not cool adiabatically +because of the CMB and Ly-α heating at early times, and we have an excess radio background above the CMB. +Next, we explore the functional constraints in the T21−z and ∆21−z planes as shown in Fig. 2. These are calculated +by taking the samples of θ21 output from our fits and using the neural network emulators to produce corresponding +global signals and power spectra. Using our theoretical models we can easily map between the constraints on the + +0.0 +-0.5 +log(T/Tr) +SARAS3 + HERA +.1.0 +HERA +SARAS3 +HERA Band +-1.5 +SARAS3 Band +-2.0 +10 +15 +20 +25 +30 +Z +26 +log(Lr /SFR) +1.5 +1.0 +22 +-2 +38 +40 +42 +log(f*) +log(Lx/SFR)5 +FIG. 2. Functional constraints on T21 and ∆2. The functional prior (purple), SARAS3 (grey), HERA (blue) and joint +(green) posteriors for the sky-averaged 21-cm signal (top row) and power spectrum (bottom row). The yellow shaded region +shows the SARAS3 band and the dashed yellow lines show the HERA redshifts. +The functional prior and posteriors are +calculated by taking representative samples from the corresponding probability distributions for the astrophysical parameters +and generating the corresponding signals using neural networks. We see that by combining the constraining power of HERA and +SARAS3, we significantly reduce the 3σ (lightest shaded regions) constraints on the magnitude of both signals from −2630 mK +to −1770 mK at z = 15 for the global signal and 103.7 mK2 to 103.2 mK2 for the power spectrum at z = 25. The figure is +produced with fgivenx [74]. +power spectra and global signals. We see that for both the power spectra and the sky-averaged signals, although more +clearly for the latter, the range of plausible models is reduced by our joint analysis. The signals that are inconsistent +with the data typically have strong power spectra and corresponding deep absorption trough in the global signal, and +belong to scenarios with weak X-ray heating and strong radio luminosity. Further, we see that the 2σ region for the +functional constraints correspond to signals with the power spectra ≲ 102.1 mK2 at z = 25, which via our modelling +maps to global 21-cm signals shallower than −277 mK. We note that the 3σ limit on the power spectrum at z = 25 of +≲ 103.2 mK2 is approximately equivalent to the expected sensitivity of NenuFAR from 1000 hours of observations [73]. +For the global 21-cm signal 3σ limit on the magnitude reduces from ≈ −2630 mK from HERA to ≈ −1770 mK from +the joint analysis. Remarkably, we find that for the sky-averaged signal, the 2σ limit is very close to the minimum +depth of the ‘standard astrophysical’ models, ≈ −165 mK [18], where the radio background is equated to the CMB, +the contributions from radio galaxies and X-ray heating sources are assumed to be negligible, while the CMB and +Ly-α heating are present. The relatively tight constraints on the global signal and power spectrum suggest that future +improved measurements will allow us to dig deeper into the models with weak excess radio background radiation. It +is clear from our analysis, that the joint constraint improves limits on the range of plausible global signals and power +spectra. +We now interpret the constraints on the temperatures in terms of constraints on the properties of the first galaxies. +The key constraints from the joint analysis are shown in the bottom panels of Fig. 1 including the limits on the high- +redshift star formation which drives the high-redshift portion of the 21-cm signal via the process of Ly-α coupling +(constraints in the planes Vc − f∗) and the constraints on the luminosity of X-ray and radio sources (Lr − LX) which +primarily regulates the depth of the absorption trough. The full marginalised 1D and 2D posteriors corresponding +to the joint analysis are shown in Fig. S1 and the key numerical results are summarized in Tab. I alongside the +individual constraints from SARAS3 and HERA. We find that the combination of the two experiments leads to + +Prior +SARAS3 +HERA +Joint +0 +[mK] +-2500 +T21 +0 +0 +SARAS3 +L250 +-250 +-250 +-5000 +HERA +10 +20 +30 +10 +20 +30 +10 +20 +30 +6 +[mK²] +4 +2 +10 +20 +30 10 +20 +30 10 +20 +30 10 +20 +30 +Z +Z +Z +Z +1g +2g +3g6 +SARAS3 +HERA +SARAS3 + HERA +Signal +Sky-averaged +Power Spectrum +Both +z +≈ 15 − 25 +≈ 8 & ≈ 10 +≈ 8, ≈ 10 & +≈ 15 − 25 +Lr/SFR +≳ 1.55 × 1025 +≳ 4.00 × 1024 +≳ 3.31 × 1024 +LX/SFR +– +≲ 7.60 × 1039 +≲ 3.71 × 1039 +Lr/SFR +& +LX/SFR +≳ 1.00 × 1025 & +≲ 1.09 × 1042 +≳ 4.00 × 1024 & +≲ 7.60 × 1039 +≳ 3.16 × 1023 & +≲ 1.00 × 1042 +M +4.40 × 105 ≲ M +≲ 1.10 × 107 +– +2.55 × 105 ≲ M +≲ 7.04 × 106 +f∗ +≳ 0.05 +– +≳ 0.06 +f∗ & M +≳ 0.03 & +≲ 8.53 × 108 +– +≳ 0.02 & +≲ 4.50 × 107 +TABLE I. Key parameter constraints from SARAS3, HERA and the joint analysis. Here the SARAS3 and HERA +limits are taken from the respective papers. In the top two rows we show the type of signal targeted by each set of analysis and the +corresponding redshifts. The joint analysis produces improved constraints on the radio and X-ray backgrounds while retaining +the constraining power of SARAS3 on the star formation properties of early galaxies. Lr is measured in W Hz−1 M−1 +⊙ +yr at a +reference frequency of 150 MHz, LX in erg s−1 M−1 +⊙ +yr, and is calculated by integrating X-ray spectral distribution of sources +between 0.2 and 95 keV assuming an X-ray SED consistent with that for X-ray binaries [71]. The halo mass, M, is measured +in solar masses. These constraints are derived from Kernel Density estimates (KDE) of the 1D and 2D posterior distributions. +stronger constraints in the two-dimensional probability distribution of Lr − LX than either of the two experiments +individually. Where HERA constrains the population of high redshift radio-luminous galaxies to be ≲ 400 times +brighter in the radio band than the current population, the combination of the data sets constrains the galaxies to be +≲ 300 times brighter when marginalising over the other parameters (Vc, f∗, fX and τ). Similarly, HERA disfavours at +68% confidence galaxies with an X-ray luminosity ≲ 0.25 times the present day value in combination with the radio +luminosity of galaxies in the early universe that is ≳ 400 times the present day value. The joint analysis provides a +stronger constraint, ruling out scenarios where the X-ray luminosity is ≲ 33 times the present day value and the radio +luminosity of the first galaxies is ≳ 32 times the present day value at 68% confidence. +We find comparable constraints on f∗ and minimum mass of star forming halos, M, as was found with SARAS3 +alone [49] when combining the data sets. Marginalising over the radio and X-ray luminosities, we disfavour at 68% +confidence galaxies in which ≳ 2% of the gas is converted into stars and the minimum mass for star forming halos is +≲ 45 million solar masses. +Further, we explore the structure of the global 21-cm signals which are consistent with the data from SARAS3, +HERA and the two sets together. In Supplementary materials and Constraints on Phemenological Parameters, we +see that when considered individually, the SARAS3 and HERA experiments allow for astrophysically motivated +sky-averaged 21-cm signals that have a minimum temperature, location and width in agreement with the EDGES +detection, while the joint analysis rules out models with a depth that is consistent with EDGES at greater than a 2σ +significance. The joint analysis has a preference for shallower (lower values of |Tmin|) and narrower signals with higher +central frequencies, as can be seen in the corresponding 1D PDFs, which is driven largely by the HERA constraints. +We also consider the impact of SARAS2 on our joint constraints as well as MWA and LOFAR in Supplementary +materials. +The latter two data sets have no effect on our results and the inclusion of SARAS2 leads to weaker +constraints on the star formation properties. We focus in the main text on SARAS3 and HERA, due to the uncertainty +in the modelling of and presence of systematic structures in the SARAS2 data which is discussed in more detail in +Supplementary materials. +CONCLUSION +Through a combination of constraints on fluctuations and the sky-averaged 21-cm signal of neutral hydrogen, we +have improved our understanding of the first galaxies that formed in the infant Universe between 200 and 700 million +years after the Big Bang. This is the first time the data from the two different 21-cm probes have been combined + +7 +to derive constraints on the astrophysical properties of the early galaxies. Even though the existing constraints are +weak, we develop novel methodology and outline the approach which will become increasingly more useful as the +next-generation experiments deliver stronger observational constraints. +Considering a wide space of plausible astrophysical models including high-redshift sources of ultraviolet, X-ray and +radio photons which affect the 21-cm signal, we calculate corresponding sky-averaged spectra as well as the power +spectra of fluctuations. Using an upper limit on the fluctuations from the HERA interferometer and non-detection of +the global 21-cm signal by the SARAS3 radiometer, we find that only 64.9+0.3 +−0.1% of the explored theoretical parameter +space is consistent with the joint SARAS3 and HERA constraint, which is a significant improvement over the individual +values of 92.3+0.3 +−0.1% and 78.7 ± 0.2% respectively. +Using the newly developed methodology we place the tightest constraints to date on the properties of cosmic +gas, such as the spin temperature of the 21-cm hydrogen line (closely related, but not equal, to the gas kinetic +temperature) and the radio background temperature, Tr, as well as on the radio and X-ray luminosities of the first +galaxies disfavouring at 68% confidence galaxies that are approximately 32 times more efficient radio emitters than +present galaxies and simultaneously are less than 33 times bright in the X-ray band. This work reports an increased +degree of confidence over a wider range of redshifts than previous works which typically extrapolate outside the +redshifts targeted by individual experiments, while here we interpolated between the observations of SARAS3 at +z = 15 − 25 and the HERA limits at lower redshifts z ∼ 8 and 10. +In this work we also considered the addition of interferometric data sets from MWA and LOFAR to our analysis, +which led to a negligible improvement in the results. Similarly to HERA, these experiments probe the physics of the +EoR covering a similar redshift range to HERA (between z ≈ 6 − 10), with the current HERA data providing the +tightest constraints on our models. Of the current global or sky-averaged 21-cm experiments only SARAS2, SARAS3 +and EDGES were able to place limits on the astrophysics of the infant Universe. 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WJH and AF were supported by Royal Society University Research Fellowships. EdLA was supported +by the STFC through the Ernest Rutherford Fellowship. RB acknowledges the support of the Israel Science Foundation +(grant No. 2359/20), The Ambrose Monell Foundation and the Institute for Advanced Study. +AUTHOR CONTRIBUTION +HTJB lead the data analysis and the writing of the manuscript. SH provided the power spectrum emulator, assisted +with the writing and data analysis. IAC performed the combined analysis of HERA, LOFAR and MWA data and +assisted with the writing of the manuscript. AF generated the idea, supervised the project and contributed to the +writing of the article. EdLA supervised the work and helped with the revision of the article. WJH supervised the +analysis, helped with revision of the article and provided advice on the Bayesian methodology. The astrophysical +signal models were provided by AF and RB and the SARAS3 data was provided by SS. RB and SS provided comments +on the manuscript and assisted with revisions. + +13 +SUPPLEMENTARY MATERIALS +Modelling the Thermal History of the Infant Universe: Power Spectrum and Global Signal Synergies +At high redshifts around z ≈ 20−30, the first stars begin to form and produce Ly-α photons that interacts with the +baryonic matter, predominantly composed of neutral hydrogen, in the Universe. Neutral hydrogen atoms absorb and +remit ambient Ly-α photons in a process known as the Wouthuysen-Field (WF) effect [69, 70] that drives the relative +number of atoms with aligned and anti-aligned proton and electron spins. This process couples the spin temperature +of the neutral hydrogen, Ts, which describes the distribution of hydrogen atom spins, to the gas temperature, Tk, +which is cooling at a faster rate than the radio background as the Universe expands. Further, interactions between +the neutral hydrogen and Ly-α emission result in the transfer of kinetic energy that raises the gas temperature, and +coupled spin temperature, in a process known as Ly-α heating [76, 77] preventing the gas from cooling adiabatically. +However, despite the heating, the dominant WF effect produces an absorption feature in the sky-averaged 21-cm +signal, which is measured relative to the radio background. Further, it leads to a peak in the power spectrum at high +z ≈ 25 on angular scales corresponding to the effective horizon of the Ly-α emission and the distribution of galaxies, +which disappears when the coupling becomes saturated [78]. +The intensity and spatial fluctuations of the Ly-α +emission evolve with the population of early galaxies and consequently it is dependent on their star formation rate +and the minimum halo mass for star formation. We parameterize these quantities with the star formation efficiency, +f∗, which quantifies the percentage of the baryonic mass in the star forming halos that is converted into stars, and +the minimum virial circular velocity, Vc, which is proportional to the cube root of the halo mass, M. +At intermediate redshifts of z ≈ 10 − 20 the gas is further heated by X-ray binaries [67, 71], continuing Ly-α +heating, CMB heating [79] and heating through structure formation. X-ray heating is dependent on processes such as +X-ray production in high redshift galaxies and black hole binary formation. This means that it has to be separately +parameterized and in our semi-numerical simulations, we model the X-ray production efficiency, fX, which is directly +proportional to the X-ray luminosity per star formation rate, LX/SFR measured in erg s−1 M−1 +⊙ yr, between 0.2 and +95 keV. The heating affects the redshift of the minimum and depth of the sky-averaged 21-cm signal. If sufficiently +efficient it can raise the temperature of the gas, and coupled 21-cm brightness temperature, above the radio background +resulting in emission in the sky-averaged signal at low redshifts. The heating rate has a direct impact on how fast +the brightness temperature transforms from absorption to 0 K or emission. In the power spectrum, due to the non- +uniformity of the heating, the various mechanisms can produce a peak in the signal at around z ≈ 15. Although this +redshift is model dependent (see [78]) and in some cases when heating is done by hard X-rays (energies > 1 keV with +long mean free paths) X-ray heating is smooth and no peak is imprinted in the power spectrum [67, 68]. +Finally, at more recent times, z ≈ 5 − 15, ultraviolet emission from the first massive galaxies begins to ionize the +neutral hydrogen, stripping the abundant gas of its electrons. This reduces the sky-averaged 21-cm signal and when +reionization is complete, the signal disappears. The process produces a peak in the power spectrum at the scales +corresponding to the typical size of ionized bubbles, but again the signal is destroyed once reionization is complete. +The process of reionization is highly dependent on the ionizing efficiency of sources, ζ, which in the models explored +here is normalized by the CMB optical depth, τ. While τ has been weakly constrained by cosmological experiments +such as Planck [80], we treat it as a free parameter and note that 21-cm cosmology offers a means by which to break +degeneracies between τ and other cosmological parameters. +Throughout our analysis, we explore a broad range of radio luminosities that produce radio excesses above the +CMB of between ≈ 0.5 − 270 times the CMB temperature at z = 20 and ≈ 1 − 32000 times the CMB at z = 10. +While moderate radio emission might be expected [81], the extreme values of radio production efficiencies are usually +invoked to explain the anomalously deep EDGES signal [14, 33–36, 72, 81] but struggle to explain the rapid star +formation and rapid heating of the gas [82] that is implied by the shape of the EDGES signal. +Combining Constraints with margarine +We use the marginal Bayesian statistics code margarine [63] to combine constraints from different data sets. +margarine uses neural networks known as Masked Autoregressive Flows (MAFs) to model the probability distribu- +tion, P(θ|D, M), of a set of samples, here θ21, marginalising over nuisance parameters describing the instrumental +systematics, θI, and foregrounds, θfg in the process. It does this by shifting and scaling a base standard normal +distribution, making the probability of the distribution easily tractable, to replicate the target samples, where the +shifting and scaling are determined by the outputs of the MAFs. + +14 +This can subsequently be used to calculate a nuisance-free likelihood given a prior distribution and the Bayesian +evidence using +L(θ21) ≡ +� +L(θ21, α)π(θ21, α)dα +� +π(θ21, α)dα += P(θ21|D, M)Z +π(θ21) +, +(2) +where α = {θI, θfg} [64]. In instances when the prior is flat then the following is true up to a normalisation constant +L(θ21) ≈ P(θ21|D, M). +(3) +With margarine we can subsequently evaluate log(L(θ21)) for any set of θ21 for any existing posterior distribution, +P(θ|D, M), from previous analysis of a data set like HERA or SARAS3 +θ = {θI, θfg, θ21} → {θ21} → margarine → log(P(θ21|D, M) → log(L(θ21)), +(4) +where the log is base 10. The likelihood evaluations can then be combined, +log(Ljoint(θ21)) = log(LHERA(θ21)) + log(LSARAS3(θ21)), +(5) +as discussed in the main text, to be sampled over using MCMC methods or in our case nested sampling implemented +with polychord [83, 84]. +MCMC sampling methods approximate the unnormalised posterior distribution by directly sampling the product +P(θ) ≈ L(θ)π(θ) and do not provide estimates of the evidence, Z. The family of algorithms typically use random +walkers to traverse the parameter space, with points accepted and rejected based on some probabilistic criteria in an +effort to explore the space fully. The HERA analysis [44] of the excess radio background models explored here used +the emcee [85] implementation of MCMC sampling [44]. +Nested sampling [86] numerically approximates the integral +Z = +� +L(θ)π(θ)δθ, +(6) +which can be derived from equation 1 and the requirement that the posterior must integrate to 1, by evolving a series +of live points to higher and higher likelihood values. By approximating the evidence, the algorithm produces samples +on the normalised posterior distribution, which we subsequently use to determine preferred and disfavoured regions +of the parameter space. A comprehensive review of nested sampling can be found in [87]. +Emulating Signals from the EoR and CD +As described in the main text, semi-numerical simulations typically have a runtime of multiple hours for a given set +of astrophysical parameters, thus performing parameter inference with the simulation directly is very costly. However, +as both the sky-averaged signal and the power spectrum change smoothly when we vary parameters, we can interpolate +values at intermediate parameters from existing simulations. An increasingly common practice to achieve this are the +fast and precise neural network based emulators we discuss in the following sections. +The sky-averaged 21-cm signal +To emulate the sky-averaged 21-cm signal in models with an excess radio background from high redshift radio- +luminous galaxies we use the publicly available emulator globalemu [88] trained on sets of astrophysical simulations. +The simulations include the effects of Ly-α and CMB heating. The mean free path of ionizing photons, Rmfp, is fixed +at 40 Mpc and the and X-ray heating is powered by a population of X-ray binaries with a realistic spectral energy +distribution [67]. The models correspond to the parameterisation detailed above and in [36]. +The emulator has previously been used in the individual analysis of data from SARAS2 [48] and SARAS3 [49] +and detailed in the corresponding papers. We therefore only briefly summarise the accuracy of the neural network +here. The original set of simulations comprise approximately 10,000 models however the explored range of τ, the +CMB optical depth, is large and when training our neural network we filter out models that have a value of τ in the +range given by Planck ±3σ. This results in a training set of approximately 4,300 models and the network is tested +on approximately 500 models. A root mean squared error (RMSE) of 5.11 mK is found and a 95 percentile RMSE of +20.53 mK indicating a high level of accuracy (see Table M.2 in [49]). + +15 +The 21-cm power spectrum +For comparing models with the HERA measurements we use the same 21-cm power spectrum emulator used in the +HERA analysis [44], based on the same suit of simulations of the sky-averaged signal emulator used in the SARAS2 +and SARAS3 analysis. Based on an input of the 5 model parameters, this emulator returns the 21-cm power spectrum +with a relative accuracy of 20% at the wave numbers and redshifts observed by HERA. For HERA the impact of τ is +more significant because the data corresponds to lower redshifts and so the full prior range on the parameter is used +for training. This results in a training set of ∼8,000 simulations and another 2,000 independent samples for testing. +Full details and tests of this emulator can be found in the HERA analysis paper [44], Appendix B. +We note that when performing our joint analysis we use the narrow prior on τ defined by Planck, since the SARAS3 +posterior is not defined in the original analysis for values outside this range. +Temperatures +To emulate physical properties such as the spin temperature and temperature of the radio background (as seen in +Figure 1) we use a globalemu-style emulator. In this framework, the emulator takes in the five astrophysical pa- +rameters and a single redshift and returns a single corresponding temperature. In practice, this means that vectorised +calls have to be made to emulate the spin temperature, Ts(z), and the radio background temperature, Tr(z), as a +function of redshift, but the method is found to be more accurate, quicker and allows for interpolation at a range of +different redshifts. We use the same training and test sets as used for the 21-cm power spectrum emulator, and a +similar architecture of 4 layers, with a reduced size of 100, 30, 10, and 5 nodes per layer. This allows us to emulate +the spin temperature Ts within ±6% accuracy (95% confidence interval), and the radio background temperature Tr +within ±4% accuracy (95% confidence interval). +Constraints on the Astrophysical Parameters +Fig. S1 shows the 1D and 2D marginal posteriors found for the joint analysis of SARAS3 and HERA. The graph +shows that the combined constraining power of the two experiments leads to a strong constraint on the combination +of the radio and X-ray luminosities per star formation rate for an early population of galaxies when marginalising +over the other astrophysical parameters. We also see constraints in the plane of Vc − f∗ when marginalizing over the +radio background and X-ray heating params. The constraints are summarized in detail in the main text. +Constraints on Phemenological Parameters +In order to explore the structure of the global 21-cm signal we look at the minimum temperature, Tmin, the +corresponding central redshift, z0, and an approximate full width at half max, ∆z, of each absorption trough as +defined in the top right of Fig. S2. More specifically, for each parameter set θ21 (either from the prior parameter +range, or from the posterior ranges consistent with SARAS3, HERA, and the joint analysis) we generate a global +signal using the neural network emulator globalemu [88]. We then measure the values of Tmin, z0, and ∆z for +each signal producing probability distributions for each parameter corresponding to the constraints from each data +and showing the extent of the prior (see results in Fig. S2). We compare these distributions with the values used to +parameterise the phemenological EDGES flattened Gaussian signal with the EDGES 99% confidence ranges shown +by the black crosses in the corner plot in Fig. S2. +Constraints on the Radio and X-ray Backgrounds +The product f∗fradio is proportional to the total radio background created by radio-luminous galaxies, and, equiv- +alently, f∗fX is a proxy for the total X-ray background created by the early population of X-ray sources. This is +demonstrably true for our model parameterisation as the star formation rate is proportional to f∗ and the radio lumi- +nosity and X-ray luminosities per star formation rate are proportional to fradio and fX respectively. Both X-ray and +radio backgrounds are responsible for regulating the depth of the absorption feature, and they can also be observed +independently by other telescopes (e.g. observations of the unresolved X-ray background by Chandra [89] and of the + +16 +FIG. S1. Parameter constraints from the joint anlaysis. The astrophysical parameter constraints on models with excess +radio background in addition to the CMB derived when combining an upper limit on the 21-cm power spectrum at z ≈ 8 +and ≈ 10 from HERA with data from the 21-cm sky-averaged experiment SARAS3 in the band z ≈ 15 − 25. Through the +combination of these two data sets probing different statistical properties of the 21-cm signal at different redshifts we are able +to improve constraints on the radio and X-ray luminosities of early radio-luminous galaxies and maintain constraints provided +by SARAS3 on the star formation properties of these early galaxies. Lr is measured in units of W Hz−1 M−1 +⊙ +yr at 150 MHz +and LX is in units of erg s−1 M−1 +⊙ +yr calculated between 0.2 and 95 keV assuming a realistic SED of an early X-ray binary +population. The pink dashed lines approximate regions that are disfavoured with 68% confidence. +low-frequency radio background by ARCADE2/LWA [90, 91]. In Fig. S3 we show constraints on the values of f∗ fX +and f∗ fradio achieved by SARAS3, HERA and the joint analysis. Since these combinations of the parameters regu- +late the absorption depth of the global 21-cm signal, we can also condition our prior on the astrophysical parameters +to produce signals with the same central frequency and depth as the absorption feature found in the EDGES data +[18, 35, 49]. In each panel of Fig. S3, we show black contours corresponding to these EDGES-like physical signals. We +see that while HERA and SARAS3 allow for combinations of fX f∗ and fradio f∗ that could partially explain EDGES, +the combination of the two experiments, which produces a tighter constraint on the X-ray and radio luminosities of +early galaxies, disfavours a large portion of the EDGES-like parameter space (i.e. most of the EDGES-like parameter +space is beyond the 95% contours of the joint constraints, while it is well within the 95% contours for SARAS3 and +HERA individually). This demonstrates further the power of combining different data sets. However, we note that the +explored theoretical signals do not fit EDGES data well as none of them closely reproduces the flattened Gaussian-like +feature found in the data [14]. + +95% Confidence +68% Confidence +1.5 +1.0 +log(PR) +42.5 +40.0 +37.5 +0.075 +0.050 +25.0 +22.5 +2 +5 +5. +0 +5. +0.050 +0.075 +25.0 +0. +4 +T +log( FR +log(sFR)17 +FIG. S2. Phenomenological constraints. The triangular plots shows the prior (purple) and posteriors (grey for SARAS3, +blue for HERA, green for joint) of the features of a typical global absorption signal: the central redshift, z0, the corresponding +minimum temperature, Tmin, and the width of the signal, ∆z, as is depicted in the top right corner. Darker shaded regions +show 1σ constraints, lighter shaded regions show 2σ constraints. Overlaid on the posterior distributions are the 99% confidence +intervals, black crosses, reported for the corresponding phemenological parameterisation of the EDGES absorption feature in +[14]. Note that this is not the same as the physical EDGES-like distribution explored in Fig. S3. We see that individually +the experiments allow for signals with depths that are consistent with EDGES. However, the combination of the two data +sets disfavours these signals with greater than 2σ significance. We do not disfavour signals with the same width or central +frequency as EDGES, but note that the joint analysis indicates a preference for shallower and narrower signals with higher +central redshifts as can be seen in the 1D PDFs. +The impact of SARAS2 +Previous analysis of the SARAS2 data revealed some weak constraints, most notably in the plane of LX − Lr +in agreement with HERA and SARAS3, on the properties of galaxies in the infant Universe [48]. SARAS2 is at +much lower redshifts than SARAS3 but overlaps with the redshifts probed by HERA having recorded observations +in the band z ≈ 7 − 12. The data is contaminated by a sinusoidal systematic and a number of different models +were fitted to this feature. The model corresponds to a signal introduced prior to the antenna possibly from ground +emission or some unknown component of the foreground and separately a signal introduced in the system electronics +potentially from cable reflections. +The sinusoidal systematic was fitted alongside a signal model generated with +globalemu and a foreground model that is conditioned to be smooth preventing it fitting out non-smooth systematics +or signals in the data [24]. Here we take the best fitting model, with a systematic from ground emission or a non- +smooth component from the foreground, with the highest evidence from the original analysis [48] and combine the +corresponding constraints on the astrophysical parameters Vc, f∗, fX, fradio and τ with the joint constraints from +HERA and SARAS3 to assess the impact. +As can be seen in Fig. S4, the addition of SARAS2 to our analysis washes out the constraint in the plane f∗ − Vc. + +Physical +EDGES +0 +Az +K +m. +-250 +(Z0, Tmin) +-500 +(Z0, Tmin) +10 +20 +30 10 +20 +30 +之 +之 +Prior +[mK] +SARAS3 +500 +HERA +SARAS3+HERA +-1000 +16 +8 +8 +16 +24 +-1000 +-500 +8 +16 +Tmin [mK] +Az +2018 +FIG. S3. Background constraints. The figure shows constraints on the radio and X-ray backgrounds, parameterized by +f∗fradio and f∗fX respectively, from SARAS3 (grey), HERA (blue) and the joint analysis (green). The SARAS3 and HERA +posteriors are based on the results presented in [49] and [44] respectively. We show each distribution individually on the top +row, overlaid pairs of distributions for comparison in the middle, and all three on the same figure in the bottom row. In +all panels, we show 68% and 95% contours (black solid and dashed lines respectively) for physical signal models that have +similar depths and central frequencies as the EDGES absorption feature as defined by the inequality in [35]. These physical +EDGES-like models have previously been explored in the literature in [35, 36]. We note that while individually both HERA +and SARAS3 allow for astrophysically motivated signal models that could explain the depth of the EDGES feature, together +they rule the corresponding parameter space out with approximately greater than 2σ confidence, although some EDGES-like +signals are still viable. We stress again, that the explored physical models cannot fully explain the shape of the EDGES signal. +One possible explanation for this is that the addition of SARAS2, while constraining the properties that affect the +signal at low redshifts, increases the envelope of possible models at higher redshifts, where star formation is more +important, that are plausible even given the constraints from the SARAS3 data. +Despite this, we note that we +maintain the constraint in the plane LX − Lr when we add SARAS2 into our analysis. +We can quantify the impact of SARAS2 on our analysis by looking at the percentage of the astrophysical prior volume +which is consistent with the different combinations of the three different data sets. To calculate this percentage, we use +margarine to calculate the marginal Kullback-Liebler divergence, D, between the flat prior on the five astrophysical +parameters in the set θ21 and the corresponding posteriors. The KL divergence is related to the percentage via +% = 100 × exp(−D) ≈ 100 × VP +Vπ +, +(7) + +Individual Constraints +SARAS3 +HERA +Joint +104 +radio. +103 +f 102 +101 +10-2 +100 +10-2 +100 +10-2 +100 +fxf* +fxf* +fxf* +Comparison of Pairs +SARAS3 vs HERA +SARAS3 vs Joint +HERA vs Joint +104 +* +103 +102 +101 +100 +10-2 +100 +10-2 +100 +10-2 +fxf* +fxf* +fxf* +Comparison with Joint +SARAS3 vs HERA vs Joint +SARAS3 +104 +HERA +Joint +103 +EDGES-like +Signals +102 +101 +10-2 +100 +fxf*19 +HERA + SARAS3 +1.0 +1.5 +log(Vc) +37.5 +40.0 +42.5 +log( LX +SFR) +0.050 +0.075 +τ +−2 +−1 +log(f∗) +22.5 +25.0 +log( Lr +SFR) +1.0 +1.5 +log(Vc) +37.5 +40.0 +42.5 +log( LX +SFR) +0.050 +0.075 +τ +22.5 +25.0 +log( Lr +SFR) +HERA + SARAS3 + SARAS2 +95% Confidence +68% Confidence +1.0 +1.5 +37.5 +40.0 +42.5 +0.050 +0.075 +−2 +−1 +log(f∗) +22.5 +25.0 +1.0 +1.5 +log(Vc) +37.5 +40.0 +42.5 +log( LX +SFR) +0.050 +0.075 +τ +22.5 +25.0 +log( Lr +SFR) +0 +max +FIG. S4. The impact of SARAS2 data on the astrophysical constraints. We show the joint posterior distributions for +HERA and SARAS3 on the left panel (identical to Figure S1, but shown here for comparison) and for HERA, SARAS3 and +SARAS2 on the right panel. SARAS2 covers the band z ≈ 7 − 12 and therefore has some overlap with HERA but not with +SARAS3. The addition of SARAS2 to the joint analysis washes out the constraint on star formation properties, Vc and f∗, +because it leads to increased uncertainty in the structure of the signals at high redshifts. However, we still see a consistent +disfavouring of a population of radio galaxies with high radio and low X-ray luminosities. The one dimensional posteriors for +τ appear to be in disagreement, however, we note that these are basically flat. We exclude SARAS2 from our main results in +the text because of uncertainty in the modelling of systematics in the data. +where Vπ is the prior volume and VP is the posterior volume. This quantity is useful as it quantifies the constraining +power of the different data sets in all five dimensions, including correlations that may not be visible in the one and +two dimensional projections used to produce the corner plots in this paper and in the literature. +We show in Fig. S5 the percentage of the astrophysical parameter prior volume that is consistent with different +combinations of the data sets discussed in this work (including additional interferometric measurements of the power +spectrum discussed in Other Power Spectrum Experiments). We see that the combination of either or both of the +SARAS data sets with HERA lead to a percentage consistency with the data of ≈ 63 − 65% and this is likely +dominated by HERA. Individually, HERA allows for ≈ 80% of the astrophysical parameter space, SARAS2 for +≈ 90% and SARAS3 for ≈ 92%. Due to the uncertainty in the modelling of the systematics in the SARAS2 analysis, +we leave SARAS2 out of the main results. +Other Power Spectrum Experiments +In Fig. S6, we show the projected posteriors derived using HERA data alone (left panel) and HERA, MWA and +LOFAR together (right panel). We note that the constraints from the different interferometers are all at low redshifts +between z ≈ 6 − 10 and varying wavenumbers or angular scales. +These are detailed in Tab. S1 along with the +constraints from the individual experiments on key parameters. +We derived parameter constraints from the MWA and LOFAR data using the approach taken in the orginal HERA +analysis. Specifically, we take the measured upper limits, the mean power spectrum and uncertainty, and treat it as a +measurement of cosmological signal plus systematics. As in HERA [44] we take this uncertainty to be Gaussian and +marginalize over a uniform prior on the systematics, yielding the likelihood + +20 +60 +65 +70 +75 +80 +85 +90 +95 +100 +% Prior Consistent with Data +SARAS2 + +HERA +SARAS2 + SARAS3 + +HERA +SARAS3 + +HERA +SARAS3 + +SARAS2 +HERA + MWA + +LOFAR +HERA +SARAS2 +SARAS3 +FIG. S5. Constraining power of different data sets. The percentage of the wide astrophysical parameter prior that is +found to be consistent with the different data sets and different combinations of data sets explored in this work. A lower +value indicates a better set of constraints, although a difference of a few percent does not necessarily translate into significant +differences in the parameter constraints as can be seen when comparing the results from HERA and HERA + LOFAR + MWA. +HERA +LOFAR +MWA +LOFAR + MWA + +HERA +z +≈ 8 & ≈ 10 +9.1 & 9.3 − 10.6 +6.5 − 8.7 +Discrete and contin- +uous ranges of z be- +tween 6.5 − 10.6 +k [h Mpc−1] +0.128 − 0.960 +0.075−0.432 & 0.053 0.070 − 3.000 +Discrete and continu- +ous ranges of k +Lr/SFR +≳ 4.00 × 1024 +≳ 1.20 × 1025 +≳ 1.58 × 1025 +≳ 4.00 × 1024 +LX/SFR +≲ 7.60 × 1039 +≲ 8.70 × 1038 +≲ 1.16 × 1039 +≲ 1.58 × 1040 +Lr/SFR +& +LX/SFR +≳ 4.00 × 1024 & ≲ +7.60 × 1039 +≳ 3.16 × 1025 & ≲ +1.00 × 1040 +≳ 1.00 × 1025 & ≲ +1.00 × 1040 +≳ 4.00 × 1024 & ≲ +1.58 × 1040 +TABLE S1. Constraints from interferometers. The table shows the various constraints on the radio and X-ray luminosities +for HERA, MWA, LOFAR and the combination of all three along with their respective wavenumbers and redshift ranges. The +joint analysis only marginally improves our understanding of the infant universe. +L(θ21) = +Nd +� +i +1 +2 +� +1 − erf +�di − mi(θ21) +√2σi +�� +, +(8) +where Nd represents the number of data points, di and σi correspond to the mean and standard deviation in each +data point, and mi(θ21) is the model prediction for that redshift and wave number. Thus a model prediction m ≫ d +gives L ≈ 0 while m ≪ d gives a constant. This likelihood is effectively a step function that disfavours models above +a given amplitude. A full discussion of its derivation can be found in [44]. +We performed this joint analysis using a full analytic likelihood approach (independent of margarine) since there +are no nuisance parameters describing the systematics. We find that each of the experiments disfavours individually +similar regions of the Lr − LX plane. However, the joint analysis does not improve the results derived from HERA +data alone (as we summarise in Tab. S1 and is further illustrated in Fig. S5) which motivates our decision to only use +HERA in the main text. + +21 +FIG. S6. The impact of MWA and LOFAR on the parameter constraints. Projected posterior distribution functions +(PDFs) for the 5 simulation parameters, obtained by assuming flat priors and combining different observations: HERA alone +(left, as in [44]) and LOFAR, HERA and MWA (right). Solid (dashed) lines correspond to regions containing the highest 68% +(95%) probability. We see that HERA constraints are not significantly improved by adding the published limits on the power +spectrum from other interferometers. + +HERA +HERA + MWA + LOFAR +95% Confidence +68% Confidence +1.5 +1.5 +0 +1.0 +1.0 +max +6'68 > +< 40.2 +42.5 +42.5 +40.0 +40.0 +37.5 +37.5 +0.075 +0.075 +0.050 +> 24.6 +0.050 +> 24.6 +25.0 +25.0 +22.5 +22.5 +37.5 +40.0 +42.5 +0.050 +0.075 +25.0 +5. +5 +5 +40.0 +42.5 +0.050 +0.075 +22.5 +3 +2 +5 +2 +0. +2 +< +log(f*) +log(f*) +log(sFR) \ No newline at end of file diff --git a/5dE1T4oBgHgl3EQfmgRl/content/tmp_files/load_file.txt b/5dE1T4oBgHgl3EQfmgRl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b4da482285575b243a87c838b7647107a0c38dbd --- /dev/null +++ b/5dE1T4oBgHgl3EQfmgRl/content/tmp_files/load_file.txt @@ -0,0 +1,2476 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf,len=2475 +page_content='Joint analysis constraints on the physics of the first galaxies with low frequency radio astronomy data Harry T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bevins1,2,∗, Stefan Heimersheim3, Irene Abril-Cabezas3, Anastasia Fialkov2,3, Eloy de Lera Acedo1,2, William Handley1,2, Saurabh Singh4, and Rennan Barkana5,6 1 Astrophysics Group, Cavendish Laboratory, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Thomson Avenue, Cambridge, CB3 0HE, UK 2Kavli Institute for Cosmology, Madingley Road, Cambridge CB3 0HA, UK 3 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK 4Raman Research Institute, C V Raman Avenue, Sadashivanagar, Bangalore 560080, India 5School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, 69978, Israel 6Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540, USA and ∗ htjb2@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='uk (Dated: January 10, 2023) Observations of the first billion years of cosmic history are currently limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We demonstrate the synergy between observations of the sky-averaged 21-cm signal from neutral hydrogen and interferometric measurements of the corresponding spatial fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' By jointly analysing data from SARAS3 (redshift z ≈ 15−25) and limits from HERA (z ≈ 8 and 10), we produce the tightest constraints to date on the astrophysics of galaxies 200 million years after the Big Bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We disfavour at 95% confidence scenarios in which power spectra are ≥ 126 mK2 at z = 25 and the sky-averaged signals are ≤ −277 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' PROBING THE INFANT UNIVERSE The infant Universe, corresponding to cosmic time between 100 and 700 million years after the Big Bang, remains largely unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This epoch in cosmic history covers the birth of primordial stars, formation of the very first black holes and assembly of the earliest galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The nature of the first bright objects to form and the exact timing of these events are yet to be constrained by observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Theoretical studies and numerical simulations suggest that first stars form between z ∼ 20 − 60, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' around 35 − 200 million years after the Big Bang[1–3], and, thus, are currently out of reach of the modern telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The newly launched James Webb Space Telescope (JWST) [4–6], with an increased sensitivity over previous in- struments such as the Hubble Space Telescope (HST), is advancing the observational frontier [7] with the prospect of infrared observations of the brightest early galaxies out to z ≈ 20 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' A complementary view of the infant Universe will be offered by radio telescopes such as the upcoming Square Kilometre Array (SKA) [8–10] which will probe the astrophysical processes at early times by measuring the redshifted 21-cm line of neutral hydrogen gas located in-between the first star forming regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This signal, sensitive to the processes of star formation, cosmic heating and reionziation, can inform us about the state of the Universe between 100 and 1000 million years after the Big Bang [11–13]) and provide an insight into the properties of the first sources of light [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This joined effort across the broad wavelength range will lift the veil on the period between the formation of the Cosmic Microwave Background (CMB), when the Universe was approximately 300, 000 years old, and the end of the Epoch of Reionization (EoR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The field of 21-cm cosmology is rapidly evolving with the very first, yet unconfirmed, detection of the sky-averaged (or global) 21-cm signal made with the EDGES Low Band antenna and reported in 2018 [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This tentative detection is much deeper than what is predicted by conventional theoretical modelling [15–19] and, thus, is hard to interpret, and the cosmological nature itself of the EDGES absorption feature has been disputed by a number of works [20–24] which suggest the existence of unaccounted for systematics in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' However, if this signal is of cosmological origin, the redshift range of the detected absorption feature implies an onset of efficient star formation at z ∼ 22 followed (with some delay) by a strong heating of the neutral gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' To explain the anomalously strong absorption, exotic mechanisms need to be invoked such as cooling of neutral gas via interactions with charged cold dark matter [25–32] or production of strong radio background in addition to the CMB [33–36], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' by radio-luminous galaxies such as considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Although the EDGES signal is the only detection of the high-redshift 21-cm signal reported to date,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' advances have been made by other existing radio telescopes with upper limits reported by both the interferometers such as PAPER [37],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' MWA [38,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 39],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' LOFAR [40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 41],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' AARTFAAC [42] and HERA [43–45] on the 21-cm power spectrum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' which quantifies the variation in the 21-cm brightness field as a function of angular scale and time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' and on the global 21-cm signal by SARAS [46–49],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' EDGES High Band [50–52] and LEDA [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' These upper limits are becoming increasingly constraining and have already being used to put limits on the properties of high-redshift luminous objects [44–49, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The ongoing and upcoming experiments including SARAS [55], MIST [56], REACH [57], LOFAR [58], NenuFAR [59], arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='03298v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='CO] 9 Jan 2023 2 HERA [60], the SKA and future proposed space based missions such as DAPPER and FARSIDE [61] aim to further improve our understanding of the infant Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Existing upper limits on the 21-cm signal provide the first very weak constraints on the astrophysical sources at a broad range of redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' we take the current tightest publicly available upper limits to date from the HERA interferometer on the magnitude of the 21-cm power spectrum at redshifts z ≈ 8 and ≈ 10 which provides a window to the EoR when ultraviolet photons emitted by first massive galaxies efficiently ionize the neutral hydrogen gas [44],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' and the SARAS3 radiometer [55] that probes the sky-averaged 21-cm emission at higher redshifts between z ≈ 15 − 25 when the first stars and X-ray emitting objects are expected to have formed in small galaxies during the Cosmic Dawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We combine these two data sets for the first time to improve constraints on the properties of the first galaxies and the state of the neutral gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We develop the methods to perform the joint analysis using a machine learning enhanced Bayesian workflow and pave the way for future discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We find that when considered together, these two experiments provide a better leverage on theoretical scenarios that bridge across the wide redshift range compared to the constraints from each individual experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In synergy, the two experiments leave only 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='1% of the explored broad theoretical parameter space to be consistent with the joint data set in comparison to 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='1% for SARAS3 and 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='2% for HERA alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We use the joint analysis to limit star formation efficiency, minimum halo mass for star formation, X-ray luminosity of early emitters and the radio luminosity of early galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The joint analysis disfavours at 68% confidence a combination of galaxies with X-ray emission that is ≲ 33 and radio emission that is ≳ 32 times as efficient as present day galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In Methodology we review the data and the specifics of the experiments incorporated in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This is followed by a discussion about the synergies between the power spectrum and sky-averaged 21-cm experiments in the same section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We present the implications of our work for the astrophysical constraints and the validity of the EDGES absorption feature as a sky-averaged 21-cm signal in Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This is followed by a summary in Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Additional information about the methodology and additional results can be found in Supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' METHODOLOGY The SARAS3 radiometer experiment took 15 hours of data, integrated in the frequency range 55 − 85 MHz, from a lake in Southern India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' After corrections were made for the receiver noise temperature, radio frequency interference and emission from the water beneath the antenna, the data was appropriately scaled by the total efficiency of the system to produce a measurement of the average sky temperature which is expected to include the Galactic and extra-galactic foregrounds as well as the 21-cm signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' SARAS3 reported a null detection of the EDGES absorption feature with a confidence of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='3% [55] and more recently the data has been used to place constraints on the properties of the first galaxies which we discuss in more detail below [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' SARAS2, the previous iteration of the instrumentation recorded data at higher frequencies (lower redshifts) of 110 − 200 MHz and was found to contain a non-smooth systematic structure possibly caused by emission from the ground that the antenna was placed on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The data has been used to derive constraints on galaxies in the infant Universe initially by fitting the foreground and systematic structure together with high order polynomials [46, 47] and subsequently by modelling the two components separately [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In Supplementary materials we combine constraints from the latter with constraints from SARAS3 and HERA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' To date, interferometric experiments have only observed upper limits of the cosmological 21-cm power spectrum, which still allow for a large range of realistic astrophysical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The tightest constraints are derived from the data from the HERA interferometer [43], followed by MWA in the redshift range z = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='7 [38] and LOFAR, which provide the tightest upper limits at redshifts z ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='1 [40] and z ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='3 − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='6 [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The HERA telescope is a radio interferometer located in the Karoo Desert of South Africa [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Already, this first public data release delivered the strongest constraints on the 21-cm power spectrum to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The publicly available HERA data[62] from the analysis of Internal Data Release 2 that we use in this paper is based on 18 nights of observations, with 39 antennas operating at science quality level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' For our analysis, we employ the publicly available spherically averaged power spectra derived from this data [43], in the wavenumber range k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='128 hMpc−1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='960 hMpc−1 and from the two bands focusing on redshifts z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In Supplementary materials we consider the implications of including the upper limits on the power spectrum from MWA and LOFAR in our joint analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In 21-cm cosmology, data analysis efforts are increasingly employing Bayes theorem P(θ|D, M) = L(θ)π(θ) Z , (1) to derive constraints on the astrophysical scenarios of the early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Here the likelihood, L(θ), represents the 3 probability that we observe the data, D, from SARAS3 or HERA, given a particular model, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The prior, π(θ), represents our assumed knowledge before we consider any data and the evidence, Z, is a normalisation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The posterior, P(θ|D, M), tells us which parts of the parameter space, θ, given the data and chosen model, are more probable than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The evaluation of Bayes theorem is usually performed with Nested Sampling or Markov Chain Monte Carlo (MCMC) algorithms (see Supplementary materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In many 21-cm experiments, θ is composed of parameters that describe instrumental effects, θI, foregrounds, θfg, and the astrophysical processes that influence the 21-cm signal, θ21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We typically refer to the set of θI and θfg as the nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Since we are only interested in the 21-cm signal, we work with the marginal or nuisance-free posteriors and nuisance-free likelihoods, L(θ21), which can be estimated using normalising flows and the marginal Bayesian statistics code margarine [63, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We give an overview of this in Supplementary materials however we note that it allows for efficient combination of the HERA and SARAS3 constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We detail the astrophysical processes included in the modelling, and the definition of θ21, next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In order to realistically model the range of time covered by the Cosmic Dawn and Epoch of Reionization, we need a consistent modelling of the cosmological and astrophysical processes from redshift 60, when star formation might have began, all the way to redshift 5 at the end of the EoR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' To that end we employ semi-numerical simulations [18, 65–68] that have previously been used in the HERA [44], SARAS2 [48] and SARAS3 [49] analyses as well as similar analysis of the LOFAR [54] and EDGES High-Band [52] data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We model the three-dimensional 21-cm field as a function of cosmic time during the infant Universe taking into account important astrophysical process including the Wouthuysen-Field (WF) effect [68–70], heating of the intergalactic medium by X-ray [67], Ly-α [18] and CMB photons [35], multiple scattering of Ly-α, relative velocity between dark matter and gas [65], feedback of Lyman-Werner radiation on star formation [66], and radio emission from galaxies [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The key parameters in the astrophysical model are: the star formation efficiency, f∗, which quantifies the percentage of the baryonic mass in the star forming halos that is converted into stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' the minimum virial circular velocity, Vc, which is proportional to the cube root of the halo mass, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' the X-ray production efficiency, fX, which is directly proportional to the X-ray luminosity per star formation rate, LX/SFR measured in erg s−1 M−1 ⊙ yr, between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='2 and 95 keV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' the CMB optical depth, τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' finally, we model the contribution of high redshift radio-luminous galaxies to the radio background by specifying a radio production efficiency, fradio, which is proportional to the radio luminosity per star formation rate, Lr, measured in W Hz−1 M−1 ⊙ yr at 150 MHz, and normalized such that it has a value of one for the present day population of radio galaxies [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The X-ray spectral energy density is modelled based on a population of X-ray binaries as in [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In our Bayesian analysis, θ21, therefore, comprises the set of parameters {f∗, Vc, fX, τ, fradio} or equivalently {f∗, M, LX/SFR, τ, Lr/SFR}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We explore wide prior ranges on all the parameters in an attempt to let the data inform us about the high-redshift astrophysical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Specifically, the model for radio-luminous galaxies that we employ here is not conventionally considered in 21-cm cosmology where typically the CMB is assumed to be the only source of radio background photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Here we expect that early galaxies will contribute to the radio background, thus increasing the amplitude of both the sky-averaged 21-cm signal [72] and the power spectrum [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Both the sky-averaged 21-cm signal and the power spectrum rely on the same underlying physics, and constraints from experiments targeting the different probes can be effectively combined to improve our knowledge of the infant Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The semi-numerical simulations take of order a few hours to produce a signal per parameter set, which is impractical for nested sampling or MCMC runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We therefore train neural networks on outputs of the detailed simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Typically, these networks take of order a few tens of milliseconds to evaluate meaning they are much more well suited for computationally intensive fitting algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The specific emulators used in this analysis and more details regarding the signal modelling can be found in Supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' RESULTS Although the constraints presented here are weak, through the novelty of combining the previously reported HERA and SARAS3 constraints we produce the tightest constraints to date on the properties of the infant Universe as detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This is the first time a joint analysis between a global signal data and interferometric limits has been attempted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' To visualize the importance of combining the constraining power of HERA and SARAS3 we show, in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 1, constraints on the ratio of the spin temperature of neutral hydrogen and background radiation temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The background radiation temperature is the sum of the CMB temperature and radio background from galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The ratio determines the maximum absorption of the sky-averaged 21-cm signal, the smaller the ratio the larger the signal can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The SARAS3 limits on the 21-cm signal (grey markers) correspond to lower limits on Ts/Tr, with the corresponding 1 and 2 σ contours shown as lines and extrapolated out of the SARAS3 band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Our simulations provide 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Key results from the joint analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Top Panel: The ratio of the spin temperature of neutral hydrogen, Ts, and the radio background temperature, Tr, as a function of redshift for the joint HERA and SARAS3 analysis in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We show the HERA and SARAS3 68% and 95% confidence constraints in blue and grey respectively as triangles at the relevant redshifts and solid and dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' As a guideline, we show the ratio for Tr = TCMB and assuming adiabatically cooled gas in an expanding Universe in the absence of any heating but with saturated coupling between Ts and the gas kinetic temperature (dashed black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bottom Left: The 2D PDF from the joint analysis on the minimum virial circular velocity, Vc, in combination with the star formation efficiency, f∗, marginalising over fradio, fX and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The solid black line shows the 68% contour, approximated by the pink dashed line, and the black dashed line shows the 95% contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The joint analysis disfavours low values of Vc and high f∗ corresponding to efficient star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bottom right: The constraint on the X-ray and radio luminosities from the joint analysis marginalising over Vc, f∗ and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The joint analysis disfavours at 68% confidence low X-ray efficiencies in combination with high radio production efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' a natural link between the power spectra and the global quantities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Ts/Tr, meaning that we can use the limits on the power spectrum from HERA to derive an equivalent constraint on Ts/Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' These constraints are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 1 (blue markers and lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The joint constraint, as shown by the green contours, provides the strongest constraints on the ratio, and in particular at redshifts z = 15 − 20, gives better constraints than either experiment alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Further, the combination of the two experimental data sets improves the constraints at intermediate redshifts over a pure extrapolation of each one of the sets of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' As a guideline, the dashed black line in the figure shows the ratio for Tr = TCMB, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' in the absence of radio emission from galaxies, and assuming adiabatically cooled gas in an expanding Universe in the absence of any astrophysical heating sources but with saturated coupling between the 21-cm spin temperature and the gas kinetic temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This limit is often used in the literature to give context to the constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [44, 45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' However, we note that for the models tested here the gas does not cool adiabatically because of the CMB and Ly-α heating at early times, and we have an excess radio background above the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Next, we explore the functional constraints in the T21−z and ∆21−z planes as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' These are calculated by taking the samples of θ21 output from our fits and using the neural network emulators to produce corresponding global signals and power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Using our theoretical models we can easily map between the constraints on the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 log(T/Tr) SARAS3 + HERA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 HERA SARAS3 HERA Band 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 SARAS3 Band 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 10 15 20 25 30 Z 26 log(Lr /SFR) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 22 2 38 40 42 log(f*) log(Lx/SFR)5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Functional constraints on T21 and ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The functional prior (purple), SARAS3 (grey), HERA (blue) and joint (green) posteriors for the sky-averaged 21-cm signal (top row) and power spectrum (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The yellow shaded region shows the SARAS3 band and the dashed yellow lines show the HERA redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The functional prior and posteriors are calculated by taking representative samples from the corresponding probability distributions for the astrophysical parameters and generating the corresponding signals using neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We see that by combining the constraining power of HERA and SARAS3, we significantly reduce the 3σ (lightest shaded regions) constraints on the magnitude of both signals from −2630 mK to −1770 mK at z = 15 for the global signal and 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='7 mK2 to 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='2 mK2 for the power spectrum at z = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The figure is produced with fgivenx [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' power spectra and global signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We see that for both the power spectra and the sky-averaged signals, although more clearly for the latter, the range of plausible models is reduced by our joint analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The signals that are inconsistent with the data typically have strong power spectra and corresponding deep absorption trough in the global signal, and belong to scenarios with weak X-ray heating and strong radio luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Further, we see that the 2σ region for the functional constraints correspond to signals with the power spectra ≲ 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='1 mK2 at z = 25, which via our modelling maps to global 21-cm signals shallower than −277 mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We note that the 3σ limit on the power spectrum at z = 25 of ≲ 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='2 mK2 is approximately equivalent to the expected sensitivity of NenuFAR from 1000 hours of observations [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' For the global 21-cm signal 3σ limit on the magnitude reduces from ≈ −2630 mK from HERA to ≈ −1770 mK from the joint analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Remarkably, we find that for the sky-averaged signal, the 2σ limit is very close to the minimum depth of the ‘standard astrophysical’ models, ≈ −165 mK [18], where the radio background is equated to the CMB, the contributions from radio galaxies and X-ray heating sources are assumed to be negligible, while the CMB and Ly-α heating are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The relatively tight constraints on the global signal and power spectrum suggest that future improved measurements will allow us to dig deeper into the models with weak excess radio background radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' It is clear from our analysis, that the joint constraint improves limits on the range of plausible global signals and power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We now interpret the constraints on the temperatures in terms of constraints on the properties of the first galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The key constraints from the joint analysis are shown in the bottom panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 1 including the limits on the high- redshift star formation which drives the high-redshift portion of the 21-cm signal via the process of Ly-α coupling (constraints in the planes Vc − f∗) and the constraints on the luminosity of X-ray and radio sources (Lr − LX) which primarily regulates the depth of the absorption trough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The full marginalised 1D and 2D posteriors corresponding to the joint analysis are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S1 and the key numerical results are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' I alongside the individual constraints from SARAS3 and HERA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We find that the combination of the two experiments leads to Prior SARAS3 HERA Joint 0 [mK] 2500 T21 0 0 SARAS3 L250 250 250 5000 HERA 10 20 30 10 20 30 10 20 30 6 [mK²] 4 2 10 20 30 10 20 30 10 20 30 10 20 30 Z Z Z Z 1g 2g 3g6 SARAS3 HERA SARAS3 + HERA Signal Sky-averaged Power Spectrum Both z ≈ 15 − 25 ≈ 8 & ≈ 10 ≈ 8, ≈ 10 & ≈ 15 − 25 Lr/SFR ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='55 × 1025 ≳ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='00 × 1024 ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='31 × 1024 LX/SFR – ≲ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='60 × 1039 ≲ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='71 × 1039 Lr/SFR & LX/SFR ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='00 × 1025 & ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='09 × 1042 ≳ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='00 × 1024 & ≲ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='60 × 1039 ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='16 × 1023 & ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='00 × 1042 M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='40 × 105 ≲ M ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='10 × 107 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='55 × 105 ≲ M ≲ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='04 × 106 f∗ ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='05 – ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='06 f∗ & M ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='03 & ≲ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='53 × 108 – ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='02 & ≲ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='50 × 107 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Key parameter constraints from SARAS3, HERA and the joint analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Here the SARAS3 and HERA limits are taken from the respective papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In the top two rows we show the type of signal targeted by each set of analysis and the corresponding redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The joint analysis produces improved constraints on the radio and X-ray backgrounds while retaining the constraining power of SARAS3 on the star formation properties of early galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Lr is measured in W Hz−1 M−1 ⊙ yr at a reference frequency of 150 MHz, LX in erg s−1 M−1 ⊙ yr, and is calculated by integrating X-ray spectral distribution of sources between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='2 and 95 keV assuming an X-ray SED consistent with that for X-ray binaries [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The halo mass, M, is measured in solar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' These constraints are derived from Kernel Density estimates (KDE) of the 1D and 2D posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' stronger constraints in the two-dimensional probability distribution of Lr − LX than either of the two experiments individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Where HERA constrains the population of high redshift radio-luminous galaxies to be ≲ 400 times brighter in the radio band than the current population, the combination of the data sets constrains the galaxies to be ≲ 300 times brighter when marginalising over the other parameters (Vc, f∗, fX and τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Similarly, HERA disfavours at 68% confidence galaxies with an X-ray luminosity ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='25 times the present day value in combination with the radio luminosity of galaxies in the early universe that is ≳ 400 times the present day value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The joint analysis provides a stronger constraint, ruling out scenarios where the X-ray luminosity is ≲ 33 times the present day value and the radio luminosity of the first galaxies is ≳ 32 times the present day value at 68% confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We find comparable constraints on f∗ and minimum mass of star forming halos, M, as was found with SARAS3 alone [49] when combining the data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Marginalising over the radio and X-ray luminosities, we disfavour at 68% confidence galaxies in which ≳ 2% of the gas is converted into stars and the minimum mass for star forming halos is ≲ 45 million solar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Further, we explore the structure of the global 21-cm signals which are consistent with the data from SARAS3, HERA and the two sets together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In Supplementary materials and Constraints on Phemenological Parameters, we see that when considered individually, the SARAS3 and HERA experiments allow for astrophysically motivated sky-averaged 21-cm signals that have a minimum temperature, location and width in agreement with the EDGES detection, while the joint analysis rules out models with a depth that is consistent with EDGES at greater than a 2σ significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The joint analysis has a preference for shallower (lower values of |Tmin|) and narrower signals with higher central frequencies, as can be seen in the corresponding 1D PDFs, which is driven largely by the HERA constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We also consider the impact of SARAS2 on our joint constraints as well as MWA and LOFAR in Supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The latter two data sets have no effect on our results and the inclusion of SARAS2 leads to weaker constraints on the star formation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We focus in the main text on SARAS3 and HERA, due to the uncertainty in the modelling of and presence of systematic structures in the SARAS2 data which is discussed in more detail in Supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' CONCLUSION Through a combination of constraints on fluctuations and the sky-averaged 21-cm signal of neutral hydrogen, we have improved our understanding of the first galaxies that formed in the infant Universe between 200 and 700 million years after the Big Bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This is the first time the data from the two different 21-cm probes have been combined 7 to derive constraints on the astrophysical properties of the early galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Even though the existing constraints are weak, we develop novel methodology and outline the approach which will become increasingly more useful as the next-generation experiments deliver stronger observational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Considering a wide space of plausible astrophysical models including high-redshift sources of ultraviolet, X-ray and radio photons which affect the 21-cm signal, we calculate corresponding sky-averaged spectra as well as the power spectra of fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Using an upper limit on the fluctuations from the HERA interferometer and non-detection of the global 21-cm signal by the SARAS3 radiometer, we find that only 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='1% of the explored theoretical parameter space is consistent with the joint SARAS3 and HERA constraint, which is a significant improvement over the individual values of 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='1% and 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='2% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Using the newly developed methodology we place the tightest constraints to date on the properties of cosmic gas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' such as the spin temperature of the 21-cm hydrogen line (closely related,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' but not equal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' to the gas kinetic temperature) and the radio background temperature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Tr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' as well as on the radio and X-ray luminosities of the first galaxies disfavouring at 68% confidence galaxies that are approximately 32 times more efficient radio emitters than present galaxies and simultaneously are less than 33 times bright in the X-ray band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This work reports an increased degree of confidence over a wider range of redshifts than previous works which typically extrapolate outside the redshifts targeted by individual experiments, while here we interpolated between the observations of SARAS3 at z = 15 − 25 and the HERA limits at lower redshifts z ∼ 8 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In this work we also considered the addition of interferometric data sets from MWA and LOFAR to our analysis, which led to a negligible improvement in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Similarly to HERA, these experiments probe the physics of the EoR covering a similar redshift range to HERA (between z ≈ 6 − 10), with the current HERA data providing the tightest constraints on our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Of the current global or sky-averaged 21-cm experiments only SARAS2, SARAS3 and EDGES were able to place limits on the astrophysics of the infant Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Our main focus is on the SARAS3 limits (although we also consider SARAS2), as there are concerns surrounding the cosmological nature of the signal reported in EDGES and a degree of uncertainty in the modelling of systematics in the SARAS2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We note that the SARAS2 data covers a similar redshift range as the HERA data and based on our analysis does not lead to a significant improvement in our constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The new methodology developed in this paper will allow for synergies between the upcoming observations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' of the power spectrum from cosmic dawn measured by the NenuFAR [59], HERA, LOFAR or the SKA [8], as well as the measurements of the global signal by the wide-band REACH experiment covering the redshift range z ≈ 7 − 28 [57], PRIZM [75], MIST [56] and missions to the moon [61] among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [1] H.' metadata={'source': 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Milvang-Jensen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Moneti, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Targett, The evolution of the galaxy UV luminosity function at redshifts z ˜8-15 from deep JWST and ground-based near-infrared imaging, arXiv e-prints , arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='12356 (2022), arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='12356 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='GA].' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Falcke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Ferrara, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Iliev, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Iocco, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Jeli´c, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Jensen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Joseph, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Labroupoulos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Meiksin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mesinger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Offringa, V.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Barkana, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' van Bemmel, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bernardi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bonaldi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Briggs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Jelic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Jones, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Lazio, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Maio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Majumdar, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mack, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mesinger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Morales, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Parsons, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Pen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Santos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Schneider, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Semelin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' de Souza, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Subrahmanyan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Takeuchi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} 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A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Fialkov, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Barkana, High-redshift radio galaxies: a potential new source of 21-cm fluctuations, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Soc.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bowman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bradley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Carilli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' DeBoer, M.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Stefan, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Walbrugh, Multiredshift Limits on the 21 cm Power Spectrum from PAPER, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 801, 51 (2015), arXiv:1408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='3389 [astro-ph.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 493, 4711 (2020), arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='02575 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [39] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='10885 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Patil, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Yatawatta, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' V.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Ciardi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Chapman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Gazagnes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Ghara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Ghosh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Giri, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Iliev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Jeli´c, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Kooistra, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mondal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Schaye, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Silva, Improved upper limits on the 21 cm signal power spectrum of neutral hydrogen at z ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='1 from LOFAR, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 493, 1662 (2020), arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='07196 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [42] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Gehlot, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mertens, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Koopmans, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Offringa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Shulevski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mevius, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Brentjens, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Kuiack, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Pandey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Rowlinson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Sardarabadi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Vedantham, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Wijers, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Yatawatta, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Zaroubi, The AARTFAAC Cosmic Explorer: observations of the 21-cm power spectrum in the EDGES absorption trough, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 499, 4158 (2020), https://academic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='oup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='com/mnras/article-pdf/499/3/4158/34068575/staa3093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [43] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Abdurashidova, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Aguirre, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Alexander, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Balfour, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Beardsley, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bernardi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Billings, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bowman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bradley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bull, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Burba, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Carey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Carilli, C.' 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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Dillon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Ely, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Ewall-Wice, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Fagnoni, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Fritz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Furlanetto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Gale-Sides, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Glendenning, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Gorthi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Greig, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Grobbelaar, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Halday, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Hazelton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Hewitt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Hickish, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Jacobs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Julius, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Kern, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Kerrigan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Kittiwisit, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Kohn, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Williams, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Zheng, and HERA Collaboration, First Results from HERA Phase I: Upper Limits on the Epoch of Reionization 21 cm Power Spectrum, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 925, 221 (2022), arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='02263 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [44] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Abdurashidova, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Aguirre, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='07282 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [45] The HERA Collaboration, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Abdurashidova, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Adams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' E.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Singh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Subrahmanyan, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Barkana, A comprehen- sive Bayesian reanalysis of the SARAS2 data from the epoch of reionization, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 513, 4507 (2022), arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='11531 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [49] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bevins, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' de Lera Acedo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Fialkov, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Handley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Singh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Subrahmanyan, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Barkana, Astrophysical constraints from the saras3 non-detection of the cosmic dawn sky-averaged 21-cm signal, Accepted for Nature Astronomy (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [50] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Monsalve, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Rogers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bowman, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mozdzen, Results from EDGES High-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Constraints on 10 Phenomenological Models for the Global 21 cm Signal, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 847, 64 (2017), arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='05817 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [51] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Monsalve, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Greig, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bowman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mesinger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Rogers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mozdzen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Kern, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mahesh, Results from EDGES High-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Constraints on Parameters of Early Galaxies, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 863, 11 (2018), arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='07774 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [52] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Monsalve, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Fialkov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bowman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Rogers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mozdzen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Cohen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Barkana, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mahesh, Results from EDGES High-Band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' New Constraints on Parameters of the Early Universe, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 875, 67 (2019), arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='10943 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [53] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bernardi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Zwart, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Price, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Schinzel, Bayesian constraints on the global 21-cm signal from the Cosmic Dawn, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 461, 2847 (2016), arXiv:1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='06006 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [54] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mondal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Fialkov, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Fling, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Iliev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Barkana, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Ciardi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mellema, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Zaroubi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Koopmans, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mertens, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Gehlot, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Ghara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Ghosh, S.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Anstey, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bevins, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Chiello, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Cumner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Josaitis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Roque, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Sims, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' H.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Grießmeier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Loh, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Tagger, The low-frequency radiotelescope nenufar, in 2018 2nd URSI Atlantic Radio Science Meeting (AT-RASC) (2018) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [60] D.' metadata={'source': 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J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bowman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Bradley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Carilli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Cheng, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' de Lera Acedo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Dillon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} 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S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Kohn, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Lekalake, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Liu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Loots, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' MacMahon, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Malan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} 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C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Burigana, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Butler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Calabrese, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Cardoso, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Carron, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Challinor, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Chiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Chluba, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Gerbino, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Ghosh, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Gonz´alez-Nuevo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' G´orski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Gratton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Gruppuso, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Gudmundsson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Hamann, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Handley, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Hansen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Herranz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Hildebrandt, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Hivon, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Huang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Jaffe, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Kisner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Knox, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Krachmalnicoff, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Kunz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Kurki-Suonio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Lagache, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Lamarre, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Lasenby, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Lattanzi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Lawrence, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Le Jeune, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Lemos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Lesgourgues, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Levrier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Lewis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Liguori, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mac´ıas-P´erez, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Maggio, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Maino, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mandolesi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mangilli, A.' 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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Rachen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Reinecke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Remazeilles, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Renzi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Rocha, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Rosset, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Roudier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Rubi˜no-Mart´ın, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Ruiz-Granados, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Salvati, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Sandri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Savelainen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Scott, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Shellard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Sirignano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Sirri, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Wehus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' White, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' White, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Zacchei, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Zonca, Planck 2018 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Cosmological parameters, A&A 641, A6 (2020), arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='06209 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [81] J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 483, 1980 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [82] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Mittal and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Kulkarni, Implications of the cosmological 21-cm absorption profile for high-redshift star formation and deep JWST surveys, Mon.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [83] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Handley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Hobson, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Lasenby, PolyChord: nested sampling for cosmology, MNRAS: Letters 450, L61 (2015), arXiv: 1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='01856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [84] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Handley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' P.' metadata={'source': 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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 453, 4385 (2015), arXiv: 1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='00171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' [85] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Foreman-Mackey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Hogg, D.' metadata={'source': 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Radio Background below 100 MHz, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 858, L9 (2018), arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='08581 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 12 ACKNOWLEDGMENTS HTJB acknowledges the support of the Science and Technology Facilities Council (STFC) through grant number ST/T505997/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' WJH and AF were supported by Royal Society University Research Fellowships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' EdLA was supported by the STFC through the Ernest Rutherford Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' RB acknowledges the support of the Israel Science Foundation (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 2359/20), The Ambrose Monell Foundation and the Institute for Advanced Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' AUTHOR CONTRIBUTION HTJB lead the data analysis and the writing of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' SH provided the power spectrum emulator, assisted with the writing and data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' IAC performed the combined analysis of HERA, LOFAR and MWA data and assisted with the writing of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' AF generated the idea, supervised the project and contributed to the writing of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' EdLA supervised the work and helped with the revision of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' WJH supervised the analysis, helped with revision of the article and provided advice on the Bayesian methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The astrophysical signal models were provided by AF and RB and the SARAS3 data was provided by SS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' RB and SS provided comments on the manuscript and assisted with revisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 13 SUPPLEMENTARY MATERIALS Modelling the Thermal History of the Infant Universe: Power Spectrum and Global Signal Synergies At high redshifts around z ≈ 20−30, the first stars begin to form and produce Ly-α photons that interacts with the baryonic matter, predominantly composed of neutral hydrogen, in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Neutral hydrogen atoms absorb and remit ambient Ly-α photons in a process known as the Wouthuysen-Field (WF) effect [69, 70] that drives the relative number of atoms with aligned and anti-aligned proton and electron spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This process couples the spin temperature of the neutral hydrogen, Ts, which describes the distribution of hydrogen atom spins, to the gas temperature, Tk, which is cooling at a faster rate than the radio background as the Universe expands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Further, interactions between the neutral hydrogen and Ly-α emission result in the transfer of kinetic energy that raises the gas temperature, and coupled spin temperature, in a process known as Ly-α heating [76, 77] preventing the gas from cooling adiabatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' However, despite the heating, the dominant WF effect produces an absorption feature in the sky-averaged 21-cm signal, which is measured relative to the radio background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Further, it leads to a peak in the power spectrum at high z ≈ 25 on angular scales corresponding to the effective horizon of the Ly-α emission and the distribution of galaxies, which disappears when the coupling becomes saturated [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The intensity and spatial fluctuations of the Ly-α emission evolve with the population of early galaxies and consequently it is dependent on their star formation rate and the minimum halo mass for star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We parameterize these quantities with the star formation efficiency, f∗, which quantifies the percentage of the baryonic mass in the star forming halos that is converted into stars, and the minimum virial circular velocity, Vc, which is proportional to the cube root of the halo mass, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' At intermediate redshifts of z ≈ 10 − 20 the gas is further heated by X-ray binaries [67, 71], continuing Ly-α heating, CMB heating [79] and heating through structure formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' X-ray heating is dependent on processes such as X-ray production in high redshift galaxies and black hole binary formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This means that it has to be separately parameterized and in our semi-numerical simulations, we model the X-ray production efficiency, fX, which is directly proportional to the X-ray luminosity per star formation rate, LX/SFR measured in erg s−1 M−1 ⊙ yr, between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='2 and 95 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The heating affects the redshift of the minimum and depth of the sky-averaged 21-cm signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' If sufficiently efficient it can raise the temperature of the gas, and coupled 21-cm brightness temperature, above the radio background resulting in emission in the sky-averaged signal at low redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The heating rate has a direct impact on how fast the brightness temperature transforms from absorption to 0 K or emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In the power spectrum, due to the non- uniformity of the heating, the various mechanisms can produce a peak in the signal at around z ≈ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Although this redshift is model dependent (see [78]) and in some cases when heating is done by hard X-rays (energies > 1 keV with long mean free paths) X-ray heating is smooth and no peak is imprinted in the power spectrum [67, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Finally, at more recent times, z ≈ 5 − 15, ultraviolet emission from the first massive galaxies begins to ionize the neutral hydrogen, stripping the abundant gas of its electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This reduces the sky-averaged 21-cm signal and when reionization is complete, the signal disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The process produces a peak in the power spectrum at the scales corresponding to the typical size of ionized bubbles, but again the signal is destroyed once reionization is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The process of reionization is highly dependent on the ionizing efficiency of sources, ζ, which in the models explored here is normalized by the CMB optical depth, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' While τ has been weakly constrained by cosmological experiments such as Planck [80], we treat it as a free parameter and note that 21-cm cosmology offers a means by which to break degeneracies between τ and other cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Throughout our analysis, we explore a broad range of radio luminosities that produce radio excesses above the CMB of between ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 − 270 times the CMB temperature at z = 20 and ≈ 1 − 32000 times the CMB at z = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' While moderate radio emission might be expected [81], the extreme values of radio production efficiencies are usually invoked to explain the anomalously deep EDGES signal [14, 33–36, 72, 81] but struggle to explain the rapid star formation and rapid heating of the gas [82] that is implied by the shape of the EDGES signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Combining Constraints with margarine We use the marginal Bayesian statistics code margarine [63] to combine constraints from different data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' margarine uses neural networks known as Masked Autoregressive Flows (MAFs) to model the probability distribu- tion, P(θ|D, M), of a set of samples, here θ21, marginalising over nuisance parameters describing the instrumental systematics, θI, and foregrounds, θfg in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' It does this by shifting and scaling a base standard normal distribution, making the probability of the distribution easily tractable, to replicate the target samples, where the shifting and scaling are determined by the outputs of the MAFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 14 This can subsequently be used to calculate a nuisance-free likelihood given a prior distribution and the Bayesian evidence using L(θ21) ≡ � L(θ21, α)π(θ21, α)dα � π(θ21, α)dα = P(θ21|D, M)Z π(θ21) , (2) where α = {θI, θfg} [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In instances when the prior is flat then the following is true up to a normalisation constant L(θ21) ≈ P(θ21|D, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' (3) With margarine we can subsequently evaluate log(L(θ21)) for any set of θ21 for any existing posterior distribution, P(θ|D, M), from previous analysis of a data set like HERA or SARAS3 θ = {θI, θfg, θ21} → {θ21} → margarine → log(P(θ21|D, M) → log(L(θ21)), (4) where the log is base 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The likelihood evaluations can then be combined, log(Ljoint(θ21)) = log(LHERA(θ21)) + log(LSARAS3(θ21)), (5) as discussed in the main text, to be sampled over using MCMC methods or in our case nested sampling implemented with polychord [83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' MCMC sampling methods approximate the unnormalised posterior distribution by directly sampling the product P(θ) ≈ L(θ)π(θ) and do not provide estimates of the evidence, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The family of algorithms typically use random walkers to traverse the parameter space, with points accepted and rejected based on some probabilistic criteria in an effort to explore the space fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The HERA analysis [44] of the excess radio background models explored here used the emcee [85] implementation of MCMC sampling [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Nested sampling [86] numerically approximates the integral Z = � L(θ)π(θ)δθ, (6) which can be derived from equation 1 and the requirement that the posterior must integrate to 1, by evolving a series of live points to higher and higher likelihood values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' By approximating the evidence, the algorithm produces samples on the normalised posterior distribution, which we subsequently use to determine preferred and disfavoured regions of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' A comprehensive review of nested sampling can be found in [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Emulating Signals from the EoR and CD As described in the main text, semi-numerical simulations typically have a runtime of multiple hours for a given set of astrophysical parameters, thus performing parameter inference with the simulation directly is very costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' However, as both the sky-averaged signal and the power spectrum change smoothly when we vary parameters, we can interpolate values at intermediate parameters from existing simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' An increasingly common practice to achieve this are the fast and precise neural network based emulators we discuss in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The sky-averaged 21-cm signal To emulate the sky-averaged 21-cm signal in models with an excess radio background from high redshift radio- luminous galaxies we use the publicly available emulator globalemu [88] trained on sets of astrophysical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The simulations include the effects of Ly-α and CMB heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The mean free path of ionizing photons, Rmfp, is fixed at 40 Mpc and the and X-ray heating is powered by a population of X-ray binaries with a realistic spectral energy distribution [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The models correspond to the parameterisation detailed above and in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The emulator has previously been used in the individual analysis of data from SARAS2 [48] and SARAS3 [49] and detailed in the corresponding papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We therefore only briefly summarise the accuracy of the neural network here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The original set of simulations comprise approximately 10,000 models however the explored range of τ, the CMB optical depth, is large and when training our neural network we filter out models that have a value of τ in the range given by Planck ±3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This results in a training set of approximately 4,300 models and the network is tested on approximately 500 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' A root mean squared error (RMSE) of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='11 mK is found and a 95 percentile RMSE of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='53 mK indicating a high level of accuracy (see Table M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='2 in [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 15 The 21-cm power spectrum For comparing models with the HERA measurements we use the same 21-cm power spectrum emulator used in the HERA analysis [44], based on the same suit of simulations of the sky-averaged signal emulator used in the SARAS2 and SARAS3 analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Based on an input of the 5 model parameters, this emulator returns the 21-cm power spectrum with a relative accuracy of 20% at the wave numbers and redshifts observed by HERA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' For HERA the impact of τ is more significant because the data corresponds to lower redshifts and so the full prior range on the parameter is used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This results in a training set of ∼8,000 simulations and another 2,000 independent samples for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Full details and tests of this emulator can be found in the HERA analysis paper [44], Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We note that when performing our joint analysis we use the narrow prior on τ defined by Planck, since the SARAS3 posterior is not defined in the original analysis for values outside this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Temperatures To emulate physical properties such as the spin temperature and temperature of the radio background (as seen in Figure 1) we use a globalemu-style emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In this framework, the emulator takes in the five astrophysical pa- rameters and a single redshift and returns a single corresponding temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In practice, this means that vectorised calls have to be made to emulate the spin temperature, Ts(z), and the radio background temperature, Tr(z), as a function of redshift, but the method is found to be more accurate, quicker and allows for interpolation at a range of different redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We use the same training and test sets as used for the 21-cm power spectrum emulator, and a similar architecture of 4 layers, with a reduced size of 100, 30, 10, and 5 nodes per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This allows us to emulate the spin temperature Ts within ±6% accuracy (95% confidence interval), and the radio background temperature Tr within ±4% accuracy (95% confidence interval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Constraints on the Astrophysical Parameters Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S1 shows the 1D and 2D marginal posteriors found for the joint analysis of SARAS3 and HERA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The graph shows that the combined constraining power of the two experiments leads to a strong constraint on the combination of the radio and X-ray luminosities per star formation rate for an early population of galaxies when marginalising over the other astrophysical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We also see constraints in the plane of Vc − f∗ when marginalizing over the radio background and X-ray heating params.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The constraints are summarized in detail in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Constraints on Phemenological Parameters In order to explore the structure of the global 21-cm signal we look at the minimum temperature, Tmin, the corresponding central redshift, z0, and an approximate full width at half max, ∆z, of each absorption trough as defined in the top right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' More specifically, for each parameter set θ21 (either from the prior parameter range, or from the posterior ranges consistent with SARAS3, HERA, and the joint analysis) we generate a global signal using the neural network emulator globalemu [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We then measure the values of Tmin, z0, and ∆z for each signal producing probability distributions for each parameter corresponding to the constraints from each data and showing the extent of the prior (see results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We compare these distributions with the values used to parameterise the phemenological EDGES flattened Gaussian signal with the EDGES 99% confidence ranges shown by the black crosses in the corner plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Constraints on the Radio and X-ray Backgrounds The product f∗fradio is proportional to the total radio background created by radio-luminous galaxies, and, equiv- alently, f∗fX is a proxy for the total X-ray background created by the early population of X-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This is demonstrably true for our model parameterisation as the star formation rate is proportional to f∗ and the radio lumi- nosity and X-ray luminosities per star formation rate are proportional to fradio and fX respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Both X-ray and radio backgrounds are responsible for regulating the depth of the absorption feature, and they can also be observed independently by other telescopes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' observations of the unresolved X-ray background by Chandra [89] and of the 16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Parameter constraints from the joint anlaysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The astrophysical parameter constraints on models with excess radio background in addition to the CMB derived when combining an upper limit on the 21-cm power spectrum at z ≈ 8 and ≈ 10 from HERA with data from the 21-cm sky-averaged experiment SARAS3 in the band z ≈ 15 − 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Through the combination of these two data sets probing different statistical properties of the 21-cm signal at different redshifts we are able to improve constraints on the radio and X-ray luminosities of early radio-luminous galaxies and maintain constraints provided by SARAS3 on the star formation properties of these early galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Lr is measured in units of W Hz−1 M−1 ⊙ yr at 150 MHz and LX is in units of erg s−1 M−1 ⊙ yr calculated between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='2 and 95 keV assuming a realistic SED of an early X-ray binary population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The pink dashed lines approximate regions that are disfavoured with 68% confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' low-frequency radio background by ARCADE2/LWA [90, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S3 we show constraints on the values of f∗ fX and f∗ fradio achieved by SARAS3, HERA and the joint analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Since these combinations of the parameters regu- late the absorption depth of the global 21-cm signal, we can also condition our prior on the astrophysical parameters to produce signals with the same central frequency and depth as the absorption feature found in the EDGES data [18, 35, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In each panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S3, we show black contours corresponding to these EDGES-like physical signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We see that while HERA and SARAS3 allow for combinations of fX f∗ and fradio f∗ that could partially explain EDGES, the combination of the two experiments, which produces a tighter constraint on the X-ray and radio luminosities of early galaxies, disfavours a large portion of the EDGES-like parameter space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' most of the EDGES-like parameter space is beyond the 95% contours of the joint constraints, while it is well within the 95% contours for SARAS3 and HERA individually).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This demonstrates further the power of combining different data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' However, we note that the explored theoretical signals do not fit EDGES data well as none of them closely reproduces the flattened Gaussian-like feature found in the data [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 95% Confidence 68% Confidence 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 log(PR) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='050 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 2 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='075 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 4 T log( FR log(sFR)17 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Phenomenological constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The triangular plots shows the prior (purple) and posteriors (grey for SARAS3, blue for HERA, green for joint) of the features of a typical global absorption signal: the central redshift, z0, the corresponding minimum temperature, Tmin, and the width of the signal, ∆z, as is depicted in the top right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Darker shaded regions show 1σ constraints, lighter shaded regions show 2σ constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Overlaid on the posterior distributions are the 99% confidence intervals, black crosses, reported for the corresponding phemenological parameterisation of the EDGES absorption feature in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Note that this is not the same as the physical EDGES-like distribution explored in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We see that individually the experiments allow for signals with depths that are consistent with EDGES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' However, the combination of the two data sets disfavours these signals with greater than 2σ significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We do not disfavour signals with the same width or central frequency as EDGES, but note that the joint analysis indicates a preference for shallower and narrower signals with higher central redshifts as can be seen in the 1D PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The impact of SARAS2 Previous analysis of the SARAS2 data revealed some weak constraints, most notably in the plane of LX − Lr in agreement with HERA and SARAS3, on the properties of galaxies in the infant Universe [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' SARAS2 is at much lower redshifts than SARAS3 but overlaps with the redshifts probed by HERA having recorded observations in the band z ≈ 7 − 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The data is contaminated by a sinusoidal systematic and a number of different models were fitted to this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The model corresponds to a signal introduced prior to the antenna possibly from ground emission or some unknown component of the foreground and separately a signal introduced in the system electronics potentially from cable reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The sinusoidal systematic was fitted alongside a signal model generated with globalemu and a foreground model that is conditioned to be smooth preventing it fitting out non-smooth systematics or signals in the data [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Here we take the best fitting model, with a systematic from ground emission or a non- smooth component from the foreground, with the highest evidence from the original analysis [48] and combine the corresponding constraints on the astrophysical parameters Vc, f∗, fX, fradio and τ with the joint constraints from HERA and SARAS3 to assess the impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S4, the addition of SARAS2 to our analysis washes out the constraint in the plane f∗ − Vc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Physical EDGES 0 Az K m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 250 (Z0, Tmin) 500 (Z0, Tmin) 10 20 30 10 20 30 之 之 Prior [mK] SARAS3 500 HERA SARAS3+HERA 1000 16 8 8 16 24 1000 500 8 16 Tmin [mK] Az 2018 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Background constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The figure shows constraints on the radio and X-ray backgrounds, parameterized by f∗fradio and f∗fX respectively, from SARAS3 (grey), HERA (blue) and the joint analysis (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The SARAS3 and HERA posteriors are based on the results presented in [49] and [44] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We show each distribution individually on the top row, overlaid pairs of distributions for comparison in the middle, and all three on the same figure in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' In all panels, we show 68% and 95% contours (black solid and dashed lines respectively) for physical signal models that have similar depths and central frequencies as the EDGES absorption feature as defined by the inequality in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' These physical EDGES-like models have previously been explored in the literature in [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We note that while individually both HERA and SARAS3 allow for astrophysically motivated signal models that could explain the depth of the EDGES feature, together they rule the corresponding parameter space out with approximately greater than 2σ confidence, although some EDGES-like signals are still viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We stress again, that the explored physical models cannot fully explain the shape of the EDGES signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' One possible explanation for this is that the addition of SARAS2, while constraining the properties that affect the signal at low redshifts, increases the envelope of possible models at higher redshifts, where star formation is more important, that are plausible even given the constraints from the SARAS3 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Despite this, we note that we maintain the constraint in the plane LX − Lr when we add SARAS2 into our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We can quantify the impact of SARAS2 on our analysis by looking at the percentage of the astrophysical prior volume which is consistent with the different combinations of the three different data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' To calculate this percentage, we use margarine to calculate the marginal Kullback-Liebler divergence, D, between the flat prior on the five astrophysical parameters in the set θ21 and the corresponding posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The KL divergence is related to the percentage via % = 100 × exp(−D) ≈ 100 × VP Vπ , (7) Individual Constraints SARAS3 HERA Joint 104 radio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 103 f 102 101 10-2 100 10-2 100 10-2 100 fxf* fxf* fxf* Comparison of Pairs SARAS3 vs HERA SARAS3 vs Joint HERA vs Joint 104 103 102 101 100 10-2 100 10-2 100 10-2 fxf* fxf* fxf* Comparison with Joint SARAS3 vs HERA vs Joint SARAS3 104 HERA Joint 103 EDGES-like Signals 102 101 10-2 100 fxf*19 HERA + SARAS3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 log(Vc) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 log( LX SFR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='075 τ −2 −1 log(f∗) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 log( Lr SFR) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 log(Vc) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 log( LX SFR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='075 τ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 log( Lr SFR) HERA + SARAS3 + SARAS2 95% Confidence 68% Confidence 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='075 −2 −1 log(f∗) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 log(Vc) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 log( LX SFR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='075 τ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='0 log( Lr SFR) 0 max FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The impact of SARAS2 data on the astrophysical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We show the joint posterior distributions for HERA and SARAS3 on the left panel (identical to Figure S1, but shown here for comparison) and for HERA, SARAS3 and SARAS2 on the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' SARAS2 covers the band z ≈ 7 − 12 and therefore has some overlap with HERA but not with SARAS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The addition of SARAS2 to the joint analysis washes out the constraint on star formation properties, Vc and f∗, because it leads to increased uncertainty in the structure of the signals at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' However, we still see a consistent disfavouring of a population of radio galaxies with high radio and low X-ray luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The one dimensional posteriors for τ appear to be in disagreement, however, we note that these are basically flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We exclude SARAS2 from our main results in the text because of uncertainty in the modelling of systematics in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' where Vπ is the prior volume and VP is the posterior volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This quantity is useful as it quantifies the constraining power of the different data sets in all five dimensions, including correlations that may not be visible in the one and two dimensional projections used to produce the corner plots in this paper and in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S5 the percentage of the astrophysical parameter prior volume that is consistent with different combinations of the data sets discussed in this work (including additional interferometric measurements of the power spectrum discussed in Other Power Spectrum Experiments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We see that the combination of either or both of the SARAS data sets with HERA lead to a percentage consistency with the data of ≈ 63 − 65% and this is likely dominated by HERA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Individually, HERA allows for ≈ 80% of the astrophysical parameter space, SARAS2 for ≈ 90% and SARAS3 for ≈ 92%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Due to the uncertainty in the modelling of the systematics in the SARAS2 analysis, we leave SARAS2 out of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Other Power Spectrum Experiments In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S6, we show the projected posteriors derived using HERA data alone (left panel) and HERA, MWA and LOFAR together (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We note that the constraints from the different interferometers are all at low redshifts between z ≈ 6 − 10 and varying wavenumbers or angular scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' These are detailed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S1 along with the constraints from the individual experiments on key parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We derived parameter constraints from the MWA and LOFAR data using the approach taken in the orginal HERA analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Specifically, we take the measured upper limits, the mean power spectrum and uncertainty, and treat it as a measurement of cosmological signal plus systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' As in HERA [44] we take this uncertainty to be Gaussian and marginalize over a uniform prior on the systematics, yielding the likelihood 20 60 65 70 75 80 85 90 95 100 % Prior Consistent with Data SARAS2 + HERA SARAS2 + SARAS3 + HERA SARAS3 + HERA SARAS3 + SARAS2 HERA + MWA + LOFAR HERA SARAS2 SARAS3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Constraining power of different data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The percentage of the wide astrophysical parameter prior that is found to be consistent with the different data sets and different combinations of data sets explored in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' A lower value indicates a better set of constraints, although a difference of a few percent does not necessarily translate into significant differences in the parameter constraints as can be seen when comparing the results from HERA and HERA + LOFAR + MWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' HERA LOFAR MWA LOFAR + MWA + HERA z ≈ 8 & ≈ 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='1 & 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='3 − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='7 Discrete and contin- uous ranges of z be- tween 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='6 k [h Mpc−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='128 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='075−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='432 & 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='070 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='000 Discrete and continu- ous ranges of k Lr/SFR ≳ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='00 × 1024 ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='20 × 1025 ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='58 × 1025 ≳ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='00 × 1024 LX/SFR ≲ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='60 × 1039 ≲ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='70 × 1038 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='16 × 1039 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='58 × 1040 Lr/SFR & LX/SFR ≳ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='00 × 1024 & ≲ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='60 × 1039 ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='16 × 1025 & ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='00 × 1040 ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='00 × 1025 & ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='00 × 1040 ≳ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='00 × 1024 & ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='58 × 1040 TABLE S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Constraints from interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The table shows the various constraints on the radio and X-ray luminosities for HERA, MWA, LOFAR and the combination of all three along with their respective wavenumbers and redshift ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The joint analysis only marginally improves our understanding of the infant universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' L(θ21) = Nd � i 1 2 � 1 − erf �di − mi(θ21) √2σi �� , (8) where Nd represents the number of data points, di and σi correspond to the mean and standard deviation in each data point, and mi(θ21) is the model prediction for that redshift and wave number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Thus a model prediction m ≫ d gives L ≈ 0 while m ≪ d gives a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' This likelihood is effectively a step function that disfavours models above a given amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' A full discussion of its derivation can be found in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We performed this joint analysis using a full analytic likelihood approach (independent of margarine) since there are no nuisance parameters describing the systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We find that each of the experiments disfavours individually similar regions of the Lr − LX plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' However, the joint analysis does not improve the results derived from HERA data alone (as we summarise in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S1 and is further illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S5) which motivates our decision to only use HERA in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' 21 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' The impact of MWA and LOFAR on the parameter constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Projected posterior distribution functions (PDFs) for the 5 simulation parameters, obtained by assuming flat priors and combining different observations: HERA alone (left, as in [44]) and LOFAR, HERA and MWA (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' Solid (dashed) lines correspond to regions containing the highest 68% (95%) probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' We see that HERA constraints are not significantly improved by adding the published limits on the power spectrum from other interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content=' HERA HERA + MWA + LOFAR 95% Confidence 68% Confidence 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE1T4oBgHgl3EQfmgRl/content/2301.03298v1.pdf'} +page_content='5 0 1.' metadata={'source': 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Mathematical Sciences Center, Tsinghua University +Haidian District, Beijing, 100084, China +4Theoretical Physics Group, The Blackett Laboratory, Imperial College London +Prince Consort Road, London, SW7 2AZ, UK +5Mathematical Institute, University of Oxford, +Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG, UK +E-mail: snawata@gmail.com, msperling@seu.edu.cn, yukawahaow@gmail.com, +zhenghao.zhong@maths.ox.ac.uk +Abstract: The study of 3d mirror symmetry has greatly enhanced our understanding of +various aspects of 3d N = 4 theories. In this paper, starting with known mirror pairs of 3d +N = 4 quiver gauge theories and gauging discrete subgroups of the flavour or topological +symmetry, we construct new mirror pairs with non-trivial 1-form symmetry. By providing +explicit quiver descriptions of these theories, we thoroughly specify their symmetries (0- +form, 1-form, and 2-group) and the mirror maps between them. +arXiv:2301.02409v1 [hep-th] 6 Jan 2023 + +Contents +1 +Introduction +2 +2 +Gauging discrete 0-form symmetries +3 +2.1 +Abelian theories +3 +2.2 +An illustrative example +9 +2.3 +T[SU(N)] theories +11 +2.4 +T σ +ρ [SU(N)] theories +16 +2.5 +T[SO(2N)] theories +22 +2.6 +Sp(k) SQCD and its orthosymplectic mirror +24 +2.7 +Sp(k) SQCD and its unitary D-type mirror quiver +25 +2.8 +Examples of non-simply laced unitary quivers and their mirrors +28 +2.9 +Magnetic quivers and gauging discrete topological symmetries +32 +2.10 Examples from 5d magnetic quivers +34 +3 +Discussion and conclusions +36 +A Notations and conventions +38 +A.1 Hilbert series +39 +A.2 Superconformal index +40 +A.3 Centre symmetries of classical Lie algebras +41 +B Discrete gauging of T[SU(N)] +41 +B.1 +Gauging discrete subgroup of topological symmetry +42 +B.2 +Gauging discrete subgroup of topological symmetry revisited +45 +C Mirror maps +49 +C.1 SQED and its abelian mirror quiver +50 +C.2 +T[SU(N)] theories +51 +C.3 Examples for T σ +ρ [SU(N)] +53 +C.4 Sp(k) SQCD and its D-type unitary mirror quiver +54 +C.5 O(2k) SQCD and its C-type unitary mirror quiver +56 +D Explicit Hilbert series results +56 +D.1 Linear Abelian quiver +57 +D.2 T[SU(N)] theories +57 +D.3 Some T[SU(N)] examples with higher charges +64 +D.4 Some T σ +ρ [SU(N)] examples +65 +D.5 T[SO(2N)] theories +67 +D.6 Sp(k) SQCD and orthosymplectic mirrors +69 +D.7 Sp(k) SQCD and D-type mirrors +71 +D.8 O(2k) SQCD and C-type mirrors +77 +– 1 – + +1 +Introduction +Supersymmetric theories with 8 supercharges in space-time dimension 3 exhibit a rich set +of intriguing features; One of the most prominent is 3d mirror symmetry [1]. Given a 3d +theory that has a mirror dual theory, 3d mirror symmetry exchanges Coulomb branch and +Higgs branch. In particular, this also implies the exchange of flavour symmetries Gf (Higgs +branch isometries) and the topological symmetries Gt (Coulomb branch isometries). +The notion of symmetries has been generalised to include novel types beyond the stan- +dard symmetries of local operators [2]. Among others, these include higher-form symme- +tries. Specifically for 3d theories, discrete 1-form symmetries can be generated by gauging +discrete 0-form symmetries. The structure of generalised symmetries in 3d supersymmet- +ric theories has been the focus of recent research, including [3–14] and references therein1. +Given the vast catalogue of 3d mirror pairs with trivial 1-form symmetry, one might wonder +what mirror symmetry implies for 3d theories with 1-form symmetry. +In this paper, we start with a known mirror pair (T , T ∨) of 3d N = 4 theories that +admit UV quiver descriptions, and gauge a discrete Γ[0] subgroup of the 0-form symmetry +to generate new theories with Γ[1] 1-form symmetry. Depending on whether Γ ≡ Γ[0] is a +subgroup of the flavour or topological symmetry, the resulting mirror pair (T /Γ, (T /Γ)∨) +changes. For Γ ≡ Γf ⊂ Gf, the field theory description of T /Γf is straightforward, but for +Γ ≡ Γt ⊂ G∨ +t , the description of T ∨/Γt is less transparent. In this paper, explicit quiver +descriptions are provided for these cases, as well as the global form of the 0-form symmetries +of (T /Γ, (T /Γ)∨). It is known that the resulting 0-form and the newly introduced discrete +1-form symmetry may not just be a direct product, but can form an extension, called +2-group symmetry [9, 17–21]. We comment on such extensions throughout this work. +The remainder of this paper is organised as follows: in Section 2, we consider known +mirror pairs and gauge discrete 0-form symmetries to generate mirror pairs with non- +trivial 1-form symmetry. We first study abelian theories, followed by non-abelian T[SU(N)] +and T σ +ρ [SU(N)] theories with non-abelian product gauge groups � +i U(Ni). This class of +examples has the benefit that all 0-form symmetries are manifest in the UV description. +Thereafter, SO(k) and Sp(k) gauge groups are considered by studying T[SO(2N)] theories, +Sp(k) SQCD, and linear orthosymplectic quivers. While the flavour 0-form symmetry is +manifest in this set of examples, the topological symmetry is at most accessible by discrete +Z2 subgroups, which turns out to be sufficient for the intents and purposes here. Lastly, we +consider mixed types: i.e. D and C-type Dynkin quivers composed of unitary gauge groups +and their mirror Sp(k) and O(2k) SQCD theories, respectively. The advantage of this class +of mirror pairs is that the flavour symmetry of the SQCD theories and the topological +symmetry of the unitary Dynkin quivers are fully manifest. Before closing, some magnetic +quiver examples are considered. Conclusions are provided in Section 3. Several appendices +complement the main text and provide computational details. +1See also [15, 16] for recent review articles. +– 2 – + +Note added. +During the course of this project, we were informed of a related work done +by Bhardwaj, Bullimore, Ferrari, and Sch¨afer-Nameki [22]. We are grateful to them for +coordinating the submission of our papers. +2 +Gauging discrete 0-form symmetries +In this section, mirror theories with non-trivial 1-form symmetry are constructed. Gauging +discrete subgroups of the 0-form symmetry, which results in 1-from symmetries and a +potential 2-group structure, has, for example, been considered in [9, 18–21]. +The principle is simple: start from a known mirror pair (T , T ∨) and gauge discrete +0-form symmetries Γ such that Γ ≡ Γt ⊂ Gt(T ) and Γ ≡ Γf ⊂ Gf(T ∨). This ensures that +the resulting theories (T /Γt, T ∨/Γf) are mirror pairs with 1-form symmetry Γ. The aims +here are (i) to provide explicit quiver descriptions for (T /Γt, T ∨/Γf) and (ii) to detail the +resulting symmetries (0-form, 1-form, and 2-group). +2.1 +Abelian theories +As a first example, consider 3d N = 4 SQED with N hypermultiplets of charge 1 and +its abelian mirror quiver theory [1], see Figure 1. The global 0-form symmetries are well- +known: for SQED one finds U(1)t × PSU(N)f, while the abelian mirror quiver enjoys a +U(1)f × PSU(N)t symmetry. +2.1.1 +SQED with higher charge +Suppose that one gauges a discrete Zq subgroup of the abelian U(1) factor of the global +0-form symmetry. The resulting theories are straightforwardly derived. Gauging a Zq ⊂ +U(1)t for SQED with N charge 1 hypermultiplets leads to SQED with N charge q hyper- +multiplets, see also [12]. Similarly, gauging a Zq ⊂ U(1)f of the abelian mirror quiver leads +to an abelian quiver with a Zq 1-form symmetry. The two theories obtained are then mirror +to each other, see Figure 1. The quiver notation is summarised in Table 3 of Appendix A. +Consistency checks. +The proposed mirror symmetry can be verified by Hilbert series +techniques [23–26]. The Higgs branch Hilbert series is insensitive to the gauging of the +Zq inside the topological symmetry of the SQED theory; similarly, the Coulomb branch +of the mirror does not perceive changes upon gauging a discrete subgroup of the flavour +symmetry. See Appendix A.1 for conventions. +Performing the discrete gauging for the SQED theory reduces to a Zq Molien-Weyl +sum over the Coulomb branch Hilbert series +HSC +SQEDq=1 +N (w|t) = +1 +1 − t +� +m∈Z +wmt +1 +2 N|m| += PE[t + (w + w−1)t +N +2 − tN] +(2.1) +HSC +SQEDq +N (z|t) = 1 +q +q−1 +� +p=0 +HSC +SQEDq′=1 +N +(w|t) +�� +w=z +1 +q (ζq)p +ζq = e +2πi +q ∈ Zq +– 3 – + +1 +N +0-form: U(1)t × PSU(N)f +1 +1 +... +1 +1 +1 +1 +0-form: PSU(N)t × U(1)f +1 +N +q +0-form: U(1)t × PSU(N)f +1-form: Zq +1 +1 +... +1 +1 +Zq +0-form: PSU(N)t × U(1)f +1-form: Zq +gauge Zq ⊂ U(1)t +gauge Zq ⊂ U(1)f +mirror +mirror +Figure 1. Gauging of discrete 0-form symmetries in SQED and its mirror. For SQED with charge 1 +hypermultiplets, a Zt +q gauging results in SQED with charge q hypermultiplets. These are indicated +by an arrow with the label q. For the abelian mirror quiver, the Zf +q gauging is realised by acting +on the flavours. The fundamental flavours that are charged under the discrete Zq are connected to +a grey node. See Appendix A for conventions. += +1 +1 − t +� +m∈Z +zmt +1 +2 N|q·m| += PE[t + (z + z−1)t +1 +2 qN − tqN] . +(2.2) +Likewise, one performs the Zq Molien-Weyl sum on the Higgs branch Hilbert series of the +mirror theory +HSH +mirror(y|t) = +N−1 +� +a=1 +� +dxa +2πixa +PE +�N−2 +� +b=1 +� xb +xb+1 ++ xb+1 +xb +� +t +1 +2 − (N−1)t +� +· PE +�� +y +1 +2 +x1 ++ x1 +y +1 +2 ++ y− 1 +2 +xN−1 ++ xN−1 +y− 1 +2 +� +t +1 +2 +� += PE[t + (y + y−1)t +N +2 − tN] +(2.3) +HSH +mirror/Zq(z|t) = 1 +q +q−1 +� +p=0 +HSH +mirror(y|t) +���� +y=z +1 +q (ζq)p += PE[t + (z + z−1)t +1 +2 qN − tqN] . +(2.4) +In summary, both results confirm the expectation and provide the explicit parameter map. +As a remark, the superconformal index is equally well suited to probe such dualities; see for +– 4 – + +instance [12] for SQED with charge q = 2 hypermultiplets. Since either Higgs or Coulomb +branch operators are unaffected by gauging a Zt/f +q +, the Hilbert series is a more convenient +tool. +Symmetries. +Using the techniques of [6], one can inspect the interplay between the +discrete 1-form symmetry Zq ⊂ U(1)t and the global 0-form symmetry PSU(N)f for the +SQED theory. The centre symmetry ZF = ZN of su(N)f is generated by αF = ζN, while +the U(1) gauge group supports a ZG = ZN·q centre generated by αG = ζN·q. The diagonal +αD = (αG, αF ) generates a E = ZN·q ⊂ ZG × ZF . The 1-form symmetry Γ[1] = Zq is +generated by αN +D = (αN +G, αN +F ) = (αN +G, 1), which acts trivially on the matter content. The +short exact sequence +0 → Γ[1] = Zq → E = Zq·N → Z = ZN → 0 +(2.5) +splits whenever gcd(q, N) = 1, i.e. q and N are co-prime. In order words, for gcd(q, N) > 1 +there exists an extension to a 2-group structure. +See for instance [9, 12] for a recent +discussion of SQED with 2 flavours of charge 2. +Comments on lines. +As explained in [9, 27, 28], 1-form symmetries and 2-group struc- +tures can be understood via equivalence classes of lines defects2. Here, we illustrate how +the higher-form symmetry is also realised on the line defects. +Consider SQED with N hypermultiplets of charge q. A Wilson line of charge h with +h ∈ {1, 2, . . . , q − 1} cannot end on a local operator because local operators are either +constructed as polynomials in the fundamental hypermultiplets of charge q or are monopole +operators, which are gauge singlets for 3d N = 4 theories. Thus, the 1-form symmetry Γ[1] +(or its Pontryagin dual) is generated by the (q − 1) Wilson lines that cannot end. Refining +with respect to the flavour symmetry shows that a Wilson line of charge q is equivalent +to a flavour Wilson line transforming as [1, 0, . . . , 0]AN−1. This however is not an allowed +representation of Gf = PSU(N), and signals the existence of a 2-group structure. In fact, +the N-th power of such a Wilson line is well-defined under Gf, because the N-th tensor +product of [1, 0, . . . , 0]AN−1 contains a singlet. Such lines generate the group �E = Zq·N. +Turning to the abelian mirror quiver, one can straightforwardly see that the fundamen- +tal Wilson lines, i.e. those of unit charge under a single U(1) gauge group factor, can end +on a local operator constructed out of the hypermultiplets. Therefore, one needs to turn +to the vortex lines to understand the 1-form symmetry. It is known [29] that the junctions +between vortex lines are significantly more challenging than those between Wilson lines. It +would be interesting to systematically address this in explicit examples. +2In brief, lines L1,2 are equivalent if there exists a local operator O at the junction between them. The +set of equivalence classes {L}/ ∼ forms the Pontryagin dual �Γ[1] of the 1-form symmetry Γ[1]. Refining +the equivalence relation by keeping track of 0-form symmetry representations R leads to the following +equivalence relation: (L1, R1) ∼ (L2, R2) iff there exists a local operator transforming as R1 ⊗ R∗ +2 (or +R∗ +1 ⊗ R2) at the junction of the lines. The equivalence classes give rise to �E (Pontryagin dual of E), which +encodes the interplay between the centres of gauge symmetry and 0-form symmetry. These groups fit into +the short exact sequence 0 → �Z → �E → �Γ[1] → 0, which is the Pontryagin dual of the sequences discussed +in the text, e.g. (2.5), (2.13), (2.15). +– 5 – + +2.1.2 +SQED with discrete gauge factor +Next, revert the logic: gauge a Zq subgroup of the PSU(N)f symmetry of SQED. Con- +versely, on the mirror side, one gauges a Zq subgroup of the PSU(N)t topological symmetry +of the abelian quiver theory. +For the abelian quiver theory, discrete gauging along a Cartan U(1)t of the topological +symmetry alters the linear quiver theory by modifying the charges of the bifundamental +hypermultiplets attached to a single gauge node. This follows from analogous arguments +as for SQED with charge q hypermultiplets or the arguments used in Appendices B.1 – +B.2. For the SQED theory, gauging of a discrete flavour 0-form symmetry affects some +of the fundamental flavours. To see this, one uses the original mirror map (C.7) between +the parameters to identify which flavour fugacities are affected by gauging along a Cartan +U(1)t factor in the abelian mirror. As a result, the flavours of the SQED split into two +sets: one charged under Zq and the other is trivial. This is shown in Figure 2. +Global symmetry: abelian mirror point of view. +The global symmetry is affected +as follows: suppose that one gauges a Zq ⊂ U(1)k ⊂ PSU(N)t subgroup of the topological +Cartam U(1) at the k-th node of the abelian quiver +1 +w1 +... +1 +wk−1 +1 +wk +1 +wk+1 +... +1 +wN−1 +1 +1 +(q, −1) +(1, −q) +. +(2.6) +The 0-form symmetry algebra after gauging is su(k) ⊕ u(1) ⊕ su(N−k). As exemplified in +Appendix D.1, the 0-form symmetry group is +Gt(2.6) = SU(k) × U(1)Q × SU(N−k) +Zk × ZN−k +(2.7) +where the centre symmetry Zℓ ⊂ SU(ℓ) acts on the fundamental representation [1, 0, . . . , 0]Aℓ−1 +with charge +1 under Zℓ, for ℓ ∈ {k, N−k}, see also Table 4. Moreover, the Zk × ZN−k +act with charge (−q mod k, q mod (N−k)) on the U(1)Q variable. Roughly Q ∼ wk, see +(C.8) for details. +The global structure (2.7) can also be inferred directly from the set of balanced nodes +in (2.6). The unbalanced gauge node U(1)k is connected to two balanced sets of gauge +nodes, forming the Ak−1 and AN−k−1 Dynkin diagram. Generalising the arguments of [30], +there are monopole operators transforming as [0, 0, . . . , q]Ak−1 × (1)Q (and its conjugate) +and [q, 0, . . . , 0] × (1)Q (and conjugate). This follows because the U(1)k node is attached +to the k − 1-th node of the Ak−1 Dynkin diagram and the 1-st node of AN−k−1 Dynkin +diagram. Compared to the standard case of unit charge bifundamental hypermultiplets, the +increased charge q modifies the appearing Aℓ representations accordingly. The existence of +these monopole operators in the Coulomb branch chiral ring leads to the isometry (2.7). +– 6 – + +1 +N +0-form: U(1)t × PSU(N)f +1 +... +1 +1 +1 +... +1 +1 +1 +(1, −1) +(1, −1) +0-form: PSU(N)t × U(1)f +1 +Zq +N − k +k +0-form: U(1)t × ( SU(k)×U(1)×SU(N−k) +Zk×ZN−k +) +f +1-form: Zq +1 +... +1 +1 +1 +... +1 +1 +1 +(q, −1) +(1, −q) +0-form: ( SU(k)×U(1)×SU(N−k) +Zk×ZN−k +) +t × U(1)f +1-form: Zq +gauge Zq ⊂ PSU(N)f +gauge Zq ⊂ PSU(N)t +mirror +mirror +Figure 2. Gauging of discrete 0-form symmetries in SQED and its mirror. The centre symmetries +act with charges (−q mod k, q mod (N−k)) on the U(1) factor. For SQED, the Zq acts on k of +the N hypermultiplets, which is indicated by k edges connected to a grey node. The remaining +hypermultiplets are uncharged under the discrete group. For the abelian linear quiver, gauging +along the Cartan U(1)t at the k-th gauge nodes leads to hypermultiplets with charge q under the +k-th U(1), while still of unit charge under the adjacent gauge factors. This is indicated by an arrow +with label. See Appendix A for conventions. +Global symmetry: SQED point of view. +To illuminate this result, it is instructive +to also consider the SQED side: +1 +Zq +N − k +Xa +˜Xd +k +, +(2.8) +where the two distinct sets of fundamentals are denoted as X and ˜X (by convention, both +have charge −1 under the U(1) gauge group). Computationally, gauging a discrete Zf +q is +realised via the following flavour fugacities, see Appendix C.1 +Xa : +a = 1, . . . , k : +ya = ζq · Q1 · +� +� +� +� +� +� +� +x1 , +a = 1 +xa +xa−1 , +1 < a < k +1 +xk−1 , +a = k +(2.9a) +– 7 – + +˜Xd : +d = k + 1, . . . , N : +yd = Q2 · +� +� +� +� +� +� +� +u1 , +d = k + 1 +ud−k +ud−k−1 , +k + 1 < d < N−k +1 +uN−k−1 , +d = N−k +(2.9b) +with ζq ∈ Zf +q . The xa and ud are weight space fugacities for su(k) and su(N−k), respec- +tively. +The first observation is that if k|q then the Zk centre symmetry of SU(k) is gauged, such +that a global PSU(k)f factor arises. Similarly, if (N−k)|q the ZN−k centre of SU(N−k) +is gauged, leading to a PSU(N−k)f factor. For the general case, one fixes the two so far +arbitrary U(1)Q1,2 symmetries3: +� +� +� +Qk +1 · QN−k +2 +!= 1 +� +Q1 +Q2 +�q +!= Q +⇒ +� +� +� +Q1 += Q +N−k +q·N +Q2 += Q +−k +q·n +(2.9c) +which agrees with (C.11) of Appendix C.1. Next, consider a gauge invariant operator O +built from the fields {Xa}k +a=1 transforming as (ζq · Q1 · [1, 0, . . . , 0]Ak−1, −1) under flavour- +gauge transformations and fields { ˜X† +d}N +d=k+1 transforming as (Q−1 +2 ·[0, . . . , 0, 1]AN−k−1, +1). +Thus, Xa ˜X† +d is U(1) gauge invariant. For Zq invariance, one also requires q-copies of Xa in +the form of Symq(Xa), which leads to the q-th symmetric representation Symq[1, 0, . . . , 0] +of SU(k)f. As a consequence, one also requires q copies of { ˜X† +d} in the form Symq( ˜Xd), +which leads to the q-th conjugate symmetric representation Symq[0, . . . , 0, 1] of SU(N−k)f. +Such a gauge invariant operator has charges +O = Symq +a1,...,aq(Xa1) · Symq +d1,...,dq( ˜Xdi) +↔ Symq[1, 0, . . . , 0]SU(k) ⊗ Symq[0, . . . , 0, 1]SU(N−k) ⊗ +�Q1 +Q2 +�q +� �� � +=Q +(2.10) +The operator O has Zk × ZN−k centre charges (q mod k, −q mod (N−k)). +Hence, the +Zk×ZN−k transformations can be compensated by a global U(1)Q rotation if Q has charges +(−q mod k, q mod (N−k)) under the centre symmetries. +This confirms (2.7) as flavour +symmetry Gf. The operator O can be detected in the Hilbert series at R-charge q·2· 1 +2 = q. +Comments on lines. +Returning to the quiver (2.6), consider a Wilson lines Wa of charge +1 under the a-th U(1) gauge group factor. For each a ̸= k, Wa can end on a local operator +composed of concatenated bifundamental hypermultiplets. For a = k, Wk cannot end since +the bifundamentals connected to the k-th gauge node are of charge q. Further, monopole +operators cannot screen gauge Wilson lines, because monopole operators are gauge singlets +for 3d N = 4 theories. Thus, the lines (Wk)h with h ∈ {0, 1, . . . , q − 1} cannot end and +generate the abelian group �Γ[1] = Zq. +3The definition of Q is a choice. Here, it is chosen such that the operator O in (2.10), as Higgs branch +operator with lowest R-charge that is charged under the U(1), has the unit charge. +– 8 – + +2.2 +An illustrative example +One of the main messages of this paper is that gauging discrete Zq subgroups of the +topological symmetry for quiver gauge theories T with unitary gauge nodes can result in +theories T /Zt +q which admit a simple quiver description. To illustrate this fact, consider +U(k) SQCD with N ≥ 2k fundamental flavours +k +N +(2.11) +with the well-known 0-form symmetries: Gf = PSU(N), Gt = U(1)t for N > 2k and +Gt = SO(3) for N = 2k. +Next, express the gauge group as U(k) ∼= U(1)×SU(k) +Zk +where Zk acts as centre on SU(k) +and via Zk charge (k−1) on the U(1) factor. Rewriting U(k) magnetic fluxes m ∈ Zk +into U(1) × SU(k) fluxes (h, l) requires the co-character lattice to be Γ = �k−1 +i=0 (Z + i +k)k. +Effectively, the SQCD theory can be understood as SU(k)×U(1) gauge theory with N copies +of bifundamentals and an “unusual” magnetic lattice Γ. One can introduce a (topological) +fugacity z that keeps track of the components of Γ. If w denotes the topological fugacity of +T , one employs w → zw +1 +k . Next, gauge a discrete Zq subgroup of the topological symmetry +by performing a discrete Molien-Weyl sum over z. It is convenient to choose either q|k or +k|q. One can show rigorously (e.g. using the superconformal index or the Coulomb branch +Hilbert series, see Appendix B) the following: +2.2.1 +Gauge a subgroup with q|k +If q|k then only the subgroup Zq ⊂ Zk is gauged. The theory becomes +� +����� +SU(k) +1 +N +� +����� +/Z k +q +with magnetic lattice +k +q −1 +� +i=0 +� +Z + i · q +k +�k +, +(2.12) +where the quotient Z k +q signals that this discrete group is not gauged, in the sense of [31]. +The resulting theory has a U(1)t × PSU(N)f 0-form symmetry and a Zq 1-form symmetry. +The potential interplay can be analysed via the action of the centre symmetries: defining +αG = ((ζk) +k +q , ζq·N) ∈ Zk × Zq·N (because only a Zq ⊂ Zk is gauged) and αF = ζN ∈ ZN, +the diagonal combination αD = (αG, αF ) generates a E = Zq·N group. +The element +N · αD = ((ζk) +k +q ·N, ζN +q·N, 1) generates a Γ(1) = Zq subgroup that acts trivial on the matter +fields. By definition, this establishes the 1-form symmetry. The short exact sequence +0 → Γ(1) = Zq → E = ZN·q → ZN → 0 +(2.13) +– 9 – + +splits if gcd(q, N) = 1. If gcd(q, N) > 1, there exists a non-trivial 2-group extension of Γ(1) +and PSU(N)f. +Symmetries via lines. +One can again illustrate this higher-form symmetry by using +line defects and their equivalence classes. A gauge Wilson line W in the representation +[0, . . . , 0]Ak−1 × (−1) cannot end on any local operator; Neither polynomials of the hyper- +multiplets nor monopole operators, because of a mismatch in gauge charges. However, W q +can end on the determinant operator O ∼ det(X), obtained by contracting hypermultiplets +X with the invariant ϵ tensor of SU(k). This operator has charges [0, . . . , 0]Ak−1 × k. Since +q|k, O has the same centre charges as W q, such that W q can end on it. Therefore, the lines +W a with a ∈ {1, 2, . . . , q − 1} cannot end on any local operator and generate the abelian +group �Γ[1] = Zq. Taking flavour charges into account, W q is equivalent to a flavour Wilson +line transforming as ∧k[0, . . . , 0, 1]AN−1, which follows from the flavour charges of O. This +is not a representation of PSU(N)f, but taking N-th tensor (W q)⊗N is equivalent to a +singlet of the flavour symmetry. Thus, these lines generate the group �E = ZN·q and the +1-form symmetry potentially forms a 2-group with the flavour symmetry (depending on +the gcd(N, q)). +2.2.2 +Gauge a discrete group Zq with k|q +If k|q then the SU(k) centre Zk is a subgroup of Zq and fully gauged. The theory becomes +SU(k) +1 +N +q +k +with magnetic lattice Zk . +(2.14) +The difference is now that the hypermultiplets transform as SU(k) fundamental with charge +q +k ∈ N under the U(1). This is indicated by the arrow, cf. Table 3. +In terms of symmetries, the theory T /Zt +q has a U(1)t topological symmetry, PSU(N)f +flavour symmetry, a Zq 1-form symmetry. Moreover, inspecting the gauge-flavour centre +symmetries shows: αG = (ζk, ζN·q) ∈ Zk × ZN·q and αF = ζN ∈ ZN. +The diagonal +generator αd = (αG, αF ) spans a E = ZN·q, and the element N ·αd = (ζN +k , ζN +N·q, 1) generates +a Γ(1) = Zk·q subgroup, using that k|q. This subgroup acts trivial on the matter fields; +thus, defining the 1-form symmetry Γ(1). The short exact sequence +0 → Γ(1) = Zq → E = ZN·q → ZN → 0 +(2.15) +splits if gcd(q, N) = 1. In all other cases, there exists a non-trivial extension giving rise to +a 2-group structure between Γ(1) and Gf = PSU(N). +Symmetries via lines. +Again, let us illustrate these structures with line defects. The +gauge Wilson line W transforming as [0, . . . , 0]Ak−1×(−1) under SU(k)×U(1) cannot end on +a local operator, which either has to be a polynomial in the hypermultiplet X transforming +as [1, 0, . . . , 0]Ak−1 × (− q +k) or has to be a monopole operator, which is gauge singlets. In +– 10 – + +contrast, the Wilson line W q can end on the local operator constructed as the determinant: +i.e. the SU(k) gauge group is equipped with the invariant ϵi1,...,ik tensor. Contracting k +hypermultiplets yields an operator O ∼ det(X) which transforms as [0, . . . , 0]Ak−1 × (−q). +Hence, the set of Wilson lines W a with a ∈ {1, 2, . . . , q − 1} cannot end and generate the +abelian group �Γ[1] = Zq. If one also keeps track of the flavour symmetry representations, one +finds that O transforms as ∧k[0, . . . , 0, 1]AN−1 which is not a representation of PSU(N)f. +Hence, this gauge Wilson line is equivalent to a flavour Wilson line and the centres of gauge +and flavour symmetry intertwine to give rise to a 2-group structure. +The following sections apply the analogous argument to other quiver gauge theories. +The relevant questions are: (i) What is the resulting theory? (ii) What are its symmetries? +(iii) What is the mirror dual theory? +2.3 +T[SU(N)] theories +Moving on to quiver theories with non-abelian gauge factors, consider the self-mirror +T[SU(N)] theories [30], see Figure 3. +The global 0-form symmetry group is given by +PSU(N)t × PSU(N)f. In the same spirit as above, one can gauge a discrete Zq 0-form +symmetry inside, say, the topological symmetry. The mirror of the resulting theory is then +obtained by gauging a Zq 0-form symmetry inside the flavour symmetry. The question is +how the Zq is embedded inside the flavour symmetry, given that the Zq is embedded into +a Cartan U(1) of the topological symmetry of the mirror. To answer this, one utilises the +mirror map (C.15). +In more detail, let us consider gauging a Zq ⊂ PSU(N)t of a T[SU(N)] theory; one +inquires about the nature of the resulting theory T[SU(N)]/Zt +q. Analogous to Section 2.2, +see also Appendix B.1, for a specific Zq embedded in the k-th topological Cartan U(1) +factor, the resulting theories T[SU(N)]/Zt +q are in fact related to versions of T[SU(N)] +encountered in [31]. These quiver theories differ from T[SU(N)] as follows: the k-th node +is replaced by U(k) → SU(k), and the flavour node becomes a U(1) gauge nodes with an +N copies of bifundamental hypermultiplets between U(N−1) and the “new” U(1) gauge +node. Restricting to the case that either q|k or k|q, the resulting theory is given by +q|k with d = k +q : +� +� +1 +w1 +2 +w2 +... +k−1 +wk−1 +SU(k) +k+1 +wk+1 +... +N−1 +wN−1 +1 +v +N +� +� /Zd +magnetic lattice: +d−1 +� +i=0 +� +Γ + i +d +� +(2.16a) +k|q with q = a · k : +1 +w1 +2 +w2 +... +k−1 +wk−1 +SU(k) +k+1 +wk+1 +... +N−1 +wN−1 +1 +v +N +a +(2.16b) +magnetic lattice: Γ +wherein Γ denotes the standard integer lattice one assigns to the quiver based on [26]. The +shifts by i +d are to be understood as in [31]. +– 11 – + +1 +2 +⋯ +k +⋯ +N−1 +N +0-form: PSU(N)f × PSU(N)t +1 +2 +⋯ +k +⋯ +N−1 +N +0-form: PSU(N)t × PSU(N)f +??? +0-form: PSU(N)f × ( SU(k)×U(1)×SU(N−k) +Zk×ZN−k +) +t +1-form: Zq +1 +2 +⋯ +k +⋯ +N−1 +Zq +N − k +k +0-form: ( SU(k)×U(1)×SU(N−k) +Zk×ZN−k +) +f × PSU(N)t +1-form: Zq +gauge Zq ⊂ PSU(N)t +gauge Zq ⊂ PSU(N)f +mirror +mirror +Figure 3. Gauging of discrete 0-form symmetries in T[SU(N)] theories. The centre symmetries act +with charges (q mod k, −q mod (N−k)) on the U(1) factor. The quiver description for T[SU(N)]/ +Zt +q, here denoted by ???, is provided in (2.16). The quiver for T[SU(N)]∨/Zf +q shows again a split +of the N fundamental flavours into two sets: k of them are charged under Zf +q , which is indicated +by an edge of multiplicity k to the grey node; the remaining N−k flavours are uncharged. +The mirror theory T[SU(N)]∨/Zf +q is obtained from T[SU(N)]∨ = T[SU(N)] by gauging +a Zq ⊂ PSU(N)f. The mirror map (C.15) dictates that this is realised by splitting the N +fundamental flavours into two sets of k and N−k flavours, and gauging the Zq symmetry +in the overall U(1) flavour symmetry of one of the two sets. For concreteness, consider +gauging the Zq on the set of k fundamental flavours: +1 +2 +... +k +... +N−1 +Zq +{yi}k +i=1 +N−k +{yj}N +j=k+1 +k +(2.17) +and Appendix D.2 provides exemplary Hilbert series computations that confirm the mirror +symmetry between the theories with non-trivial 1-form symmetry. +The mirror map between the parameters of T[SU(N)]/Zt +q in (2.16) and T[SU(N)]∨/Zf +q +in (2.17) can be derived exactly. For concreteness, consider the case q = k, then the map +between the parameters in (2.16) and (2.17) is established via +� +wi += +yi +yi+1 , i ̸= k , +v += yN +N . +(2.18) +– 12 – + +Further details on this map are provided in Appendix C.2. +Global symmetry. +Building on the understanding of the 0-form symmetry group (2.7) +for (balanced) abelian quivers, one can utilise a similar logic for the balanced T[SU(N)] +theories. Consider the quiver (2.16) the topological symmetry algebra is su(k) ⊕ u(1) ⊕ +su(N−k). The global form is then given by +Gt = SU(k) × U(1)Q × SU(N−k) +Zk × ZN−k +(2.19) +with Q has Zk × ZN−k charges (−q mod k, q mod (N−k)) +where the centre symmetries Zℓ act in the standard way on SU(ℓ). Note that for k|q there +is a PSU(k) factor in the global symmetry. The examples in the next paragraph, as well +as the explicit character decomposition in Appendix D.2, confirm this structure. +This structure (2.19) is also apparent from the Higgs branch isometry of the mirror +(2.17), i.e. denote the two distinct sets of fundamentals by +1 +... +N−1 +Zq +N−k +Xa +˜Xd +. +(2.20) +Analogously to (2.9), one can perform the Zf +q gauging by assigning (c.f. Appendix C.2) +Xa : +a = 1, . . . , k : +ya = ζq · Q1 · +� +� +� +� +� +� +� +x1 , +a = 1 +xa +xa+1 , +1 < a < k +1 +xk−1 , +a = k +(2.21a) +˜Xd : +d = k + 1, . . . , N : +yd = Q2 · +� +� +� +� +� +� +� +u1 , +d = k + 1 +ud−k +ud−k−1 , +k + 1 < d < N−k +1 +uN−k−1 , +d = N−k +(2.21b) +with ζq ∈ Zf +q . The xi and uj are weight space fugacities of su(k) and su(N−k), respectively. +The two appearing U(1) fugacities Q1,2 effectively reduce to a single U(1)Q; for instance +by imposing � +i yi = 1, i.e. +� +� +� +Qk +1 · QN−k +2 +!= 1 +� +Q1 +Q2 +�q +!= Q +⇒ +� +� +� +Q1 = Q +N−k +N·q +Q2 = Q− +k +N·q +(2.21c) +which agrees4 with (C.20) of Appendix C.2. Note also that for q|k the (ζq) +k +q ∈ Zf +q acts +as the Zk ⊂ SU(k)xi centre symmetry; thus the global factor is PSU(k)xi in this case. +4Again, the definition of Q is a choice. It is motivated by assigning the unit charge to the Higgs branch +operator O in (2.22), which is the operator with the lowest R-charge that is charged under the U(1)Q. +– 13 – + +The U(1)Q may transform non-trivially under the Zℓ ⊂ SU(ℓ) centre symmetries, de- +pending on the charge of Q. +To determine the charge, one again considers a specific +gauge-invariant operator O build out of the two sets of fundamentals: X transforms as +(ζq · Q1 · [1, 0, . . . , 0]Ak−1, N−1) and ˜X† transforms as (Q−1 +2 +· [0, . . . , 0, 1]AN−k−1, N−1). +U(N−1) gauge invariance imposes Tr(X ˜X†), wherein the trace is taken over the gauge +indices. Zq gauge invariance requires O = SymqTr(X ˜X†), where the symmetrisation acts +on the flavour indices. The resulting operator transforms as +O : +Symq[1, 0, . . . , 0]k ⊗ Symq[0, . . . , 0, 1]N−k ⊗ +�Q1 +Q2 +�q +� �� � +=Q +(2.22) +such that the Zk × ZN−k centre charges are (q mod k, −q mod (N−k)). +These can be +compensated by a global U(1)Q rotation provided the centre charges of Q are (−q mod +k, q mod (N−k)). This confirms (2.19) as flavour symmetry for the quiver (2.17). +As a remark, the operator O can be detected in the Hilbert series as the first non-trivial +term in Q. The R-charge of O is simply q ×2· 1 +2 = q. The appendix D.2 provides examples +that illustrate this point. +By analogous arguments as in Section 2.2, one can verify that theories (2.16) indeed +have the expected Γ(1) = Zq 1-form symmetry. One finds that the centre generators of +the combined gauge-flavour symmetry span a E = Zq·N group, such that there exists a +non-trivial 2-group extension between Γ(1) and Gf = PSU(N) whenever gcd(q, N) > 1. +Similarly, the same conclusion is reached by inspecting the screening of Wilson lines. +Example 1. +For an illustrative purpose, let us consider N = 4. Gauging a specific Z2 +0-form symmetry leads to a mirror pair: +1 +SU(2) +3 +1 +4 +mirror +←−−−−−−→ +1 +2 +3 +2 +Z2 +. +(2.23) +The Hilbert series in (D.15) confirms that the Coulomb branch symmetry algebra for the left +quiver (and the Higgs branch isometry algebra of the right quiver) is g = su(2)⊕su(2)⊕u(1). +Moreover, the appearing SU(2) representations are all of integer spin; thus, suggesting the +global form G = SO(3) × SO(3) × U(1). +Choosing to gauge a specific discrete Z3 subgroup of the 0-form symmetry results in +the pair: +1 +2 +SU(3) +1 +4 +mirror +←−−−−−−→ +1 +2 +3 +1 +Z3 +. +(2.24) +– 14 – + +The explicit Hilbert series in (D.12) shows that the Coulomb branch symmetry algebra +of the left quiver (and the Higgs branch isometry algebra of the right theory) is g = +su(3)⊕u(1). Moreover, all appearing characters are neutral under the Z3 centre symmetry +of SU(3); hence, the global form is G = PSU(3) × U(1). +Example 2. +Considering discrete symmetries of the type Zq with q = a · k allows us +to uncover equivalent descriptions. Consider T[SU(5)] and gauge a Z6 0-form symmetry. +Among the choices considered here, gauging a Z6 ⊂ PSU(5)t is realised by turning the +U(3) gauge node into SU(3) together with charge 2 for the “new” U(1) node +1 +w1 +2 +w2 +SU(3) +4 +w4 +1 +v +5 +2 +mirror +←−−−−−−→ +1 +2 +3 +4 +Z6 +2 +3 +, +(2.25) +or by turing the U(2) node into SU(2) together with charge 3 for the “new” U(1) gauge +factor +1 +w1 +SU(2) +3 +w3 +4 +w4 +1 +v +5 +3 +mirror +←−−−−−−→ +1 +2 +3 +4 +Z6 +3 +2 +. +(2.26) +Without the additional charges the theories are clearly distinct, for instance by the 0-form +and 1-form symmetries, see (2.19) and Figure 3. However, with the modification, both +become equivalent as, for example, the monopole formula in (D.39) confirms. +Equivalently, one the mirror, one gauges a Z6 subgroup of the flavour 0-form symmetry, +but one time acting on three fundamental hypermultiplets and one time on two. For (2.25) +and (2.26), this is realised by +{yi}5 +i=1 → +� +� +� +� +� +� +� +{y1, y2, ζ6 y3, ζ6 y4, ζ6 y5}, +for (2.25) +or +{ζ6 y1, ζ6 y2, y3, y4, y5}, +for (2.26) +(2.27) +with ζ6 ∈ Zf +6. But in both cases, the Z2 × Z3 centre symmetries are gauged by the discrete +gauging of the Zf +6 0-form symmetry. +Hence, the global symmetry is simply PSU(2) × +U(1)Q × PSU(3) for both. +Comment. +The considerations so far implicitly assume that the gauge node U(k) at +which the discrete subgroup of the Cartan U(1)t of the topological symmetry is gauged +has k > 1, see for instance Appendix B.1 and B.2. Gauging the Cartan U(1)t of the U(1) +gauge node of T[SU(N)] is in spirit similar to Section 2.1.1. Concretely, after gauging Zt +q +at the U(1) node, the bifundamental between U(1) and U(2) is modified to have charge q +– 15 – + +under the U(1). Thus, the mirror pair becomes +1 +w1 +2 +w2 +... +N−1 +wN−1 +N +q +mirror +←−−−→ +1 +2 +... +N−1 +Zq +y1 +N−1 +{yj}N +j=2 +(2.28) +and the global symmetry becomes +Gt(left quiver (2.28)) = Gf(right quiver (2.28)) = U(1)Q × SU(N−1) +ZN−1 +(2.29) +where Q has ZN−1 charge q mod (N−1). +2.4 +T σ +ρ [SU(N)] theories +The class of linear quiver gauge theories with unitary gauge groups and fundamental or +bifundamental hypermultiplets is given by the T σ +ρ [SU(N)] theories [30], where ρ, σ are +two partitions of N. For σ = ρ = (1, . . . , 1) ≡ (1N), the corresponding theory is simply +T (1N) +(1N) [SU(N)] = T[SU(N)] and the partition data can be dropped. +Mirror symmetry +exchanges the partitions σ and ρ, i.e. T σ +ρ [SU(N)]∨ = T ρ +σ[SU(N)]. +Analogous to the cases considered so far, the gauging of a discrete 0-form symmetry +(either inside the topological symmetry group Gt or the flavour symmetry group Gf) leads +to a theory with non-trivial 1-form symmetry. Again, consider the two options in turn. +While the process of gauging a discrete subgroup of Gt is by now understood (see Section +2.3 and Appendix B), determining the action of the discrete group on the flavour symmetry +of the mirror theory becomes more challenging when Gf is a generic product group. Thus, +special attention is paid to determining the mirror theory of T σ +ρ [SU(N)]/Zt +q. +Gauging a ZNk ⊂ Gt. +A T σ +ρ [SU(N)] is a linear quiver theory with gauge/flavour groups +specified by a sequence of integers {Ni} and {Mi}, respectively. The partitions determine +the integers as detailed in [30] and the quiver becomes +T σ +ρ [SU(N)] : +N1 +N2 +... +Nk +... +Nn−1 +Nn +M1 +M2 +Mk +Mn−1 +Mn +. +(2.30) +For concreteness, take the node Nk, with Nk > 1, and gauge a ZNk ⊂ U(1)k ⊂ Gt inside +the Cartan factor of the topological symmetry associated to the k-th node. By the same +– 16 – + +arguments as in Appendices B.1 and B.2, one straightforwardly derives the resulting theory +T σ +ρ [SU(N)]/Zt +Nk : +N1 +N2 +... +SU(Nk) +... +Nn−1 +Nn +1 +M1 +Mn +(2.31) +which has a non-trivial ZNk 1-form symmetry. Now, one constructs the mirror theory. +Gauging a ZNk ⊂ G∨ +f . +The mirror quiver gauge theory of (2.30) is given by +T σ +ρ [SU(N)]∨ = T ρ +σ[SU(N)] : +N∨ +1 +N∨ +2 +... +N∨ +n′−2 +N∨ +n′−1 +N∨ +n′ +M∨ +1 +M∨ +2 +M∨ +n′−2 M∨ +n′−1 +M∨ +n′ +. +(2.32) +and the integers {N∨ +i } and {M∨ +i } are determined by the partition data ρ, σ. +To determine which ZNk ⊂ G∨ +f subgroup needs to be gauged, one has two options: one +could derive the mirror map of parameters for the specific pair (T σ +ρ [SU(N)], T ρ +σ[SU(N)]) +and compute which flavours are charged under Zf +Nk. In principle, this is straightforward +but likely to be tedious. Alternatively, one can employ the following train of thought: The +mirror theory T ρ +σ[SU(N)] can be rewritten in an unframed form +N∨ +1 +N∨ +2 +... +N∨ +n′−2 +N∨ +n′−1 +N∨ +n′ +M∨ +1 +M∨ +2 +M∨ +n′−2 M∨ +n′−1 +M∨ +n′ +∼= +N∨ +1 +N∨ +2 +... +N∨ +n′−2 +N∨ +n′−1 +N∨ +n′ +1 +M∨ +1 +M∨ +n′ +///U(1)diag +(2.33) +where no explicit flavour group appears. For such a theory, it is implied that an overall +U(1)diag subgroup decouples so the two quiver diagrams express the same theory. +The next step is to turn the unitary gauge group U(Nk) in (2.30) into a special unitary +gauge group SU(Nk). This theory still has a trivial 1-form symmetry due to the flavour +groups. +However, the 3d mirror theory can be found by using the algorithm in [32]. +Schematically, one finds +N1 +... +SU(Nk) +... +Nn +M1 +Mk +Mn +mirror +←−−→ +N∨ +1 +N∨ +2 +... +N∨ +n′−2 +N∨ +n′−1 +N∨ +n′ +1 +1 +M∨ +1 +M∨ +n′ +///U(1)diag +(2.34) +Turning U(Nk) into SU(Nk) means the 3d mirror has an additional U(1) gauge group. The +– 17 – + +additional U(1) gauge group in the unframed quiver is the result of gauging the flavour +symmetry. Now, there are two U(1) gauge groups connected to the rest of the linear quiver. +The number of bonds M∨ +i attached to each U(1) depends precisely on the choice of SU(Nk), +i.e. which k. The splitting can also occur where the same gauge node, for example, U(N∨ +2 ) +is connected to one U(1) gauge group with an edge of multiplicity M∨ +2 −x and to the other +U(1) with an edge of multiplicity x; see for instance Example 2 below. +The final step is to gauge the diagonal U(1) flavour symmetry in the left quiver of +(2.34) to obtain T σ +ρ [SU(N)]/Zt +Nk. However, simply introducing a new U(1) gauge node +leads to the ambiguity of the global form of the product gauge group, which can either be +G = U(1) × SU(Nk) × � +i̸=k U(Ni) or G removed by a subgroup of its centre. For G/ZNk, +with ZNk embedded into SU(Nk) as centre and into the diagonal U(1) ⊂ U(Ni) of each of +the other gauge group factors, one obtains back the original theory T σ +ρ [SU(N)], see [31] +or Appendix B. But this theory exhibits a Zt +Nk topological symmetry, see also Section +2.9. +In order to generate the desired T σ +ρ [SU(N)]/Zt +Nk theory with gauge group G, one +needs to gauge the discrete Zt +Nk symmetry, which effectively reduces the magnetic lattice +to the standard integer lattice. For the 3d mirror, this means that one first gauges a U(1)t +topological symmetry, which effectively removes a U(1) gauge degree of freedom. But one +also needs to gauge a Zf +Nk in a subsequent step. This Zf +Nk can be thought of as embedded +in the U(1) that one has to be removed. Hence, the intermediate step is given by +T σ +ρ [SU(N)]/Zt +Nk +mirror +←−−→ +N∨ +1 +N∨ +2 +... +N∨ +n′−2 +N∨ +n′−1 +N∨ +n′ +1 +1 +M∨ +1 +M∨ +n′ +/// +� +U(1)diag×U(1) +Zf +Nk +� +(2.35) +From the unframed quiver on the right, one has to ungauge a U(1)diag ×U(1) and also keep +a Zf +Nk gauged. The natural choice is to ungauge the two U(1) gauge groups on top; thus, +turning them into flavour groups up to a choice of Zf +Nk. The last step is to choose in which +of the two U(1)s one embeds the Zf +Nk. This is because, as with the T[SU(N)] theories, +one knows the only difference between T σ +ρ [SU(N)]∨ and +� +T σ +ρ [SU(N)]/Zt +Nk +�∨ +should be the +splitting of the flavour groups along with a discrete quotient. Schematically, one finds +N∨ +1 +N∨ +2 +... +N∨ +n′−2 +N∨ +n′−1 +N∨ +n′ +1 +1 +M∨ +1 +M∨ +n′ +/// +� +U(1)diag×U(1) +Zf +Nk +� +(2.36) +∼= +N∨ +1 +N∨ +2 +... +N∨ +n′−2 +N∨ +n′−1 +N∨ +n′ +M∨ +1 +M∨ +2 +ZNk +M∨ +n′−2 +M∨ +n′ +∼= +N∨ +1 +N∨ +2 +... +N∨ +n′−2 +N∨ +n′−1 +N∨ +n′ +ZNk +M∨ +n′−2 M∨ +n′−1 +M∨ +n′ +M∨ +1 +M∨ +2 +– 18 – + +and the two framed mirrors show that the discrete quotient can be applied diagonally on +either one of the two sets of flavour hypermultiplets. This is also clear from Sections 2.1.2 +and 2.3, and Appendices C.1 and C.2, as an overall U(1) rotation can be used to shuffle +the discrete ZNk charges from one set of fundamental flavours to another. +1 +2 +2 +1 +3 +2 +2 +2 +2 +2 +2 +1 +1 +2 +1 +1 +1 +≅ +2 +2 +2 +2 +1 +1 +1 +///U(1)diag +1 +2 +SU(2) +1 +3 +2 +2 +1 +2 +SU(2) +1 +1 +2 +2 +2 +2 +1 +1 +1 +1 +///U(1)diag +2 +2 +2 +2 +1 +1 +1 +1 +/// U(1)diag×U(1) +Z2 +≅ +2 +2 +2 +2 +1 +1 +2 +1 +Z2 +≅ +2 +2 +2 +2 +1 +1 +Z2 +1 +1 +U → SU +gauge U(1) +ungauge U(1) +mirror +mirror +mirror +Figure 4. Starting from the mirror pair T σ +ρ [SU(15)] and T ρ +σ[SU(15)] with σ = (33, 22, 12) and +ρ = (6, 4, 3, 12), one can gauge a discrete Z2 0-form symmetry to create a new mirror pair with Z2 +1-form symmetry. See also Appendix C.3 for the choice of Zf +2 gauging. +– 19 – + +Example 1. +One can apply the above procedure to T σ +ρ [SU(15)] where σ = (33, 22, 12) +and ρ = (6, 4, 3, 12) for the example as in Figure 4. The global form of the 0-form symmetry +is expected to be +Gt(bottom left quiver of Figure 4) = Gf(bottom right quiver(s) of Figure 4) += SU(2) × U(1)1 +Z2 +× U(1)2 × U(1)3 ∼= U(2) × U(1)2 × U(1)3 +(2.37) +and one can explicitly verify this structure as demonstrated in (D.42). Alternatively, the +Coulomb branch quiver indicates this isometry group as follows: only the leftmost U(1) +is balanced, leading to a topological su(2)t because there are monopole operators of U(1) +magnetic flux ±1 at R-charge 1 (see also [30]). The remaining U(2) and U(1) gauge nodes +provide one U(1)t +i=1,2,3 topological symmetry factor each. Let the one associated with U(2) +be denoted by U(1)t +1. Since this node is connected to the balanced node, arguments similar +to [30] show the existence of a chiral ring operator that transforms as a spinor under su(2)t +and has charge ±1 under U(1)t +1. Therefore, the Z2 centre action can be absorbed into +U(1)t +1, resulting in a U(2)t topological symmetry factor. +One can also choose the other SU(2) in T σ +ρ [SU(15)] which gives the mirror pairs dis- +played in Figure 5. Comparing Figure 4 and 5, one observes that the global form of the +1 +SU(2) +2 +1 +1 +2 +2 +2 +2 +1 +1 +1 +1 +/// U(1)diag×U(1) +Z2 +≅ +2 +2 +2 +2 +1 +1 +2 +Z2 +≅ +2 +2 +2 +2 +1 +1 +Z2 +1 +1 +1 +mirror +Figure 5. Again starting from the mirror pair T σ +ρ [SU(15)] and T ρ +σ[SU(15)] with σ = (33, 22, 12) +and ρ = (6, 4, 3, 12), one can gauge a different discrete Z2 0-form symmetry to generate a another +mirror pair with Z2 1-form symmetry. See Appendix C.3 for the choice of Zf +2 in the mirror. +– 20 – + +0-form symmetry in Figure 5 is simply +PSU(2) × +3 +� +i=1 +U(1)i , +(2.38) +which is supported by the explicit calculations in (D.45). +This conclusion can also be +drawn by examining the Coulomb branch quiver. Since the balanced U(1) gauge node is +not directly connected to any of the U(2) or U(1) gauge groups, there is no expectation +on a chiral ring operator that transforms non-trivially under the Z2 centre of the SU(2)t +topological symmetry. +Example 2. +Consider the mirror pair T σ +ρ [SU(9)] with ρ = (3, 23) and σ = (32, 13) +2 +2 +2 +3 +2 +←→ +1 +2 +2 +1 +1 +3 +(2.39) +whose symmetry algebra is su(3) ⊕ u(1), as apparent from the balanced set of nodes. The +global form is evaluated to be +Gt(LHS (2.39)) = Gf(RHS (2.39)) = SU(3) × U(1) +Z3 +∼= U(3) +(2.40) +because the U(1) has charge −1 under the Z3 centre symmetry. See (D.46) for details. Al- +ternatively, the left-hand-side quiver in (2.39) allows us to derive this by using the balanced +set of nodes. Since the unbalanced gauge nodes connect to the A2 Dynkin diagram (formed +by the balanced nodes) on its first node, there exists a chiral ring operator transforming +as [1, 0] × (+1) (plus conjugate) under the topological SU(3)t × U(1)t. Thus, the Z3 centre +can be compensated by suitable embedding into the U(1)t factor. +To create a new mirror pair, we can gauge a Z2 symmetry on both sides of the dual +theories. For example, we can gauge the topological Zt +2 symmetry on w3. The mirror map, +as shown in (C.24), indicates that gauging the Zf +2 symmetry leads to the following mirror +pair +2 +2 +SU(2) +1 +←→ +1 +2 +2 +1 +1 +Z2 +(2.41) +whose symmetry algebra is su(2) ⊕ u(1) ⊕ u(1). The Hilbert series (D.49) then suggests a +symmetry group of +Gt(LHS (2.41)) = Gf(RHS (2.41)) = SU(2) × U(1) × U(1) +Z2 +∼= U(2) × U(1) +(2.42) +– 21 – + +because the centre Z2 acts trivial on one U(1) factor and with charge −1 on the other. This +can also be read off from the Coulomb branch quiver. As there is a U(2) node connected +to the balanced U(2) node, there exists a chiral ring operator transforming as [1]A1 × (±1) +under the associated SU(2)t×U(1)t topological symmetry factors. Therefore, the Z2 centre +symmetry then gives rise to a U(2)t isometry factor. The other topological Cartan U(1)t +is uncharged under the Z2 centre, as the gauge nodes are not connected to each other. +Gauging Zt +q on a U(1) node. +Analogous to Section 2.3, one can also gauge discrete +subgroups of the topological symmetry associated to a U(1) gauge node. From the examples +considered, it is clear what the theory after gauge the Zt +q is: the same quiver as before, +but all hypermultiplets connected to the specific U(1) gauge node have now charge q. The +question is then, what the corresponding mirror theory is. This can be determined by +utilising the mirror map between the fugacities, as demonstrated in Appendix C. +2.5 +T[SO(2N)] theories +In a similar vein to T[SU(N)], one can consider the self-mirror theory T[SO(2N)] [30], see +Figure 6. For quiver theories composed of alternating SO(n) and Sp(m) gauge nodes, only +the Z2 factors of the SO(n) gauge nodes are the discrete parts of the topological symmetry +visible in the UV description. If we gauge any of these, we get a T[SO(2N)]-type quiver with +a single replacement SO(2k) → Spin(2k)5. The corresponding mirror theory is obtained +from T[SO(2N)] by gauging a suitable Z2 inside the flavour symmetry. This leads to a +splitting of the flavour node as indicated in Figure 6. Appendix D.5 provides examples and +consistency checks for T[SO(6)] and T[SO(8)]. +Considering the theory T[SO(2N)]/Zt +2 obtained from gauging Zt +2, the quiver descrip- +tion allows us to use the techniques of [6] to verify the 1-form symmetry and its interplay +with the flavour 0-form symmetry. One finds the discrete groups summarised in Table 1 +which constitute the short exact sequence +0 → Γ[1] → E → Z → 0 . +(2.43) +As expected, the flavour 0-form symmetry is always PSO(2N) since the flavour centre Z +is maximal. Moreover, only for T[SO(4N)]/Zt +2 with a Spin(4l + 2) gauge node does the +1-form symmetry and the flavour 0-form symmetry form a non-trivial extension hinting to +a 2-group symmetry. +Following [21, 28], it is straightforward to illustrate the 1-form symmetry and 2-group +structure via line operators. For the Spin(2l) gauge group, a Wilson line Ws in the spinor +representation cannot end on a local operator, because all half-hypermultiplets transform +in the vector representation. +For l =even, the tensor product of the spinor with itself +contains a singlet; therefore, W 2 +s is equivalent to the identity line without the need for any +local operator. The lines that cannot end generate the (Pontryagin dual of the) 1-form +symmetry and there is no 2-group structure. For l =odd, the tensor product of the spinor +5This follows as gauging the Z2 topological symmetry of an SO(2n) gauge group leads to an Spin(2k) +gauge group. Conversely [21, 33], gauging the Z2 1-form symmetry in Spin(2k) recovers the SO(2k) theory. +– 22 – + +2 +2 +... +2k +... +2N−2 +2N +0-form: PSO(2N)f × PSO(2N)t +2 +2 +... +2k +... +2N−2 +2N +0-form: PSO(2N)t × PSO(2N)f +2 +2 +... +Spin(2k) +... +2N−2 +2N +0-form: PSO(2N)f × P(O(2k) × O(2N − 2k))t +1-form: Z2 +2 +2 +... +2k +... +2N−2 +Z2 +2N − 2k +k +0-form: P(O(2k) × O(2N − 2k))f × PSO(2N)t +1-form: Z2 +gauge Z2 ⊂ PSO(2N)t +gauge Z2 ⊂ PSO(2N)f +mirror +mirror +Figure 6. Gauging of discrete 0-form symmetries in T[SO(2N)] theories. Gauging the Zt +2 for an +SO(2k) gauge leads to a Spin(2k) gauge group. Gauging the mirror dual Zf +2 is realised by splitting +the fundamental flavours into two sets: one set is uncharged and the other set is charged under Zf +2, +indicated by an edge with multiplicity k connected to a grey node. +theory +Γ[1] +E +Z +T[SO(4N)]/Zt +2 with Spin(4l) +Z2 +Z2 × Z2 × Z2 +Z2 × Z2 +T[SO(4N)]/Zt +2 with Spin(4l + 2) +Z2 +Z2 × Z4 +Z2 × Z2 +T[SO(4N + 2)]/Zt +2 with Spin(4l) +Z2 +Z2 × Z4 +Z4 +T[SO(4N + 2)]/Zt +2 with Spin(4l + 2) +Z2 +Z2 × Z4 +Z4 +Table 1. Interplay of 1-form symmetry and the flavour centre for T[SO(2N)]/Zt +2 theories. +with itself contains the vector. Now W 2 +s is equivalent to a flavour Wilson line because +it can end on a local operator build from the half-hypermultiplets. However, the vector +representation is not an allowed representation of PSO(2N), which means that the Z2 1- +form symmetry forms a 2-group with the flavour symmetry. This is consistent with Table +1. +On the other hand, in the theory T[SO(2N)]/Zf +2 obtained by gauging the Zf +2 symmetry, +there are two distinct sets of flavour hypermultiplets, each forming half-hypermultiplets H +and h in the vector-vector representation of SO(2N−2k)×Sp(N−1) and SO(2k)×Sp(N−1), +– 23 – + +respectively, i.e. +2 +2 +... +2N−2 +Z2 +2N−2k +H +k +h +. +(2.44) +The only difference is that h is also charged under Z2. As in Sections 2.1.2 and 2.3, to +study the global form of the flavour symmetry of this theory, one can consider gauge- +invariant operators. Using the invariant Sp(N−1) anti-symmetric tensor J, the standard +mesons-type invariants are HJH and hJh, both of which then transform in the adjoint +representation [0, 1, . . . , 0]D of so(2N−2k) and so(2k), respectively. +Likewise, one can +consider hJH, which is Sp(N−1) gauge invariant, but not Z2 invariant due to the Z2 +charge of h. Hence, O = Sym2(hJH) is indeed a gauge invariant operator transforming as +[2, 0, . . . , 0]Dk ⊗ [2, 0, . . . , 0]DN−k. All of these gauge-invariant Higgs branch operators have +trivial charges under the so(2k) or so(2N−2k) centre symmetries. This suggests that the +global form of the flavour symmetry is PSO(2k) × PSO(2N−2k). +2.6 +Sp(k) SQCD and its orthosymplectic mirror +The lessons learnt can be readily applied to other orthosymplectic quivers, such as Sp(k) +SQCD with N fundamental hypermultiplets and its orthosymplectic mirror quiver [34]. +Focusing on N ≥ 2k+1, the SQCD theory admits a manifest flavour symmetry, while there +is no topological symmetry for N > 2k + 1 and a U(1)t symmetry for N = 2k + 1. Thus, +it is quite natural to consider gauging discrete subgroups of the flavour 0-form symmetry. +Conversely, the mirror orthosymplectic quiver does not have a continuous flavour symmetry +for N > 2k1 (i.e. no mass parameter) and an SO(2)f symmetry for N = 2k + 1 (i.e. one +mass parameter). While the topological symmetry is not manifest in the UV description, +certain remnants are: each SO(l) gauge group admits a manifest Zt +2 symmetry. +Therefore, one can gauge a Z2 topological symmetry of a specific SO(ℓ) gauge node and +inquire about the implications. It is straightforward to observe that this gauging modifies +the particular gauge group SO(ℓ) → Spin(ℓ), see for instance [26, 35]. On the mirror side, +one gauges a Z2 ⊂ SO(2N) flavour symmetry, which then leads to a split of the flavour +symmetry. This is summarised in Figure 7. Exemplary cases with explicit calculations are +provided in Appendix D.6. +The interplay of the discrete Z2 1-form symmetry with the continuous 0-form symmetry +is simple here. Consider the linear orthosymplectic mirror quiver. For N > 2k + 1, there +is no continuous 0-form flavour symmetry that could mix with the 1-form symmetry Z2. +For N = 2k +1, there exists an enhance U(1) 0-form symmetry, but the 1-form and 0-form +symmetry are simply a product of each other. +– 24 – + +2k +2N +0-form: PSO(2N)f × Ht +2 +2 +... +2k +2k+1 +... +2k+1 +2k +... +2 +2 +1 +1 +(N−2k−1) × SO(2k + 1) nodes +(N−2k) × Sp(k) nodes +0-form: PSO(2N)t × Hf +2k +Z2 +2N − 2ℓ +ℓ +0-form: (SO(2ℓ) × SO(2N − 2ℓ))f × Ht +1-form: Z2 +2 +... +2ℓ−2 Spin(2ℓ) +2ℓ +... +2k +... +2k +... +2 +1 +1 +0-form: (SO(2ℓ) × SO(2N − 2ℓ))t × Hf +1-form: Z2 +gauge Z2 ⊂ PSO(2N)f +gauge Z2 ⊂ PSO(2N)t +mirror +mirror +Figure 7. Gauging of discrete 0-form symmetries in Sp(k) SQCD with N fundamentals and its +linear orthosymplectic mirror quiver. Here, the isometry group Ht/f is trivial for N > 2k + 1 and +U(1) for N = 2k + 1. +2.7 +Sp(k) SQCD and its unitary D-type mirror quiver +It is well-known that Sp(k) SQCD with N fundamental flavours admits a second mirror +description [36], based on a DN-type Dynkin quiver: +2k +2N +←→ +1 +2 +... +2k−1 +2k +2k +... +2k +k +k +1 +N − 2k − 1 nodes +. +(2.45) +This mirror pair has the advantage that the PSO(2N) global symmetry is manifest as +Higgs branch isometry in the SQCD theory and as Coulomb branch isometry in the D- +type Dynkin quiver. It is, hence, natural to study gaugings of discrete Zq symmetries in +this manifest 0-form symmetry. +Starting with the Dynkin quiver, there are two distinct choices: Firstly, gauging a Zl +– 25 – + +on a U(l) node which satisfies 2 < l < 2k, one obtains6 +1 +... +l−1 +SU(l) +l+1 +... +2k−1 +2k +2k +... +2k +k +k +1 +N − 2k − 1 nodes +(2.46) +which has a Zl 1-form symmetry and the Coulomb branch isometry algebra is su(l) ⊕ +u(1)Q ⊕ so(N − l). For the global form, one can study the action of the centre symmetries +of the non-abelian factors. One finds +Gt(2.46) = PSU(l) × U(1)Q × Spin(2N − 2l) +ZDN−l +(2.47) +where the ZDN−l charges of Q are given by the charges of the congruence class of the j-th +fundamental representation [0, . . . , 0, 1, 0, . . . , 0]D with j = 2k − l, see Appendix A.3 for +details. For explicit examples including Hilbert series computations see Appendix D.7. +Alternatively, gauging a Zl on a U(l) node which satisfies l ≥ 2k, one obtains +1 +... +2k−1 +2k +... +2k +SU(2k) +2k +... +2k +k +k +1 +l − 2k nodes +(2.48) +and the Coulomb branch isometry algebra is the same as in (2.46). However, the “extra” +U(1) node is now attached to the balanced A-type Dynkin diagram such that the global +form is given by +Gt(2.48) = PSU(l) × U(1)Q +Zl +× PSO(2N − 2l) +(2.49) +where Q carries Zl charge 2k mod l, i.e. the charges of the congruence class of the 2k-th +fundamental representation, see Appendix A.3. Explicit examples for this discrete gauging +are given in Appendix D.7. +Global form via the mirror. +Analogous to the discussion in Sections 2.1.2, 2.3, and +2.5, one can confirm this global symmetry via the Higgs branch of the mirror theory. The +starting point is the mirror map (C.27) between the flavour fugacities of Sp(k) SQCD and +its unitary D-type Dynkin quiver, see Appendix C.1. This allows us to identify which +flavour fugacities are involved in the discrete Zf +l gauging on the SQCD side. +• l < 2k: The familiar argument then proceeds by splitting the fundamental flavours +into two distinct groups: the first l fundamental flavours are grouped as X, transform- +ing as ζlQ− 1 +l [1, 0, . . . , 0]Al, and the remaining N −l fundamental flavours, transform- +6The cases l = 1, 2 are addressed separately below. +– 26 – + +ing as [1, 0, . . . , 0]DN−l. Building a gauge invariant Higgs branch operator proceeds +in two steps: firstly, using the Sp(k) invariant tensors J on constructs operators of +the form XJ ˜X, which transform as ζl under the discrete symmetry. Secondly, Zl in- +variance is achieved via O = Syml(XJ ˜X), which transforms as Q−1[l, 0, . . . , 0]Al−1 ⊗ +[l, 0, . . . , 0]DN−l. The Zl centre surely acts trivial on [l, 0, . . . , 0]Al−1, while the centre +charges of [l, 0, . . . , 0]DN−l are (0, l mod 2) if N − l is even or (2l mod 4) if N − l is +odd. Thus, the non-trivial transformations under the centres can be compensated if +Q transforms as follows: +N − l = even +Zl × Z2 × Z2 charges of Q: +(0, 0, l mod 2) +(2.50a) +N − l = odd +Zl × Z4 charges of Q: +(0, 2l mod 4) +(2.50b) +which confirms (2.47). To see this, recall from Appendix A.3 that the congruence +class of the j-th fundamental representation of DN−l with j = 2k − l is (0, l mod 2) +for N − l even and 2l mod 4 for N − l odd. +• l > 2k: The argument is slightly modified: the first set X of flavours transforms +as ζ2kQ− 1 +2k [1, 0, . . . , 0]Al−1, while the second set ˜X transforms as [1, 0, . . . , 0]DN−l. +The Higgs branch operator O = Sym2k(XJ ˜X) transforms as Q−1[2k, 0, . . . , 0]Al−1 ⊗ +[2k, 0, . . . , 0]DN−l, which has trivial D-type centre charges. To see this, for N −l even, +the Z2 × Z2 charges are (0, 2k mod 2) = (0, 0); while for N − l odd, the Z4 charge +is 2 · 2k mod 4 = 0. Thus, to compensate potential irreps that are non-trivial under +Z2k, one requires that Q has the following charges: +N − l = even +Zl × Z2 × Z2 charges of Q: +(2k mod l, 0, 0) +(2.51a) +N − l = odd +Zl × Z4 charges of Q: +(2k mod l, 0) +(2.51b) +which then confirms (2.49). +Two special cases. +In the l = 2 case of (2.47), a symmetry enhancement is observed +in the explicit computations (D.66) and (D.81). These show that there is not only the +expected su(2)t, but the topological Cartan symmetry of the “new” U(1) gauge node is +also enhanced to a non-abelian su(2)t. These two su(2)t symmetries can both be interpreted +as PSO(4)t ∼= PSO(3)t × PSO(3)t. +As in previous sections, one can also gauge a discrete Zt +q along the topological fugacity +w1 associated to the first U(1) gauge node. The D-type Dynkin quiver is modified in the +by now familiar way: the bifundamental of U(1) × U(2) turns into a hypermultiplet that +transforms as fundamental under U(2) but is of U(1) charge q. In the mirror theory, the Zf +q +acts on a single fundamental flavour, as dictated by the mirror map (C.27). In summary, +– 27 – + +the mirror pair with Zq 1-form symmetry is +2k +2N−2 +Zq +←→ +1 +2 +... +2k−1 +2k +2k +... +2k +k +k +1 +N − 2k − 1 nodes +q +. +(2.52) +and the global Higgs / Coulomb branch isometry is +G = U(1)Q × Spin(2N − 2) +ZDN−1 +, +ZDN−1 charges of Q +� +(0, q mod 2) , +N = even +2q mod 4 , +N = odd +(2.53) +where Q is the topological fugacity of the left-most U(1) gauge node. +2.8 +Examples of non-simply laced unitary quivers and their mirrors +The last class of quiver theories considered here are non-simply laced unitary quivers7, +whose monopole formula has been proposed in [39]. Consider the following example +N1 +N2 +N3 +N4 +M1 +M2 +M3 +M4 +κ +(2.54) +with nodes U(N1,2) on the “short” side and U(N3,4) on the “long” side; wherein the naming +is borrowed from Dynkin diagrams. The multiplicity of the non-simply laced edge is denoted +by κ. Even though these quiver theories are non-Lagrangian (hence superconformal index +and Higgs branch Hilbert series are not computable by the standard methods), we can still +study their Coulomb branch using Hilbert series techniques. This allows us to investigate +the effects of gauging a discrete Zt +q symmetry. +Gauging at the long side. +To begin with, attempt to gauge a discrete Zt +N3 topological +symmetry associated to the U(N3) node at the long side, with N3 > 1. As a first step, one +rewrites (2.54) by expressing U(N3) ∼= (SU(N3) × U(1))/ZN3. By analogous arguments as +in Appendix B, one arrives at +� +�������� +N1 +N2 +SU(N3) +N4 +1 +m1 +ℓ +m2 +m4 +h +κ +M1 +κ +M2 +M4 +M3 +� +�������� +/ZN3 +with (m1, m2, l, m4, h) ∈ �Γ +(2.55) +7See, for example, [37, 38] for the appearance of such quiver theories via branes and ON planes. +– 28 – + +�Γ := +N3−1 +� +a=0 +� +Z + κ·a +N3 +�N1 × +� +Z + κ·a +N3 +�N2 × +� +Z + +a +N3 +�N3−1 +× +� +Z + +a +N3 +�N4 × +� +Z + +a +N3 +� +and the Coulomb branch moduli space is the same as that of (2.54). The green edges +transform in the fundamental representation of U(N1,2) and with charge κ under the U(1). +Next, the ZN3 symmetry is gauged. One obtains the following quiver description: +N1 +N2 +SU(N3) +N4 +1 +m1 +ℓ +m2 +m4 +h +κ +M1 +κ +M2 +M4 +M3 +with (m1, m2, l, m4, h) ∈ Γ +(2.56) +Γ := +N3−1 +� +a=0 +ZN1 × ZN2 × ZN3−1 × ZN4 × Z +where Γ is again short-hand for the integer magnetic lattice. This theory exhibits a ZN3 +1-form symmetry, by construction. +Gauging at the short side. +Now, consider gauging a Zt +N2 on the topological fugacity +associated to the U(N2) gauge node, with N2 > 1. Again, the first step is to simply rewrite +U(N2) ∼= (SU(N2) × U(1))/ZN2. By adopting the arguments of Appendix B, one finds +� +�������� +N1 +SU(N2) +N3 +N4 +1 +m1 +ℓ +m3 +m4 +h +κ +M1 +κ +M2 +M4 +M3 +� +�������� +/ZN2 +with (m1, l, m3, m4, h) ∈ �Γ +(2.57) +�Γ = +N2·κ−1 +� +a=0 +� +Z + +a +N2 +�N1 × +� +Z + +a +N2 +�N2−1 +× +� +Z + +a +N2·κ +�N3 × +� +Z + +a +N2·κ +�N4 × +� +Z + +a +N2·κ +� +whose Coulomb branch coincides with that of (2.54). Moreover, the edges highlighted in +green transform in the fundamental representation of U(N1,2) and with charge κ under the +U(1) node. As a next step, gauging the ZN2 results in the following theory: +N1 +SU(N2) +N3 +N4 +1 +m1 +ℓ +m3 +m4 +h +κ +M1 +κ +M2 +M4 +M3 +with (m1, l, m3, m4, h) ∈ Γ +(2.58) +Γ = +κ−1 +� +a=0 +(Z + a)N1 × (Z + a)N2−1 × +� +Z + a +κ +�N3 × +� +Z + a +κ +�N4 × +� +Z + a +κ +� +– 29 – + +where Γ is a short-hand notation for several shifted copies of the standard integer lattice +of the magnetic charges. +Comment. +One could also gauge a discrete Zt +q along the topological Cartan U(1) of a +U(1) gauge node. In this case, the connected hypermultiplets are modified to have charge +q under the U(1), but no other changes to the quiver occur. +2.8.1 +C-type quivers +A representative example is the mirror pair of O(2k) SQCD with N hypermultiplets in the +vector representation and its C-type Dynkin mirror quiver +O(2k) +2N +←→ +1 +2 +... +2k +2k +... +2k +2k +1 +N gauge nodes +(2.59) +which can be realised by a systems of D3-D5-NS5 branes with O5 and ON planes, re- +spectively. The logic is the same as before: Choose a Zt +q in the C-type Dynkin quiver, +by selecting a gauge node and its associated topological fugacity. Using the mirror map +(C.33) for (2.59) one identifies how the Zf +q acts on the vectors. For concreteness, consider +examples for k = 1 and N = 4: +Example: gauging on the long side. +Gauging a Zt +2 on the fourth node yields the +mirror pair (i.e. using (C.33) with discrete variable on w4) +O(2) +Z2 +8 +←→ +1 +2 +2 +SU(2) +1 +2 +(2.60) +where the ‘new’ U(1) node is connected with a hypermultiplet of charge 2. The global +symmetry algebra is su(4) ⊕ u(1), as read off from the balanced set of nodes. Explicit +Hilbert series (D.100) show that the global form is +Gf(LHS (2.60)) = Gt(RHS (2.60)) = PSU(4) × U(1)Q +(2.61) +because the Z4 centre acts trivial on all appearing representations. +– 30 – + +Example: gauging on the short side. +Gauging a Zt +2 on the third gauge node results +in the new mirror pair (i.e. using (C.33) with discrete variable on w3) +O(2) +2 +Z2 +6 +←→ +1 +2 +SU(2) +2 +1 +2 +(2.62) +the global symmetry algebra is su(3) ⊕ u(1) ⊕ sp(1), as suggested by the balanced nodes in +the unitary quiver. Recalling the maximal subalgebra su(3) ⊕ u(1) ⊂ sp(3), an analysis of +the Hilbert series then suggests that the global form is +Gf(LHS (2.62)) = Gt(RHS (2.62)) = PSp(3) × PSp(1) . +(2.63) +See (D.104) for explicit computations. +Example: gauging on the short side. +Gauging a Zt +2 on the second gauge node results +in the new mirror pair (i.e. using (C.33) with discrete variable on w2) +O(2) +4 +Z2 +4 +←→ +1 +SU(2) +2 +2 +1 +2 +(2.64) +and the balanced set of nodes suggests the symmetry algebra su(2)⊕u(1)⊕sp(2). A Hilbert +series computation (D.108) then indicates the following symmetry group +Gf(LHS (2.64)) = Gt(RHS (2.64)) = PSp(2) × PSp(2) . +(2.65) +This suggests that the su(2) ⊕ u(1) realise a maximal subalgebra in one sp(2) factor. +2.8.2 +B-type quivers +Alternatively, we could consider an Sp(k) gauge theory with SO(2n + 1) flavour symmetry. +However, to prevent a parity anomaly, we would need to include a suitable Chern-Simons +term. The Higgs branch, which is not affected by Chern-Simons levels, is known to be the +closure of a B-type nilpotent orbit. Therefore, a natural mirror theory would be a B-type +Dynkin quiver, for which analogous arguments apply as above. +2.8.3 +A comment on F4 Coulomb branch quivers +The reasoning can be also applied to other non-simply laced Coulomb branch quivers, even +if there may not exist a known mirror. Such an example is the F4 Coulomb branch quiver +of [40]. Table 2 summarises the resulting theories after a suitable Zt +n is gauged, following +the prescriptions (2.58) and (2.56). +Here, a few remarks in comparison to the “ungauging scheme” of [41] are in order. +The ungauging scheme involves removing a U(1) factor from a selected U(n) gauge group, +– 31 – + +which in the context of the monopole formula means setting one of the magnetic charges to +zero. For simply-laced quivers, this procedure leads to the same consequence as replacing +a U(n) gauge group with an SU(n) and quotienting out a diagonal Zn. +However, the ungauging scheme becomes problematic when applied to a node on the +short side of non-simply laced quivers. If the short node is a U(1) gauge group, then the +ungauging simply converts it into a flavour group. In [41], the ungauging of the short +U(1) node in the F4 quiver leads to a Coulomb branch that is the next-to-next-to minimal +nilpotent orbit closure of so(9). In contrast, if the short node is non-abelian, such as the +U(2) node in the F4 quiver, the resulting moduli space cannot be identified with any known +space and the procedure has been argued to be “invalid” in [41]. +On the other hand, by replacing the short U(2) node with an SU(2) and following +the prescriptions in (2.58) and (2.56), one is able to obtain consistent results, as shown in +the fourth row of Table 2. The resulting Coulomb branch is the next-to-next-to minimal +nilpotent orbit closure of so(9) as well.8 It is to be noted, that if one uses the prescriptions +(2.57) and (2.55), then one recovers the original minimal nilpotent orbit closure of F4. +2.9 +Magnetic quivers and gauging discrete topological symmetries +Suppose that one is given an unframed unitary magnetic quiver T with only simply-laced +edges (i.e. bifundamental hypermultiplets between the unitary gauge nodes). To evaluate +the Hilbert series or the index, it is necessary to remove an overall U(1) gauge group factor. +In [31], it was emphasised that choosing this U(1) from a U(k) gauge node leads to an SU(k) +gauge node, but the magnetic lattice Γ is extended to include shifted versions of the form +�k−1 +i=0 +� +Γ + i +k +� +. This situation can also be understood from a complementary perspective. +Given an unframed unitary magnetic quiver, pick a U(k) gauge node and rewrite it +as U(k) ∼= SU(k)×U(1) +Zk +, with fluxes (l, h) ∈ �k−1 +i=0 +�� +Z + i +k +�k−1 , +� +Z + i +k +�� +. The aim is to +remove this U(1) factor. As demonstrated in Appendix B, this rewriting shifts all other +magnetic fluxes m by the flux h associated to the U(1); as a result, all magnetic fluxes +receive the shifts Γ + i +k simultaneously. Now, removing this U(1) means treating it as a +background vector multiplet. Nevertheless, all remaining magnetic fluxes are still subject +to the shifts Γ + i +k. Hence, the Coulomb branch Hilbert series, as well as the index for T , +have the form +FT = +� +(l,m)∈�k−1 +i=0 (Γ+ i +k ) +f(l, m) +(2.66) +which is message conveyed in [31]. +It turns out that one can refine FT by introducing a Zk-valued fugacity z as follows: +the U(1)t topological symmetry of the U(k) node appears in both the monopole formula +and the index through the factor w +�k +a=1 ma +k +. Upon rewriting into magnetic fluxes (l, h) +for (SU(k) × U(1)) /Zk, this becomes wk·h +k . Since h ∈ �k−1 +i=0 (Z + i +k), one has wk·n+i +k +for +h = n + i +k ∈ (Z + i +k) and some n ∈ Z. This means that one can introduce a discrete +8In general, for non-simply laced quivers, the prescriptions (2.58) do not always provide the same +Coulomb branch for all the short nodes. +– 32 – + +quiver +symmetry +Coulomb branch Hilbert series +2 +3 +2 +1 +1 +F4 +1 + χ1,0,0,0t + χ2,0,0,0t2 + χ3,0,0,0t3 + . . . += 1 + 52t + 1053t2 + 12376t3 + . . . +SU(2) +3 +2 +1 +1 +A1 × C3 +1+t(χ2,0,0+φ2)+t2(1+χ4,0,0+χ0,2,0+χ2,0,0φ2+ +χ0,0,2φ2 + φ4) + . . . += 1 + 24t + 537t2 + . . . +2 +SU(3) +2 +1 +1 +A2 × A2 +1 + t(χ1,1 + φ1,1) + t2(1 + χ1,1 + χ2,2 + χ1,1φ1,1 + +χ2,2φ1,1 + φ2,2 + φ1,1) + . . . += 1 + 16t + 351t2 + . . . +2 +3 +SU(2) +1 +1 +A3 × A1 ⊂ B4 +1+t(χ2+χ2φ0,1,0+φ1,0,1)+t2(1+χ4+χ2φ2,0,0+ +φ0,1,0 + χ2φ0,1,0 + χ4φ0,1,0 + φ0,2,0 + χ4φ0,2,0 + +φ1,0,1+2χ2φ1,0,1+χ2φ1,1,1+χ2φ0,0,2+φ2,2)+. . . += 1 + 36t + 621t2 + . . . +2 +3 +2 +1 +1 +2 +C3 × U1 ⊂ C4 +1 + t(Qχ1,0,0 + χ1,0,0 +Q ++ χ0,1,0 + 1) + t2(Q2χ2,0,0 + +χ2,0,0 +Q2 +Qχ0,0,2+Qχ1,0,0+Qχ1,1,0+ χ0,0,2 +Q ++ χ1,0,0 +Q ++ +χ1,1,0 +Q ++ χ0,0,2 + 2χ0,1,0 + χ0,2,0 + χ2,0,0 + 1) + . . . += 1 + 36t + 621t2 + . . . +2 +3 +2 +1 +1 +3 +C3 × U1 +1 + t(χ0,1,0 + 1) + t2(Qχ1,0,1 + χ1,0,1 +Q ++ χ0,0,2 + +2χ0,1,0 + χ0,2,0 + χ2,0,0 + 1) + . . . += 1 + 22t + 369t2 + . . . +2 +3 +2 +1 +1 +4 +C3 × U1 +1 + t(χ0,1,0 + 1) + t2(Qχ2,0,0 + χ2,0,0 +Q ++ χ0,0,2 + +2χ0,1,0 + χ0,2,0 + χ2,0,0 + 1) + . . . += 1 + 22t + 327t2 + . . . +Table 2. The F4 Coulomb branch quiver and its Zq gaugings. The first row is the standard F4 +quiver proposed in [40]. Rows 2 - 4 display different choices of gauging a Zt +N of a U(N) node in the +Coulomb branch quiver for the minimal nilpotent orbit closure of F4. The gaugings on the “long” +side produce global symmetries given by the balanced set of nodes. For the gauging on the “short” +side, the global so(9) symmetry is only visible via the subalgebra su(4) ⊕ su(2). Rows 5 - 7 display +the effects of gauging a Zt +q inside the topological Cartan factor of the U(1) gauge node. For q = 2, +the symmetry algebra is enhance from sp(3) × u(1) to sp(4); while for q > 2, the algebra is simply +sp(3)×u(1). In the Hilbert series expressions, χ and φ are characters for the non-abelian symmetry +factors and Q is a U(1) fugacity. +fugacity z to keep track of the Zk centre symmetry, setting wk → z such that wk·n+i +k += zi. +This fugacity remains even if the U(1) is taken to be non-dynamical. One ends up with +FT (z) = +k−1 +� +i=0 +zi +� +(l,m)∈(Γ+ i +k ) +f(l, m) . +(2.67) +– 33 – + +It is now clear what happens if this discrete Zt +k topological symmetry is gauged: the entire +range of the summation collapses to the i = 0 sector, i.e., the integer lattice +FT /Zt +k = 1 +k +k−1 +� +i=0 +FT +� +z = (ζk)i� += +� +(l,m)∈Γ +f(l, m) . +(2.68) +Consequently, the quiver theory, in which U(k) is replaced by an SU(k) and the magnetic +lattice is simply the integer lattice, is obtained from the unframed unitary quiver T by +gauging a discrete Zt +k topological symmetry. This Zk distinguishes between SU(k)×U(1) +Zk +∼= +U(k) and SU(k) × U(1). Additionally, the gauging of the Zt +k symmetry has introduced a +Zk 1-form symmetry into T /Zt +k. +2.10 +Examples from 5d magnetic quivers +One can demonstrate gauging discrete subgroups of the topological symmetry on known +magnetic quivers9. It is most suitable to choose quivers whose Coulomb branches have a +known Higgs branch realisation. +E5 quiver. +The infinite coupling magnetic quiver for 5d Sp(1) SQCD with 4 flavours +realises O +E5 +min ∼= O +D5 +min, which is also the Higgs branch of Sp(1) with 5 flavours. Thus, one +arrives at [48] +� +����� +2 +2 +4 +2 +2 +1 +� +����� +/Z2 +←→ +2 +10 +(2.69) +It is worth recalling that the magnetic lattice for the left-hand side quiver has the form +Γ ∪ (Γ + 1 +2) with Γ being the standard GNO integer lattice, as can be found in [31]. The +corresponding discrete Zt +2 topological symmetry of the magnetic quiver can be gauged in +the same vein as before. On the level of the magnetic quiver, this just reduces the relevant +magnetic lattice to the integer lattice Γ. Equivalently, one can gauge a Zf +2 on the Sp(2) +SQCD side, which then gives rise to the following pair of theories +2 +2 +4 +2 +2 +1 +←→ +2 +8 +Z2 +2 +(2.70) +and it is straightforward to verify that the Coulomb / Higgs branch Hilbert series reproduce +the results of [31, Tab. 9]. The global form of the 0-form symmetry is PSO(8) × U(1). +E4 quiver. +Similarly, the infinite coupling magnetic quiver for 5d Sp(1) SQCD with 3 +flavours realises O +E4 +min ∼= O +A4 +min via its Coulomb branch. Of course, this moduli space admits +9See also [42–47] for magnetic quivers of theories with 1-form symmetries. +– 34 – + +a known Higgs branch realisation and one arrives at +� +�������� +2 +2 +2 +1 +1 +2 +� +�������� +/Z2 +←→ +1 +5 +(2.71) +The magnetic lattice for the magnetic quiver is of the form Γ ∪ (Γ + 1 +2), so the associated +Zt +2 symmetry can be gauged. The question then becomes what Zf +2 symmetry is realised on +the SQED side. Through explicit calculations, one verifies that +2 +2 +2 +1 +1 +2 +←→ +1 +4 +Z2 +(2.72) +reproduces the known Hilbert series [31, Tab. 10]. +The isometry group in this case is +SO(6) × U(1). +Folded E6 quiver. +The infinite coupling magnetic quiver for 5d Sp(1) SQCD with 5 +flavours admits a Z2 outer automorphism. +Folding the corresponding magnetic quiver +leads to O +E6 +min → O +D5 +min on the Coulomb branch [49]. Since there is a known Higgs branch +realisation for D-type minimal nilpotent orbit closures, one arrives at +� +1 +4 +4 +2 +2 +� +/Z2 +←→ +2 +10 +(2.73) +where again the left-hand side quiver has a magnetic lattice of the form Γ ∪ (Γ + 1 +2). +Gauging this Zt +2 has a by now clear consequence on the magnetic quiver, as the GNO +lattice is reduced to the integer lattice. On the Sp(1) SQCD side, the corresponding Zf +2 is +realised as follows: +1 +4 +4 +2 +2 +←→ +2 +Z2 +(2.74) +– 35 – + +and one straightforwardly verifies the agreement of the Coulomb branch / Higgs branch +Hilbert series, which is given by +HS = 1 + 25t + 400t2 + 3864t3 + 26600t4 + 141672t5 + 621480t6 + 2337280t7 +(2.75) ++ 7763283t8 + 23265515t9 + 63954800t10 + O +� +t11� +and the global symmetry group is PSU(5) × U(1). +3 +Discussion and conclusions +In this paper, mirror pairs with non-trivial 1-form symmetry have been studied. Starting +from known mirror pairs with trivial 1-form symmetry, gauging of discrete Zq subgroups +of the 0-form symmetry allowed us to construct new mirror pairs with non-trivial 1-form +symmetry. +The main results are as follows: +1. It has been shown that theories T /Zt +q, obtained by gauging a discrete subgroup Zt +q +of the topological symmetry, may admit quiver descriptions if the discrete subgroup +is suitably chosen. +2. The mirror theories +� +T /Zt +q +�∨ can be constructed using T ∨/Zf +q , but the precise choice +of Zf +q in the flavour symmetry of T ∨ can be subtle. This paper provides a simple +algorithm for specifying Zf +q . +3. The global form of the 0-form symmetries of (T /Zt +q, T ∨/Zf +q ) have been derived using +both field theory methods and monopole operators (via the balanced set of nodes), +and the resulting symmetry groups have been verified through explicit Hilbert series +computations. +4. The interplay between continuous 0-form and discrete 1-form symmetries has been +studied using established field theory techniques and the equivalence classes of lines. +5. On the technical side, the gauging of discrete subgroups of the topological symmetry +on non-simply laced quivers has been proposed and tested on both long and short-side +gauge nodes. +A comment on the moduli spaces. +The maximal branches of the moduli space of +vacua in a theory T are the Coulomb branch C(T ) and the Higgs branch H(T ). These +are symplectic singularities that can be resolved when the theory T is given either an FI +parameter (for the Higgs branch) or a mass parameter (for the Coulomb branch). For +instance, consider SQED with N hypermultiplets of charge 1. This theory admits N − 1 +mass parameters that resolve the C2/ZN Coulomb branch, and a single FI parameter that +resolves the Higgs branch, specifically the minimal nilpotent orbit closure O +su(N) +min . If we +gauge a Zt +q 0-form symmetry in this theory, the resulting SQED with charge q hypers has +the same Higgs branch, but the Coulomb branch is modified to be C2/ZN·q. However, +– 36 – + +there are no additional mass parameters in the theory, which means that the singularity +cannot be fully resolved even though a symplectic resolution exists. +More generally, one can perform a simple test10 via Hilbert series that shows +T −→ +� +� +� +� +� +� +� +� +� +� +� +T /Zt +q : +limt→1 +HSC(T /Ztq)(t) +HSC(T )(t) += 1 +q +or +T /Zf +q : +limt→1 +HSH(T /Zf +q )(t) +HSH(T )(t) += 1 +q +(3.1) +and the presence of a +1 +q fraction in the expression suggests (at least locally) that the +Coulomb branch C(T /Zt +q) is a Zq orbifold of C(T ), and a similar relationship holds for the +Higgs branches H(T /Zf +q ) and H(T ). Again, no additional deformation parameter appears. +In contrast, consider T to be U(2) SQCD with 4 fundamental flavours. The maximal +branches are H(T ) = O +su(4) +(22) and C(T ) = S(22) ∩ Nsu(4), i.e. the Slodowy slice to the su(4) +nilpotent orbit defined by partition (22). There are 3 masses resolving the Coulomb branch +and 1 FI term resolving the Higgs branch. If we gauge the topological U(1)t symmetry +in this theory, the resulting theory is SU(2) SQCD with 4 fundamental flavours. Then, +the Coulomb branch of this theory is C(T /U(1)t) = C2/D4 while the Higgs branch is +H(T /U(1)t) = O +so(8) +min . In this case, the Coulomb branch can be resolved by the 3 + 1 mass +parameters, while the minimal orbit closure of so(8) does not admit a symplectic resolution, +which is consistent with the absence of an FI parameter in this theory. These symplectic +resolutions can also be studied via Hilbert series techniques, see for instance [51, 52]. +Generalisations and open questions. +In this work, a single Zq factor of the 0-form +symmetry has been gauged. One straightforward generalisation is to consider orthosym- +plectic quivers and gauge several Zt +2 topological symmetry factors associated to SO(ni) +nodes. The resulting theory is simply obtained by replacing the relevant SO(ni) → Spin(ni) +and the 1-form symmetry is the product group � +i(Zt +2)i. Similarly, one could also enter- +tain the thought of gauging several Zqi inside distinct topological Cartan factors of, say, +T[SU(N)]. It is a priori not clear if a simple quiver description exists. +Another aspect of 3d mirror symmetry is the exchange of Wilson and vortex line +defects [29, 45, 53]. Given the central role of line defects in the understanding of 1-form +and 2-group symmetries, it would be interesting to systematically analyse the exchange of +Wilson and vortex lines under mirror symmetry systematically for the theories with 1-form +symmetry. +Acknowledgments +We would like to thank Fabio Apruzzi, Lakshya Bhardwaj, Mathew Bullimore, Andrea Fer- +rari, Heeyeon Kim, Noppadol Mekareeya, Matteo Sacchi, and Sakura Sch¨afer-Nameki for +discussions. The research of S.N. is supported by the National Science Foundation of China +10Following [50], the volume of the Sasakian base S of H or C is evaluated via Vol(S) = limt→1(1−t)dHS(t), +where d = dimC(H or C). +– 37 – + +under Grant No. 12050410234 and Shanghai Foreign Expert grant No. 22WZ2502100. M.S. +is grateful to Ryo Suzuki for use of his computing facilities. M.S. is also grateful to Rudolph +Kalveks for invaluable help with Mathematica. +A +Notations and conventions +node +vector +n +U(n) +SU(n) +SU(n) +n +SO(n) +Spin(n) +Spin(n) +2n +Sp(n) +Zq +Zq +(a) +edge +hyper +n +k +bifundamental n ⊗ k +n +SU(k) +bifundamental n ⊗ [0, . . . , 0, 1]A +n +2k +half-hyper [1, 0, . . . , 0]D/B ⊗ [1, 0, . . . , 0]C +Spin(n) +2k +half-hyper in vector × vector +n +k +N +N copies of bifundamental +n +Zq +N +N copies of fundamental +n +1 +Q +fundamental of U(n) but charge Q of U(1) +(b) +Table 3. Notation for nodes and links in the quiver diagrams. +A quiver diagram, composed of nodes and edges, encodes a 3d N = 4 theory as follows: +• Gauge nodes ⃝ denote dynamical vector multiplets, while flavour nodes □ denote +background vector multiplets. The notations are summarised in Table 3a. +• An edge between two nodes corresponds to a hypermultiplet H = (X, Y †), with X, Y +two N = 2 chiral multiplets. The notation is summarised in Table 3b. +• An exception are so-called non-simply laced edges in a quiver theory. Between unitary +gauge node, such an edge has been proposed purely on the level of the conformal +dimension of the monopole formula [39] +n +k +κ +←→ +1 +2 +n +� +i=1 +k +� +j=1 +|m1,i − κ · m2,j| +(A.1) +and it is to stress that this does not correspond to a representation of the gauge +groups. For the special case of U(N = 1), such a non-simply laced edge is effectively +the same as a U(1) gauge group with a charge κ hypermultiplet. +– 38 – + +Between orthosymplectic nodes, the conformal dimension has been proposed in [49] +n +2k +κ +←→ +1 +2 · 2 +� +ρ∈[1,0,...,0]B/D +� +λ∈[1,0,...,0]C +|ρ(m) − κ · λ(n)| +(A.2) +with m, n the magnetic fluxes which are evaluated on the weights ρ, λ, respectively. +A.1 +Hilbert series +A.1.1 +Monopole formula +The Hilbert series for the 3d N = 4 Coulomb branch is known as the monopole formula +[26]. Schematically, the Hilbert series is computed as a sum over magnetic fluxes m valued +in the GNO lattice Γ of the gauge group G. +HSC = +� +m∈Γ/W +P(t, m)wmt∆(m) +(A.3) +and W denotes the Weyl group of G. A bare monopole operator is characterised by the flux +m as well as its conformal dimension ∆(m), which coincides with the third component of +the SU(2)R spin. The factors P(t, m) dress a bare monopole operator by gauge invariants +formed by the adjoint chiral multiplet of the residual gauge group H(m). Lastly, w denotes +the fugacity of the topological symmetry, assuming that G contains U(1) factors. +A.1.2 +Higgs branch Hilbert series +The Higgs branch Hilbert series [23–25] for the 3d N = 4 quiver gauge theory relevant here +is schematically obtained by a Molien-Weyl integral of the form +HSH = +� +G +dµG +PE[χG +Adj t] +PE[χG +R · χF +F t +1 +2 ] +(A.4) +where the numerator contains the character χG +Adj of the adjoint representation of the gauge +group, while the denominator contains all matter fields characterised by their representa- +tions R under the gauge group G and the representations F under the flavour symmetry +F. +A.1.3 +Gauging a discrete 0-form symmetry. +Suppose one is given a generating function H(z|t) which is a power series in t with coeffi- +cients that are Laurent polynomials in a U(1) fugacity z. Next, embed a Zq �→ U(1) via +(ζq)p = e +2πip +q +with p = 0, 1, . . . , q − 1. Gauging this discrete Zq 0-form symmetry is realised +in terms of the generating function via a discrete Molien-Weyl sum +1 +q +q−1 +� +p=0 +H +� +(ζq)p · y +1 +q |t +� +(A.5) +where y is the fugacity for the residual U(1)/Zq ∼= U(1) symmetry. +– 39 – + +A.2 +Superconformal index +The 3d superconformal index can be computed as partition function on S2 × S1 via local- +isation techniques, see [54–60] for details. Schematically, one arrives at +Z = +� +m +1 +|Wm| +� +Trk(G) +rk(G) +� +i=1 +dsi +2πisi +Icl · Ivec · Imatter +(A.6) +where s denotes the gauge fugacities, which are valued in a maximal torus of the gauge +group G. The magnetic fluxes m take values in the GNO-lattice of G. A flux m breaks +G to the residual gauge group Hm (the stabiliser subgroup of m inside G) with Weyl +group WHm ≡ Wm. The integration contour is chosen to be the unit circle T for each si. +The integrand is composed of classical contributions and the 1-loop determinants of the +supermulitpelts. For concreteness, the G = U(N) case is reviewed: +The classical contribution is given by +IU(N) +cl +(w, m; n) = +N +� +a=1 +(sa)n w +�N +a=1 ma +(A.7) +with w the fugacity of the topological U(1)t symmetry. +The N = 2 multiplets have the following 1-loop determinants: +• 3d N = 2 Chiral multiplet of R-charge r coupled with unit charge to a gauge field: +Ir +chi(z, m| x) = +� +x1−rz−1� |m| +2 +∞ +� +j=0 +1 − (−1)mz−1x|m|+2−r+2j +1 − (−1)mzx|m|+r+2j +(A.8) += +� +x1−rz−1� |m| +2 +� +(−1)mz−1x|m|+2−r; x2� +∞ +� +(−1)mzx|m|+r; x2� +∞ +with a U(1) holonomy z around S1 and the Z-valued magnetic flux m on S2. Here, +the definition (z; q)∞ = �∞ +j=0(1 − zqj) has been used. +• 3d N = 4 Hypermultiplet transforming as bifundamental of U(N) × U(M) +IU(N)×U(M) +hyp +(s1, m1; s2, m2| x) = +N +� +a=1 +M +� +b=1 +I +1 +2 +chi +� +s1,as−1 +2,b, m1,a − m2,b| x +� +(A.9) +· I +1 +2 +chi +� +s−1 +1,as2,b, m2,b − m1,a| x +� +• 3d N = 2 vector multiplet for a U(N) gauge group: +IU(N) +vec +(s, m| x) = +� +a 1 and gauge +a discrete subgroup Zd ⊂ U(1)t of the k-th Cartan subgroup of the topological sym- +metry, provided d|k. One can repeat all steps as above, i.e. rewriting all contributions +as U(k) ∼= (U(1) × SU(k)) /Zk. The only step that requires modifications is (B.4). Re- +call h ∈ �k−1 +i=0 +� +Z + i +k +� +and non-trivial contributions arise for d|(k · h). Since d|k, rele- +vant fluxes h need to satisfy k +d · h ∈ Z, the summation range after Zd gauging becomes +h ∈ � k +d −1 +i=0 +� +Z + i +k +d +� +. Therefore, the resulting theory is given by (B.6), but the magnetic +fluxes take values in (h, l, {na}a̸=k) ∈ � k +d −1 +p=0 +� +Γ + d +k · p +� +. See also [31] for a related discus- +sion. +B.2 +Gauging discrete subgroup of topological symmetry revisited +In this appendix, the aim is to consider a more general choice of discrete subgroup and to +gain further evidence on the resulting theory. Starting from +1 +s1 +m1 +w1 +2 +{s2,j} +{m2,j} +w2 +... +k−1 +{sk−1,j} +{mk−1,j} +wk−1 +k +{sk,j} +{mk,j} +wk +k+1 +{sk+1,j} +{mk+1,j} +wk+1 +... +N−1 +{sN−1,j} +{mN−1,j} +wN−1 +N +{yj} +{kj} +(B.24) +define the superconformal index of T[SU(N)] as +ZT[SU(N)] = +N−1 +� +ℓ=1 +� +� � +mℓ∈Zℓ +1 +|Wmℓ| +� +Tℓ +ℓ� +a=1 +dsℓ,a +2πisℓ,a +� +� IT[SU(N)] +� +{sℓ}N−1 +ℓ=1 ; {mℓ}N−1 +ℓ=1 +� +(B.25) +IT[SU(N)] = +N−2 +� +ℓ=1 +IU(ℓ)×U(ℓ+1) +hyp +(sℓ, mℓ; sℓ+1, mℓ+1| x) +(B.26) +· IU(N−1)×U(N) +hyp +(sN−1, mN−1; y, k| x) +· +N−1 +� +ℓ=1 +IU(ℓ) +cl +(wℓ, mℓ; nℓ) IU(ℓ) +vec (sℓ, mℓ| x) +and repeat the analogous step as in the monopole formula. Pick a gauge node U(k) and +relabel the magnetic fluxes mk into a (U(1) × SU(k)) /Zk fluxes (h, l), see (B.9) and (B.2). +Likewise, the U(k) gauge fugacities sk are transformed into U(1) × SU(k) fugacities (S, σ) +– 45 – + +via +� +� +� +� +� +� +� +sk,1 = +S−1 · σ1 , +sk,i = +S−1 · σi · σ−1 +i−1 , +1 < i < k , +sk,k = +S−1 · σ−1 +k−1 . +(B.27) +Consider the vector multiplet +IU(k) +cl +(wk, sk, mk; nk) = +� +S−k�nk w−k·h +k +=IU(1) +cl +(wk +k, S−k, −h; nk) +(B.28) +IU(k) +vec (sk, mk| x) = +� +a f − 1 +0 , +j ≤ f − 1 +(C.6) +for f = 1, . . . , N and j = 1, . . . , N−1. Hence, the mirror map becomes +yf = +N−1 +� +j=1 +w +�N−1 +i=1 M[N] +fi C−1 +ij +j += +f−1 +� +i=1 +w +− i +N +i +N−1 +� +j=f +w +1− j +N +j +(C.7) +for f = 1, . . . , N. +– 50 – + +C.1.2 +Mirror map after gauging +Suppose that one gauges a Zq on the U(1)t generated by wk and employs the following +parameter map +� +� +� +� +� +� +� +� +� +� +� +wi = �k−1 +j=1 x +C +Ak−1 +ij +j +, +i = 1, 2, . . . , k − 1 +wk = +Q +(xk−1 u1)q +wi+k = �N−k−1 +j=1 +u +C +AN−k−1 +ij +j +, +i = 1, 2, . . . , N−k − 1 +(C.8) +using the Cartan matrices of Ak−1 and AN−k−1, respectively. Using (C.7) for Ak−1 and +AN−k−1, the map for wk can be expressed as +wk = ζq +Q +1 +q +�k−1 +i=1 w +i +k +i +�N−1 +j=k+1 w +1− j−k +N−k +j +. +(C.9) +A straightforward computation yields the new mirror map +yf = +� +� +� +ζ +1− k +N +q +Q +N−k +q·N · �f−1 +i=1 w +− i +k +i +· �k−1 +j=f w +1− j +k +j +, +f ≤ k +ζ +− k +N +q +Q +−k +N·q · �f−k−1 +i=1 +w +− +i +N−k +k+i +· �N−k−1 +j=f−k w +1− +j +N−k +k+j +, +f ≥ k + 1 +(C.10a) += +� +� +� +� +� +ζ +1− k +N +q +Q +N−k +q·N · �k−1 +i=1 x +M[k] +fi +i +, +f ≤ k +ζ +− k +N +q +Q +−k +N·q · �N−k−1 +i=1 +u +M[N−k] +fi +i +, +f ≥ k + 1 +(C.10b) +using the parametrisation (C.8) for the Ak−1 fugacities xi and the AN−k−1 fugacities ui. +Remark. +The map (C.10) assigns ζq charges to each yf; however, one can remove any +overall U(1) phase by a gauge transformation. This leads to two convenient choices: either +the first k fundamental flavours are charged under Zq +yf = +� +� +� +� +� +ζq · Q +N−k +q·N · �k−1 +i=1 x +M[k] +fi +i +, +f ≤ k +Q +−k +N·q · �N−k−1 +i=1 +u +M[N−k] +fi +i +, +f ≥ k + 1 +(C.11) +by rotating via ζ +k +N +q . Alternatively, one rotates via ζ +−1+ k +N +q +such that only the last N−k +fundamental flavours are non-trivially charged under Zq. In principle, one could also extend +the overall rotation to include Q, but there is no need to do so. +C.2 +T[SU(N)] theories +One can construct the mirror map explicitly. +– 51 – + +C.2.1 +Standard mirror map +Denote the Coulomb branch root space fugacities of T[SU(N)] by {wi}N−1 +i=1 . These can be +mapped to the Coulomb branch weight space fugacities via the AN−1 Cartan matrix Cij: +wi = +� +j +ωCij +j +. +(C.12) +The Higgs branch U(N) flavour fugacities are {ya}N +a=1, which are reduced to SU(N) fugac- +ities {ηi}N−1 +i=1 +via +yf = +N−1 +� +i=1 +η +M[N] +fi +i +with +M[N] +fi += δf,i − δf,i+1 +for +� +i += 1, . . . , N−1 , +f += 1, . . . , N . +(C.13) +The self-mirror property of T[SU(N)] is the reflection in the exchange ωi ↔ ηi. +The aim is to express the natural flavour fugacities {ya} of the theory in terms of the +Coulomb branch fugacities of the mirror. The first step is +(y1, y2, . . . , yN) → +� +η1, η2 +η1 +, . . . , +1 +ηN−1 +� +ωi↔ηi +−−−−→ +� +ω1, ω2 +ω1 +, . . . , +1 +ωN−1 +� +ωi→� +j w +C−1 +ij +j +−−−−−−−−→ (f1(wi), f2(wi), . . . , fN(wi)) . +(C.14) +This map can be made explicit by using the inverse of the AN−1 Cartan matrix (C.2). +Analogous to the abelian case, the combined map reads +yf = +N−1 +� +j=1 +w +�N−1 +i=1 M[N] +fi C−1 +ij +j +with +N−1 +� +i=1 +M[N] +fi C−1 +ij += min(f, j) − min(f − 1, j) − j +N += +f−1 +� +i=1 +w +− i +N +i +N−1 +� +j=f +w +1− j +N +j +, +(C.15) +which is the explicit form of (C.14). +C.2.2 +Mirror map after gauging +The next step is utilising the parameter map (B.21) established in Appendix B.1 +wk = ζq +� +� +� +v−1 +�N−1 +i=1 +i̸=k +wi +i +� +� +� +1 +k +. +(C.16) +– 52 – + +Lastly, to make contact with the global symmetries, one uses the Cartan matrix for Ak−1 +and AN−1−k in a standard fashion +wi = +k−1 +� +j=1 +x +C +Ak−1 +ij +j +, +i = 1, . . . , k − 1 , +wi+k = +N−1−k +� +j=1 +u +C +AN−1−k +ij +j +, +i = 1, . . . , N−1 − k , +(C.17) +and one needs to redefine +v = +Q−1 +(uN−1−k)N = +Q−1 +�N−1 +r=k w +r−k +N−k ·N +r +. +(C.18) +Applying this to (C.15), one finds +yf = +� +� +� +ζ +1− k +N +q +· Q +N−k +k·N · �f−1 +i=1 w +− i +k +i +· �k−1 +j=f w +1− j +k +j +, +for f ≤ k +ζ +− k +N +q +· Q− 1 +N · �f−k−1 +i=1 +w +− +i +N−k +i+k +· �N−k−1 +j=f−k w +1− +j +N−k +j+k +, +for f ≥ k + 1 +(C.19a) += +� +� +� +� +� +ζ +1− k +N +q +· Q +N−k +k·N · �k−1 +j=1 x +M[k] +fj +j +, +for f ≤ k +ζ +− k +N +q +· Q− 1 +N · �N−k−1 +j=1 +u +M[N−k] +fj +j +for f ≥ k + 1 +(C.19b) +which displays the split into Ak−1 fugacities xj and AN−k−1 fugacities uj. +Remark. +Analogously to the SQED case, one can simplify the ζq dependence in (C.19) +by a suitable overall U(1) rotation. A convenient choice is then given by +yf = +� +� +� +� +� +ζq · Q +N−k +k·N · �k−1 +j=1 x +M[k] +fj +j +, +for f ≤ k +Q− 1 +N · �N−k−1 +j=1 +u +M[N−k] +fj +j +for f ≥ k + 1 +(C.20) +such that only the first k fundamental flavours are charged under Zq. +C.3 +Examples for T σ +ρ [SU(N)] +C.3.1 +Example 1 +Consider the example T σ +ρ [SU(15)] with σ = [32, 22, 12] and ρ = [6, 4, 3, 12] of Section 2.4. +Using the labelling +1 +w1 +2 +w2 +2 +w3 +1 +w4 +3 +2 +2 +←→ +2 +2 +2 +2 +1 +1 +2 +y1,2 +1 +Q1 +1 +Q2 +1 +Q3 +(C.21) +– 53 – + +the mirror map is given by +� +� +� +y1 += w4/5 +1 +w3/5 +2 +w2/5 +3 +5√w4 , +y2 += w3/5 +2 +w2/5 +3 +5√w4 +5√w1 +, +� +� +� +� +� +� +� +� +� +� +� +Q1 += w2/5 +3 +5√w4 +5√w1w2/5 +2 +, +Q2 += +5√w4 +5√w1w2/5 +2 +w3/5 +3 +, +Q3 += +1 +5√w1w2/5 +2 +w3/5 +3 +w4/5 +4 +(C.22) +such that �2 +i=1 yi · �3 +j=1 Qj = 1 holds. +Gauging on w2. +The example of Figure 5 is realised by a Z2 gauging on w2. Inspecting +the mirror map (C.22) shows that one has two options for the Zf +2 gauging in the mirror +• The y1, y2 are charged as ζ2, while the Q1,2,3 are trivial under Z2. +• The y1,2 are trivial under Z2, and the Q1,2,3 transform with ζ2. +This reflects the two choices in Figure 5. +Gauging on w3. +Turning to Figure 4, one performs a Z2 gauging on w3. The mirror +map (C.22) indicates two options for the Zf +2 gauging in the mirror +• The y1, y2, Q1 are charged as ζ2, while the Q2,3 are trivial under Z2. +• The y1,2, Q1 are trivial under Z2, and the Q2,3 transform with ζ2. +Again, this confirms the two choices in Figure 4. +C.3.2 +Example 2 +The labelling for the T σ +ρ [SU(9)] example with ρ = (3, 23) and σ = (32, 13) of Section 2.4 is +defined by +2 +w1 +2 +w2 +2 +w3 +3 +2 +←→ +1 +2 +2 +1 +1 +Q +3 +y1,2,3 +(C.23) +and the mirror map is given by +y1 = w +1 +2 +2 w +1 +4 +3 +w +1 +4 +1 +, +y2 = +w +1 +4 +3 +w +1 +4 +1 w +1 +2 +2 +, +y3 = +1 +w +1 +4 +1 w +1 +2 +2 w +3 +4 +3 +, +Q = w +3 +4 +1 w +1 +2 +2 w +1 +4 +3 . +(C.24) +Analogous to the example above, gauging the Zf +2 associated to w3 has two convenient +realisations in the mirror theory: either (y1, y2, Q) transform non-trivial under Z2 and y3 +is trivial or vice versa. +C.4 +Sp(k) SQCD and its D-type unitary mirror quiver +The closed formula for the inverse Cartan matrix of DN is provided in [65]. +– 54 – + +C.4.1 +Standard mirror map +For the balanced DN-type unitary quiver, the mirror map to the flavour symmetry of the +Sp(k) SQCD mirror with N fundamental flavours is given by +• Let yi denote the U(N) flavour fugacities. +• Denote by xi the SO(2N) weight space fugacities. +The relation between both is +established via +yi = +� +� +� +� +� +� +� +� +� +� +� +� +� +x1 , +for i = 1 +xi +xi−1 , +for 1 < i < N − 2 +xN−1xN +xN−2 +, +for i = N−1 +xN−1 +xN +, +for i = N +(C.25) +• Denote by wi the root space fugacities of DN, which are related to the weight space +fugacities xi via the DN Cartan matrix +wi = +� +j +x +CDN +ij +j +(C.26) +• Thus, one finds the map between fundamental flavour fugacities yi and root space +fugacities wi to be +yf = +� +� +� +� +� +� +� +��N−2 +i=f wi +� √wN−1 · √wN , +for 1 ≤ f < N−1 +√wN−1 · √wN , +for f = N−1 +√wN−1 +√wN +, +for f = N +(C.27) +C.4.2 +Mirror map after gauging +Suppose that one gauges a discrete Zq symmetry at the gauge node with topological fugacity +wl. Then, analogous to the T[SU(N)] derivation, the fugacity map to the quiver after +gauging is simply given by +wl = +� +� +� +v−1 +�N +i=1 +i̸=l +(wi)Ni +� +� +� +1 +l +(C.28) +where Ni denotes the rank of the i-th node. The remaining topological fugacities wi̸=l are +identified before and after gauging. +Lastly, one needs to redefine v such that Al−1 and DN−l representations become man- +ifest. For this, one uses +v = +Q +weight space fugacity at extra U(1) +(C.29) +here, the weight space fugacity is either the Al−1 fugacity xi, if the extra U(1) intersects +– 55 – + +the balanced Al−1 Dynkin diagram at node i, or it is the DN−l weight space fugacity uj, if +the extra U(1) is attached at the j-th node of the balanced D-type Dynkin diagram. See +Appendix D.7 for examples. +For l ≤ N − 2, the mirror map is given by13 +yf = +� +� +� +� +� +� +� +� +� +� +� +� +� +Q− 1 +l �f−1 +i=1 w +− i +l +i +�l−1 +j=f w +1− j +l +j +for 1 ≤ f ≤ l +��N−2 +i=f wi +� √wN−1 · √wN , +for l + 1 ≤ f < N−1 +√wN−1 · √wN , +for f = N−1 +√wN−1 +√wN +, +for f = N +(C.30) +which clearly displays that the first l fundamental flavours transform under su(l) × U(1), +and the remaining N − l fundamental flavours transform under so(2N − 2l). +C.5 +O(2k) SQCD and its C-type unitary mirror quiver +Consider the O(2k) SQCD with N fundamental hypermultiplets. The mirror theory is a +balanced C-type Dynkin quiver with N gauge nodes. +• Denote the U(N) flavour fugacities by yi. +• Denote the Sp(N) flavour fugacities by xi. These are related via the transformation +y1 = x1 +and +yj = +xj +xj−1 +for j = 2, . . . , n . +(C.31) +• Denote the topological fugacities of the C-type quiver by wi. Then the CN Cartan +matrix mediates the transformation between root and weight space fugacities +wi = +N +� +j=1 +x +CCN +ij +j +for i = 1, . . . , N . +(C.32) +Combining the above leads to the mirror map between the unitary flavour fugacities and +the root space topological fugacities +� +yi = √wN +�N−1 +j=i wj +i = 1, . . . , N−1 +yN = √wN +(C.33) +D +Explicit Hilbert series results +In this appendix, some exemplary Hilbert series calculations are presented. Matching the +Hilbert series can be used as the stringent test to check the dualities and find the global +topological or flavour symmetry groups. +13The choice (C.29) implies that the U(1)Q charges are negative here. This choice is convenient because +the charges of Q under the centre symmetries are directly read off from the Coulomb branch quiver, see +(2.46) and (2.48). +– 56 – + +D.1 +Linear Abelian quiver +Consider the abelian quiver gauge theory +1 +w1 +... +1 +wk−1 +1 +wk +1 +wk+1 +... +1 +wN−1 +1 +1 +(q, −1) +(1, −q) +(D.1) +Explicit character expansions indicate the following global forms +N=3 +k=1 +Gt = +� +SO(3)y × U(1)Q , w/ Q of Z2-charge 0 +q = 2, 4, 6, 8 +SU(2)y×U(1)Q +Z2 +, w/ Q of Z2-charge +1 +q = 3, 5, 7, 9 +(D.2) +N=4 +k=1 +Gt = +� +� +� +� +� +� +� +PSU(3)y × U(1)Q , w/ Q of Z3-charge 0 +q = 3, 6 +SU(3)u×U(1)Q +Z3 +, w/ Q of Z3-charge +1 +q = 4, 7 +SU(3)u×U(1)Q +Z3 +, w/ Q of Z3-charge +2 +q = 2, 5 +(D.3) +N=4 +k=2 +Gt = +� +SO(3)x × U(1)Q × SO(3)u , w/ Q of Z2 × Z2-charge (0, 0) +q = 2, 4 +SU(2)x×U(1)Q×SU(2)u +Z2×Z2 +, w/ Q of Z2 × Z2-charge (+1, +1) +q = 3, 5 +(D.4) +N=5 +k=1 +Gt = +� +� +� +� +� +� +� +� +� +� +� +� +� +SU(4)u×U(1)Q +Z4 +, w/ Q of Z4-charge (+2) +q = 2 +SU(4)u×U(1)Q +Z4 +, w/ Q of Z4-charge (+3) +q = 3 +PSU(4)u × U(1)Q , w/ Q of Z4-charge (0) +q = 4 +SU(4)u×U(1)Q +Z4 +, w/ Q of Z4-charge (+1) +q = 5 +(D.5) +N=5 +k=2 +Gt = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +SU(2)x×U(1)Q×SU(3)u +Z2×Z3 +, w/ Q of Z2 × Z3-charge (0, +2) +q = 2 +SU(2)x×U(1)Q +Z2 +× PSU(3)u , w/ Q of Z2 × Z3-charge (+1, 0) +q = 3 +PSU(2)x × U(1)Q×SU(3)u +Z3 +, w/ Q of Z2 × Z3-charge (0, +1) +q = 4 +SU(2)x×U(1)Q×SU(3)u +Z2×Z3 +, w/ Q of Z2 × Z3-charge (+1, +2) +q = 5 +PSU(2)x × U(1)Q × PSU(3)u , w/ Q of Z2 × Z3-charge (0, 0) +q = 6 +SU(2)x×U(1)Q×SU(3)y +Z2×Z3 +, w/ Q of Z2 × Z3-charge (+1, +1) +q = 7 +(D.6) +which then confirms the general formula (2.7). +D.2 +T[SU(N)] theories +We move on to the examples of T[SU(N)] theories. The employed fugacity maps follow +from (2.18) in combination with (2.21). +D.2.1 +T[SU(3)] theories +1 +w1 +SU(2) +1 +v +3 +mirror +←−−−−−−→ +1 +2 +Z2 +1 +y3 +y1,2 +. +(D.7) +– 57 – + +Use the fugacity map +C : +w1 → x2 +1 , +v → Q−1 +H : +y1 → Q +1 +6 x1 , +y2 → Q +1 +6 x−1 +1 +, +y3 → Q− 1 +3 +(D.8) +to an x1 weight space fugacity for A1 and a U(1) variable Q. The Coulomb branch Hilbert +series of the left quiver (and Higgs branch Hilbert series of the right quiver) reads +HS = 1 + t (χ2 + 1) + t2 �� +Q + Q−1� +χ2 + 2χ2 + χ4 + 2 +� +(D.9) ++ t3 +� � +Q + Q−1� +(2χ2 + χ4 + 1) + 4χ2 + 2χ4 + χ6 + 2 +� ++ . . . +here χn1 are the SU(2)x1 characters for irreps with Dynkin labels [n1]. The term in red +corresponds to the operator O in (2.22). The symmetry algebra is su(2)x1 ⊕ u(1)Q. The +SU(2) centre symmetry Z2 acts trivial on irreps [2 · n1] for n1 ∈ N. Thus (D.9) suggests +that the symmetry group is SO(3)x1 × U(1)Q. +D.2.2 +T[SU(4)] theories +Gauging a Z3. +1 +w1 +2 +w2 +SU(3) +1 +v +4 +mirror +←−−−−−−→ +1 +2 +3 +Z3 +1 +y4 +y1,2,3 +. +(D.10) +Use the fugacity map +C : +w1 → x2 +1 +x2 +, +w2 → x2 +2 +x1 +, +v → Q−1 +H : +y1 → Q +1 +12 x1 , +y2 → Q +1 +12 x2 +x1 +, +y3 → Q +1 +12 1 +x2 +, +y4 → Q− 1 +4 +(D.11) +to xi weight space fugactites for A2 and a U(1) variable Q. The Coulomb branch Hilbert +series of the left quiver (and Higgs branch Hilbert series of the right quiver) reads +HS =1 + t (χ1,1 + 1) + t2 (3χ1,1 + χ2,2 + 2) +(D.12) ++ t3 +� +Qχ3,0 + Q−1χ0,3 + 2χ0,3 + 6χ1,1 + 3χ2,2 + 2χ3,0 + χ3,3 + 3 +� ++ t4 +� +Q (χ1,1 + χ2,2 + 2χ3,0 + χ4,1) + Q−1 (2χ0,3 + χ1,1 + χ1,4 + χ2,2) ++ 4χ0,3 + 11χ1,1 + 2χ1,4 + 9χ2,2 + 4χ3,0 + 3χ3,3 + 2χ4,1 + χ4,4 + 4 +� ++ . . . +here χk,n are the SU(3)x1x2 characters for irreps with Dynkin labels [k, n]. The terms in +red corresponds to the operator O (and its conjugate) in (2.22). The symmetry algebra +is su(3)x1,x2 ⊕ u(1)Q. The SU(3) centre symmetry Z3 acts trivial on irreps [n1, n2] with +n1−n2 = 0 mod 3. Thus (D.12) suggests that the symmetry group is PSU(3)x1,x2 ×U(1)Q. +– 58 – + +Gauging a Z2. +1 +w1 +SU(2) +3 +w3 +1 +v +4 +mirror +←−−−−−−→ +1 +2 +3 +Z2 +2 +y3,4 +y1,2 +. +(D.13) +Use the fugacity map +C : +w1 → x2 +1 , +w3 → u2 +1 , +v → Q−1 +(u1)4 +H : +y1 → Q +1 +4 x1 , +y2 → Q +1 +4 x−1 +1 +, +y3 → Q− 1 +4 u1 , +y4 → Q− 1 +4 u−1 +1 +(D.14) +to an x1 weight space fugactity of A1, u1 the weight space fugacity of another A1, and a +U(1) variable Q. The Coulomb branch Hilbert series of the left quiver (and Higgs branch +Hilbert series of the right quiver) reads +HS = 1 + t (ϕ2 + χ2 + 1) +(D.15) ++ t2 �� +Q + Q−1� +(1 + ϕ2χ2) + 2ϕ2 + ϕ4 + 2ϕ2χ2 + 2χ2 + χ4 + 3 +� ++ t3 +� � +Q + Q−1� +(3χ2ϕ2 + χ4ϕ2 + 2ϕ2 + ϕ4χ2 + 2χ2 + 1) ++ 6ϕ2 + 2ϕ4 + ϕ6 + 6ϕ2χ2 + 2ϕ4χ2 + 6χ2 + 2ϕ2χ4 + 2χ4 + χ6 + 4 +� ++ . . . +here χn1 are the SU(2)x1 characters for irreps with Dynkin label [n1]. While ϕk1 are the +SU(2)u1 characters for irreps [k1]. The term in red corresponds to the operator O in (2.22). +The symmetry algebra is su(2)x1 ⊕ su(2)u1 ⊕ u(1)Q. The SU(2) centre symmetries Z2 act +trivial on irreps [n1] with n1 = 0 mod 2 and [k1] with k1 = 0 mod 2, respectively. Thus +(D.15) suggests that the symmetry group is SO(3)x1 × SO(3)u1 × U(1)Q. +D.2.3 +T[SU(5)] theories +Gauging a Z4. +1 +w1 +2 +w2 +3 +w3 +SU(4) +1 +v +5 +mirror +←−−−−−−→ +1 +2 +3 +4 +Z4 +1 +y5 +y1,...,4 +. +(D.16) +Use the fugacity map +C : +w1 → x2 +1 +x2 +, +w2 → +x2 +2 +x1x3 +, +w3 → x2 +3 +x2 +, +v → Q−1 +(D.17) +H : +y1 → Q +1 +20 x1 , +y2 → Q +1 +20 x2 +x1 +, +y3 → Q +1 +20 x3 +x2 +, +y4 → Q +1 +20 1 +x3 +, +y5 → Q− 1 +5 +– 59 – + +to xi weight space fugacities of A3, and a U(1) variable Q. The Coulomb branch Hilbert +series of the left quiver (and Higgs branch Hilbert series of the right quiver) reads +HS = 1 + t (χ1,0,1 + 1) + t2 (χ0,2,0 + 3χ1,0,1 + χ2,0,2 + 2) +(D.18) ++ t3 (2χ0,1,2 + 2χ0,2,0 + 7χ1,0,1 + χ1,2,1 + 3χ2,0,2 + 2χ2,1,0 + χ3,0,3 + 3) ++ t4 +� +Q−1χ0,0,4 + Qχ4,0,0 + 6χ0,1,2 + 6χ0,2,0 + χ0,4,0 + 13χ1,0,1 + 2χ1,1,3 + 4χ1,2,1 ++ 10χ2,0,2 + 6χ2,1,0 + χ2,2,2 + 3χ3,0,3 + 2χ3,1,1 + χ4,0,4 + 5 +� ++ . . . +here χn1,n2,n3 are the SU(4)x1,x2,x3 characters for irreps with Dynkin labels [n1, n2, n3]. +The terms in red correspond to the operator O (and its conjugate) in (2.22). The sym- +metry algebra is su(4)x1,x2,x3 ⊕ u(1)Q. The SU(4) centre symmetry Z4 act trivial on irreps +[n1, n2, n3] with n1 + 2n2 − n3 = 0 mod 4. Thus (D.18) suggests that the symmetry group +is PSU(4)x1,x2,x3 × U(1)Q. +Gauging a Z3. +1 +w1 +2 +w2 +SU(3) +4 +w4 +1 +v +5 +mirror +←−−−−−−→ +1 +2 +3 +4 +Z3 +2 +y4,5 +y1,2,3 +. +(D.19) +Use the fugacity map +C : +w1 → x2 +1 +x2 +, +w2 → x2 +2 +x1 +, +w4 → u2 +1 , +v → Q−1 +(u1)5 +(D.20) +H : +y1 → Q +2 +15 x1 , +y2 → Q +2 +15 x2 +x1 +, +y3 → Q +2 +15 1 +x2 +, +y4 → Q− 1 +5 u1 , +y5 → Q− 1 +5 1 +u1 +to xi weight space fugacities of A2, u1 the weight space fugacity of A1, and a U(1) variable +Q. The Hilbert series are +HS = 1 + t (χ1,1 + φ2 + 1) + t2 (3χ1,1 + χ2,2 + 2φ2χ1,1 + 2φ2 + φ4 + 3) +(D.21) ++ t3 +� +Q−1 (φ1χ1,1 + φ3χ3,0) + Q (φ3χ0,3 + φ1χ1,1) ++ 2χ0,3 + 8χ1,1 + 3χ2,2 + 2χ3,0 + χ3,3 + φ2χ0,3 + 8φ2χ1,1 + 2φ4χ1,1 ++ 2φ2χ2,2 + φ2χ3,0 + 6φ2 + 2φ4 + φ6 + 5 +� ++ . . . +here χn1,n2 are the SU(3)x1,x2 characters for irreps with Dynkin labels [n1, n2]. The φk1 are +the SU(2)u1 characters for irreps with Dynkin label [k1]. The terms in red correspond to +the operator O (and its conjugate) in (2.22). The symmetry algebra is su(3)x1,x2⊕su(2)u1⊕ +u(1)Q. The SU(3) centre symmetry Z3 act trivial on irreps [n1, n2] with n1−n2 = 0 mod 3; +while the SU(2) centre Z2 acts trivial on irreps [k1] with k1 = 0 mod 2. Thus (D.21) +suggests that the symmetry group is PSU(3)x1,x2 × (SU(2)u1 × U(1)Q) /Z2. Z2 ⊂ U(1)Q +– 60 – + +such that Q has charge 1. +Gauging a Z2. +1 +w1 +SU(2) +3 +w3 +4 +w4 +1 +v +5 +mirror +←−−−−−−→ +1 +2 +3 +4 +Z2 +3 +y3,4,5 +y1,2 +. +(D.22) +Use the fugacity map +C : +w1 → x2 +1 , +w3 → u2 +1 +u2 +, +w4 → u2 +2 +u1 +, +v → Q−1 +(u2)5 +(D.23) +H : +y1 → Q +3 +10 x1 , +y2 → Q +3 +10 1 +x1 +, +y3 → Q− 1 +5 u1 , +y4 → Q− 1 +5 u2 +u1 +, +y5 → Q− 1 +5 1 +u2 +to an x1 weight space fugacity of A1, ui the weight space fugacities of A2, and a U(1) variable +Q. The Coulomb branch Hilbert series of the left quiver (and Higgs branch Hilbert series +of the right quiver) reads +HS = 1 + (χ2 + φ1,1 + 1) t +(D.24) ++ t2 +� +Q (χ2φ0,2 + φ1,0) + Q−1 (φ0,1 + χ2φ2,0) + 2χ2 + χ4 + 2χ2φ1,1 + 3φ1,1 + φ2,2 + 3 +� ++ t3 +� +Q (3χ2φ0,2 + χ4φ0,2 + 2φ0,2 + 2χ2φ1,0 + 2φ1,0 + χ2φ1,3 + χ2φ2,1 + φ2,1) ++ Q−1 (2χ2φ0,1 + 2φ0,1 + χ2φ1,2 + φ1,2 + 3χ2φ2,0 + χ4φ2,0 + 2φ2,0 + χ2φ3,1) ++ 6χ2 + 2χ4 + χ6 + χ2φ0,3 + 2φ0,3 + 8χ2φ1,1 + 2χ4φ1,1 + 8φ1,1 + 2χ2φ2,2 ++ 3φ2,2 + χ2φ3,0 + 2φ3,0 + φ3,3 + 5 +� ++ . . . +here χn1 are the SU(2)x1 characters for irreps with Dynkin labels [n1]. The φk1,k2 are the +SU(3)u1,u2 characters for irreps with Dynkin label [k1, k2]. The terms in red correspond to +the operator O (and its conjugate) in (2.22). The symmetry algebra is su(2)x1⊕su(3)u1,u2⊕ +u(1)Q. The SU(3) centre symmetry Z3 act trivial on irreps [k1, k2] with k1−k2 = 0 mod 3; +while the SU(2) centre Z2 acts trivial on irreps [n1] with n1 = 0 mod 2. Thus (D.24) +suggests that the symmetry group is PSU(2)x1 × (SU(3)u1,u2 × U(1)Q) /Z3. Z3 ⊂ U(1)Q +such that Q has charge 2. +D.2.4 +T[SU(6)] theories +Gauging a Z5. +1 +w1 +2 +w2 +3 +w3 +4 +w4 +SU(5) +1 +v +6 +mirror +←−−−→ +1 +2 +3 +4 +5 +Z5 +1 +y6 +y1,...,5 +. +(D.25) +– 61 – + +Use the fugacity map +C : +w1 → x2 +1 +x2 +, +w2 → +x2 +2 +x1x3 +, +w3 → +x2 +3 +x2x4 +, +w4 → x2 +4 +x3 +, +v → Q−1 +(D.26) +H : +y1 → Q +1 +30 x1 , +y2 → Q +1 +30 x2 +x1 +, +y3 → Q +1 +30 x3 +x2 +, +y4 → Q +1 +30 x4 +x3 +, +y5 → Q +1 +30 1 +x4 +, +y6 → Q− 1 +6 +to {xi} the weight space fugacities of A4, and a U(1) variable Q. +The perturbative +Coulomb/Higgs branch Hilbert series reads +HS = 1 + t (χ1,0,0,1 + 1) + t2 (χ0,1,1,0 + 3χ1,0,0,1 + χ2,0,0,2 + 2) +(D.27) ++ t3 (2χ0,1,0,2 + 3χ0,1,1,0 + 7χ1,0,0,1 + χ1,1,1,1 + 3χ2,0,0,2 + 2χ2,0,1,0 + χ3,0,0,3 + 3) ++ t4 +� +2χ0,0,2,1 + 6χ0,1,0,2 + 8χ0,1,1,0 + χ0,2,2,0 + 14χ1,0,0,1 + 2χ1,1,0,3 + 5χ1,1,1,1 ++ 2χ1,2,0,0 + 10χ2,0,0,2 + 6χ2,0,1,0 + χ2,1,1,2 + 3χ3,0,0,3 + 2χ3,0,1,1 + χ4,0,0,4 + 5 +� ++ t5 +� +Q−1χ0,0,0,5 + Qχ5,0,0,0 + χ0,0,1,3 + 6χ0,0,2,1 + 17χ0,1,0,2 + 17χ0,1,1,0 + 2χ0,2,1,2 ++ 3χ0,2,2,0 + 2χ0,3,0,1 + 25χ1,0,0,1 + 2χ1,0,2,2 + 2χ1,0,3,0 + 9χ1,1,0,3 + 18χ1,1,1,1 ++ 6χ1,2,0,0 + χ1,2,2,1 + 23χ2,0,0,2 + 17χ2,0,1,0 + 2χ2,1,0,4 + 5χ2,1,1,2 + 2χ2,1,2,0 ++ 2χ2,2,0,1 + 10χ3,0,0,3 + 9χ3,0,1,1 + χ3,1,0,0 + χ3,1,1,3 + 3χ4,0,0,4 + 2χ4,0,1,2 ++ χ5,0,0,5 + 7 +� ++ . . . +here χn1,n2,n3,n4 are the SU(5)xi characters for irreps with Dynkin labels [n1, n2, n3, n4]. +The terms in red correspond to the operator O (and its conjugate) in (2.22). The algebra +is su(5)xi ⊕ U(1)Q. The SU(4) centre symmetry Z5 acts with charge 1 in the fundamental +[1, 0, 0, 0]. +The appearing characters in (D.27) are all neutral under the centre, which +suggests the global form PSU(5)xi × U(1)Q. +Gauging a Z4. +1 +w1 +2 +w2 +3 +w3 +SU(4) +5 +w5 +1 +v +6 +mirror +←−−−→ +1 +2 +3 +4 +5 +Z4 +2 +y5,6 +y1,...,4 +. +(D.28) +Use the fugacity map +C : +w1 → x2 +1 +x2 +, +w2 → +x2 +2 +x1x3 +, +w3 → x2 +3 +x2 +, +w5 → u2 +1 , +v → Q−1 +(u1)6 +(D.29) +H : +y1 → Q +1 +12 x1 , +y2 → Q +1 +12 x2 +x1 +, +y3 → Q +1 +12 x3 +x2 +, +y4 → Q +1 +12 1 +x3 +, +y5 → Q− 1 +6 u1 , +y6 → Q− 1 +6 1 +u1 +– 62 – + +to {xi} the weight space fugacities of A3, u1 the weight space fugacity of A1, and a U(1) +variable Q. The Hilbert series reads +HS = 1 + t (φ1,0,1 + χ2 + 1) +(D.30) ++ t2 (2χ2φ1,0,1 + φ0,2,0 + 3φ1,0,1 + φ2,0,2 + 2χ2 + χ4 + 3) ++ t3 +� +χ2φ0,1,2 + 2χ2φ0,2,0 + 8χ2φ1,0,1 + 2χ2φ2,0,2 + χ2φ2,1,0 + 2χ4φ1,0,1 + 2φ0,1,2 ++ 2φ0,2,0 + 9φ1,0,1 + φ1,2,1 + 3φ2,0,2 + 2φ2,1,0 + φ3,0,3 + 6χ2 + 2χ4 + χ6 + 5 +� ++ t4 +� +Q−1 (χ4φ0,0,4 + χ2φ0,1,2 + φ0,2,0) + Q (χ2φ2,1,0 + χ4φ4,0,0 + φ0,2,0) + 8χ2φ0,1,2 ++ 7χ2φ0,2,0 + 26χ2φ1,0,1 + χ2φ1,1,3 + 3χ2φ1,2,1 + 10χ2φ2,0,2 + 8χ2φ2,1,0 ++ 2χ2φ3,0,3 + χ2φ3,1,1 + χ4φ0,1,2 + 2χ4φ0,2,0 + 9χ4φ1,0,1 + 2χ6φ1,0,1 + 3χ4φ2,0,2 ++ χ4φ2,1,0 + 7φ0,1,2 + 9φ0,2,0 + φ0,4,0 + 21φ1,0,1 + 2φ1,1,3 + 4φ1,2,1 + 12φ2,0,2 ++ 7φ2,1,0 + φ2,2,2 + 3φ3,0,3 + 2φ3,1,1 + φ4,0,4 + 12χ2 + 7χ4 + 2χ6 + χ8 + 11 +� ++ . . . +The symmetry algebra is su(4)x1,x2,x3 × su(2)u1 × U(1)Q. The terms in red correspond to +the operator O (and its conjugate) in (2.22). The appearing characters suggest that the +global form is PSU(4)x1,x2,x3,x4 × SO(3)u1 × U(1)Q, i.e. the centre symmetries act trivially. +Gauging a Z3. +1 +w1 +2 +w2 +SU(3) +4 +w4 +5 +w5 +1 +v +6 +mirror +←−−−→ +1 +2 +3 +4 +5 +Z3 +3 +y4,5,6 +y1,2,3 +. +(D.31) +Use the fugacity map +C : +w1 → x2 +1 +x2 +, +w2 → x2 +2 +x1 +, +w4 → u2 +1 +u2 +, +w5 → u2 +2 +u1 +, +v → Q−1 +(u2)5 +(D.32) +H : +y1 → Q +1 +6 x1 , +y2 → Q +1 +6 x2 +x1 +, +y3 → Q +1 +6 1 +x2 +y4 → Q− 1 +6 u1 , +y5 → Q− 1 +6 u2 +u1 +, y6 → Q− 1 +6 1 +u2 +to xi weight space fugacities of A2, ui the weight space fugacities of another A2, and a +U(1) variable Q. The perturbative Coulomb/Higgs branch Hilbert series is evaluated to +HS = 1 + t (χ1,1 + φ1,1 + 1) +(D.33) ++ t2 (3χ1,1 + χ2,2 + 2χ1,1φ1,1 + 3φ1,1 + φ2,2 + 3) ++ t3 +� +Q−1 (χ0,3φ3,0 + χ1,1φ1,1 + 1) + Q (χ1,1φ1,1 + χ3,0φ0,3 + 1) ++ 2χ0,3 + 8χ1,1 + 3χ2,2 + 2χ3,0 + χ3,3 + χ1,1φ0,3 + χ0,3φ1,1 + 10χ1,1φ1,1 + 2χ1,1φ2,2 +– 63 – + ++ χ1,1φ3,0 + 2χ2,2φ1,1 + χ3,0φ1,1 + 2φ0,3 + 8φ1,1 + 3φ2,2 + 2φ3,0 + φ3,3 + 6 +� ++ . . . +The symmetry algebra is su(3)x1,x2 ⊕ su(3)u1,u2 ⊕ u(1). The terms in red correspond to +the operator O (and its conjugate) in (2.22). The appearing characters indicate that all +irreps are trivial under the centre symmetries, such that the global form is PSU(3)x1,x2 × +PSU(3)u1,u2 × U(1)Q. +Gauging a Z2. +1 +w1 +SU(2) +3 +w3 +4 +w4 +5 +w5 +1 +v +6 +mirror +←−−−→ +1 +2 +3 +4 +5 +Z2 +4 +y3,...,6 +y1,2 +. +(D.34) +Use the fugacity map +C : +w1 → x2 +1 , +w3 → u2 +1 +u2 +, +w4 → +u2 +2 +u2u3 +, +w5 → u2 +3 +u2 +, +v → Q−1 +(u3)5 +(D.35) +H : +y1 → Q +1 +3 x1 , +y2 → Q +1 +3 1 +x1 +, +y3 → Q− 1 +6 u1 , +y4 → Q− 1 +6 u2 +u1 +, +y5 → Q− 1 +6 u3 +u2 +, +y6 → Q− 1 +6 1 +u3 +to an x1 weight space fugacity of A1, ui the weight space fugacities of A3, and a U(1) +variable Q. The Coulomb branch (or Higgs branch) Hilbert series reads +HS = 1 + t (φ1,0,1 + χ2 + 1) + t2 +� +Q (χ2φ0,0,2 + φ0,1,0) + Q−1 (χ2φ2,0,0 + φ0,1,0) +(D.36) ++ 2χ2φ1,0,1 + φ0,2,0 + 3φ1,0,1 + φ2,0,2 + 2χ2 + χ4 + 3 +� ++ . . . +The terms in red correspond to the operator O (and its conjugate) in (2.22). The global +symmetry is PSU(2)x1 × (SU(4)u1,u2,u3 × U(1)Q) /Z4 and Q has Z4 charge 2 mod 4. +D.3 +Some T[SU(N)] examples with higher charges +Consider the quiver theories in (2.25) and (2.26). Redefine fugacities as +(2.25) : +w1 → y2 +1 +y2 +, +w2 → y2 +2 +y1 +, +w4 → x2 +1 , +v → Q +x10 +1 +(D.37) +(2.26) : +w1 → x2 +1 , +w3 → y2 +1 +y2 +, +w4 → y2 +2 +y1 +, +v → Q−1 +y15 +2 +, +(D.38) +such that x1 is an A1 fugacity and {y1,2} are A2 fugacities. The perturbative monopole +formula for the left-hand-side quivers reads +HS = 1 + t (χ2 + φ1,1 + 1) +(D.39) ++ t2 (2φ1,1χ2 + 2χ2 + χ4 + 3φ1,1 + φ2,2 + 3) +– 64 – + ++ t3 +� +φ0,3χ2 + 8φ1,1χ2 + 2φ2,2χ2 + φ3,0χ2 + 6χ2 + 2χ4 + χ6 + 2φ0,3 + 2χ4φ1,1 ++ 8φ1,1 + 3φ2,2 + 2φ3,0 + φ3,3 + 5 +� ++ t4 +� +7φ0,3χ2 + 24φ1,1χ2 + φ1,4χ2 + 10φ2,2χ2 + 7φ3,0χ2 + 2φ3,3χ2 + φ4,1χ2 ++ 12χ2 + 7χ4 + 2χ6 + χ8 + χ4φ0,3 + 5φ0,3 + 9χ4φ1,1 + 2χ6φ1,1 + 19φ1,1 ++ 2φ1,4 + 3χ4φ2,2 + 11φ2,2 + χ4φ3,0 + 5φ3,0 + 3φ3,3 + 2φ4,1 + φ4,4 + 10 +� ++ t5 +� +22φ0,3χ2 + 60φ1,1χ2 + 9φ1,4χ2 + 38φ2,2χ2 + φ2,5χ2 + 22φ3,0χ2 + 10φ3,3χ2 ++ 9φ4,1χ2 + 2φ4,4χ2 + φ5,2χ2 + 25χ2 + 15χ4 + 7χ6 + 2χ8 + χ10 + 9χ4φ0,3 ++ χ6φ0,3 + 16φ0,3 + 30χ4φ1,1 + 9χ6φ1,1 + 2χ8φ1,1 + 40φ1,1 + 2χ4φ1,4 + 8φ1,4 ++ 15χ4φ2,2 + 3χ6φ2,2 + 28φ2,2 + 2φ2,5 + 9χ4φ3,0 + χ6φ3,0 + 16φ3,0 + 3χ4φ3,3 ++ 11φ3,3 + 2χ4φ4,1 + 8φ4,1 + 3φ4,4 + 2φ5,2 + φ5,5 + 15 +� ++ t6 +� +Q (χ6φ0,6 + χ4φ1,4 + χ2φ2,2 + φ3,0) + φ0,3 + χ2φ2,2 + χ4φ4,1 + χ6φ6,0 +Q ++ 62φ0,3χ2 + 2φ0,6χ2 + 132φ1,1χ2 + 37φ1,4χ2 + 107φ2,2χ2 + 9φ2,5χ2 + 62φ3,0χ2 ++ 41φ3,3χ2 + φ3,6χ2 + 37φ4,1χ2 + 10φ4,4χ2 + 9φ5,2χ2 + 2φ5,5χ2 + 2φ6,0χ2 + φ6,3χ2 ++ 44χ2 + 33χ4 + 16χ6 + 7χ8 + 2χ10 + χ12 + 31χ4φ0,3 + 9χ6φ0,3 + χ8φ0,3 + 36φ0,3 ++ χ4φ0,6 + 3φ0,6 + 81χ4φ1,1 + 31χ6φ1,1 + 9χ8φ1,1 + 2χ10φ1,1 + 77φ1,1 + 16χ4φ1,4 ++ 2χ6φ1,4 + 25φ1,4 + 59χ4φ2,2 + 16χ6φ2,2 + 3χ8φ2,2 + 70φ2,2 + 2χ4φ2,5 + 8φ2,5 ++ 31χ4φ3,0 + 9χ6φ3,0 + χ8φ3,0 + 36φ3,0 + 17χ4φ3,3 + 4χ6φ3,3 + 32φ3,3 + 2φ3,6 ++ 16χ4φ4,1 + 2χ6φ4,1 + 25φ4,1 + 3χ4φ4,4 + 11φ4,4 + 2χ4φ5,2 + 8φ5,2 + 3φ5,5 ++ χ4φ6,0 + 3φ6,0 + 2φ6,3 + φ6,6 + 28 +� ++ . . . +wherein φm1,m2 are the characters of A2 irreps [m1, m2] and χn1 are the A1 characters +for [n1] irreps. The global form of the Coulomb branch isometry group is PSU(2)x1 × +PSU(3)y1,2 × U(1)Q. +D.4 +Some T σ +ρ [SU(N)] examples +1st example. +Consider the example +1 +w1 +2 +w2 +SU(2) +1 +w3 +1 +w4 +(D.40) +and use the fugacity map +v1 → x2 +1 , +v2 → Q1 +x1 +, +v3 → Q2 , +v3 → Q3 . +(D.41) +– 65 – + +The Coulomb branch Hilbert series reads +HS = 1 + t (χ2 + 3) + t2 +� � +Q1 + Q−1 +1 +� +χ1 + +� +Q2 + Q−1 +2 +� ++ 3χ2 + χ4 + 8 +� +(D.42) ++ t3 +� � +Q1 + Q−1 +1 +� +(4χ1 + χ3) + +� +Q2 + Q−1 +2 +� +(χ2 + 3) ++ Q2Q3Q2 +1 + +1 +Q2 +1Q2Q3 ++ 8χ2 + 3χ4 + χ6 + 17 +� ++ t4 +� +Q2 +1 (Q2Q3 (χ2 + 4) + Q3 + χ2) + +χ2+4 +Q2Q3 + +1 +Q3 + χ2 +Q2 +1 ++ Q1 +� χ1 +Q2 ++ Q2 (Q3χ1 + χ1) + 12χ1 + 4χ3 + χ5 +� ++ +χ1 +Q3 +χ1 +Q2 ++ Q2χ1 + 12χ1 + 4χ3 + χ5 +Q1 ++ Q2 (3χ2 + χ4 + 9) + 3χ2 + χ4 + 9 +Q2 ++ Q2 +2 + 1 +Q2 +2 ++ 18χ2 + 8χ4 + 3χ6 + χ8 + 34 +� ++ . . . +and the global symmetry is given by (SU(2)x × U(1)Q1) /Z2 × U(1)Q2 × U(1)Q3 where the +Z2 centre symmetry acts with charge +1 on Q1 and trivial on all other Qi. +2nd example. +Next, modify the example slightly and consider +1 +w1 +SU(2) +2 +w2 +1 +w3 +1 +w4 +(D.43) +together with the fugacity map +v1 → x2 +1 , +v2 → Q1 , +v3 → Q2 , +v3 → Q3 . +(D.44) +The Coulomb branch Hilbert series reads +HS = 1 + t (χ2 + 3) + t3/2 +� +Q1 + 1 +Q1 +� ++ t2 +� +Q2 + 1 +Q2 ++ 3χ2 + χ4 + 8 +� +(D.45) ++ t5/2 +� 1 +Q2 + χ2 + 4 +Q1 ++ Q1 (Q2 + χ2 + 4) +� ++ t3 +� +Q2 (χ2 + 3) + χ2 + 3 +Q2 ++ Q2 +1 + 1 +Q2 +1 ++ 8χ2 + 3χ4 + χ6 + 17 +� ++ t7/2 +� χ2+4 +Q2 ++ Q2 + 4χ2 + χ4 + 11 +Q1 ++ Q1 +� +Q2 (χ2 + 4) + 1 +Q2 ++ 4χ2 + χ4 + 11 +�� ++ t4 +� +Q2 +1 (Q2 (Q3χ2 + 1) + χ2 + 4) + +χ2 +Q3 +1 +Q2 ++ χ2 + 4 +Q2 +1 ++ Q2 +2 + 1 +Q2 +2 ++ Q2 (3χ2 + χ4 + 9) + 3χ2 + χ4 + 9 +Q2 ++ 18χ2 + 8χ4 + 3χ6 + χ8 + 34 +� ++ . . . +– 66 – + +and the global symmetry is given by PSU(2)x×�3 +i=1 U(1)Qi where the Z2 centre symmetry +acts trivial on all Qi. +3rd example. +The Hilbert series for the example in (2.39) with notation (C.23) +HS = 1 + t (χ1,1 + 1) + t +3 +2 +�χ0,1 +Q + Qχ1,0 +� ++ t2 (3χ1,1 + χ2,2 + 3) +(D.46) ++ t +5 +2 +�3χ0,1 + χ1,2 + χ2,0 +Q ++ Q (χ0,2 + 3χ1,0 + χ2,1) +� ++ . . . +with χn1,n2 the A2 characters for irreps [n1, n2]. The Z3 centre charge of the U(1)Q is +determined to be −1 mod 3, such that the symmetry group is (SU(3) × U(1))/Z3 ∼= U(3). +After gauging a discrete Z2 0-form symmetry, the labelling becomes +2 +w1 +2 +w2 +SU(2) +1 +v +←→ +1 +2 +2 +1 +1 +y3 +Z2 +Q +y1,2 +(D.47) +with fugacity map +w1 = Q1 +u1 +, +w2 = x2 +1 , +v = Q0 +(D.48) +with x1 an A1 weight space fugacity and Q0,1 two U(1) fugacities. +The Hilbert series +becomes +HS = 1 + t (χ2 + 2) + t3/2 +� +Q1 + 1 +Q1 +� +χ1 +(D.49) ++ t2 +�� +Q0Q2 +1 + +1 +Q0Q2 +1 ++ 4 +� +χ2 + χ4 + 7 +� ++ t5/2 +�� +Q0Q1 + 5Q1 + +1 +Q0Q1 ++ 5 +Q1 +� +χ1 + +� +Q1 + 1 +Q1 +� +χ3 +� ++ . . . +where χn1 denotes A1 characters for irreps [n1]. It is apparent that the Z2 centre charges +of (Q0, Q1) are (0, −1 mod 2). +D.5 +T[SO(2N)] theories +In this appendix, computational details for the T[SO(2N)] theories are provided. For or- +thosymplectic quivers, the topological symmetries visible in the UV Lagrangian are severely +limited. For an SO(k) gauge group, there is only a Z2 valued topological 0-form symmetry. +For SO(2), there exists a whole U(1) topological 0-form symmetry. Thus, to confirm mirror +symmetry after such a Z2 is gauged, one needs to identify the Z2 in the original mirror +pair. Therefore, it is sufficient to provide the Z2-refined Hilbert series of the original mirror +pair to demonstrate agreement after gauging. The Hilbert series after gauging the discrete +Z2 symmetry is simply obtained by averaging over Z2. +– 67 – + +D.5.1 +T[SO(6)] theories +For T[SO(6)], one can gauge a Zt +2 ⊂ PSO(6)t, which corresponds to gauging the Zf +2 factor +inside the flavour symmetry of the mirror theory, which is identified by a “2+1” splitting of +the fundamental flavours. Before gauging, the discrete fugacities are attributed as follows: +2 +z1 +2 +4 +z2 +4 +6 +←→ +2 +2 +4 +4 +4 +2, a +(D.50) +The Coulomb branch Hilbert series of the left-hand side (which equals the Higgs branch +Hilbert series of the right theory) reads +HS = 1 + (7 + 8z2)t + (63 + 56z2)t2 + (328 + 336z2)t3 + (1476 + 1448z2)t4 +(D.51) ++ (5390 + 5424z2)t5 + (17500 + 17416z2)t6 + . . . +with the Z2 fugacity z2 = a. +D.5.2 +T[SO(8)] theories +For T[SO(8)], one can gauge Z2 ⊂ PSO(8)t, which corresponds to gauge a Z2 factor inside +the flavour symmetry for the mirror theory. Now one can choose to gauge this Zt +2 for the +SO(4) or SO(6) gauge node. +Zt +2 of SO(4). +Gauging the Zt +2 of the SO(4) gauge node leads to a “2+2” splitting of the +fundamental flavours. Before gauging, the discrete fugacities are attributed as follows: +2 +z1 +2 +z2 +4 +4 +6 +z3 +6 +8 +(D.52) +←→ +2 +2 +4 +4 +6 +6 +4 +4, a +The Coulomb branch Hilbert series of the left-hand side (which equals the Higgs branch +Hilbert series of the right theory) reads +HS = 1 + (12 + 16z2)t + (213 + 192z2)t2 + (1984 + 2048z2)t3 + . . . +with the Z2 fugacity z2 = a. +Zt +2 of SO(6). +Gauging the Zt +2 of the SO(6) gauge node leads to “3+1” splitting of the +fundamental flavours in the mirror theory. +Before gauging, the discrete fugacities are +attributed as follows: +2 +z1 +2 +z2 +4 +4 +6 +z3 +6 +8 +(D.53) +– 68 – + +←→ +2 +2 +4 +4 +6 +6 +6 +2, a +The Coulomb branch Hilbert series of the left-hand side (and the Higgs branch Hilbert +series of the right-hand theory) reads +HS = 1 + (12z3 + 16)t + (213 + 192z3)t2 + (1984z3 + 2048)t3 + . . . +with the Z2 fugacity z3 = a. +D.6 +Sp(k) SQCD and orthosymplectic mirrors +The logic is as in Appendix D.5, to evaluate the Hilbert series of the theories after Z2 +gauging, one evaluates the Z2-refined Hilbert series of Sp(k) SQCD and its mirror theory. +Once agreement is found, the theories after gauging have agreeing the Hilbert series by +construction. +D.6.1 +Sp(2) SQCD, 5 flavours +Consider Sp(2) with 5 fundamental flavours. +Zt +2 of SO(2). +Gauging the Zt +2 of one of the SO(2) gauge nodes corresponds to a “1+4”- +splitting of the fundamental flavours in the mirror theory. Before gauging, the discrete +fugacities are assigned as follows: +4 +a1 +2 +8 +←→ +a1 +2 +2 +4 +4 +4 +2 +2 +2 +(D.54) +The Coulomb branch Hilbert series of the right theory (which agrees with the Higgs branch +Hilbert series of the left theory) reads +HS1+4 = 1 + (16a1 + 29)t + (448a1 + 532)t2 + . . . +(D.55) +with the Z2-fugacity a1. +Zt +2 inside SO(4). +Gauging the Zt +2 of one of the SO(4) gauge nodes corresponds to a +“2+3”-splitting of the fundamental flavours in the mirror theory. +Before gauging, the +discrete fugacities are assigned as follows: +4 +a2 +4 +6 +←→ +2 +2 +a2 +4 +4 +4 +2 +2 +2 +(D.56) +– 69 – + +The Coulomb branch Hilbert series of the right theory (which agrees with the Higgs branch +Hilbert series of the left theory) reads +HS2+3 = 1 + (24a2 + 21)t + (480a2 + 500)t2 + . . . +(D.57) +with Z2-fugacity a2. +D.6.2 +Sp(2) SQCD, 6 flavours +Zt +2 of SO(2). +Gauging the Zt +2 of one of the SO(2) gauge nodes corresponds to a “1+5”- +splitting of the fundamental flavours in the mirror theory. Before gauging, the discrete +fugacities are assigned as follows: +4 +a1 +2 +10 +←→ +a1 +2 +2 +4 +4 +5 +4 +4 +2 +2 +1 +1 +(D.58) +The Coulomb branch Hilbert series of the right theory (which agrees with the Higgs branch +Hilbert series of the left theory) reads +HS1+5 = 1 + (20a1 + 46)t + (900a1 + 1233)t2 + . . . +(D.59) +with the Z2-fugacity a1. +Zt +2 of SO(4). +Gauging the Zt +2 of one of the SO(4) gauge nodes corresponds to a “2+4”- +splitting of the fundamental flavours in the mirror theory. Before gauging, the discrete +fugacities are assigned as follows: +4 +a2 +4 +8 +←→ +2 +2 +a2 +4 +4 +5 +4 +4 +2 +2 +1 +1 +(D.60) +The Coulomb branch Hilbert series of the right theory (which agrees with the Higgs branch +Hilbert series of the left theory) reads +HS2+4 = 1 + (32a2 + 34)t + (1056a2 + 1077)t2 + . . . +(D.61) +with the Z2-fugacity a2. +Zt +2 of SO(5). +Gauging the Zt +2 of the SO(5) gauge node corresponds to a “3+3”-splitting +of the fundamental flavours in the mirror theory. Before gauging, the discrete fugacities +– 70 – + +are assigned as follows: +4 +a3 +6 +6 +←→ +2 +2 +4 +4 +a3 +5 +4 +4 +2 +2 +1 +1 +(D.62) +and the Coulomb branch Hilbert series of the right theory (which agrees with the Higgs +branch Hilbert series of the left theory) reads +HS3+3 = 1 + (36a3 + 30)t + (1044a3 + 1089)t2 + . . . +(D.63) +with the Z2-fugacity a3. +D.7 +Sp(k) SQCD and D-type mirrors +In this appendix, computational evidence for the general results of Section 2.7 is provided. +The relevant mirror map is discussed in Appendix D.6. +D.7.1 +D7 Dynkin quiver +Consider Sp(2) SQCD with N = 7 flavours and its D7 Dynkin mirror quiver, see (2.45). +2nd node. +The mirror pairs is defined by +4 +10 +y3,...,7 +Z2 +y1,2 +←→ +1 +w1 +SU(2) +3 +w3 +4 +w4 +4 +w5 +2 +w6 +2 +w7 +1 +v +. +(D.64) +and the fugacity map is +w1 = u2 +1 , +(w3, w4, w5, w6, w7)i = +5 +� +j=1 +x +CD5 +ij +j +, +v = u2 +2 +x2 +(D.65) +with the weight space fugacities xi, ua of so(10) and so(4), respectively. +(See the end +of Section 2.7 for the global symmetry enhancement.) The flavour fugacities follow from +(C.30). The Higgs/Coulomb branch Hilbert series then evaluates to +HS = 1 + t (χ0,1,0,0,0 + φ0,2 + φ2,0) +(D.66) ++ t2 +� +χ0,0,0,1,1 + χ0,2,0,0,0 + χ2,0,0,0,0 + 2χ0,1,0,0,0φ0,2 + 2χ0,1,0,0,0φ2,0 ++ χ2,0,0,0,0φ2,2 + φ0,4 + φ2,2 + φ4,0 + 2 +� ++ . . . +with χ, φ denoting so(10), so(4) characters, respectively. The global form of the isometry +group is PSO(4) × PSO(10). +– 71 – + +3rd node. +The mirror pair is +4 +8 +y4,...,7 +Z2 +y1,2,3 +←→ +1 +w1 +2 +w2 +SU(3) +4 +w4 +4 +w5 +2 +w6 +2 +w7 +1 +v +. +(D.67) +and the fugacity map is +(w1, w2)a = +2 +� +b=1 +u +CA2 +ab +b +, +(w4, w5, w6, w7)i = +4 +� +j=1 +x +CD4 +ij +j +, +v = Q +x1 +(D.68) +with the weight space fugacities xi, ua of so(8) and su(3), respectively. The flavour fugac- +ities follow from (C.30). The Higgs/Coulomb branch Hilbert series then evaluates to +HS = 1 + t (χ0,1,0,0 + φ1,1 + 1) +(D.69) ++ t2 +� +Qχ1,0,0,0 + χ1,0,0,0 +Q ++ Qχ1,0,0,0φ1,1 + χ1,0,0,0φ1,1 +Q ++ χ0,0,0,2 + χ0,0,2,0 + 2χ0,1,0,0 + χ0,2,0,0 + χ2,0,0,0 ++ 2χ0,1,0,0φ1,1 + χ2,0,0,0φ1,1 + 3φ1,1 + φ2,2 + 3 +� ++ . . . +with χ, φ denoting so(8), su(3) characters, respectively. +The global form is given by +PSU(3) × U(1)Q×Spin(8) +Z2×Z2 +with Q having Z2 × Z2 centre charges (0, 1 mod 2). +Thus, the +isometry group is PSU(3) × U(1)Q×SO(8) +Z2 +. +4th node. +The mirror pair is +4 +6 +y5,...,7 +Z2 +y1,...,4 +←→ +1 +w1 +2 +w2 +3 +w3 +SU(4) +4 +w5 +2 +w6 +2 +w7 +1 +v +. +(D.70) +and the fugacity map is +(w1, w2, w3)a = +3 +� +b=1 +u +CA3 +ab +b +, +(w5, w6, w7)i = +3 +� +j=1 +x +CD3 +ij +j +, +v = Q +(D.71) +with the weight space fugacities xi, ua of so(6) and su(4), respectively. The flavour fugac- +ities follow from (C.30). The Higgs/Coulomb branch Hilbert series then evaluates to +HS = 1 + t (χ0,1,1 + φ1,0,1 + 1) +(D.72) +– 72 – + ++ t2 +� +Qφ0,2,0 + φ0,2,0 +Q ++ 3χ0,1,1 + χ0,2,2 + χ2,0,0 + 2χ0,1,1φ1,0,1 ++ χ2,0,0φ1,0,1 + 2φ0,2,0 + 3φ1,0,1 + φ2,0,2 + Q + 1 +Q + 3 +� ++ . . . +with χ, φ denoting so(6), su(4) characters, respectively. +The global form is PSU(4) × +U(1)Q × PSO(6). +5th node. +The mirror pair is +4 +4 +y6,7 +Z2 +y1,...,5 +←→ +1 +w1 +2 +w2 +3 +w3 +4 +w4 +SU(4) +2 +w6 +2 +w7 +1 +v +. +(D.73) +and the fugacity map is +(w1, w2, w3, w4)a = +4 +� +b=1 +u +CA4 +ab +b +, +(w6, w7)i = +2 +� +j=1 +x +CD2 +ij +j +, +v = Q +u4 +(D.74) +with the weight space fugacities xi, ua of so(4) and su(5), respectively. The flavour fugac- +ities follow from (C.30). The Higgs/Coulomb branch Hilbert series then evaluates to +HS = 1 + t (χ0,2 + χ2,0 + φ1,0,0,1 + 1) +(D.75) ++ t2 +�φ0,0,0,1 +Q ++ Qφ1,0,0,0 + Qφ0,0,2,0 + φ0,2,0,0 +Q ++ 2χ0,2 + χ0,4 ++ 2χ2,0 + χ2,2 + χ4,0 + 2χ0,2φ1,0,0,1 + 2χ2,0φ1,0,0,1 ++ χ2,2φ1,0,0,1 + 2φ0,1,1,0 + 3φ1,0,0,1 + φ2,0,0,2 + 4 +� ++ . . . +with χ, φ denoting so(4), su(5) characters, respectively. The global form is SU(5)×U(1)Q +Z5 +× +PSO(4) with Q having Z5 centre charge 4 mod 5; i.e. the isometry group is U(5)×PSO(4). +6th node. +The mirror pair reads +4 +Z2 +y1,...,7 +←→ +1 +w1 +2 +w2 +3 +w3 +4 +w4 +4 +w5 +SU(2) +2 +w7 +1 +v +. +(D.76) +– 73 – + +and the fugacity map is +(w1, w2, w3, w4, w5, w7)a = +6 +� +b=1 +u +CA6 +ab +b +, +v = Q +u4 +(D.77) +with the weight space fugacities ua of su(7), respectively. The flavour fugacities follow from +(C.30). The Higgs/Coulomb branch Hilbert series then evaluates to +HS = 1 + t (φ1,0,0,0,0,1 + 1) +(D.78) ++ t2 +� +Qφ0,0,0,0,2,0 + φ0,0,0,1,0,0 +Q ++ Qφ0,0,1,0,0,0 + φ0,2,0,0,0,0 +Q ++ 2φ0,1,0,0,1,0 + 2φ1,0,0,0,0,1 + φ2,0,0,0,0,2 + 2 +� ++ . . . +with φ denoting su(7) characters. The global symmetry is SU(7)×U(1)Q +Z7 +with Q having Z7 +centre charges 4 mod 7. +D.7.2 +D8 Dynkin quiver +Consider Sp(2) SQCD with N = 8 flavours and its D8 Dynkin mirror quiver, see (2.45). +2nd node. +The mirror pair is defined by +4 +12 +y3,...,8 +Z2 +y1,2 +←→ +1 +w1 +SU(2) +3 +w3 +4 +w4 +4 +w5 +4 +w6 +2 +w7 +2 +w8 +1 +v +. +(D.79) +and the fugacity map is +w1 = u2 +1 , +(w3, w4, w5, w6, w7, w8)i = +6 +� +j=1 +x +CD6 +ij +j +, +v = u2 +2 +x2 +(D.80) +with the weight space fugacities ua and xj of so(4) and so(12), respectively. The flavour +fugacities follow from (C.30). The Higgs/Coulomb branch Hilbert series then evaluates to +HS = 1 + t (χ0,1,0,0,0,0 + φ0,2 + φ2,0) +(D.81) ++ t2 +� +χ0,0,0,1,0,0 + χ0,2,0,0,0,0 + χ2,0,0,0,0,0 + 2χ0,1,0,0,0,0φ0,2 ++ 2χ0,1,0,0,0,0φ2,0 + χ2,0,0,0,0,0φ2,2 + φ0,4 + φ2,2 + φ4,0 + 2 +� ++ . . . +and φ are so(4) characters and χ are so(12) characters. The global form is read off to be +PSO(4) × PSO(12). +– 74 – + +3rd node. +The mirror pair is +4 +10 +y4,...,8 +Z2 +y1,2,3 +←→ +1 +w1 +2 +w2 +SU(3) +4 +w4 +4 +w5 +4 +w6 +2 +w7 +2 +w8 +1 +v +. +(D.82) +and the fugacity map is +(w1, w2)a = +2 +� +b=1 +u +CA2 +ab +b +, +(w4, w5, w6, w7, w8)i = +5 +� +j=1 +x +CD5 +ij +j +, +v = Q +x1 +(D.83) +with the weight space fugacities ua and xj of su(3) and so(10), respectively. The flavour +fugacities follow from (C.30). The Higgs/Coulomb branch Hilbert series then evaluates to +HS = 1 + t (χ0,1,0,0,0 + φ1,1 + 1) +(D.84) ++ t2 +� +Qχ1,0,0,0,0 + χ1,0,0,0,0 +Q ++ Qχ1,0,0,0,0φ1,1 + χ1,0,0,0,0φ1,1 +Q ++ χ0,0,0,1,1 + 2χ0,1,0,0,0 + χ0,2,0,0,0 + χ2,0,0,0,0 + 2χ0,1,0,0,0φ1,1 ++ χ2,0,0,0,0φ1,1 + 3φ1,1 + φ2,2 + 3 +� ++ . . . +and φ denotes su(3) characters, while χ are so(10) characters. The global symmetry is +PSU(3) × U(1)Q×Spin(10) +Z4 +where the Q can be assigned Z4 charge 2 mod 4. +4th node. +The mirror pair reads +4 +8 +y5,...,8 +Z2 +y1,...,4 +←→ +1 +w1 +2 +w2 +3 +w3 +SU(4) +4 +w5 +4 +w6 +2 +w7 +2 +w8 +1 +v +. +(D.85) +and the fugacity map is +(w1, w2, w3)a = +3 +� +b=1 +u +CA3 +ab +b +, +(w5, w6, w7, w8)i = +4 +� +j=1 +x +CD4 +ij +j +, +v = Q +(D.86) +with the weight space fugacities ua and xj of su(4) and so(8), respectively. The flavour +fugacities follow from (C.30). The Higgs/Coulomb branch Hilbert series then evaluates to +HS = 1 + t (χ0,1,0,0 + φ1,0,1 + 1) +(D.87) +– 75 – + ++ t2 +� +Qφ0,2,0 + φ0,2,0 +Q ++ χ0,0,0,2 + χ0,0,2,0 + 2χ0,1,0,0 + χ0,2,0,0 + χ2,0,0,0 ++ 2χ0,1,0,0φ1,0,1 + χ2,0,0,0φ1,0,1 + 2φ0,2,0 + 3φ1,0,1 + φ2,0,2 + Q + 1 +Q + 3 +� ++ . . . +with φ denoting su(4) characters and χ are so(8) characters. +The global symmetry is +PSU(4) × U(1)Q × PSO(8). +5th node. +The mirror pair is +4 +6 +y6,...,8 +Z2 +y1,...,5 +←→ +1 +w1 +2 +w2 +3 +w3 +4 +w4 +SU(4) +4 +w6 +2 +w7 +2 +w8 +1 +v +. +(D.88) +and the fugacity map is +(w1, w2, w3, w4)a = +4 +� +b=1 +u +CA4 +ab +b +, +(w6, w7, w8)i = +3 +� +j=1 +x +CD3 +ij +j +, +v = Q +u4 +(D.89) +with the weight space fugacities ua and xj of su(5) and so(6), respectively. The flavour +fugacities follow from (C.30). The Higgs/Coulomb branch Hilbert series then evaluates to +HS = 1 + t (χ0,1,1 + φ1,0,0,1 + 1) +(D.90) ++ t2 +�φ0,0,0,1 +Q ++ Qφ1,0,0,0 + Qφ0,0,2,0 + φ0,2,0,0 +Q ++ 3χ0,1,1 + χ0,2,2 + χ2,0,0 ++ 2χ0,1,1φ1,0,0,1 + χ2,0,0φ1,0,0,1 + 2φ0,1,1,0 + 3φ1,0,0,1 + φ2,0,0,2 + 3 +� ++ . . . +and φ, χ denote su(5), so(6) characters, respectively. The isometry group is SU(5)×U(1)Q +Z5 +× +PSO(6) where the Z5 charge of Q is 4 mod 5. The global form is then U(5) × PSO(6). +6th node. +The mirror pair is +4 +4 +y7,8 +Z2 +y1,...,6 +←→ +1 +w1 +2 +w2 +3 +w3 +4 +w4 +4 +w5 +SU(4) +2 +w7 +2 +w8 +1 +v +. +(D.91) +and the fugacity map is +(w1, w2, w3, w4, w5)a = +5 +� +b=1 +u +CA5 +ab +b +, +(w7, w8)i = +2 +� +j=1 +x +CD2 +ij +j +, +v = Q +u4 +(D.92) +– 76 – + +with the weight space fugacities ua and xj of su(6) and so(2), respectively. The flavour +fugacities follow from (C.30). The Higgs/Coulomb branch Hilbert series then evaluates to +HS = 1 + t (χ0,2 + χ2,0 + φ1,0,0,0,1 + 1) +(D.93) ++ t2 +�φ0,0,0,1,0 +Q ++ Qφ0,1,0,0,0 + Qφ0,0,0,2,0 + φ0,2,0,0,0 +Q ++ 2χ0,2 + χ0,4 + 2χ2,0 ++ χ2,2 + χ4,0 + 2χ0,2φ1,0,0,0,1 + 2χ2,0φ1,0,0,0,1 + χ2,2φ1,0,0,0,1 + 2φ0,1,0,1,0 ++ 3φ1,0,0,0,1 + φ2,0,0,0,2 + 4 +� ++ . . . +here φ denote su(6) characters and χ denote so(4) characters. The global form is SU(6)×U(1)Q +Z6 +× +PSO(4) where the Z6 charge of Q is 4 mod 6. +7th node. +The mirror pair is given by +4 +Z2 +y1,...,8 +←→ +1 +w1 +2 +w2 +3 +w3 +4 +w4 +4 +w5 +4 +w6 +SU(2) +2 +w8 +1 +v +. +(D.94) +and the fugacity map is +(w1, w2, w3, w4, w5, w6, w8)a = +7 +� +b=1 +u +CA7 +ab +b +, v = Q +u4 +(D.95) +with the weight space fugacities ua of su(8). The flavour fugacities follow from (C.30). The +Higgs/Coulomb branch Hilbert series then evaluates to +HS = 1 + t (φ1,0,0,0,0,0,1 + 1) +(D.96) ++ t2 +� +Qφ0,0,0,0,0,2,0 + Qφ0,0,0,1,0,0,0 + φ0,0,0,1,0,0,0 +Q ++ φ0,2,0,0,0,0,0 +Q ++ 2φ0,1,0,0,0,1,0 + 2φ1,0,0,0,0,0,1 + φ2,0,0,0,0,0,2 + 2 +� ++ . . . +where φ denotes su(8) characters. The global form is SU(8)×U(1)Q +Z8 +and Q has Z8 charge +4 mod 8. +D.8 +O(2k) SQCD and C-type mirrors +This appendix contains explicit Hilbert series for non-simply laced Dynkin quivers and their +O(2k) mirror SQCD theories. For concreteness, O(2) SQCD with 4 fundamental flavours +is considered. The mirror is a C4 balanced Dynkin quiver. +– 77 – + +Example 1. +Gauging the topological symmetry of the gauge node at the long edge leads +to the mirror pair +O(2) +Z2 +y1,2,3,4 +←→ +1 +w1 +m1 +2 +w2 +m2 +2 +w3 +m3 +SU(2) +− +l +1 +v +h +2 +(D.97) +and the magnetic fluxes for the right-hand side quiver in (D.97) take values in +(m1, m2, m3, l, h) ∈ Z × Z2 × Z2 × Z × Z . +(D.98) +The fugacity map is given by +w1 = x2 +1 +x2 +, +w2 = +x2 +2 +x1x3 +, +w3 = x2 +3 +x2 +, +v = Q−1 +x2 +2 +(D.99a) +y1 = Q +1 +4 x1 , +y2 = Q +1 +4 x2 +x1 +, +y3 = Q +1 +4 x3 +x2 +, +y4 = Q +1 +4 1 +x3 +(D.99b) +with the A3 weight space fugacities xi. One evaluates the Hilbert series to read +HS = 1 + t (φ1,0,1 + 1) +(D.100) ++ t2 +�φ0,0,4 + φ0,2,0 +Q ++ Q (φ0,2,0 + φ4,0,0) + φ0,2,0 + 2φ1,0,1 + 2φ2,0,2 + 2 +� ++ t3 +�2φ0,0,4 + φ0,1,2 + φ0,2,0 + φ1,0,5 + φ1,1,3 + φ1,2,1 +Q ++ Q (φ0,2,0 + φ1,2,1 + φ2,1,0 + φ3,1,1 + 2φ4,0,0 + φ5,0,1) ++ φ0,1,2 + φ0,2,0 + 3φ1,0,1 + φ1,1,3 + φ1,2,1 + 4φ2,0,2 + φ2,1,0 + 2φ3,0,3 + φ3,1,1 + 2 +� ++ . . . +where φn1,n2,n3 denotes A3 characters for irreps [n1, n2, n3]. The U(1)Q is trivial under the +Z4 centre of SU(4), such that the global symmetry group becomes PSU(4) × U(1)Q. +Example 2. +Gauging the topological symmetry for the node closest to the non-simply +laced edge on the short side leads to +O(2) +2 +y1 +Z2 +y2,3,4 +←→ +1 +w1 +m1 +2 +w2 +m2 +SU(2) +− +l +2 +w4 +m4 +1 +v +h +2 +(D.101) +– 78 – + +and the magnetic fluxes for the right-hand side quiver in (D.101) are defined via +(m1, m2, l, m4, h) ∈ +1� +i=0 +� +Z × Z2 × Z × +� +Z + i +2 +�2 +× +� +Z + i +2 +�� +. +(D.102) +The fugacity map is given by +w1 = x2 +1 +x2 +, +w2 = +x2 +2 +x1x3 +, +v = 1 +x2 +2 +, +w4 = u2 +1 +(D.103a) +y1 = u1 , +y2 = x1 , +y3 = x2 +x1 +, +y4 = x3 +x2 +(D.103b) +where xi are C3 weight space fugacities and u1 is a C1 weight space fugacity. the Hilbert +series can be evaluated to read +HS = 1 + t (χ2,0,0 + φ2) + t2 (χ0,1,0 + χ0,2,0 + χ4,0,0 + 2φ2χ2,0,0 + φ4 + 1) +(D.104) ++ t3 +� +χ2,0,0 + χ2,1,0 + χ2,2,0 + χ6,0,0 + φ2χ0,1,0 + φ2χ0,2,0 + φ2χ2,0,0 + φ2χ2,1,0 ++ 2φ2χ4,0,0 + 2φ4χ2,0,0 + φ2 + φ6 +� ++ . . . +with χn1,n2,n3 the C3 characters for irreps [n1, n2, n3]C and φk1 the C1 characters for irreps +[k1]C. As all appearing irreps are invariant under the Z2 centre symmetries for C3 and C1, +the global symmetry group s PSp(3) × PSp(1). +Example 3. +Gauging the topological symmetry of the other U(2) gauge node on the +short side leads to +O(2) +4 +y1,2 +Z2 +y3,4 +←→ +1 +w1 +m1 +SU(2) +− +l +2 +w3 +m3 +2 +w4 +m4 +1 +v +h +2 +(D.105) +and the magnetic fluxes for the right-hand side quiver in (D.105) take values in +(m1, l, m3, m4, h) ∈ +1� +i=0 +� +Z × Z × Z2 × +� +Z + i +2 +�2 +× +� +Z + i +2 +�� +(D.106) +The relevant fugacity map is given by +w1 = x2 +1 +x2 +, +v = 1 +x2 +2 +, +w3 = u2 +1 +u2 +, +w4 = u2 +2 +u2 +1 +(D.107a) +y1 = u1 , +y2 = u2 +u1 +, +y3 = x1 , +y4 = x2 +x1 +(D.107b) +– 79 – + +with xi and ui two sets of C2 weight space fugacities. 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Zou, Inverses of cartan matrices of lie algebras and lie superalgebras, +Linear Algebra and its Applications 521 (may, 2017) 283–298. +– 83 – + diff --git a/9dE0T4oBgHgl3EQffwCf/content/tmp_files/load_file.txt b/9dE0T4oBgHgl3EQffwCf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5dc5677854d48de1c2be3f0c8e0d3ceacdc8a5e4 --- /dev/null +++ b/9dE0T4oBgHgl3EQffwCf/content/tmp_files/load_file.txt @@ -0,0 +1,3170 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf,len=3169 +page_content='3d N = 4 mirror symmetry with 1-form symmetry Satoshi Nawata1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Marcus Sperling2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Hao Ellery Wang3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Zhenghao Zhong4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='5 1Department of Physics and Center for Field Theory and Particle Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 220,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Handan Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 200433 Shanghai,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Tsinghua University Haidian District,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 100084,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' China 4Theoretical Physics Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The Blackett Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Imperial College London Prince Consort Road,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' OX2 6GG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' UK E-mail: snawata@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='com, msperling@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='cn, yukawahaow@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='com, zhenghao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='zhong@maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='uk Abstract: The study of 3d mirror symmetry has greatly enhanced our understanding of various aspects of 3d N = 4 theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In this paper, starting with known mirror pairs of 3d N = 4 quiver gauge theories and gauging discrete subgroups of the flavour or topological symmetry, we construct new mirror pairs with non-trivial 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' By providing explicit quiver descriptions of these theories, we thoroughly specify their symmetries (0- form, 1-form, and 2-group) and the mirror maps between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='02409v1 [hep-th] 6 Jan 2023 Contents 1 Introduction 2 2 Gauging discrete 0-form symmetries 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 Abelian theories 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 An illustrative example 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3 T[SU(N)] theories 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='4 T σ ρ [SU(N)] theories 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='5 T[SO(2N)] theories 22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='6 Sp(k) SQCD and its orthosymplectic mirror 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='7 Sp(k) SQCD and its unitary D-type mirror quiver 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='8 Examples of non-simply laced unitary quivers and their mirrors 28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='9 Magnetic quivers and gauging discrete topological symmetries 32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='10 Examples from 5d magnetic quivers 34 3 Discussion and conclusions 36 A Notations and conventions 38 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 Hilbert series 39 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 Superconformal index 40 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3 Centre symmetries of classical Lie algebras 41 B Discrete gauging of T[SU(N)] 41 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 Gauging discrete subgroup of topological symmetry 42 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 Gauging discrete subgroup of topological symmetry revisited 45 C Mirror maps 49 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 SQED and its abelian mirror quiver 50 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 T[SU(N)] theories 51 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3 Examples for T σ ρ [SU(N)] 53 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='4 Sp(k) SQCD and its D-type unitary mirror quiver 54 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='5 O(2k) SQCD and its C-type unitary mirror quiver 56 D Explicit Hilbert series results 56 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 Linear Abelian quiver 57 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 T[SU(N)] theories 57 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3 Some T[SU(N)] examples with higher charges 64 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='4 Some T σ ρ [SU(N)] examples 65 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='5 T[SO(2N)] theories 67 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='6 Sp(k) SQCD and orthosymplectic mirrors 69 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='7 Sp(k) SQCD and D-type mirrors 71 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='8 O(2k) SQCD and C-type mirrors 77 – 1 – 1 Introduction Supersymmetric theories with 8 supercharges in space-time dimension 3 exhibit a rich set of intriguing features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' One of the most prominent is 3d mirror symmetry [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Given a 3d theory that has a mirror dual theory, 3d mirror symmetry exchanges Coulomb branch and Higgs branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In particular, this also implies the exchange of flavour symmetries Gf (Higgs branch isometries) and the topological symmetries Gt (Coulomb branch isometries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The notion of symmetries has been generalised to include novel types beyond the stan- dard symmetries of local operators [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Among others, these include higher-form symme- tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Specifically for 3d theories, discrete 1-form symmetries can be generated by gauging discrete 0-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The structure of generalised symmetries in 3d supersymmet- ric theories has been the focus of recent research, including [3–14] and references therein1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Given the vast catalogue of 3d mirror pairs with trivial 1-form symmetry, one might wonder what mirror symmetry implies for 3d theories with 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In this paper, we start with a known mirror pair (T , T ∨) of 3d N = 4 theories that admit UV quiver descriptions, and gauge a discrete Γ[0] subgroup of the 0-form symmetry to generate new theories with Γ[1] 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Depending on whether Γ ≡ Γ[0] is a subgroup of the flavour or topological symmetry, the resulting mirror pair (T /Γ, (T /Γ)∨) changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For Γ ≡ Γf ⊂ Gf, the field theory description of T /Γf is straightforward, but for Γ ≡ Γt ⊂ G∨ t , the description of T ∨/Γt is less transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In this paper, explicit quiver descriptions are provided for these cases, as well as the global form of the 0-form symmetries of (T /Γ, (T /Γ)∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' It is known that the resulting 0-form and the newly introduced discrete 1-form symmetry may not just be a direct product, but can form an extension, called 2-group symmetry [9, 17–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' We comment on such extensions throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The remainder of this paper is organised as follows: in Section 2, we consider known mirror pairs and gauge discrete 0-form symmetries to generate mirror pairs with non- trivial 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' We first study abelian theories, followed by non-abelian T[SU(N)] and T σ ρ [SU(N)] theories with non-abelian product gauge groups � i U(Ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This class of examples has the benefit that all 0-form symmetries are manifest in the UV description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Thereafter, SO(k) and Sp(k) gauge groups are considered by studying T[SO(2N)] theories, Sp(k) SQCD, and linear orthosymplectic quivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' While the flavour 0-form symmetry is manifest in this set of examples, the topological symmetry is at most accessible by discrete Z2 subgroups, which turns out to be sufficient for the intents and purposes here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Lastly, we consider mixed types: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' D and C-type Dynkin quivers composed of unitary gauge groups and their mirror Sp(k) and O(2k) SQCD theories, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The advantage of this class of mirror pairs is that the flavour symmetry of the SQCD theories and the topological symmetry of the unitary Dynkin quivers are fully manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Before closing, some magnetic quiver examples are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Conclusions are provided in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Several appendices complement the main text and provide computational details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 1See also [15, 16] for recent review articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 2 – Note added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' During the course of this project, we were informed of a related work done by Bhardwaj, Bullimore, Ferrari, and Sch¨afer-Nameki [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' We are grateful to them for coordinating the submission of our papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2 Gauging discrete 0-form symmetries In this section, mirror theories with non-trivial 1-form symmetry are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging discrete subgroups of the 0-form symmetry, which results in 1-from symmetries and a potential 2-group structure, has, for example, been considered in [9, 18–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The principle is simple: start from a known mirror pair (T , T ∨) and gauge discrete 0-form symmetries Γ such that Γ ≡ Γt ⊂ Gt(T ) and Γ ≡ Γf ⊂ Gf(T ∨).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This ensures that the resulting theories (T /Γt, T ∨/Γf) are mirror pairs with 1-form symmetry Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The aims here are (i) to provide explicit quiver descriptions for (T /Γt, T ∨/Γf) and (ii) to detail the resulting symmetries (0-form, 1-form, and 2-group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 Abelian theories As a first example, consider 3d N = 4 SQED with N hypermultiplets of charge 1 and its abelian mirror quiver theory [1], see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The global 0-form symmetries are well- known: for SQED one finds U(1)t × PSU(N)f, while the abelian mirror quiver enjoys a U(1)f × PSU(N)t symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 SQED with higher charge Suppose that one gauges a discrete Zq subgroup of the abelian U(1) factor of the global 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The resulting theories are straightforwardly derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging a Zq ⊂ U(1)t for SQED with N charge 1 hypermultiplets leads to SQED with N charge q hyper- multiplets, see also [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Similarly, gauging a Zq ⊂ U(1)f of the abelian mirror quiver leads to an abelian quiver with a Zq 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The two theories obtained are then mirror to each other, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The quiver notation is summarised in Table 3 of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Consistency checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The proposed mirror symmetry can be verified by Hilbert series techniques [23–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The Higgs branch Hilbert series is insensitive to the gauging of the Zq inside the topological symmetry of the SQED theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' similarly, the Coulomb branch of the mirror does not perceive changes upon gauging a discrete subgroup of the flavour symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 for conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Performing the discrete gauging for the SQED theory reduces to a Zq Molien-Weyl sum over the Coulomb branch Hilbert series HSC SQEDq=1 N (w|t) = 1 1 − t � m∈Z wmt 1 2 N|m| = PE[t + (w + w−1)t N 2 − tN] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1) HSC SQEDq N (z|t) = 1 q q−1 � p=0 HSC SQEDq′=1 N (w|t) �� w=z 1 q (ζq)p ζq = e 2πi q ∈ Zq – 3 – 1 N 0-form: U(1)t × PSU(N)f 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 1 1 1 1 0-form: PSU(N)t × U(1)f 1 N q 0-form: U(1)t × PSU(N)f 1-form: Zq 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 1 1 Zq 0-form: PSU(N)t × U(1)f 1-form: Zq gauge Zq ⊂ U(1)t gauge Zq ⊂ U(1)f mirror mirror Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging of discrete 0-form symmetries in SQED and its mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For SQED with charge 1 hypermultiplets, a Zt q gauging results in SQED with charge q hypermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' These are indicated by an arrow with the label q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For the abelian mirror quiver, the Zf q gauging is realised by acting on the flavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The fundamental flavours that are charged under the discrete Zq are connected to a grey node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' See Appendix A for conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' = 1 1 − t � m∈Z zmt 1 2 N|q·m| = PE[t + (z + z−1)t 1 2 qN − tqN] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2) Likewise, one performs the Zq Molien-Weyl sum on the Higgs branch Hilbert series of the mirror theory HSH mirror(y|t) = N−1 � a=1 � dxa 2πixa PE �N−2 � b=1 � xb xb+1 + xb+1 xb � t 1 2 − (N−1)t � PE �� y 1 2 x1 + x1 y 1 2 + y− 1 2 xN−1 + xN−1 y− 1 2 � t 1 2 � = PE[t + (y + y−1)t N 2 − tN] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3) HSH mirror/Zq(z|t) = 1 q q−1 � p=0 HSH mirror(y|t) ���� y=z 1 q (ζq)p = PE[t + (z + z−1)t 1 2 qN − tqN] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='4) In summary, both results confirm the expectation and provide the explicit parameter map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' As a remark, the superconformal index is equally well suited to probe such dualities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' see for – 4 – instance [12] for SQED with charge q = 2 hypermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Since either Higgs or Coulomb branch operators are unaffected by gauging a Zt/f q , the Hilbert series is a more convenient tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Using the techniques of [6], one can inspect the interplay between the discrete 1-form symmetry Zq ⊂ U(1)t and the global 0-form symmetry PSU(N)f for the SQED theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The centre symmetry ZF = ZN of su(N)f is generated by αF = ζN, while the U(1) gauge group supports a ZG = ZN·q centre generated by αG = ζN·q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The diagonal αD = (αG, αF ) generates a E = ZN·q ⊂ ZG × ZF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The 1-form symmetry Γ[1] = Zq is generated by αN D = (αN G, αN F ) = (αN G, 1), which acts trivially on the matter content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The short exact sequence 0 → Γ[1] = Zq → E = Zq·N → Z = ZN → 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='5) splits whenever gcd(q, N) = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' q and N are co-prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In order words, for gcd(q, N) > 1 there exists an extension to a 2-group structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' See for instance [9, 12] for a recent discussion of SQED with 2 flavours of charge 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Comments on lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' As explained in [9, 27, 28], 1-form symmetries and 2-group struc- tures can be understood via equivalence classes of lines defects2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Here, we illustrate how the higher-form symmetry is also realised on the line defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Consider SQED with N hypermultiplets of charge q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' A Wilson line of charge h with h ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , q − 1} cannot end on a local operator because local operators are either constructed as polynomials in the fundamental hypermultiplets of charge q or are monopole operators, which are gauge singlets for 3d N = 4 theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Thus, the 1-form symmetry Γ[1] (or its Pontryagin dual) is generated by the (q − 1) Wilson lines that cannot end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Refining with respect to the flavour symmetry shows that a Wilson line of charge q is equivalent to a flavour Wilson line transforming as [1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]AN−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This however is not an allowed representation of Gf = PSU(N), and signals the existence of a 2-group structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In fact, the N-th power of such a Wilson line is well-defined under Gf, because the N-th tensor product of [1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]AN−1 contains a singlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Such lines generate the group �E = Zq·N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Turning to the abelian mirror quiver, one can straightforwardly see that the fundamen- tal Wilson lines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' those of unit charge under a single U(1) gauge group factor, can end on a local operator constructed out of the hypermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Therefore, one needs to turn to the vortex lines to understand the 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' It is known [29] that the junctions between vortex lines are significantly more challenging than those between Wilson lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' It would be interesting to systematically address this in explicit examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2In brief, lines L1,2 are equivalent if there exists a local operator O at the junction between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The set of equivalence classes {L}/ ∼ forms the Pontryagin dual �Γ[1] of the 1-form symmetry Γ[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Refining the equivalence relation by keeping track of 0-form symmetry representations R leads to the following equivalence relation: (L1, R1) ∼ (L2, R2) iff there exists a local operator transforming as R1 ⊗ R∗ 2 (or R∗ 1 ⊗ R2) at the junction of the lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The equivalence classes give rise to �E (Pontryagin dual of E), which encodes the interplay between the centres of gauge symmetry and 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' These groups fit into the short exact sequence 0 → �Z → �E → �Γ[1] → 0, which is the Pontryagin dual of the sequences discussed in the text, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='13), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 5 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 SQED with discrete gauge factor Next, revert the logic: gauge a Zq subgroup of the PSU(N)f symmetry of SQED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Con- versely, on the mirror side, one gauges a Zq subgroup of the PSU(N)t topological symmetry of the abelian quiver theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For the abelian quiver theory, discrete gauging along a Cartan U(1)t of the topological symmetry alters the linear quiver theory by modifying the charges of the bifundamental hypermultiplets attached to a single gauge node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This follows from analogous arguments as for SQED with charge q hypermultiplets or the arguments used in Appendices B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 – B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For the SQED theory, gauging of a discrete flavour 0-form symmetry affects some of the fundamental flavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' To see this, one uses the original mirror map (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='7) between the parameters to identify which flavour fugacities are affected by gauging along a Cartan U(1)t factor in the abelian mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' As a result, the flavours of the SQED split into two sets: one charged under Zq and the other is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Global symmetry: abelian mirror point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The global symmetry is affected as follows: suppose that one gauges a Zq ⊂ U(1)k ⊂ PSU(N)t subgroup of the topological Cartam U(1) at the k-th node of the abelian quiver 1 w1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 1 wk−1 1 wk 1 wk+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 1 wN−1 1 1 (q, −1) (1, −q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='6) The 0-form symmetry algebra after gauging is su(k) ⊕ u(1) ⊕ su(N−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' As exemplified in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1, the 0-form symmetry group is Gt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='6) = SU(k) × U(1)Q × SU(N−k) Zk × ZN−k (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='7) where the centre symmetry Zℓ ⊂ SU(ℓ) acts on the fundamental representation [1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]Aℓ−1 with charge +1 under Zℓ, for ℓ ∈ {k, N−k}, see also Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Moreover, the Zk × ZN−k act with charge (−q mod k, q mod (N−k)) on the U(1)Q variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Roughly Q ∼ wk, see (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='8) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The global structure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='7) can also be inferred directly from the set of balanced nodes in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The unbalanced gauge node U(1)k is connected to two balanced sets of gauge nodes, forming the Ak−1 and AN−k−1 Dynkin diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Generalising the arguments of [30], there are monopole operators transforming as [0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , q]Ak−1 × (1)Q (and its conjugate) and [q, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0] × (1)Q (and conjugate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This follows because the U(1)k node is attached to the k − 1-th node of the Ak−1 Dynkin diagram and the 1-st node of AN−k−1 Dynkin diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Compared to the standard case of unit charge bifundamental hypermultiplets, the increased charge q modifies the appearing Aℓ representations accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The existence of these monopole operators in the Coulomb branch chiral ring leads to the isometry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 6 – 1 N 0-form: U(1)t × PSU(N)f 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 1 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 1 1 1 (1, −1) (1, −1) 0-form: PSU(N)t × U(1)f 1 Zq N − k k 0-form: U(1)t × ( SU(k)×U(1)×SU(N−k) Zk×ZN−k ) f 1-form: Zq 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 1 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 1 1 1 (q, −1) (1, −q) 0-form: ( SU(k)×U(1)×SU(N−k) Zk×ZN−k ) t × U(1)f 1-form: Zq gauge Zq ⊂ PSU(N)f gauge Zq ⊂ PSU(N)t mirror mirror Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging of discrete 0-form symmetries in SQED and its mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The centre symmetries act with charges (−q mod k, q mod (N−k)) on the U(1) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For SQED, the Zq acts on k of the N hypermultiplets, which is indicated by k edges connected to a grey node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The remaining hypermultiplets are uncharged under the discrete group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For the abelian linear quiver, gauging along the Cartan U(1)t at the k-th gauge nodes leads to hypermultiplets with charge q under the k-th U(1), while still of unit charge under the adjacent gauge factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This is indicated by an arrow with label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' See Appendix A for conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Global symmetry: SQED point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' To illuminate this result, it is instructive to also consider the SQED side: 1 Zq N − k Xa ˜Xd k , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='8) where the two distinct sets of fundamentals are denoted as X and ˜X (by convention, both have charge −1 under the U(1) gauge group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Computationally, gauging a discrete Zf q is realised via the following flavour fugacities, see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 Xa : a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , k : ya = ζq · Q1 · � � � � � � � x1 , a = 1 xa xa−1 , 1 < a < k 1 xk−1 , a = k (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='9a) – 7 – ˜Xd : d = k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , N : yd = Q2 · � � � � � � � u1 , d = k + 1 ud−k ud−k−1 , k + 1 < d < N−k 1 uN−k−1 , d = N−k (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='9b) with ζq ∈ Zf q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The xa and ud are weight space fugacities for su(k) and su(N−k), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The first observation is that if k|q then the Zk centre symmetry of SU(k) is gauged, such that a global PSU(k)f factor arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Similarly, if (N−k)|q the ZN−k centre of SU(N−k) is gauged, leading to a PSU(N−k)f factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For the general case, one fixes the two so far arbitrary U(1)Q1,2 symmetries3: � � � Qk 1 · QN−k 2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='= 1 � Q1 Q2 �q !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='= Q ⇒ � � � Q1 = Q N−k q·N Q2 = Q −k q·n (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='9c) which agrees with (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='11) of Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Next, consider a gauge invariant operator O built from the fields {Xa}k a=1 transforming as (ζq · Q1 · [1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]Ak−1, −1) under flavour- gauge transformations and fields { ˜X† d}N d=k+1 transforming as (Q−1 2 ·[0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0, 1]AN−k−1, +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Thus, Xa ˜X† d is U(1) gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For Zq invariance, one also requires q-copies of Xa in the form of Symq(Xa), which leads to the q-th symmetric representation Symq[1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0] of SU(k)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' As a consequence, one also requires q copies of { ˜X† d} in the form Symq( ˜Xd), which leads to the q-th conjugate symmetric representation Symq[0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0, 1] of SU(N−k)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Such a gauge invariant operator has charges O = Symq a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=',aq(Xa1) · Symq d1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=',dq( ˜Xdi) ↔ Symq[1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]SU(k) ⊗ Symq[0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0, 1]SU(N−k) ⊗ �Q1 Q2 �q � �� � =Q (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='10) The operator O has Zk × ZN−k centre charges (q mod k, −q mod (N−k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Hence, the Zk×ZN−k transformations can be compensated by a global U(1)Q rotation if Q has charges (−q mod k, q mod (N−k)) under the centre symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This confirms (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='7) as flavour symmetry Gf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The operator O can be detected in the Hilbert series at R-charge q·2· 1 2 = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Comments on lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Returning to the quiver (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='6), consider a Wilson lines Wa of charge 1 under the a-th U(1) gauge group factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For each a ̸= k, Wa can end on a local operator composed of concatenated bifundamental hypermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For a = k, Wk cannot end since the bifundamentals connected to the k-th gauge node are of charge q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Further, monopole operators cannot screen gauge Wilson lines, because monopole operators are gauge singlets for 3d N = 4 theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Thus, the lines (Wk)h with h ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , q − 1} cannot end and generate the abelian group �Γ[1] = Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 3The definition of Q is a choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Here, it is chosen such that the operator O in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='10), as Higgs branch operator with lowest R-charge that is charged under the U(1), has the unit charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 8 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 An illustrative example One of the main messages of this paper is that gauging discrete Zq subgroups of the topological symmetry for quiver gauge theories T with unitary gauge nodes can result in theories T /Zt q which admit a simple quiver description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' To illustrate this fact, consider U(k) SQCD with N ≥ 2k fundamental flavours k N (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='11) with the well-known 0-form symmetries: Gf = PSU(N), Gt = U(1)t for N > 2k and Gt = SO(3) for N = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Next, express the gauge group as U(k) ∼= U(1)×SU(k) Zk where Zk acts as centre on SU(k) and via Zk charge (k−1) on the U(1) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Rewriting U(k) magnetic fluxes m ∈ Zk into U(1) × SU(k) fluxes (h, l) requires the co-character lattice to be Γ = �k−1 i=0 (Z + i k)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Effectively, the SQCD theory can be understood as SU(k)×U(1) gauge theory with N copies of bifundamentals and an “unusual” magnetic lattice Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' One can introduce a (topological) fugacity z that keeps track of the components of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' If w denotes the topological fugacity of T , one employs w → zw 1 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Next, gauge a discrete Zq subgroup of the topological symmetry by performing a discrete Molien-Weyl sum over z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' It is convenient to choose either q|k or k|q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' One can show rigorously (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' using the superconformal index or the Coulomb branch Hilbert series, see Appendix B) the following: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 Gauge a subgroup with q|k If q|k then only the subgroup Zq ⊂ Zk is gauged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The theory becomes � ����� SU(k) 1 N � ����� /Z k q with magnetic lattice k q −1 � i=0 � Z + i · q k �k , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='12) where the quotient Z k q signals that this discrete group is not gauged, in the sense of [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The resulting theory has a U(1)t × PSU(N)f 0-form symmetry and a Zq 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The potential interplay can be analysed via the action of the centre symmetries: defining αG = ((ζk) k q , ζq·N) ∈ Zk × Zq·N (because only a Zq ⊂ Zk is gauged) and αF = ζN ∈ ZN, the diagonal combination αD = (αG, αF ) generates a E = Zq·N group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The element N · αD = ((ζk) k q ·N, ζN q·N, 1) generates a Γ(1) = Zq subgroup that acts trivial on the matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' By definition, this establishes the 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The short exact sequence 0 → Γ(1) = Zq → E = ZN·q → ZN → 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='13) – 9 – splits if gcd(q, N) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' If gcd(q, N) > 1, there exists a non-trivial 2-group extension of Γ(1) and PSU(N)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Symmetries via lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' One can again illustrate this higher-form symmetry by using line defects and their equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' A gauge Wilson line W in the representation [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]Ak−1 × (−1) cannot end on any local operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Neither polynomials of the hyper- multiplets nor monopole operators, because of a mismatch in gauge charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' However, W q can end on the determinant operator O ∼ det(X), obtained by contracting hypermultiplets X with the invariant ϵ tensor of SU(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This operator has charges [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]Ak−1 × k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Since q|k, O has the same centre charges as W q, such that W q can end on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Therefore, the lines W a with a ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , q − 1} cannot end on any local operator and generate the abelian group �Γ[1] = Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Taking flavour charges into account, W q is equivalent to a flavour Wilson line transforming as ∧k[0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0, 1]AN−1, which follows from the flavour charges of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This is not a representation of PSU(N)f, but taking N-th tensor (W q)⊗N is equivalent to a singlet of the flavour symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Thus, these lines generate the group �E = ZN·q and the 1-form symmetry potentially forms a 2-group with the flavour symmetry (depending on the gcd(N, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 Gauge a discrete group Zq with k|q If k|q then the SU(k) centre Zk is a subgroup of Zq and fully gauged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The theory becomes SU(k) 1 N q k with magnetic lattice Zk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='14) The difference is now that the hypermultiplets transform as SU(k) fundamental with charge q k ∈ N under the U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This is indicated by the arrow, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In terms of symmetries, the theory T /Zt q has a U(1)t topological symmetry, PSU(N)f flavour symmetry, a Zq 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Moreover, inspecting the gauge-flavour centre symmetries shows: αG = (ζk, ζN·q) ∈ Zk × ZN·q and αF = ζN ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The diagonal generator αd = (αG, αF ) spans a E = ZN·q, and the element N ·αd = (ζN k , ζN N·q, 1) generates a Γ(1) = Zk·q subgroup, using that k|q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This subgroup acts trivial on the matter fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' thus, defining the 1-form symmetry Γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The short exact sequence 0 → Γ(1) = Zq → E = ZN·q → ZN → 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='15) splits if gcd(q, N) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In all other cases, there exists a non-trivial extension giving rise to a 2-group structure between Γ(1) and Gf = PSU(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Symmetries via lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Again, let us illustrate these structures with line defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The gauge Wilson line W transforming as [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]Ak−1×(−1) under SU(k)×U(1) cannot end on a local operator, which either has to be a polynomial in the hypermultiplet X transforming as [1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]Ak−1 × (− q k) or has to be a monopole operator, which is gauge singlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In – 10 – contrast, the Wilson line W q can end on the local operator constructed as the determinant: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' the SU(k) gauge group is equipped with the invariant ϵi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=',ik tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Contracting k hypermultiplets yields an operator O ∼ det(X) which transforms as [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]Ak−1 × (−q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Hence, the set of Wilson lines W a with a ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , q − 1} cannot end and generate the abelian group �Γ[1] = Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' If one also keeps track of the flavour symmetry representations, one finds that O transforms as ∧k[0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0, 1]AN−1 which is not a representation of PSU(N)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Hence, this gauge Wilson line is equivalent to a flavour Wilson line and the centres of gauge and flavour symmetry intertwine to give rise to a 2-group structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The following sections apply the analogous argument to other quiver gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The relevant questions are: (i) What is the resulting theory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (ii) What are its symmetries?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (iii) What is the mirror dual theory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3 T[SU(N)] theories Moving on to quiver theories with non-abelian gauge factors, consider the self-mirror T[SU(N)] theories [30], see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The global 0-form symmetry group is given by PSU(N)t × PSU(N)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In the same spirit as above, one can gauge a discrete Zq 0-form symmetry inside, say, the topological symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The mirror of the resulting theory is then obtained by gauging a Zq 0-form symmetry inside the flavour symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The question is how the Zq is embedded inside the flavour symmetry, given that the Zq is embedded into a Cartan U(1) of the topological symmetry of the mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' To answer this, one utilises the mirror map (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In more detail, let us consider gauging a Zq ⊂ PSU(N)t of a T[SU(N)] theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' one inquires about the nature of the resulting theory T[SU(N)]/Zt q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Analogous to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2, see also Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1, for a specific Zq embedded in the k-th topological Cartan U(1) factor, the resulting theories T[SU(N)]/Zt q are in fact related to versions of T[SU(N)] encountered in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' These quiver theories differ from T[SU(N)] as follows: the k-th node is replaced by U(k) → SU(k), and the flavour node becomes a U(1) gauge nodes with an N copies of bifundamental hypermultiplets between U(N−1) and the “new” U(1) gauge node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Restricting to the case that either q|k or k|q, the resulting theory is given by q|k with d = k q : � � 1 w1 2 w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' k−1 wk−1 SU(k) k+1 wk+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' N−1 wN−1 1 v N � � /Zd magnetic lattice: d−1 � i=0 � Γ + i d � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='16a) k|q with q = a · k : 1 w1 2 w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' k−1 wk−1 SU(k) k+1 wk+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' N−1 wN−1 1 v N a (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='16b) magnetic lattice: Γ wherein Γ denotes the standard integer lattice one assigns to the quiver based on [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The shifts by i d are to be understood as in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 11 – 1 2 ⋯ k ⋯ N−1 N 0-form: PSU(N)f × PSU(N)t 1 2 ⋯ k ⋯ N−1 N 0-form: PSU(N)t × PSU(N)f ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 0-form: PSU(N)f × ( SU(k)×U(1)×SU(N−k) Zk×ZN−k ) t 1-form: Zq 1 2 ⋯ k ⋯ N−1 Zq N − k k 0-form: ( SU(k)×U(1)×SU(N−k) Zk×ZN−k ) f × PSU(N)t 1-form: Zq gauge Zq ⊂ PSU(N)t gauge Zq ⊂ PSU(N)f mirror mirror Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging of discrete 0-form symmetries in T[SU(N)] theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The centre symmetries act with charges (q mod k, −q mod (N−k)) on the U(1) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The quiver description for T[SU(N)]/ Zt q, here denoted by ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=', is provided in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The quiver for T[SU(N)]∨/Zf q shows again a split of the N fundamental flavours into two sets: k of them are charged under Zf q , which is indicated by an edge of multiplicity k to the grey node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' the remaining N−k flavours are uncharged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The mirror theory T[SU(N)]∨/Zf q is obtained from T[SU(N)]∨ = T[SU(N)] by gauging a Zq ⊂ PSU(N)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The mirror map (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='15) dictates that this is realised by splitting the N fundamental flavours into two sets of k and N−k flavours, and gauging the Zq symmetry in the overall U(1) flavour symmetry of one of the two sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For concreteness, consider gauging the Zq on the set of k fundamental flavours: 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' N−1 Zq {yi}k i=1 N−k {yj}N j=k+1 k (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='17) and Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 provides exemplary Hilbert series computations that confirm the mirror symmetry between the theories with non-trivial 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The mirror map between the parameters of T[SU(N)]/Zt q in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='16) and T[SU(N)]∨/Zf q in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='17) can be derived exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For concreteness, consider the case q = k, then the map between the parameters in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='16) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='17) is established via � wi = yi yi+1 , i ̸= k , v = yN N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='18) – 12 – Further details on this map are provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Global symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Building on the understanding of the 0-form symmetry group (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='7) for (balanced) abelian quivers, one can utilise a similar logic for the balanced T[SU(N)] theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Consider the quiver (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='16) the topological symmetry algebra is su(k) ⊕ u(1) ⊕ su(N−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The global form is then given by Gt = SU(k) × U(1)Q × SU(N−k) Zk × ZN−k (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='19) with Q has Zk × ZN−k charges (−q mod k, q mod (N−k)) where the centre symmetries Zℓ act in the standard way on SU(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Note that for k|q there is a PSU(k) factor in the global symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The examples in the next paragraph, as well as the explicit character decomposition in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2, confirm this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This structure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='19) is also apparent from the Higgs branch isometry of the mirror (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='17), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' denote the two distinct sets of fundamentals by 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' N−1 Zq N−k Xa ˜Xd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='20) Analogously to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='9), one can perform the Zf q gauging by assigning (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2) Xa : a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , k : ya = ζq · Q1 · � � � � � � � x1 , a = 1 xa xa+1 , 1 < a < k 1 xk−1 , a = k (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='21a) ˜Xd : d = k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , N : yd = Q2 · � � � � � � � u1 , d = k + 1 ud−k ud−k−1 , k + 1 < d < N−k 1 uN−k−1 , d = N−k (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='21b) with ζq ∈ Zf q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The xi and uj are weight space fugacities of su(k) and su(N−k), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The two appearing U(1) fugacities Q1,2 effectively reduce to a single U(1)Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' for instance by imposing � i yi = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' � � � Qk 1 · QN−k 2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='= 1 � Q1 Q2 �q !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='= Q ⇒ � � � Q1 = Q N−k N·q Q2 = Q− k N·q (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='21c) which agrees4 with (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='20) of Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Note also that for q|k the (ζq) k q ∈ Zf q acts as the Zk ⊂ SU(k)xi centre symmetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' thus the global factor is PSU(k)xi in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 4Again, the definition of Q is a choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' It is motivated by assigning the unit charge to the Higgs branch operator O in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='22), which is the operator with the lowest R-charge that is charged under the U(1)Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 13 – The U(1)Q may transform non-trivially under the Zℓ ⊂ SU(ℓ) centre symmetries, de- pending on the charge of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' To determine the charge, one again considers a specific gauge-invariant operator O build out of the two sets of fundamentals: X transforms as (ζq · Q1 · [1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]Ak−1, N−1) and ˜X† transforms as (Q−1 2 [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0, 1]AN−k−1, N−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' U(N−1) gauge invariance imposes Tr(X ˜X†), wherein the trace is taken over the gauge indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Zq gauge invariance requires O = SymqTr(X ˜X†), where the symmetrisation acts on the flavour indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The resulting operator transforms as O : Symq[1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]k ⊗ Symq[0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0, 1]N−k ⊗ �Q1 Q2 �q � �� � =Q (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='22) such that the Zk × ZN−k centre charges are (q mod k, −q mod (N−k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' These can be compensated by a global U(1)Q rotation provided the centre charges of Q are (−q mod k, q mod (N−k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This confirms (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='19) as flavour symmetry for the quiver (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' As a remark, the operator O can be detected in the Hilbert series as the first non-trivial term in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The R-charge of O is simply q ×2· 1 2 = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 provides examples that illustrate this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' By analogous arguments as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2, one can verify that theories (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='16) indeed have the expected Γ(1) = Zq 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' One finds that the centre generators of the combined gauge-flavour symmetry span a E = Zq·N group, such that there exists a non-trivial 2-group extension between Γ(1) and Gf = PSU(N) whenever gcd(q, N) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Similarly, the same conclusion is reached by inspecting the screening of Wilson lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For an illustrative purpose, let us consider N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging a specific Z2 0-form symmetry leads to a mirror pair: 1 SU(2) 3 1 4 mirror ←−−−−−−→ 1 2 3 2 Z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='23) The Hilbert series in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='15) confirms that the Coulomb branch symmetry algebra for the left quiver (and the Higgs branch isometry algebra of the right quiver) is g = su(2)⊕su(2)⊕u(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Moreover, the appearing SU(2) representations are all of integer spin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' thus, suggesting the global form G = SO(3) × SO(3) × U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Choosing to gauge a specific discrete Z3 subgroup of the 0-form symmetry results in the pair: 1 2 SU(3) 1 4 mirror ←−−−−−−→ 1 2 3 1 Z3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='24) – 14 – The explicit Hilbert series in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='12) shows that the Coulomb branch symmetry algebra of the left quiver (and the Higgs branch isometry algebra of the right theory) is g = su(3)⊕u(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Moreover, all appearing characters are neutral under the Z3 centre symmetry of SU(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' hence, the global form is G = PSU(3) × U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Considering discrete symmetries of the type Zq with q = a · k allows us to uncover equivalent descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Consider T[SU(5)] and gauge a Z6 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Among the choices considered here, gauging a Z6 ⊂ PSU(5)t is realised by turning the U(3) gauge node into SU(3) together with charge 2 for the “new” U(1) node 1 w1 2 w2 SU(3) 4 w4 1 v 5 2 mirror ←−−−−−−→ 1 2 3 4 Z6 2 3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='25) or by turing the U(2) node into SU(2) together with charge 3 for the “new” U(1) gauge factor 1 w1 SU(2) 3 w3 4 w4 1 v 5 3 mirror ←−−−−−−→ 1 2 3 4 Z6 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='26) Without the additional charges the theories are clearly distinct, for instance by the 0-form and 1-form symmetries, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='19) and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' However, with the modification, both become equivalent as, for example, the monopole formula in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='39) confirms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Equivalently, one the mirror, one gauges a Z6 subgroup of the flavour 0-form symmetry, but one time acting on three fundamental hypermultiplets and one time on two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='25) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='26), this is realised by {yi}5 i=1 → � � � � � � � {y1, y2, ζ6 y3, ζ6 y4, ζ6 y5}, for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='25) or {ζ6 y1, ζ6 y2, y3, y4, y5}, for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='26) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='27) with ζ6 ∈ Zf 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' But in both cases, the Z2 × Z3 centre symmetries are gauged by the discrete gauging of the Zf 6 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Hence, the global symmetry is simply PSU(2) × U(1)Q × PSU(3) for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The considerations so far implicitly assume that the gauge node U(k) at which the discrete subgroup of the Cartan U(1)t of the topological symmetry is gauged has k > 1, see for instance Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging the Cartan U(1)t of the U(1) gauge node of T[SU(N)] is in spirit similar to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Concretely, after gauging Zt q at the U(1) node, the bifundamental between U(1) and U(2) is modified to have charge q – 15 – under the U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Thus, the mirror pair becomes 1 w1 2 w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' N−1 wN−1 N q mirror ←−−−→ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' N−1 Zq y1 N−1 {yj}N j=2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='28) and the global symmetry becomes Gt(left quiver (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='28)) = Gf(right quiver (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='28)) = U(1)Q × SU(N−1) ZN−1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='29) where Q has ZN−1 charge q mod (N−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='4 T σ ρ [SU(N)] theories The class of linear quiver gauge theories with unitary gauge groups and fundamental or bifundamental hypermultiplets is given by the T σ ρ [SU(N)] theories [30], where ρ, σ are two partitions of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For σ = ρ = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 1) ≡ (1N), the corresponding theory is simply T (1N) (1N) [SU(N)] = T[SU(N)] and the partition data can be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Mirror symmetry exchanges the partitions σ and ρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' T σ ρ [SU(N)]∨ = T ρ σ[SU(N)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Analogous to the cases considered so far, the gauging of a discrete 0-form symmetry (either inside the topological symmetry group Gt or the flavour symmetry group Gf) leads to a theory with non-trivial 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Again, consider the two options in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' While the process of gauging a discrete subgroup of Gt is by now understood (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3 and Appendix B), determining the action of the discrete group on the flavour symmetry of the mirror theory becomes more challenging when Gf is a generic product group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Thus, special attention is paid to determining the mirror theory of T σ ρ [SU(N)]/Zt q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging a ZNk ⊂ Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' A T σ ρ [SU(N)] is a linear quiver theory with gauge/flavour groups specified by a sequence of integers {Ni} and {Mi}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The partitions determine the integers as detailed in [30] and the quiver becomes T σ ρ [SU(N)] : N1 N2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Nk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Nn−1 Nn M1 M2 Mk Mn−1 Mn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='30) For concreteness, take the node Nk, with Nk > 1, and gauge a ZNk ⊂ U(1)k ⊂ Gt inside the Cartan factor of the topological symmetry associated to the k-th node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' By the same – 16 – arguments as in Appendices B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2, one straightforwardly derives the resulting theory T σ ρ [SU(N)]/Zt Nk : N1 N2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' SU(Nk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Nn−1 Nn 1 M1 Mn (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='31) which has a non-trivial ZNk 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Now, one constructs the mirror theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging a ZNk ⊂ G∨ f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The mirror quiver gauge theory of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='30) is given by T σ ρ [SU(N)]∨ = T ρ σ[SU(N)] : N∨ 1 N∨ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' N∨ n′−2 N∨ n′−1 N∨ n′ M∨ 1 M∨ 2 M∨ n′−2 M∨ n′−1 M∨ n′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='32) and the integers {N∨ i } and {M∨ i } are determined by the partition data ρ, σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' To determine which ZNk ⊂ G∨ f subgroup needs to be gauged, one has two options: one could derive the mirror map of parameters for the specific pair (T σ ρ [SU(N)], T ρ σ[SU(N)]) and compute which flavours are charged under Zf Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In principle, this is straightforward but likely to be tedious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Alternatively, one can employ the following train of thought: The mirror theory T ρ σ[SU(N)] can be rewritten in an unframed form N∨ 1 N∨ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' N∨ n′−2 N∨ n′−1 N∨ n′ M∨ 1 M∨ 2 M∨ n′−2 M∨ n′−1 M∨ n′ ∼= N∨ 1 N∨ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' N∨ n′−2 N∨ n′−1 N∨ n′ 1 M∨ 1 M∨ n′ ///U(1)diag (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='33) where no explicit flavour group appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For such a theory, it is implied that an overall U(1)diag subgroup decouples so the two quiver diagrams express the same theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The next step is to turn the unitary gauge group U(Nk) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='30) into a special unitary gauge group SU(Nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This theory still has a trivial 1-form symmetry due to the flavour groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' However, the 3d mirror theory can be found by using the algorithm in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Schematically, one finds N1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' SU(Nk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Nn M1 Mk Mn mirror ←−−→ N∨ 1 N∨ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' N∨ n′−2 N∨ n′−1 N∨ n′ 1 1 M∨ 1 M∨ n′ ///U(1)diag (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='34) Turning U(Nk) into SU(Nk) means the 3d mirror has an additional U(1) gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The – 17 – additional U(1) gauge group in the unframed quiver is the result of gauging the flavour symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Now, there are two U(1) gauge groups connected to the rest of the linear quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The number of bonds M∨ i attached to each U(1) depends precisely on the choice of SU(Nk), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' which k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The splitting can also occur where the same gauge node, for example, U(N∨ 2 ) is connected to one U(1) gauge group with an edge of multiplicity M∨ 2 −x and to the other U(1) with an edge of multiplicity x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' see for instance Example 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The final step is to gauge the diagonal U(1) flavour symmetry in the left quiver of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='34) to obtain T σ ρ [SU(N)]/Zt Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' However, simply introducing a new U(1) gauge node leads to the ambiguity of the global form of the product gauge group, which can either be G = U(1) × SU(Nk) × � i̸=k U(Ni) or G removed by a subgroup of its centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For G/ZNk, with ZNk embedded into SU(Nk) as centre and into the diagonal U(1) ⊂ U(Ni) of each of the other gauge group factors, one obtains back the original theory T σ ρ [SU(N)], see [31] or Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' But this theory exhibits a Zt Nk topological symmetry, see also Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In order to generate the desired T σ ρ [SU(N)]/Zt Nk theory with gauge group G, one needs to gauge the discrete Zt Nk symmetry, which effectively reduces the magnetic lattice to the standard integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For the 3d mirror, this means that one first gauges a U(1)t topological symmetry, which effectively removes a U(1) gauge degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' But one also needs to gauge a Zf Nk in a subsequent step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This Zf Nk can be thought of as embedded in the U(1) that one has to be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Hence, the intermediate step is given by T σ ρ [SU(N)]/Zt Nk mirror ←−−→ N∨ 1 N∨ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' N∨ n′−2 N∨ n′−1 N∨ n′ 1 1 M∨ 1 M∨ n′ /// � U(1)diag×U(1) Zf Nk � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='35) From the unframed quiver on the right, one has to ungauge a U(1)diag ×U(1) and also keep a Zf Nk gauged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The natural choice is to ungauge the two U(1) gauge groups on top;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' thus, turning them into flavour groups up to a choice of Zf Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The last step is to choose in which of the two U(1)s one embeds the Zf Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This is because, as with the T[SU(N)] theories, one knows the only difference between T σ ρ [SU(N)]∨ and � T σ ρ [SU(N)]/Zt Nk �∨ should be the splitting of the flavour groups along with a discrete quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Schematically, one finds N∨ 1 N∨ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' N∨ n′−2 N∨ n′−1 N∨ n′ 1 1 M∨ 1 M∨ n′ /// � U(1)diag×U(1) Zf Nk � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='36) ∼= N∨ 1 N∨ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' N∨ n′−2 N∨ n′−1 N∨ n′ M∨ 1 M∨ 2 ZNk M∨ n′−2 M∨ n′ ∼= N∨ 1 N∨ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' N∨ n′−2 N∨ n′−1 N∨ n′ ZNk M∨ n′−2 M∨ n′−1 M∨ n′ M∨ 1 M∨ 2 – 18 – and the two framed mirrors show that the discrete quotient can be applied diagonally on either one of the two sets of flavour hypermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This is also clear from Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3, and Appendices C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2, as an overall U(1) rotation can be used to shuffle the discrete ZNk charges from one set of fundamental flavours to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 1 2 2 1 3 2 2 2 2 2 2 1 1 2 1 1 1 ≅ 2 2 2 2 1 1 1 ///U(1)diag 1 2 SU(2) 1 3 2 2 1 2 SU(2) 1 1 2 2 2 2 1 1 1 1 ///U(1)diag 2 2 2 2 1 1 1 1 /// U(1)diag×U(1) Z2 ≅ 2 2 2 2 1 1 2 1 Z2 ≅ 2 2 2 2 1 1 Z2 1 1 U → SU gauge U(1) ungauge U(1) mirror mirror mirror Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Starting from the mirror pair T σ ρ [SU(15)] and T ρ σ[SU(15)] with σ = (33, 22, 12) and ρ = (6, 4, 3, 12), one can gauge a discrete Z2 0-form symmetry to create a new mirror pair with Z2 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' See also Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3 for the choice of Zf 2 gauging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 19 – Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' One can apply the above procedure to T σ ρ [SU(15)] where σ = (33, 22, 12) and ρ = (6, 4, 3, 12) for the example as in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The global form of the 0-form symmetry is expected to be Gt(bottom left quiver of Figure 4) = Gf(bottom right quiver(s) of Figure 4) = SU(2) × U(1)1 Z2 × U(1)2 × U(1)3 ∼= U(2) × U(1)2 × U(1)3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='37) and one can explicitly verify this structure as demonstrated in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Alternatively, the Coulomb branch quiver indicates this isometry group as follows: only the leftmost U(1) is balanced, leading to a topological su(2)t because there are monopole operators of U(1) magnetic flux ±1 at R-charge 1 (see also [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The remaining U(2) and U(1) gauge nodes provide one U(1)t i=1,2,3 topological symmetry factor each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Let the one associated with U(2) be denoted by U(1)t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Since this node is connected to the balanced node, arguments similar to [30] show the existence of a chiral ring operator that transforms as a spinor under su(2)t and has charge ±1 under U(1)t 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Therefore, the Z2 centre action can be absorbed into U(1)t 1, resulting in a U(2)t topological symmetry factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' One can also choose the other SU(2) in T σ ρ [SU(15)] which gives the mirror pairs dis- played in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Comparing Figure 4 and 5, one observes that the global form of the 1 SU(2) 2 1 1 2 2 2 2 1 1 1 1 /// U(1)diag×U(1) Z2 ≅ 2 2 2 2 1 1 2 Z2 ≅ 2 2 2 2 1 1 Z2 1 1 1 mirror Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Again starting from the mirror pair T σ ρ [SU(15)] and T ρ σ[SU(15)] with σ = (33, 22, 12) and ρ = (6, 4, 3, 12), one can gauge a different discrete Z2 0-form symmetry to generate a another mirror pair with Z2 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3 for the choice of Zf 2 in the mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 20 – 0-form symmetry in Figure 5 is simply PSU(2) × 3 � i=1 U(1)i , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='38) which is supported by the explicit calculations in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This conclusion can also be drawn by examining the Coulomb branch quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Since the balanced U(1) gauge node is not directly connected to any of the U(2) or U(1) gauge groups, there is no expectation on a chiral ring operator that transforms non-trivially under the Z2 centre of the SU(2)t topological symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Consider the mirror pair T σ ρ [SU(9)] with ρ = (3, 23) and σ = (32, 13) 2 2 2 3 2 ←→ 1 2 2 1 1 3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='39) whose symmetry algebra is su(3) ⊕ u(1), as apparent from the balanced set of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The global form is evaluated to be Gt(LHS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='39)) = Gf(RHS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='39)) = SU(3) × U(1) Z3 ∼= U(3) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='40) because the U(1) has charge −1 under the Z3 centre symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' See (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='46) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Al- ternatively, the left-hand-side quiver in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='39) allows us to derive this by using the balanced set of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Since the unbalanced gauge nodes connect to the A2 Dynkin diagram (formed by the balanced nodes) on its first node, there exists a chiral ring operator transforming as [1, 0] × (+1) (plus conjugate) under the topological SU(3)t × U(1)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Thus, the Z3 centre can be compensated by suitable embedding into the U(1)t factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' To create a new mirror pair, we can gauge a Z2 symmetry on both sides of the dual theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For example, we can gauge the topological Zt 2 symmetry on w3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The mirror map, as shown in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='24), indicates that gauging the Zf 2 symmetry leads to the following mirror pair 2 2 SU(2) 1 ←→ 1 2 2 1 1 Z2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='41) whose symmetry algebra is su(2) ⊕ u(1) ⊕ u(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The Hilbert series (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='49) then suggests a symmetry group of Gt(LHS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='41)) = Gf(RHS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='41)) = SU(2) × U(1) × U(1) Z2 ∼= U(2) × U(1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='42) – 21 – because the centre Z2 acts trivial on one U(1) factor and with charge −1 on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This can also be read off from the Coulomb branch quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' As there is a U(2) node connected to the balanced U(2) node, there exists a chiral ring operator transforming as [1]A1 × (±1) under the associated SU(2)t×U(1)t topological symmetry factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Therefore, the Z2 centre symmetry then gives rise to a U(2)t isometry factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The other topological Cartan U(1)t is uncharged under the Z2 centre, as the gauge nodes are not connected to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging Zt q on a U(1) node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Analogous to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3, one can also gauge discrete subgroups of the topological symmetry associated to a U(1) gauge node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' From the examples considered, it is clear what the theory after gauge the Zt q is: the same quiver as before, but all hypermultiplets connected to the specific U(1) gauge node have now charge q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The question is then, what the corresponding mirror theory is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This can be determined by utilising the mirror map between the fugacities, as demonstrated in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='5 T[SO(2N)] theories In a similar vein to T[SU(N)], one can consider the self-mirror theory T[SO(2N)] [30], see Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For quiver theories composed of alternating SO(n) and Sp(m) gauge nodes, only the Z2 factors of the SO(n) gauge nodes are the discrete parts of the topological symmetry visible in the UV description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' If we gauge any of these, we get a T[SO(2N)]-type quiver with a single replacement SO(2k) → Spin(2k)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The corresponding mirror theory is obtained from T[SO(2N)] by gauging a suitable Z2 inside the flavour symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This leads to a splitting of the flavour node as indicated in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='5 provides examples and consistency checks for T[SO(6)] and T[SO(8)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Considering the theory T[SO(2N)]/Zt 2 obtained from gauging Zt 2, the quiver descrip- tion allows us to use the techniques of [6] to verify the 1-form symmetry and its interplay with the flavour 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' One finds the discrete groups summarised in Table 1 which constitute the short exact sequence 0 → Γ[1] → E → Z → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='43) As expected, the flavour 0-form symmetry is always PSO(2N) since the flavour centre Z is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Moreover, only for T[SO(4N)]/Zt 2 with a Spin(4l + 2) gauge node does the 1-form symmetry and the flavour 0-form symmetry form a non-trivial extension hinting to a 2-group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Following [21, 28], it is straightforward to illustrate the 1-form symmetry and 2-group structure via line operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For the Spin(2l) gauge group, a Wilson line Ws in the spinor representation cannot end on a local operator, because all half-hypermultiplets transform in the vector representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For l =even, the tensor product of the spinor with itself contains a singlet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' therefore, W 2 s is equivalent to the identity line without the need for any local operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The lines that cannot end generate the (Pontryagin dual of the) 1-form symmetry and there is no 2-group structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For l =odd, the tensor product of the spinor 5This follows as gauging the Z2 topological symmetry of an SO(2n) gauge group leads to an Spin(2k) gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Conversely [21, 33], gauging the Z2 1-form symmetry in Spin(2k) recovers the SO(2k) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 22 – 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2N−2 2N 0-form: PSO(2N)f × PSO(2N)t 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2N−2 2N 0-form: PSO(2N)t × PSO(2N)f 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Spin(2k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2N−2 2N 0-form: PSO(2N)f × P(O(2k) × O(2N − 2k))t 1-form: Z2 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2N−2 Z2 2N − 2k k 0-form: P(O(2k) × O(2N − 2k))f × PSO(2N)t 1-form: Z2 gauge Z2 ⊂ PSO(2N)t gauge Z2 ⊂ PSO(2N)f mirror mirror Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging of discrete 0-form symmetries in T[SO(2N)] theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging the Zt 2 for an SO(2k) gauge leads to a Spin(2k) gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging the mirror dual Zf 2 is realised by splitting the fundamental flavours into two sets: one set is uncharged and the other set is charged under Zf 2, indicated by an edge with multiplicity k connected to a grey node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' theory Γ[1] E Z T[SO(4N)]/Zt 2 with Spin(4l) Z2 Z2 × Z2 × Z2 Z2 × Z2 T[SO(4N)]/Zt 2 with Spin(4l + 2) Z2 Z2 × Z4 Z2 × Z2 T[SO(4N + 2)]/Zt 2 with Spin(4l) Z2 Z2 × Z4 Z4 T[SO(4N + 2)]/Zt 2 with Spin(4l + 2) Z2 Z2 × Z4 Z4 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Interplay of 1-form symmetry and the flavour centre for T[SO(2N)]/Zt 2 theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' with itself contains the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Now W 2 s is equivalent to a flavour Wilson line because it can end on a local operator build from the half-hypermultiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' However, the vector representation is not an allowed representation of PSO(2N), which means that the Z2 1- form symmetry forms a 2-group with the flavour symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This is consistent with Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' On the other hand, in the theory T[SO(2N)]/Zf 2 obtained by gauging the Zf 2 symmetry, there are two distinct sets of flavour hypermultiplets, each forming half-hypermultiplets H and h in the vector-vector representation of SO(2N−2k)×Sp(N−1) and SO(2k)×Sp(N−1), – 23 – respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2N−2 Z2 2N−2k H k h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='44) The only difference is that h is also charged under Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' As in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3, to study the global form of the flavour symmetry of this theory, one can consider gauge- invariant operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Using the invariant Sp(N−1) anti-symmetric tensor J, the standard mesons-type invariants are HJH and hJh, both of which then transform in the adjoint representation [0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]D of so(2N−2k) and so(2k), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Likewise, one can consider hJH, which is Sp(N−1) gauge invariant, but not Z2 invariant due to the Z2 charge of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Hence, O = Sym2(hJH) is indeed a gauge invariant operator transforming as [2, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]Dk ⊗ [2, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]DN−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' All of these gauge-invariant Higgs branch operators have trivial charges under the so(2k) or so(2N−2k) centre symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This suggests that the global form of the flavour symmetry is PSO(2k) × PSO(2N−2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='6 Sp(k) SQCD and its orthosymplectic mirror The lessons learnt can be readily applied to other orthosymplectic quivers, such as Sp(k) SQCD with N fundamental hypermultiplets and its orthosymplectic mirror quiver [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Focusing on N ≥ 2k+1, the SQCD theory admits a manifest flavour symmetry, while there is no topological symmetry for N > 2k + 1 and a U(1)t symmetry for N = 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Thus, it is quite natural to consider gauging discrete subgroups of the flavour 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Conversely, the mirror orthosymplectic quiver does not have a continuous flavour symmetry for N > 2k1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' no mass parameter) and an SO(2)f symmetry for N = 2k + 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' one mass parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' While the topological symmetry is not manifest in the UV description, certain remnants are: each SO(l) gauge group admits a manifest Zt 2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Therefore, one can gauge a Z2 topological symmetry of a specific SO(ℓ) gauge node and inquire about the implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' It is straightforward to observe that this gauging modifies the particular gauge group SO(ℓ) → Spin(ℓ), see for instance [26, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' On the mirror side, one gauges a Z2 ⊂ SO(2N) flavour symmetry, which then leads to a split of the flavour symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This is summarised in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Exemplary cases with explicit calculations are provided in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The interplay of the discrete Z2 1-form symmetry with the continuous 0-form symmetry is simple here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Consider the linear orthosymplectic mirror quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For N > 2k + 1, there is no continuous 0-form flavour symmetry that could mix with the 1-form symmetry Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For N = 2k +1, there exists an enhance U(1) 0-form symmetry, but the 1-form and 0-form symmetry are simply a product of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 24 – 2k 2N 0-form: PSO(2N)f × Ht 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k 2k+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k+1 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2 2 1 1 (N−2k−1) × SO(2k + 1) nodes (N−2k) × Sp(k) nodes 0-form: PSO(2N)t × Hf 2k Z2 2N − 2ℓ ℓ 0-form: (SO(2ℓ) × SO(2N − 2ℓ))f × Ht 1-form: Z2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2ℓ−2 Spin(2ℓ) 2ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2 1 1 0-form: (SO(2ℓ) × SO(2N − 2ℓ))t × Hf 1-form: Z2 gauge Z2 ⊂ PSO(2N)f gauge Z2 ⊂ PSO(2N)t mirror mirror Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging of discrete 0-form symmetries in Sp(k) SQCD with N fundamentals and its linear orthosymplectic mirror quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Here, the isometry group Ht/f is trivial for N > 2k + 1 and U(1) for N = 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='7 Sp(k) SQCD and its unitary D-type mirror quiver It is well-known that Sp(k) SQCD with N fundamental flavours admits a second mirror description [36], based on a DN-type Dynkin quiver: 2k 2N ←→ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k−1 2k 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k k k 1 N − 2k − 1 nodes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='45) This mirror pair has the advantage that the PSO(2N) global symmetry is manifest as Higgs branch isometry in the SQCD theory and as Coulomb branch isometry in the D- type Dynkin quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' It is, hence, natural to study gaugings of discrete Zq symmetries in this manifest 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Starting with the Dynkin quiver, there are two distinct choices: Firstly, gauging a Zl – 25 – on a U(l) node which satisfies 2 < l < 2k, one obtains6 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' l−1 SU(l) l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k−1 2k 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k k k 1 N − 2k − 1 nodes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='46) which has a Zl 1-form symmetry and the Coulomb branch isometry algebra is su(l) ⊕ u(1)Q ⊕ so(N − l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For the global form, one can study the action of the centre symmetries of the non-abelian factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' One finds Gt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='46) = PSU(l) × U(1)Q × Spin(2N − 2l) ZDN−l (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='47) where the ZDN−l charges of Q are given by the charges of the congruence class of the j-th fundamental representation [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]D with j = 2k − l, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For explicit examples including Hilbert series computations see Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Alternatively, gauging a Zl on a U(l) node which satisfies l ≥ 2k, one obtains 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k−1 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k SU(2k) 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k k k 1 l − 2k nodes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='48) and the Coulomb branch isometry algebra is the same as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' However, the “extra” U(1) node is now attached to the balanced A-type Dynkin diagram such that the global form is given by Gt(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='48) = PSU(l) × U(1)Q Zl × PSO(2N − 2l) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='49) where Q carries Zl charge 2k mod l, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' the charges of the congruence class of the 2k-th fundamental representation, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Explicit examples for this discrete gauging are given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Global form via the mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Analogous to the discussion in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='5, one can confirm this global symmetry via the Higgs branch of the mirror theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The starting point is the mirror map (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='27) between the flavour fugacities of Sp(k) SQCD and its unitary D-type Dynkin quiver, see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This allows us to identify which flavour fugacities are involved in the discrete Zf l gauging on the SQCD side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' l < 2k: The familiar argument then proceeds by splitting the fundamental flavours into two distinct groups: the first l fundamental flavours are grouped as X, transform- ing as ζlQ− 1 l [1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]Al, and the remaining N −l fundamental flavours, transform- 6The cases l = 1, 2 are addressed separately below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 26 – ing as [1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]DN−l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Building a gauge invariant Higgs branch operator proceeds in two steps: firstly, using the Sp(k) invariant tensors J on constructs operators of the form XJ ˜X, which transform as ζl under the discrete symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Secondly, Zl in- variance is achieved via O = Syml(XJ ˜X), which transforms as Q−1[l, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]Al−1 ⊗ [l, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]DN−l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The Zl centre surely acts trivial on [l, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]Al−1, while the centre charges of [l, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]DN−l are (0, l mod 2) if N − l is even or (2l mod 4) if N − l is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Thus, the non-trivial transformations under the centres can be compensated if Q transforms as follows: N − l = even Zl × Z2 × Z2 charges of Q: (0, 0, l mod 2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='50a) N − l = odd Zl × Z4 charges of Q: (0, 2l mod 4) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='50b) which confirms (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' To see this, recall from Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3 that the congruence class of the j-th fundamental representation of DN−l with j = 2k − l is (0, l mod 2) for N − l even and 2l mod 4 for N − l odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' l > 2k: The argument is slightly modified: the first set X of flavours transforms as ζ2kQ− 1 2k [1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]Al−1, while the second set ˜X transforms as [1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]DN−l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The Higgs branch operator O = Sym2k(XJ ˜X) transforms as Q−1[2k, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]Al−1 ⊗ [2k, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]DN−l, which has trivial D-type centre charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' To see this, for N −l even, the Z2 × Z2 charges are (0, 2k mod 2) = (0, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' while for N − l odd, the Z4 charge is 2 · 2k mod 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Thus, to compensate potential irreps that are non-trivial under Z2k, one requires that Q has the following charges: N − l = even Zl × Z2 × Z2 charges of Q: (2k mod l, 0, 0) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='51a) N − l = odd Zl × Z4 charges of Q: (2k mod l, 0) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='51b) which then confirms (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Two special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In the l = 2 case of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='47), a symmetry enhancement is observed in the explicit computations (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='66) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' These show that there is not only the expected su(2)t, but the topological Cartan symmetry of the “new” U(1) gauge node is also enhanced to a non-abelian su(2)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' These two su(2)t symmetries can both be interpreted as PSO(4)t ∼= PSO(3)t × PSO(3)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' As in previous sections, one can also gauge a discrete Zt q along the topological fugacity w1 associated to the first U(1) gauge node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The D-type Dynkin quiver is modified in the by now familiar way: the bifundamental of U(1) × U(2) turns into a hypermultiplet that transforms as fundamental under U(2) but is of U(1) charge q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In the mirror theory, the Zf q acts on a single fundamental flavour, as dictated by the mirror map (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In summary, – 27 – the mirror pair with Zq 1-form symmetry is 2k 2N−2 Zq ←→ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k−1 2k 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k k k 1 N − 2k − 1 nodes q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='52) and the global Higgs / Coulomb branch isometry is G = U(1)Q × Spin(2N − 2) ZDN−1 , ZDN−1 charges of Q � (0, q mod 2) , N = even 2q mod 4 , N = odd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='53) where Q is the topological fugacity of the left-most U(1) gauge node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='8 Examples of non-simply laced unitary quivers and their mirrors The last class of quiver theories considered here are non-simply laced unitary quivers7, whose monopole formula has been proposed in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Consider the following example N1 N2 N3 N4 M1 M2 M3 M4 κ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='54) with nodes U(N1,2) on the “short” side and U(N3,4) on the “long” side;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' wherein the naming is borrowed from Dynkin diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The multiplicity of the non-simply laced edge is denoted by κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Even though these quiver theories are non-Lagrangian (hence superconformal index and Higgs branch Hilbert series are not computable by the standard methods), we can still study their Coulomb branch using Hilbert series techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This allows us to investigate the effects of gauging a discrete Zt q symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging at the long side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' To begin with, attempt to gauge a discrete Zt N3 topological symmetry associated to the U(N3) node at the long side, with N3 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' As a first step, one rewrites (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='54) by expressing U(N3) ∼= (SU(N3) × U(1))/ZN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' By analogous arguments as in Appendix B, one arrives at � �������� N1 N2 SU(N3) N4 1 m1 ℓ m2 m4 h κ M1 κ M2 M4 M3 � �������� /ZN3 with (m1, m2, l, m4, h) ∈ �Γ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='55) 7See, for example, [37, 38] for the appearance of such quiver theories via branes and ON planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 28 – �Γ := N3−1 � a=0 � Z + κ·a N3 �N1 × � Z + κ·a N3 �N2 × � Z + a N3 �N3−1 × � Z + a N3 �N4 × � Z + a N3 � and the Coulomb branch moduli space is the same as that of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The green edges transform in the fundamental representation of U(N1,2) and with charge κ under the U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Next, the ZN3 symmetry is gauged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' One obtains the following quiver description: N1 N2 SU(N3) N4 1 m1 ℓ m2 m4 h κ M1 κ M2 M4 M3 with (m1, m2, l, m4, h) ∈ Γ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='56) Γ := N3−1 � a=0 ZN1 × ZN2 × ZN3−1 × ZN4 × Z where Γ is again short-hand for the integer magnetic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This theory exhibits a ZN3 1-form symmetry, by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging at the short side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Now, consider gauging a Zt N2 on the topological fugacity associated to the U(N2) gauge node, with N2 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Again, the first step is to simply rewrite U(N2) ∼= (SU(N2) × U(1))/ZN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' By adopting the arguments of Appendix B, one finds � �������� N1 SU(N2) N3 N4 1 m1 ℓ m3 m4 h κ M1 κ M2 M4 M3 � �������� /ZN2 with (m1, l, m3, m4, h) ∈ �Γ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='57) �Γ = N2·κ−1 � a=0 � Z + a N2 �N1 × � Z + a N2 �N2−1 × � Z + a N2·κ �N3 × � Z + a N2·κ �N4 × � Z + a N2·κ � whose Coulomb branch coincides with that of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Moreover, the edges highlighted in green transform in the fundamental representation of U(N1,2) and with charge κ under the U(1) node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' As a next step, gauging the ZN2 results in the following theory: N1 SU(N2) N3 N4 1 m1 ℓ m3 m4 h κ M1 κ M2 M4 M3 with (m1, l, m3, m4, h) ∈ Γ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='58) Γ = κ−1 � a=0 (Z + a)N1 × (Z + a)N2−1 × � Z + a κ �N3 × � Z + a κ �N4 × � Z + a κ � – 29 – where Γ is a short-hand notation for several shifted copies of the standard integer lattice of the magnetic charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' One could also gauge a discrete Zt q along the topological Cartan U(1) of a U(1) gauge node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In this case, the connected hypermultiplets are modified to have charge q under the U(1), but no other changes to the quiver occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 C-type quivers A representative example is the mirror pair of O(2k) SQCD with N hypermultiplets in the vector representation and its C-type Dynkin mirror quiver O(2k) 2N ←→ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2k 2k 1 N gauge nodes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='59) which can be realised by a systems of D3-D5-NS5 branes with O5 and ON planes, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The logic is the same as before: Choose a Zt q in the C-type Dynkin quiver, by selecting a gauge node and its associated topological fugacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Using the mirror map (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='33) for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='59) one identifies how the Zf q acts on the vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For concreteness, consider examples for k = 1 and N = 4: Example: gauging on the long side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging a Zt 2 on the fourth node yields the mirror pair (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' using (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='33) with discrete variable on w4) O(2) Z2 8 ←→ 1 2 2 SU(2) 1 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='60) where the ‘new’ U(1) node is connected with a hypermultiplet of charge 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The global symmetry algebra is su(4) ⊕ u(1), as read off from the balanced set of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Explicit Hilbert series (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='100) show that the global form is Gf(LHS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='60)) = Gt(RHS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='60)) = PSU(4) × U(1)Q (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='61) because the Z4 centre acts trivial on all appearing representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 30 – Example: gauging on the short side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging a Zt 2 on the third gauge node results in the new mirror pair (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' using (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='33) with discrete variable on w3) O(2) 2 Z2 6 ←→ 1 2 SU(2) 2 1 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='62) the global symmetry algebra is su(3) ⊕ u(1) ⊕ sp(1), as suggested by the balanced nodes in the unitary quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Recalling the maximal subalgebra su(3) ⊕ u(1) ⊂ sp(3), an analysis of the Hilbert series then suggests that the global form is Gf(LHS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='62)) = Gt(RHS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='62)) = PSp(3) × PSp(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='63) See (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='104) for explicit computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Example: gauging on the short side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging a Zt 2 on the second gauge node results in the new mirror pair (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' using (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='33) with discrete variable on w2) O(2) 4 Z2 4 ←→ 1 SU(2) 2 2 1 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='64) and the balanced set of nodes suggests the symmetry algebra su(2)⊕u(1)⊕sp(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' A Hilbert series computation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='108) then indicates the following symmetry group Gf(LHS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='64)) = Gt(RHS (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='64)) = PSp(2) × PSp(2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='65) This suggests that the su(2) ⊕ u(1) realise a maximal subalgebra in one sp(2) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 B-type quivers Alternatively, we could consider an Sp(k) gauge theory with SO(2n + 1) flavour symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' However, to prevent a parity anomaly, we would need to include a suitable Chern-Simons term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The Higgs branch, which is not affected by Chern-Simons levels, is known to be the closure of a B-type nilpotent orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Therefore, a natural mirror theory would be a B-type Dynkin quiver, for which analogous arguments apply as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3 A comment on F4 Coulomb branch quivers The reasoning can be also applied to other non-simply laced Coulomb branch quivers, even if there may not exist a known mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Such an example is the F4 Coulomb branch quiver of [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Table 2 summarises the resulting theories after a suitable Zt n is gauged, following the prescriptions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='58) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Here, a few remarks in comparison to the “ungauging scheme” of [41] are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The ungauging scheme involves removing a U(1) factor from a selected U(n) gauge group, – 31 – which in the context of the monopole formula means setting one of the magnetic charges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For simply-laced quivers, this procedure leads to the same consequence as replacing a U(n) gauge group with an SU(n) and quotienting out a diagonal Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' However, the ungauging scheme becomes problematic when applied to a node on the short side of non-simply laced quivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' If the short node is a U(1) gauge group, then the ungauging simply converts it into a flavour group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In [41], the ungauging of the short U(1) node in the F4 quiver leads to a Coulomb branch that is the next-to-next-to minimal nilpotent orbit closure of so(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In contrast, if the short node is non-abelian, such as the U(2) node in the F4 quiver, the resulting moduli space cannot be identified with any known space and the procedure has been argued to be “invalid” in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' On the other hand, by replacing the short U(2) node with an SU(2) and following the prescriptions in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='58) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='56), one is able to obtain consistent results, as shown in the fourth row of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The resulting Coulomb branch is the next-to-next-to minimal nilpotent orbit closure of so(9) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='8 It is to be noted, that if one uses the prescriptions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='57) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='55), then one recovers the original minimal nilpotent orbit closure of F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='9 Magnetic quivers and gauging discrete topological symmetries Suppose that one is given an unframed unitary magnetic quiver T with only simply-laced edges (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' bifundamental hypermultiplets between the unitary gauge nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' To evaluate the Hilbert series or the index, it is necessary to remove an overall U(1) gauge group factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In [31], it was emphasised that choosing this U(1) from a U(k) gauge node leads to an SU(k) gauge node, but the magnetic lattice Γ is extended to include shifted versions of the form �k−1 i=0 � Γ + i k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This situation can also be understood from a complementary perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Given an unframed unitary magnetic quiver, pick a U(k) gauge node and rewrite it as U(k) ∼= SU(k)×U(1) Zk , with fluxes (l, h) ∈ �k−1 i=0 �� Z + i k �k−1 , � Z + i k �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The aim is to remove this U(1) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' As demonstrated in Appendix B, this rewriting shifts all other magnetic fluxes m by the flux h associated to the U(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' as a result, all magnetic fluxes receive the shifts Γ + i k simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Now, removing this U(1) means treating it as a background vector multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Nevertheless, all remaining magnetic fluxes are still subject to the shifts Γ + i k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Hence, the Coulomb branch Hilbert series, as well as the index for T , have the form FT = � (l,m)∈�k−1 i=0 (Γ+ i k ) f(l, m) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='66) which is message conveyed in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' It turns out that one can refine FT by introducing a Zk-valued fugacity z as follows: the U(1)t topological symmetry of the U(k) node appears in both the monopole formula and the index through the factor w �k a=1 ma k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Upon rewriting into magnetic fluxes (l, h) for (SU(k) × U(1)) /Zk, this becomes wk·h k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Since h ∈ �k−1 i=0 (Z + i k), one has wk·n+i k for h = n + i k ∈ (Z + i k) and some n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This means that one can introduce a discrete 8In general, for non-simply laced quivers, the prescriptions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='58) do not always provide the same Coulomb branch for all the short nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 32 – quiver symmetry Coulomb branch Hilbert series 2 3 2 1 1 F4 1 + χ1,0,0,0t + χ2,0,0,0t2 + χ3,0,0,0t3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' = 1 + 52t + 1053t2 + 12376t3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' SU(2) 3 2 1 1 A1 × C3 1+t(χ2,0,0+φ2)+t2(1+χ4,0,0+χ0,2,0+χ2,0,0φ2+ χ0,0,2φ2 + φ4) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' = 1 + 24t + 537t2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2 SU(3) 2 1 1 A2 × A2 1 + t(χ1,1 + φ1,1) + t2(1 + χ1,1 + χ2,2 + χ1,1φ1,1 + χ2,2φ1,1 + φ2,2 + φ1,1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' = 1 + 16t + 351t2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2 3 SU(2) 1 1 A3 × A1 ⊂ B4 1+t(χ2+χ2φ0,1,0+φ1,0,1)+t2(1+χ4+χ2φ2,0,0+ φ0,1,0 + χ2φ0,1,0 + χ4φ0,1,0 + φ0,2,0 + χ4φ0,2,0 + φ1,0,1+2χ2φ1,0,1+χ2φ1,1,1+χ2φ0,0,2+φ2,2)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' = 1 + 36t + 621t2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2 3 2 1 1 2 C3 × U1 ⊂ C4 1 + t(Qχ1,0,0 + χ1,0,0 Q + χ0,1,0 + 1) + t2(Q2χ2,0,0 + χ2,0,0 Q2 +Qχ0,0,2+Qχ1,0,0+Qχ1,1,0+ χ0,0,2 Q + χ1,0,0 Q + χ1,1,0 Q + χ0,0,2 + 2χ0,1,0 + χ0,2,0 + χ2,0,0 + 1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' = 1 + 36t + 621t2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2 3 2 1 1 3 C3 × U1 1 + t(χ0,1,0 + 1) + t2(Qχ1,0,1 + χ1,0,1 Q + χ0,0,2 + 2χ0,1,0 + χ0,2,0 + χ2,0,0 + 1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' = 1 + 22t + 369t2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2 3 2 1 1 4 C3 × U1 1 + t(χ0,1,0 + 1) + t2(Qχ2,0,0 + χ2,0,0 Q + χ0,0,2 + 2χ0,1,0 + χ0,2,0 + χ2,0,0 + 1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' = 1 + 22t + 327t2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The F4 Coulomb branch quiver and its Zq gaugings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The first row is the standard F4 quiver proposed in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Rows 2 - 4 display different choices of gauging a Zt N of a U(N) node in the Coulomb branch quiver for the minimal nilpotent orbit closure of F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The gaugings on the “long” side produce global symmetries given by the balanced set of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For the gauging on the “short” side, the global so(9) symmetry is only visible via the subalgebra su(4) ⊕ su(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Rows 5 - 7 display the effects of gauging a Zt q inside the topological Cartan factor of the U(1) gauge node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For q = 2, the symmetry algebra is enhance from sp(3) × u(1) to sp(4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' while for q > 2, the algebra is simply sp(3)×u(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In the Hilbert series expressions, χ and φ are characters for the non-abelian symmetry factors and Q is a U(1) fugacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' fugacity z to keep track of the Zk centre symmetry, setting wk → z such that wk·n+i k = zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This fugacity remains even if the U(1) is taken to be non-dynamical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' One ends up with FT (z) = k−1 � i=0 zi � (l,m)∈(Γ+ i k ) f(l, m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='67) – 33 – It is now clear what happens if this discrete Zt k topological symmetry is gauged: the entire range of the summation collapses to the i = 0 sector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=', the integer lattice FT /Zt k = 1 k k−1 � i=0 FT � z = (ζk)i� = � (l,m)∈Γ f(l, m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='68) Consequently, the quiver theory, in which U(k) is replaced by an SU(k) and the magnetic lattice is simply the integer lattice, is obtained from the unframed unitary quiver T by gauging a discrete Zt k topological symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This Zk distinguishes between SU(k)×U(1) Zk ∼= U(k) and SU(k) × U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Additionally, the gauging of the Zt k symmetry has introduced a Zk 1-form symmetry into T /Zt k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='10 Examples from 5d magnetic quivers One can demonstrate gauging discrete subgroups of the topological symmetry on known magnetic quivers9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' It is most suitable to choose quivers whose Coulomb branches have a known Higgs branch realisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' E5 quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The infinite coupling magnetic quiver for 5d Sp(1) SQCD with 4 flavours realises O E5 min ∼= O D5 min, which is also the Higgs branch of Sp(1) with 5 flavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Thus, one arrives at [48] � ����� 2 2 4 2 2 1 � ����� /Z2 ←→ 2 10 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='69) It is worth recalling that the magnetic lattice for the left-hand side quiver has the form Γ ∪ (Γ + 1 2) with Γ being the standard GNO integer lattice, as can be found in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The corresponding discrete Zt 2 topological symmetry of the magnetic quiver can be gauged in the same vein as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' On the level of the magnetic quiver, this just reduces the relevant magnetic lattice to the integer lattice Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Equivalently, one can gauge a Zf 2 on the Sp(2) SQCD side, which then gives rise to the following pair of theories 2 2 4 2 2 1 ←→ 2 8 Z2 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='70) and it is straightforward to verify that the Coulomb / Higgs branch Hilbert series reproduce the results of [31, Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The global form of the 0-form symmetry is PSO(8) × U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' E4 quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Similarly, the infinite coupling magnetic quiver for 5d Sp(1) SQCD with 3 flavours realises O E4 min ∼= O A4 min via its Coulomb branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Of course, this moduli space admits 9See also [42–47] for magnetic quivers of theories with 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 34 – a known Higgs branch realisation and one arrives at � �������� 2 2 2 1 1 2 � �������� /Z2 ←→ 1 5 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='71) The magnetic lattice for the magnetic quiver is of the form Γ ∪ (Γ + 1 2), so the associated Zt 2 symmetry can be gauged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The question then becomes what Zf 2 symmetry is realised on the SQED side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Through explicit calculations, one verifies that 2 2 2 1 1 2 ←→ 1 4 Z2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='72) reproduces the known Hilbert series [31, Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The isometry group in this case is SO(6) × U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Folded E6 quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The infinite coupling magnetic quiver for 5d Sp(1) SQCD with 5 flavours admits a Z2 outer automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Folding the corresponding magnetic quiver leads to O E6 min → O D5 min on the Coulomb branch [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Since there is a known Higgs branch realisation for D-type minimal nilpotent orbit closures, one arrives at � 1 4 4 2 2 � /Z2 ←→ 2 10 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='73) where again the left-hand side quiver has a magnetic lattice of the form Γ ∪ (Γ + 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging this Zt 2 has a by now clear consequence on the magnetic quiver, as the GNO lattice is reduced to the integer lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' On the Sp(1) SQCD side, the corresponding Zf 2 is realised as follows: 1 4 4 2 2 ←→ 2 Z2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='74) – 35 – and one straightforwardly verifies the agreement of the Coulomb branch / Higgs branch Hilbert series, which is given by HS = 1 + 25t + 400t2 + 3864t3 + 26600t4 + 141672t5 + 621480t6 + 2337280t7 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='75) + 7763283t8 + 23265515t9 + 63954800t10 + O � t11� and the global symmetry group is PSU(5) × U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 3 Discussion and conclusions In this paper, mirror pairs with non-trivial 1-form symmetry have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Starting from known mirror pairs with trivial 1-form symmetry, gauging of discrete Zq subgroups of the 0-form symmetry allowed us to construct new mirror pairs with non-trivial 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The main results are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' It has been shown that theories T /Zt q, obtained by gauging a discrete subgroup Zt q of the topological symmetry, may admit quiver descriptions if the discrete subgroup is suitably chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The mirror theories � T /Zt q �∨ can be constructed using T ∨/Zf q , but the precise choice of Zf q in the flavour symmetry of T ∨ can be subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This paper provides a simple algorithm for specifying Zf q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The global form of the 0-form symmetries of (T /Zt q, T ∨/Zf q ) have been derived using both field theory methods and monopole operators (via the balanced set of nodes), and the resulting symmetry groups have been verified through explicit Hilbert series computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The interplay between continuous 0-form and discrete 1-form symmetries has been studied using established field theory techniques and the equivalence classes of lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' On the technical side, the gauging of discrete subgroups of the topological symmetry on non-simply laced quivers has been proposed and tested on both long and short-side gauge nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' A comment on the moduli spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The maximal branches of the moduli space of vacua in a theory T are the Coulomb branch C(T ) and the Higgs branch H(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' These are symplectic singularities that can be resolved when the theory T is given either an FI parameter (for the Higgs branch) or a mass parameter (for the Coulomb branch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For instance, consider SQED with N hypermultiplets of charge 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' This theory admits N − 1 mass parameters that resolve the C2/ZN Coulomb branch, and a single FI parameter that resolves the Higgs branch, specifically the minimal nilpotent orbit closure O su(N) min .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' If we gauge a Zt q 0-form symmetry in this theory, the resulting SQED with charge q hypers has the same Higgs branch, but the Coulomb branch is modified to be C2/ZN·q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' However, – 36 – there are no additional mass parameters in the theory, which means that the singularity cannot be fully resolved even though a symplectic resolution exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' More generally, one can perform a simple test10 via Hilbert series that shows T −→ � � � � � � � � � � � T /Zt q : limt→1 HSC(T /Ztq)(t) HSC(T )(t) = 1 q or T /Zf q : limt→1 HSH(T /Zf q )(t) HSH(T )(t) = 1 q (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1) and the presence of a 1 q fraction in the expression suggests (at least locally) that the Coulomb branch C(T /Zt q) is a Zq orbifold of C(T ), and a similar relationship holds for the Higgs branches H(T /Zf q ) and H(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Again, no additional deformation parameter appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In contrast, consider T to be U(2) SQCD with 4 fundamental flavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The maximal branches are H(T ) = O su(4) (22) and C(T ) = S(22) ∩ Nsu(4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' the Slodowy slice to the su(4) nilpotent orbit defined by partition (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' There are 3 masses resolving the Coulomb branch and 1 FI term resolving the Higgs branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' If we gauge the topological U(1)t symmetry in this theory, the resulting theory is SU(2) SQCD with 4 fundamental flavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Then, the Coulomb branch of this theory is C(T /U(1)t) = C2/D4 while the Higgs branch is H(T /U(1)t) = O so(8) min .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In this case, the Coulomb branch can be resolved by the 3 + 1 mass parameters, while the minimal orbit closure of so(8) does not admit a symplectic resolution, which is consistent with the absence of an FI parameter in this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' These symplectic resolutions can also be studied via Hilbert series techniques, see for instance [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Generalisations and open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' In this work, a single Zq factor of the 0-form symmetry has been gauged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' One straightforward generalisation is to consider orthosym- plectic quivers and gauge several Zt 2 topological symmetry factors associated to SO(ni) nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The resulting theory is simply obtained by replacing the relevant SO(ni) → Spin(ni) and the 1-form symmetry is the product group � i(Zt 2)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Similarly, one could also enter- tain the thought of gauging several Zqi inside distinct topological Cartan factors of, say, T[SU(N)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' It is a priori not clear if a simple quiver description exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Another aspect of 3d mirror symmetry is the exchange of Wilson and vortex line defects [29, 45, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Given the central role of line defects in the understanding of 1-form and 2-group symmetries, it would be interesting to systematically analyse the exchange of Wilson and vortex lines under mirror symmetry systematically for the theories with 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Acknowledgments We would like to thank Fabio Apruzzi, Lakshya Bhardwaj, Mathew Bullimore, Andrea Fer- rari, Heeyeon Kim, Noppadol Mekareeya, Matteo Sacchi, and Sakura Sch¨afer-Nameki for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The research of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' is supported by the National Science Foundation of China 10Following [50], the volume of the Sasakian base S of H or C is evaluated via Vol(S) = limt→1(1−t)dHS(t), where d = dimC(H or C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 37 – under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 12050410234 and Shanghai Foreign Expert grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 22WZ2502100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' is grateful to Ryo Suzuki for use of his computing facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' is also grateful to Rudolph Kalveks for invaluable help with Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' A Notations and conventions node vector n U(n) SU(n) SU(n) n SO(n) Spin(n) Spin(n) 2n Sp(n) Zq Zq (a) edge hyper n k bifundamental n ⊗ k n SU(k) bifundamental n ⊗ [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0, 1]A n 2k half-hyper [1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]D/B ⊗ [1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , 0]C Spin(n) 2k half-hyper in vector × vector n k N N copies of bifundamental n Zq N N copies of fundamental n 1 Q fundamental of U(n) but charge Q of U(1) (b) Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Notation for nodes and links in the quiver diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' A quiver diagram, composed of nodes and edges, encodes a 3d N = 4 theory as follows: Gauge nodes ⃝ denote dynamical vector multiplets, while flavour nodes □ denote background vector multiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The notations are summarised in Table 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' An edge between two nodes corresponds to a hypermultiplet H = (X, Y †), with X, Y two N = 2 chiral multiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The notation is summarised in Table 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' An exception are so-called non-simply laced edges in a quiver theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Between unitary gauge node, such an edge has been proposed purely on the level of the conformal dimension of the monopole formula [39] n k κ ←→ 1 2 n � i=1 k � j=1 |m1,i − κ · m2,j| (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1) and it is to stress that this does not correspond to a representation of the gauge groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For the special case of U(N = 1), such a non-simply laced edge is effectively the same as a U(1) gauge group with a charge κ hypermultiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 38 – Between orthosymplectic nodes, the conformal dimension has been proposed in [49] n 2k κ ←→ 1 2 · 2 � ρ∈[1,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=',0]B/D � λ∈[1,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=',0]C |ρ(m) − κ · λ(n)| (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2) with m, n the magnetic fluxes which are evaluated on the weights ρ, λ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 Hilbert series A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1 Monopole formula The Hilbert series for the 3d N = 4 Coulomb branch is known as the monopole formula [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Schematically, the Hilbert series is computed as a sum over magnetic fluxes m valued in the GNO lattice Γ of the gauge group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' HSC = � m∈Γ/W P(t, m)wmt∆(m) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3) and W denotes the Weyl group of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' A bare monopole operator is characterised by the flux m as well as its conformal dimension ∆(m), which coincides with the third component of the SU(2)R spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The factors P(t, m) dress a bare monopole operator by gauge invariants formed by the adjoint chiral multiplet of the residual gauge group H(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Lastly, w denotes the fugacity of the topological symmetry, assuming that G contains U(1) factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 Higgs branch Hilbert series The Higgs branch Hilbert series [23–25] for the 3d N = 4 quiver gauge theory relevant here is schematically obtained by a Molien-Weyl integral of the form HSH = � G dµG PE[χG Adj t] PE[χG R · χF F t 1 2 ] (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='4) where the numerator contains the character χG Adj of the adjoint representation of the gauge group, while the denominator contains all matter fields characterised by their representa- tions R under the gauge group G and the representations F under the flavour symmetry F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='3 Gauging a discrete 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Suppose one is given a generating function H(z|t) which is a power series in t with coeffi- cients that are Laurent polynomials in a U(1) fugacity z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Next, embed a Zq �→ U(1) via (ζq)p = e 2πip q with p = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' , q − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Gauging this discrete Zq 0-form symmetry is realised in terms of the generating function via a discrete Molien-Weyl sum 1 q q−1 � p=0 H � (ζq)p · y 1 q |t � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='5) where y is the fugacity for the residual U(1)/Zq ∼= U(1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' – 39 – A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='2 Superconformal index The 3d superconformal index can be computed as partition function on S2 × S1 via local- isation techniques, see [54–60] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Schematically, one arrives at Z = � m 1 |Wm| � Trk(G) rk(G) � i=1 dsi 2πisi Icl · Ivec · Imatter (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='6) where s denotes the gauge fugacities, which are valued in a maximal torus of the gauge group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The magnetic fluxes m take values in the GNO-lattice of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' A flux m breaks G to the residual gauge group Hm (the stabiliser subgroup of m inside G) with Weyl group WHm ≡ Wm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The integration contour is chosen to be the unit circle T for each si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The integrand is composed of classical contributions and the 1-loop determinants of the supermulitpelts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' For concreteness, the G = U(N) case is reviewed: The classical contribution is given by IU(N) cl (w, m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' n) = N � a=1 (sa)n w �N a=1 ma (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='7) with w the fugacity of the topological U(1)t symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' The N = 2 multiplets have the following 1-loop determinants: 3d N = 2 Chiral multiplet of R-charge r coupled with unit charge to a gauge field: Ir chi(z, m| x) = � x1−rz−1� |m| 2 ∞ � j=0 1 − (−1)mz−1x|m|+2−r+2j 1 − (−1)mzx|m|+r+2j (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='8) = � x1−rz−1� |m| 2 � (−1)mz−1x|m|+2−r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' x2� ∞ � (−1)mzx|m|+r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' x2� ∞ with a U(1) holonomy z around S1 and the Z-valued magnetic flux m on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' Here, the definition (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' q)∞ = �∞ j=0(1 − zqj) has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' 3d N = 4 Hypermultiplet transforming as bifundamental of U(N) × U(M) IU(N)×U(M) hyp (s1, m1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content=' s2, m2| x) = N � a=1 M � b=1 I 1 2 chi � s1,as−1 2,b, m1,a − m2,b| x � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE0T4oBgHgl3EQffwCf/content/2301.02409v1.pdf'} +page_content='9) I 1 2 chi � s−1 1,as2,b, m2,b − m1,a| x � 3d N = 2 vector multiplet for a U(N) gauge group: IU(N) vec (s, m| x) = � a +0.4 +20 +-40 +0.2 +-40 +-20 +0 +20 +40 +u (m)MkillI +NPOI +1.0 + CHARA +0.8 +0.6 - +0.4 - +0.2 +0.0 +0 +5 +10 +15 +20 +25 +30 +35 +Baseline (m)8 +those using filtered bandpasses, and those in dispersed light. Furthermore, higher sensitivity observations, paired with +polarimetric capabilities would allow for better constraints, if not a direct measure of, radiative processes displaying +polarized emission in the disk, as was attempted by Rousselet-Perraut et al. (1997). +These observations were performed using two facilities: the T1M and M´eO that had not been used for interfero- +metric observations prior to this report. The portability of the T1M provides the capability to optimize the baseline +configuration for a given target, or similarly perform multiple configurations for a given target as done here. This +technical accomplishment also illustrates the potential for performing up to 4-telescope II measurements on the Calern +Plateau enabling 6 simultaneous baselines by including the 2 additional C2PU telescopes. +We acknowledge the financial support of the R´egion PACA (project I2C), the French National Research Agency +(ANR, project I2C, ANR-20-CE31-0003), OCA, Doeblin federation, UCA science councils grants, and the LABEX +Cluster of Excellence FIRST-TF (ANR-10-LABX-48-01), within the program Investissements d’Avenir operated by the +ANR. The authors would like to thank Jacques Belin and Damien Pesce for their installment of reference markers, and +to the members of the M`eO team including Hervey Mariey, Mourad Aimar, Herv´e Viot, Gr´egoire Martinot-Lagarde, +Julien Scariot, Nicolas Maurice, Duy-H`a Phung, and Nils Raymond for their assistance during the observations. +REFERENCES +Abeysekara, A. 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T., et al. 2006, AJ, +131, 2710. doi:10.1086/502679 +Wiener, N. 1930, AcMa, 55, 117 + diff --git a/ANE4T4oBgHgl3EQfEgyT/content/tmp_files/load_file.txt b/ANE4T4oBgHgl3EQfEgyT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b13384a1df997d7471cf3f7b7d081c7ae373c712 --- /dev/null +++ b/ANE4T4oBgHgl3EQfEgyT/content/tmp_files/load_file.txt @@ -0,0 +1,540 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf,len=539 +page_content='Draft version January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2023 Typeset using LATEX default style in AASTeX62 Intensity Interferometry observations of the Hα envelope of γ Cas with M´eO and a portable telescope Nolan Matthews,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='1 Jean-Pierre Rivet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='2 David Vernet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='3 Mathilde Hugbart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='1 Guillaume Labeyrie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='1 Robin Kaiser,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='1 Julien Chab´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='4 Cl´ement Courde,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='4 Olivier Lai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='2 Farrokh Vakili,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='2 Olivier Garde,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 6 and William Guerin1 1Universit´e Cˆote d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Institut de Physique de France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' France 2Universit´e Cˆote d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Observatoire de la Cˆote d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Laboratoire Lagrange,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' France 3Universit´e Cˆote d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Observatoire de la Cˆote d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' UMS Galil´ee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' France 4Universit´e Cˆote d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Observatoire de la Cˆote d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Laboratoire G´eoazur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' France 52SPOT (Southern Spectrocopic Project Observatory Team) 6Observatoire de la Tourbi`ere - 38690 Chˆabons - France (Received January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2023) ABSTRACT We report on observations of the extended environment of the bright Be star γ-Cas performed using intensity interferometry measurements within its Hα emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' These observations were performed using a modified version of the I2C intensity interferometry instrument installed onto the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='54 meter M´eO optical metrology telescope and a portable 1-meter telescope (T1M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' In order to better constrain the extent of the H α envelope, observations were performed for two different positions of the T1M telescope, corresponding to an intermediate and long baselines in which the extended region was partially and fully resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' We find that the observed data are consistent with past interferometric observations of γ-Cas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' These observations demonstrate the capability to equip optical telescopes of different optical designs with intensity interferometry capabilities and illustrate the potential to scale a similar system onto many additional telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Keywords: stars: emission-line, Be — instrumentation: high angular resolution — techniques: inter- ferometric 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' INTRODUCTION Recognized as the first stellar object displaying emission line spectra (Secchi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 1867), γ-Cas is the prototype of the Be stellar class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The emission line features originate from radiative processes with up to X-ray energies (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2012) occurring in an extended disc surrounding the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The disc formation is primarily attributed to mass ejection from the central star enabled from a combination of strong radiative pressure, and low effective surface gravity near the equatorial latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The latter is a consequence of the extremely high rotation rate that is nearly critical, in which the outward centrifugal force is equal to the inward gravitational force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Due to its bright stellar magnitude and characteristic stellar size, optical interferometry has been extensively used to study the disk emission of γ-Cas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The extended atmosphere of γ-Cas was first resolved with the I2T interferometer (Thom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 1986) and subsequently by the GI2T interferometer showing that the Hα region could be fit by a disk model and was in Keplerian motion (Mourard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Observations by Quirrenbach A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (1997) with the MkIII interferometer demonstrated that the emission-line region were not compatible with circularly symmetric models and required the assumption of an elongated profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Density and velocity relationships in the equatorial plane were constrained and accounted for by a radiative wind driven model in Stee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Subsequently, spectro- interferometric measurements of the envelope size were performed across both the Hα and Hβ lines, as well as in the near-by continuum emission (Stee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 1998) leading to a measurement of the disk mass and opening angle (Stee et Corresponding author: Nolan Matthews nolankmatthews@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='com arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='04878v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='IM] 12 Jan 2023 2 al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' In addition, the Navy Precision Optical Interferometer (NPOI) was used to characterize the disc geometry and further confirmed the oblateness of the disc (Tycner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The CHARA interferometric array measured the disk extent in the K’ photometric band for the first time, found to be slightly smaller than previous observations in Hα (Gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Finally, high sensitivity spectro-interferometric measurements with CHARA were performed in the near-infrared, as well across the Hα line and near-by continuum suggesting a larger disk size than prior measurements and linking the origin of X-ray emission to a compact binary companion due to the absence of one-armed spiral structures or secondary star (Stee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' In this work we present the first known intensity interferometry (II) measurements of the extended atmosphere of γ-Cas using a modified version of our intensity interferometry instrument (I2C) installed onto the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='54-meter telescope of the M´eO laser ranging facility and a mobile 1-meter telescope (hereafter T1M), both located on the Calern Plateau site of the Observatoire de la Cˆote d’Azur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' While the I2C instrument shares similarities with past II observations using the telescopes of the Centre P´edagogique Plan`ete Univers (C2PU) (Guerin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2017, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Rivet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' de Almeida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2022), there were several modifications required to outfit these telescopes with II capabilities, and to be compatible with each other in an interferometric mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The experimental setup is thus described in Section 2 with additional details also presented in Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The observations and results are shown in Section 3 with an analysis presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Finally, we discuss the results and present an outlook for future intensity interferometry measurements in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' EXPERIMENTAL SETUP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Principles of Intensity Interferometry An intensity interferometer correlates the intensity fluctuations of starlight between separated telescopes in order to measure the squared visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' For two telescopes with a projected baseline r between them, the second order coherence function is g(2)(r, τ) = ⟨I1(t)I2(r, t + τ)⟩ ⟨I1⟩⟨I2⟩ (1) where I1 and I2 are the intensities recorded at each of the two telescopes, τ is the relative time-lag between the signals, and the brackets indicate an average over time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The Siegert relation (Siegert 1943;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Ferreira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2020) relates the second-order coherence function g(2) to the first order coherence function g(1) by g(2)(r, τ) = 1 + |g(1)(r, τ)|2 (2) where the first-order coherence function can be separated into spatial and temporal components g(1)(r, τ) = V (r)g(1)(τ) (3) where V (r) is the interferometric visibility of the source, given by the Fourier transform of the source sky brightness distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' For an unresolved point-like source V (r) = 1, and the resulting second order coherence function will depend only on the temporal component g(1)(τ) given by the Fourier transform of the measured light spectral den- sity (Wiener 1930;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Khintchine 1934).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' For linearly polarized thermal light at zero optical path delay g(1)(τ = 0) = 1 where for time-lags much greater than the coherence or correlation time the first order coherence function should be equal to zero such that there is a “bunching peak” centered about zero optical path delay with an effective temporal width given by the coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The peak amplitude at zero time-lag of g(2) thus measures the squared visibility at some projected baseline assuming that the instrumental resolving time is shorter than the light coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The coherence time can be defined by the integral of the squared first-order coherence function (Mandel, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=', & Wolf, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 1995), Tc = � |g(1)(τ)|2dτ = � |s(ν)|2dν, (4) which is equal to the integral of the squared normalized spectral density s(ν) by Parseval’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' For visible light with a bandpass of ∆λ ∼ 1 nm the corresponding coherence time is of order 1 ps, much shorter than what can be achieved with conventional detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' In this case, a measurement averages over many coherence times and reduces the value of the g(2) peak amplitude at τ = 0 by a factor of ∼ Tc/Td where Td is the effective time-resolution of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The amplitude of the g(2) peak therefore measures this loss of contrast times the squared visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The squared visibility can be extracted by dividing the value of g(2)(r) − 1 peak amplitude measured between telescopes 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Photographs of the coupling assemblies mounted on the Nasmyth arm of the M´eO telescope (left) and at the Newtonian focus of the T1M portable telescope (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' to the g(2)(r = 0) − 1 peak amplitude measured at zero-baseline under the assumption that the profile of the g(2)(τ) peak is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' In practice, we measure the ratio of the area of the g(2)(r) − 1 peak to the area of the g(2)(r = 0) − 1 peak for the squared visibility, |V (r)|2 = � � g(2)(r, τ) − 1 � dτ � � g(2)(r = 0, τ) − 1 � dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (5) The denominator, or equivalently the area of the g(2) peak at zero baseline, corresponds to the coherence time that can be calculated from the measured spectrum as given by Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The equivalence between the coherence time from intensity interferometry and spectral measurements assumes that the spectral resolution is narrower than any spectral lines within the instrumental bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Since intensity interferometry measurements probe the intrinsic spectrum it is a useful method for characterizing narrow spectral lines present in, for example, studies of light scattering off of atomic clouds (Dussaux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2016), with potential applications in astrophysics (Tan & Kurtsiefer 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Telescopes The II observations presented in this paper were performed by outfitting two telescopes located on the Calern Plateau site of the Observatoire de la Cˆote d’Azur with individual coupling assemblies (CAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' A CA is mounted near the focus of each telescope, both shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The first facility was the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='54 m diameter M´eO (M´etrologie Optique) telescope primarily used for satellite laser ranging (Bertrand, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2021), lunar ranging measurements (Bourgoin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2021) and low Earth orbit satellite laser communication (Giggenbach, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The optical design is based upon a Ritchey-Chr´etien configuration on an altitude-azimuth mount with a primary focal length of 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 m giving an approximate focal ratio of f/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' In typical operation the light is brought to a Coud´e focus, but for II observations the light is redirected to the CA along the Nasmyth arm using a removable 45 degree mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' In addition, a f=150 mm lens is inserted before the CA in order to decrease the effective focal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The second facility is the portable 1 m diameter T1M, a Newtonian telescope on a Dobson-type fully motorized azimuthal mount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The portability of the telescope enables configurable baselines to expand the accessible coverage of the uv-plane where the telescope can be disassembled and moved in just a few hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The telescope has a primary focal length of 3 m, and a Barlow lens is included in order to expand the effective length at the input of the CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' While both telescopes are azimuthal, there will be a relative field rotation due to the Newtonian versus Nasmyth optical designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' However, each CA utilizes polarizing filters that must be aligned with respect to one another, although not with respect to the target as astrophysical polarization effects are not investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' To compensate, the M´eO telescope CA is mounted into a rotation stage that orients the CA such that the polarization axes are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The 地4 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Schematic of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The light collected by both telescopes are brought to individualized coupling assemblies (CA), shown in detail in the right inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' A tip-tilt corrects the beam with respect to transverse displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The converging beam is collimated using a diverging lens (L1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' A dichroic (D) reflects short wavelengths to a guiding camera (GC) used in a closed loop with the tip-tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The transmitted light passes through a narrowband filter (NF) centered on Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' A polarizing beam splitter (PBS) separates the light into orthogonal polarizations where each polarization is injected into a graded-index multimode fiber (GRIN-MMF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' A linear polarizer (P) is included on the reflected arm to improve polarization purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Not shown is the rotation stage used for the M´eO telescope, and additional focal reducers/extenders, which are described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The light for each polarization mode for each telescope is split by a 50/50 fiber beamsplitter, and passed to single photon resolving detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The photon arrival times are recorded by the TDC that also produces intensity correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' stage is actively controlled throughout the observation where the amount of rotation is determined from the target sky position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The relative position of both telescopes must be known with a precision less than a few centimeters for optical path delay corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' For M´eO, the position was previously determined to millimeter accuracy in terrestrial coordinates due to its use in geodetic surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' To determine an absolute position of the mobile T1M telescope, geodetic markers were installed by the National Institute of Geographic and Forest Information at the ground level for several positions and their positions were measured from differential GPS methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The T1M was installed above these markers and the offset between the marker and the T1M reference point was estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The estimated cumulative error on the reference position is ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='5 cm in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Instrumental Setup The primary function of the CAs (shown in Figure 2) is to perform spectral/polarization filtering, and fiber injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The current version includes an automated tip-tilt device that provides stable fiber injection over several hours without manual intervention (Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The detector signal output is fed to a time-to-digital converter (TDC) that measures photon arrival times, and produces the correlation between all relevant pairs of detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Across both telescopes, there are 4 independent measurements of the zero baseline correlation g(2)(r = 0, τ) enabled by using fiber splitters in each polarization mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' These allow normalization of the spatial correlations across telescopes to measure visibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Spatial intensity correlations are obtained by calculating the correlation across telescopes for all pairs of detectors in the same polarization mode corresponding to a total of 8 measurements of g(2)(r, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' During one night, a time delay monitoring system was used on the electronic cables connecting the detectors from the M´eO telescope to Coupling Assembly (CA) MéO T1M IP D NF △α ~ 1nm PBS L2 GRIN-MMF TDC GC5 the TDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The drift throughout the whole night was significantly less than the characteristic jitter of the detectors of ∼ 500 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' OBSERVATIONS The observations of γ-Cas were performed between the nights of January 17th, 2022 to January 21st, 2022 in the short baseline configuration, and then from January 24, 2022 to January 27th, 2022 in the longer baseline configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' At the beginning of each night images were recorded in both telescopes of the visual binary system γ-Ari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The measured position angle of the binary for both telescopes, and thus of the polarization axes, were found to be always within 5 degrees, corresponding to a loss of visibility of less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Temporal Intensity Correlations The coherence time obtained from zero baseline intensity correlations were compared to expected values from the spectral throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' These temporal intensity correlation functions were computed for each polarization state and for each telescope by summing all individually acquired correlations acquired over the entire observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Each of the resulting correlation functions were then shifted by the instrumental delay and then co-added together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The resulting correlation function is displayed on the left in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The peak is fit by a Gaussian with free parameters for the amplitude and width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The coherence time, given by the integral of the peak, is extracted via the fit values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Before fitting, there is a choice of the time-lag range to fit over and the number of time-lags to bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Here, we fit and show the data over a range of ± 20 ns binned into 50 ps bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Under these parameters, we find an amplitude of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='05)×10−3 and a full-width at half-maximum of 885±40 ps corresponding to a measured coherence time of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='05 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Systematics of the fitting process were studied by fitting the data varying the fit range from ±10 ns to ±40 ns and additionally the binning size from 10 ps to 80 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Within these parameters we find a maximum difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='015 ps in the extracted coherence time, notably less than the measurement error, In the previously quoted coherence time, the correlations from different polarization states and telescopes were co-added and subsequently fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' This procedure requires that shape of the bunching peak in each correlation, given by the temporal response of the detectors, are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' To test systematics, each correlation function for both polarization states and telescopes were fit by a Gaussian to extract the coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Each individual fit was within 1 σ of the quoted coherence time, and furthermore the weighted mean of fits (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='35±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='05 ps) is in perfect agreement with a single fit of co-added correlations indicating that within our measurement precision there are no significant systematics that preclude us from combining individual zero-baseline correlations from separate detector pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The coherence time measured from the zero-baseline correlations can then be compared to expectation from the recorded spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The spectral transmission of the Hα filter was measured in the laboratory with a high resolution spectrograph (Matthews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Spectra of γ-Cas with resolution R=25000 were recorded contemporaneously with Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The left plot shows the measured zero-baseline correlation in the blue points with a Gaussian fit shown by the dashed black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The right image displays the measured spectrum of γ-Cas, along with the theoretical (black) and measured (blue) spectral transmission of the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='5 )-1(×103) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='5 20 15 10 5 0 5 10 15 20 Time-lag (ns)100 NominalTransmission MeasuredTransmission Transmission (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=') GammaCasSpectra 80 60 Normalized 40 20 0 655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='5 656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='5 657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='5 658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 Wavelength (nm)6 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Spatial intensity cor- relation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Baseline Range Peak Area m ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='35±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='8 < r ≤ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='17 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 < r ≤ 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='14 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 < r ≤ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='08 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 < r ≤ 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='07±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='04 our observations using a Whoppshel echelle spectrograph provided through collaboration with the 2SPOT1 association of amateur astronomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' This is especially important as the width of the temporally variable emission line is narrower than the filter bandpass thus affecting the coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The right side of Figure 3 shows the filter transmission, and the emission line spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Through Equation 4 we extract the expected coherence time to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='41 ps which is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='2 σ larger than the value from zero-baseline correlations when taking only the measurement uncertainty and thus in fair agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' In contrast, the coherence time that would be expected for a flat stellar spectrum would be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='16 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' This is considerably less than the measured value by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='8 σ illustrating the importance of including the emission line profile in calculations of the coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The general agreement of the coherence time measured between intensity interferometry and spectral measurements indicate that there are no systematic effects arising from the presence of unidentified narrow spectral lines due to a lack of spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Spatial Intensity Correlations The spatial intensity correlations correspond to the correlations between all detector pairs on separate telescopes that are observing in the same polarization mode corresponding to a total of 8 cross-correlations across telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' All computed correlations are shifted in time by instrumental and geometrical delays, and then co-added together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The full data set corresponds to a wide projected baseline and position angle range and so the data was divided into smaller baseline ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' For each sub-division, we compute the averaged correlation function and then fit a Gaussian function to the resulting bunching peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The measured areas of the g(2) peak for each subdivision are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Squared visibilities are extracted by computing the ratio of the integral of the Gaussian peak of the cross-correlation to the computed value from the measured spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' ANALYSIS OF RESULTS The reduced II data resulted in 4 measurements of the squared visibility, each averaged over a range of baselines required to significantly resolve a bunching peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' In turn, the limited sampling does not allow any reasonable in- dependent visibility modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Nevertheless, it is interesting to compare the measured values to past results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Past interferometric observations generally characterize the angular brightness distribution of γ-Cas with a parameterized geometrical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' A common assumption is a two-component system consisting of a photosphere and disk, with some flux ratio between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The photosphere is typically approximated as a uniform disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' This is an oversimplified model of Be stars as it does not take into strong temperature gradients and equatorial flattening from the near critical rotation (Domiciano de Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' However, to resolve these effects requires an angular resolution at the characteristic diameter of the photosphere (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='5 mas for γ-Cas (Stee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 1998)), whereas the effective resolution for our observations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='22λ/D) at the largest baseline is ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='3 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Furthermore, these observations were conducted within the Hα line in which the disk emission is much stronger, such that photospheric contributions are significantly minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' For the disk emission, several geometric models were tested, including elongated uniform disks, Gaussian disks, and uniform rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Tycner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Stee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (2012) showed that a Gaussian disk profile best described the extended emission relative to the other assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 1 www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='2spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='org 7 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Reported Gaussian disk fit values in prior γ-Cas observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' θGD is the full-width at half-maximum, φ is the position angle, and r is the axial ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Observatory Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' θGD φ r (mas) (◦) MkIII Quirrenbach A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (1997) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='02 19±2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='70±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='02 NPOI Tycner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (2006) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='04 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='58±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='03 CHARA Stee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (2012) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='4 19±5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='74 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The left image shows the uv-plane coverage, over-plotted on expected squared visibilities formed from a Gaussian disk model of γ-Cas using reported parameters from Stee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Each of the colors represent the range of sampled points averaged together in order to measure squared visibilities, as correspondingly plotted on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Additionally, we plot the expected squared visibilities for each of the models in Table 2 at our sampled uv-plane points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' This two component model of a uniform disk + elongated Gaussian disk was applied to γ-Cas data in three prior reported observations using the MkIII interferometer (Quirrenbach A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 1997), NPOI (Tycner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2006), and CHARA (Stee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The reported parameters for these observations are summarized and presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Figure 4 shows our squared visibilities, along with expected values obtained from a Gaussian disk model from the previous observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The comparison of our results with the models produced using reported values from literature tends to align with the values given by Tycner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (2006) and Quirrenbach A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (1997) over Stee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (2012) who suggest a smaller extent of the Hα region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Within our measurement precision this is not strongly conclusive and can also be a result of instrumental differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Already, Stee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (2012) noted the larger angular extent could be explained in that they used high resolution spectro-interferometry, in contrast to the other observations including our own, that utilize narrowband filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The reasoning is that the filters detect more of the less resolved continuum resulting in an effectively smaller angular extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' DISCUSSION AND OUTLOOK We reported here on II measurements of the extended Hα emitting region of γ-Cas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The observed angular extent of the emission was found to be consistent with past direct interferometry measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Following our previous observations of Rivet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (2020) and de Almeida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (2022) this extends the work of II measurements in emission lines to another system and complements recent on-sky results of other intensity interferometry facilities (Acciari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Abeysekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Horch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Future improvements to the system will aim to improve the sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The most significant gain comes from simultaneously performing II correlations in many spectral channels that can be co-added to improve the signal to noise ratio by a factor of the square root of the number of channels (Trippe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' One could also imagine recording many independent spectral channels across the Hα line in order to perform intensity spectro-interferometry to test the discrepancy seen in past observations between Model Visibilities and (u,v)Sampling 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='8 20 (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='6 0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='4 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='2 40 20 0 20 40 u (m)MkillI NPOI 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 CHARA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content='0 0 5 10 15 20 25 30 35 Baseline (m)8 those using filtered bandpasses, and those in dispersed light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' Furthermore, higher sensitivity observations, paired with polarimetric capabilities would allow for better constraints, if not a direct measure of, radiative processes displaying polarized emission in the disk, as was attempted by Rousselet-Perraut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' These observations were performed using two facilities: the T1M and M´eO that had not been used for interfero- metric observations prior to this report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The portability of the T1M provides the capability to optimize the baseline configuration for a given target, or similarly perform multiple configurations for a given target as done here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' This technical accomplishment also illustrates the potential for performing up to 4-telescope II measurements on the Calern Plateau enabling 6 simultaneous baselines by including the 2 additional C2PU telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' We acknowledge the financial support of the R´egion PACA (project I2C), the French National Research Agency (ANR, project I2C, ANR-20-CE31-0003), OCA, Doeblin federation, UCA science councils grants, and the LABEX Cluster of Excellence FIRST-TF (ANR-10-LABX-48-01), within the program Investissements d’Avenir operated by the ANR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE4T4oBgHgl3EQfEgyT/content/2301.04878v1.pdf'} +page_content=' The authors would like to thank Jacques Belin and Damien Pesce for their installment of reference markers, and to the members of the M`eO team including Hervey Mariey, Mourad Aimar, Herv´e Viot, Gr´egoire Martinot-Lagarde, Julien Scariot, Nicolas Maurice, Duy-H`a Phung, and Nils Raymond for their assistance during the observations.' metadata={'source': 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CPUs. In this paper we explore the reasons +why and propose a simple, yet effective approach based on +the well-known Divide-and-Conquer Principle to tackle this +problem of great practical importance. Given an inference +job, instead of using all available computing resources (i.e., +CPU cores) for running it, the idea is to break the job into +independent parts that can be executed in parallel, each with +the number of cores according to its expected computational +cost. We implement this idea in the popular OnnxRuntime +framework and evaluate its effectiveness with several use +cases, including the well-known models for optical character +recognition (PaddleOCR) and natural language processing +(BERT). +1 +Introduction +We live in the era of unprecedented attention to machine +learning (ML) from researchers and practitioners alike. New +ML models across a variety of domains (or modalities, such +as video, images and text) are proposed nearly daily, the +models grow bigger and more sophisticated, and their com- +ponents are continuously revised to achieve better accuracy +scores on various tasks. While lots of attention is given to +training efficiency and prediction accuracy, seemingly less +effort is focused on making sure those models perform well +when deployed in practice, i.e., during inference [11]. As we +demonstrate in this paper, some models scale poorly (and at +times, even worse!) when the number of available cores in a +CPU-based deployment is increased. +Why does not the inference on CPUs scale? There are a +variety of reasons, and we devote the entire section of this +paper to look into some of them. Briefly, they range from +the micro-level, such as the use of non-scalable operators +inside ML architectures, to macro-level, such as employing +ML architectures that process input iteratively. +To mitigate those scalability challenges, one might con- +sider redesigning their ML architecture or reimplementing its +non-scalable operations with a more efficient version. Such +approaches, however, require either substantial ML domain +specific expertise, exceptional engineering skills and famil- +iarity with ML frameworks used for inference, significant +investments (e.g., to retrain a new model, with a potential +risk to the accuracy metrics), or all of the above. +In this paper, we take a different approach and propose to +leverage the poor scalability of ML models by applying the +Divide-and-Conquer Principle, a well-known algorithm de- +sign technique in Computer Science [8]. Specifically, instead +of allocating all available computing resources (CPU cores) +to the entire problem, we propose to divide the problem into +smaller chunks1, let the framework decide how the comput- +ing resources should be allocated among those chunks and +then run their respective computations in parallel. We argue +that in many use cases, such a division is natural and requires +only trivial changes in the user code. We also describe a sim- +ple mechanism that allocates computing resources based on +the expected computational intensity (or weight) of each +chunk. +Consider, for instance, a model for solving a natural lan- +guage processing (NLP) task such as tweet classification. Our +approach allows efficient batching of inference requests of +various sizes, eliminating the need for padding (a common, +but wasteful solution to deal with batches of requests of +variable size) and letting the framework allocate comput- +ing resources proportionally to the length of each sequence. +We implement the aforementioned allocation mechanism in +OnnxRuntime [24], a popular framework for training and +inferencing ML models, and extend its inference API to al- +low user code to invoke parallel inference on multiple inputs. +We demonstrate the effectiveness of our approach with sev- +eral use cases, including highly popular models for image +processing (PaddleOCR [14]) and NLP tasks (BERT [10]). +The remainder of this paper is organized as following. +In Section 2 we elaborate on various reasons for why the +inference (on CPUs) commonly does not scale well. Next, +we describe in Section 3 the concept and implementation +details of the Divide-and-Conquer Principle as it applies to +inference. Following that, we present in Section 4 several +use cases of ML models where this principle can be applied, +along with the performance evaluation of its benefits. We +discuss related work in Section 5 and conclude the paper in +Section 6. +1We note that unlike the classical Divide-and-Conquer Principle [8], we +divide the problem only once, although it might be possible in some cases to +divide it recursively into increasingly smaller chunks that can be executed +by one thread each. +1 +arXiv:2301.05099v1 [cs.LG] 12 Jan 2023 + +2 +Why is Inference Slow? +There are numerous reasons for this lack of scalability. In +this section we survey some of them. +2.1 +Not “enough” work +One reason is simply because the amount of computation +required by a model during inference is not “enough” for +efficient parallelization. As noted by Aminabadi et al. [3], +kernel implementations of various ML operations are often +geared towards training, which tends to consist of sizable +batches of large inputs (e.g., sentences of 512 tokens). During +inference, however, the batches tend to be much smaller, and +often include just one input (e.g., for real-time / interactive +inference). Besides, the inputs themselves can be small, e.g., +a tweet or chatbot interaction consisting of just a few words. +Consider, for instance, highly popular Transformer-based [28] +models for NLP tasks, such as BERT [10] or GPT-3 [5], which +rely mostly (but not solely) on matrix multiplication prim- +itives. Those primitives are known to scale well for large +matrices [11, 22, 30]. However, when the actual input to +the model during inference is short, matrix multiplications +involve smaller and therefore, less amendable to efficient +parallelization, matrices [11, 23, 30]. +2.2 +Non-Scalable Operators +Another reason for poor scalability of some ML models is +the use of non-scalable (and often, sequential) operators in +their architecture. Typically, the overhead of those operators +would be negligible compared to other, more scalable parts +of the model. Yet, as the number of cores increases and fol- +lowing directly from the Amdahl’s Law [2], their negative +impact of non-scalable operators on the overall inference per- +formance would grow. Considering again the Transformer- +based [28] models mentioned above, Dice and Kogan have +observed that while matrix multiplication scales well, at least +for long inputs, other operations such as layer normaliza- +tion and softmax do not, contributing to the overall poor +scalability of those models [11]. In this paper, we consider +a vision-based model, which employs sequentially imple- +mented functions for internal format conversions, which +similarly cause the entire model not to scale. +We note that some of those cases could be considered a +performance bug in the underlying ML framework, which +could be fixed by reimplementing the respective operators +with more efficient (and parallel) alternatives. This, however, +requires lots of engineering effort, which includes perfor- +mance analysis and deep understanding of corresponding +framework implementation details. Besides, some of the ML +operators, such as layer normalization [4], require careful +coordination among computing threads (e.g., to compute +variance and standard deviation of all the hidden units in a +layer and then use those statistics to normalize the values +of the units) and therefore do not lend themselves naturally +for efficient parallelization. +2.3 +Framework Overhead +Somewhat related to the prior point, an ML framework might +add small but measurable overhead in invoking model op- +erations. Most popular ML frameworks, such as PyTorch, +Tensorflow or OnnxRuntime, support multiple backends for +executing ML operations, targeting different hardware archi- +tectures (CPU, GPU, TPU), utilizing different BLAS libraries +(MKL, OpenBLAS, oneDNN, etc.), different threading infras- +tructure (Intel TBB, pthreads, custom implementation, etc.), +etc. Dispatching appropriate kernel (implementation) for +every operator is efficient, but is sequential and requires +non-trivial amount of work, especially when the model is +executed interactively [11] (the default execution mode in +PyTorch). This overhead becomes substantial as the actual +execution time of the kernels reduces with the increased +number of cores. +In addition to the above, various kernels might require +specific memory layout for its input parameters (tensors), +and the framework would add appropriate dummy operators +for input/output conversion or data preparation [30]. As we +demonstrate later in this paper, these operators might add +substantial overhead as well. +2.4 +Model Architecture +Quite often the high-level architecture of an ML model itself +plays a substantial role in causing inference not to scale. For +instance, some ML models, especially ones built for video +and image processing (e.g., [14, 21, 29]), are composed as a +multi-phase pipeline. The first phase of the pipeline would +typically identify the points of interest in the input (e.g., text +boxes in an image or a moving object in a video), while sub- +sequent phases would process those points (iteratively or as +a batch) to solve the predefined problem (e.g., identify text +in the boxes or classify the moving object in the video). The +inference latency of such models might grow linearly with +the number of objects identified in the first phase. Further- +more, if even one phase of the pipeline does not scale well, +the scalability of the entire pipeline is impaired. +2.5 +Padding +Batching multiple inputs and processing them at once is a +well-known way of improving inference throughput [1, 3, 9, +15, 32]. In fact, multiple serving system for machine learning +models (such as TensorFlow Serving [16] or TorchServe [7]) +include tunable parameters that configure how long an in- +ference server can wait in order to batch as many input +requests as possible. However, when inputs in a batch do not +have exactly the same shape, they need to be padded to be +processed efficiently, since underlying kernels typically an- +ticipate batches of homogeneous inputs. The padding leads +2 + +to reduced computational efficiency, since it is treated by ker- +nels as the rest of the input, even though the corresponding +output produced by the model is dismissed. +3 +Divide-and-Conquer Principle Applied +to Inference +In this section, we describe the application of the Divide- +and-Conquer Principle [8] to the inference of ML models at +the conceptual level and as a concrete realization by imple- +menting it in the OnnxRuntime framework. We note that +applying this principle does not directly address the reasons +for poor scalability detailed in the previous section. In fact, +the advantage of our approach is that one does not have to +identify and/or fix any scalability bottlenecks in their models +to rip the benefits of its underlying idea. +3.1 +Concept +The basic idea is pretty straightforward — consider a compu- +tation job 𝐽, which can be broken into𝑘 independent parts, 𝑗1, +𝑗2, ..., 𝑗𝑘, which can be executed in parallel. Assume we have +an oracle assigning relative weight 𝑤𝑖 ∈ (0, 1] correspond- +ing to, e.g., the number of required floating point operations +(FLOPs) or single-thread latency of the computation job part +𝑗𝑖. Finally, assume we have 𝐶 computing cores available. We +strive to allocate to each part the number of cores relative +to its weight, namely, we assign 𝑐𝑖 = 𝑚𝑎𝑥{1, ⌊𝑤𝑖 ∗𝐶⌋} cores +for the part 𝑗𝑖. This effectively means allocating 𝑐𝑖 worker +threads for 𝑗𝑖 since we later create one worker thread per core +(as common in ML frameworks, including in OnnxRuntime). +Note that �𝑘 +𝑖=1 𝑐𝑖 might be larger than C. This is obvious +when the number of job parts, 𝑘, is larger than C, but it is +possible even when 𝑘 ≤ 𝐶. This does not create a problem +other than implying that some job parts will be run after +other job parts have finished (rather than running them all +in parallel). At the same time, due to the rounding-down +(floor) function intended to reduce the above possibility of +oversubscription, some unallocated cores might remain. To +avoid this waste of available resources, we sort all the job +parts by their remaining unallocated weight, i.e., by 𝑤𝑖 ∗𝐶 − +⌊𝑤𝑖 ∗ 𝐶⌋, and assign one core to each part in the descending +order, up until all cores are allocated. The C++-like pseudo- +code for the entire algorithm is given in Listing 1. +Naturally, the idea described above raises the question of +how to assign relative weight to a job part 𝑗𝑖. In all our cases +considered in Section 4, the weight is simply set proportion- +ally to the size of input tensors. Specifically, let 𝑠𝑖 be the size +of the input tensor for job part 𝑗𝑖. We set 𝑤𝑖 to +𝑠𝑖 +�𝑘 +𝑖=1 𝑠𝑖 , essen- +tially assuming that the amount of computation (expressed +as the number of required FLOPs) grows roughly linearly +with the input tensors’ size. In general, however, assigning +weight can be done with the help of a profiling phase and a +lightweight classification mechanism, which associates job +parts of the same (or similar) shape (as the one encountered +1 vector allocate(vector inputs, int numCores) { +2 +vector threadAllocation; +3 +vector> threadUnallocatedWeight; +4 +int numInputs = inputs.size(); +5 +int allocatedCores = 0; +6 +int index = 0; +7 +int totalSize = 0; +8 +for (auto j_i : inputs) totalSize += j_i.size() +9 +for (auto j_i : inputs) { +10 +int numThreadsToUse = 1; +11 +if (numInputs <= numCores) { +12 +int size = j_i.size(); +13 +float w_i = ((float)size) / totalSize; +14 +int numThreadsToUse = floor(w_i * numCores); +15 +// this may happen due to flooring +16 +if (numThreadsToUse < 1) numThreadsToUse = 1; +17 +unallocatedWeight.add( +18 +make_tuple(index, w_i * numCores - numThreadsToUse)); +19 +} +20 +threadAllocation.add(numThreadsToUse); +21 +allocatedCores += numThreadsToUse; +22 +index++; +23 +} +24 +if (allocatedCores < numCores) { +25 +// sort the vector in decreasing order by +26 +// comparing the second field in each tuple +27 +sort(unallocatedWeight, bySecondField); +28 +int nextToAdjust = 0; +29 +while (allocatedCores < numCores) { +30 +// fetch the first field in the `nextToAdjust` tuple +31 +index = +32 +unallocatedWeight[nextToAdjust % numInputs].get(0); +33 +threadAllocation[index]++; +34 +allocatedCores++; +35 +nextToAdjust++; +36 +} +37 +} +38 +return threadAllocation; +39 } +Listing 1. Thread allocation algorithm +during the profiling phase) to the relative weight obtained +during profiling. +3.2 +Implementation Details +We extend the API of the InferenceSession class of On- +nxRuntime with a new prun method. This method is modeled +3 + +1 class TextRecognizer(object): +2 +def __init__(self, args): +3 +... +4 +self.predictor = ort.InferenceSession(args.file_path) +5 +self.postprocess_op = build_post_process(args) +6 +... +7 +def __call__(self, img_list): +8 +img_num = len(img_list) +9 +for beg_img_no in range(0, img_num, batch_num): +10 +end_img_no = min(img_num, beg_img_no + batch_num) +11 +inputs = prepare(img_list, beg_img_no, end_img_no) +12 +outputs = self.predictor.run(inputs) +13 +preds = outputs[0] +14 +rec_result = self.postprocess_op(preds) +15 +all_results.add(rec_result) +16 +return all_results +Listing 2. Original (shortened and edited for clarity) +TextRecognizer class implementation from PaddleOCR +1 class TextRecognizer(object): +2 +def __init__(self, args): +3 +... +4 +self.predictor = ort.InferenceSession(args.file_path) +5 +self.postprocess_op = build_post_process(args) +6 +... +7 +def __call__(self, img_list): +8 +img_num = len(img_list) +9 +for beg_img_no in range(0, img_num, batch_num): +10 +end_img_no = min(img_num, beg_img_no + batch_num) +11 +inputs = prepare(img_list, beg_img_no, end_img_no) +12 +all_inputs.append(inputs) +13 +all_outputs = self.predictor.prun(all_inputs) +14 +for outputs in all_outputs: +15 +preds = outputs[0] +16 +rec_result = self.postprocess_op(preds) +17 +all_results.add(rec_result) +18 +return all_results +Listing 3. Modified TextRecognizer class implementation +(uses prun). Added or modified lines are in red +after the existing run method used as the main entry point +when running inference. The main difference is that prun +accepts a list of inputs (instead of just one) and returns a list +of outputs. +Internally, the implementation of prun iterates over the +list of inputs, calculates their size (after validating those are +tensors) and corresponding relative weight, and applies the +allocation algorithm described in Listing 1 to associate the +number of worker threads with each input (job part). Follow- +ing that, the implementation creates one worker thread for +each input, and runs them in parallel. Each worker thread, in +turn, creates a thread pool of the size calculated by the alloca- +tion algorithm (the thread pool includes the worker thread it- +self), and invokes the run method of the InferenceSession +object with that thread pool. The entire patch of the On- +nxRuntime codebase to implement the prun functionality +and other minor internal changes (such as having the run +method to accept a thread pool as an optional argument in- +stead of always using the default pool) consisted of around +200 lines of code. +On the user side, the code also has to change to make +use of the new prun API. Those changes, however, are quite +straightforward. Instead of invoking run for every job, a user +needs to create a list of job parts and call prun. In addition, the +user needs to rearrange the post-processing code to iterate +over the results of prun, and apply any post-processing to +each returned output (object). As an example of what the +user code changes entail, we show the original Python code +(edited for brevity and clarity) of the TextRecognizer class in +PaddleOCR (Listing 2) alongside the modified version that +makes use of the new prun API (Listing 3). +4 +Use Cases +Before we detail the use cases where the Divide-and-Conquer +Principle is beneficial and report on our performance find- +ings, we give a brief summary of our evaluation setup and +methodology. We run all our experiments on a 16-core AMD- +based VM in Oracle Cloud (aka OCI VM.Standard.E3.Flex). +(We also ran some experiments on a newer E4 shape, but +have not noticed substantial differences). To reduce perfor- +mance variability, especially as we create separate thread +pools for the variants that use prun, we use thread binding +(pinning), for all the evaluated variants. Every experiment +was repeated 5 times, and we report the mean. We note that +the standard deviation of all reported results, except for one +specific case discussed below, was extremely low (typically, +less than 1% of the mean). For our experiments, we use the +latest release versions (as of the date of writing this paper) +of the corresponding software, specifically OnnxRuntime +v1.11.1 and PaddleOCR v2.5. +4.1 +Sequential Pipeline +Our first example of where applying the Divide-and-Conquer +Principle is extremely useful is PaddleOCR [14]. PaddleOCR +is a lightweight OCR system, which consists of three parts: +Text Detection, Text Classification (called Detection Boxes +Rectify in [14]) and Text Recognition. Each of those parts +corresponds to a separate ML model. +4 + +Figure 1. PaddleOCR 3-phase pipeline (edited version of Figure 2 from [14]). +The OCR pipeline accepts an image file and passes it first +through the text detection phase whose objective is to locate +text areas in the image. The output of this phase is a list of +potential text boxes’ coordinates. Next, the list is iterated +over, and each item in that list (i.e., a text box) is sent to +the text classification model, which decides whether the +box needs to be transformed into a horizontal rectangle box +before the actual text recognition takes place. Based on the +classifier’s decision, each box is altered respectively. Finally, +the list is iterated over again, and each item is sent to the text +recognition model for inference, which recognizes the text +in the given box and produces the actual character sequence +based on the supplied character dictionary. This process is +depicted in Figure 1, which is a redacted version of Figure 2 +from [14]. +In our experiments with PaddleOCR, we observe that the +system does not scale well with the increase in the number +of available cores. We demonstrate that in Figure 2 depicting +inference latency as a function of available cores (which +directly translates into the number of worker threads used +by the runtime). For all experiments in this section, including +the one in Figure 2, we use a subset of images from the +OpenImages dataset [17], selected according to a criterium +described below. +In Figure 2, we break the total latency into time spans cor- +responding to the three phases of the OCR pipeline discussed +above. As one can notice, the average inference latency goes +down from 554 ms for 1 thread to 364 ms for 4 cores and then +back up to 435 ms for 16 cores. Interestingly, the Text Clas- +sification phase shows negative scalability, where it takes +27 ms to process an image, on average, with 1 thread, but it +takes 38 ms to do the same with 16 threads — a slowdown +of 1.4x. This shows an example of a system where, beyond a +certain point, adding more threads not only does not help, +but actually harms performance. Discussing concrete rea- +sons for the lack of scalability of these specific models is +not in the scope of this paper. For a curious reader, however, +we note that a built-in OnnxRuntime profiling tool shows +inflated execution times for the output reordering operators +(which are inserted by the framework, along with the input +reordering operator, to convert the memory layouts of input +arguments for various kernels). +We apply the Divide-and-Conquer Principle to the last +two phases of the OCR pipeline, namely the Text Classifica- +tion and Recognition. To that end, instead of invoking the +corresponding models for each text box produced by Text +Detection, we send all the boxes to the runtime (by invoking +the prun API) and effectively let the runtime decide how +many cores / worker threads to allocate each box based on +its relative size. The required changes to implement this +functionality in the Text Recognition phase are depicted in +Listing 3; the changes to the Text Classification phase are +similar. +For our performance evaluation, we compare the prun +implementation as discussed in Section 3 (and depicted in +Listing 1), which we denote as prun-def on the charts, to +a few simple variants. The first variant, denoted as prun-1, +simply allocates one worker thread to each input in the +list given to prun. The second variant, denoted as prun-eq, +allocates an equal number of cores for each input (but at +least one), i.e., sets 𝑐𝑖 = 𝑚𝑎𝑥{1, ⌊𝑘/𝐶⌋}. Our motivation is +to show that trivial solutions might also be useful in certain +scenarios (as discussed below), yet they tend to underperform +compared to prun-def. +We note that the benefit of prun in this use case is possible +only when there are at least two text boxes identified in the +Text Detection phase. Otherwise, the other two phases would +not be used (if no text boxes detected) or the prun-def vari- +ant will use the same (maximum) number of cores as the base +(unmodified) version (if only one text box is detected). As a +result, the subset of images used for performance evaluation +in this section includes images with at least two identified +text boxes. The pie chart in Figure 3 shows the distribution +of the actual number of boxes detected in the first phase of +the OCR pipeline for the entire dataset. The total number of +images in the dataset was 500 – this number was chosen to +keep the evaluation times reasonably short. (We note that we +also ran evaluations on a larger dataset that includes images +with less than two text boxes and confirmed that the use of +prun does not create any overhead in those cases.) +5 + +营养护发器 +ODM OEM +ODMOEM +ODMOEM +ODMOEM +ODMOEM +Image +Text Detection +Detection Boxes Rectify +Text Recognition +(db_mv3_slim,1.4M) +(dir_cls_mv3_slim, 0.5M) +(crnn_mv3_slim, 1.6M) +OutputFigure 2. Inference latency of PaddleOCR with a varying +number of threads, broken down by the three phases of the +pipeline. +In light of the discussion above, we break down the com- +parison of the latency results by the number of detected +boxes, as depicted in Figure 4. The latency numbers in this +figure were collected with 16 cores; we discuss the over- +all scalability trends later on. We also break down the per- +formance in two of the phases where we have used prun, +namely Text Classification (Figure 4 (a)) and Recognition +(Figure 4 (b)). +Considering the results in Figure 4, one can notice that, +as expected, the benefit of prun increases with the number +of detected text boxes. For instance, when considering the +total end-to-end latency (Figure 4 (c)), with only two boxes +prun-def outperforms base by 1.28x. However, with 9 and +10+ boxes, prun-def outperforms base by 2.33x and 1.81x, +respectively. +It is interesting to compare the performance of prun-def +with other pun-based variants. As one can notice in Fig- +ure 4 (a), the prun-1 variant produces the lowest latency +when the number of detected boxes is small. In fact, the base +variant also performs better than prun-def in this case. We +attribute this to two factors. First, this specific phase of the +pipeline shows negative scalability, which can be also seen in +Figure 2. Therefore, best performance is achieved when fewer +threads per box is used in this phase, which is what prun-1 +effectively achieves. Second, prun-def (and prun-eq) cre- +ate and destroy more threads than prun-1 in those cases as +they create thread pools containing more threads for each +prun invocation. This adds small, but non-negligible over- +head given that the the execution time of this phase is short. +In the future work, we intend to experiment with reusing +thread pools between prun invocations. As the number of +detected boxes increases, however, all prun variants allocate +less threads (or even just 1) per each box, and they allocate +a similar number of threads for their pools, thus closing the +gap with the prun-1 variant. +When the Text Recognition phase is concerned (cf. Fig- +ure 4 (b)), however, it is apparent from Figure 2 that one +Figure 3. Distribution of the number of detected text boxes +in the input dataset. +can improve its latency by using more than one thread. We +note that, quantitively, this phase is also far more dominant +than the Text Classification one. Here, prun-def manages +to achieve best or close to best result across all counts of +detected boxes, which translates to overall highly competi- +tive end-to-end inference performance (cf. Figure 4 (c)). In +general, the results in Figure 4 call for a dynamic mechanism, +which would choose the best thread allocation strategy based +on the given workload and available resources. Devising and +experimenting with such a strategy is left for future work. +Finally, we shed more light on how the scalability im- +proves with the use of prun in Figure 5, where we vary the +number of cores (and therefore, the total number of worker +threads) available for OnnxRuntime. Once again, we include +the latency of each of the two last phases of PaddleOCR +(denoted as Rec for Text Recognition and Cls for Text Clas- +sification) along with the end-to-end (Total) latency. We +include only the results of the base and prun-def variants +(denoted simply as prun in Figure 5), for clarity. +Overall, one can notice similar trends to the ones discussed +above. In the base version, the Text Recognition phase does +scale up to 4 threads, but then its performance suffers as the +number of threads increases. The prun variant avoids this +performance degradation, and in fact, continues to scale up +to 16 threads. Indeed, when considering the Text Recognition +phase only, the prun variant outperforms base by more than +2.4x at 16 threads. However, since both variants have an +identical Text Detection phase, which according to Figure 2 +subsumes a substantial part of the total latency, the end-to- +end speedup of prun is only 1.5x at 16 threads. +4.2 +Batching of Heterogeneous Inputs +Our next example concerns with the Transformer architec- +ture [28], which revolutionized the domain of NLP when it +was introduced in 2017 and has been applied to other do- +mains since then (e.g., [12, 18]). This architecture consists +of a stack of layers, each composed of a self-attention block +followed by a fully connected network [28]. Past work has +6 + +700 +Detection +Recognition +600 +Classification +Other +500 +Latency (ms) +400 +300 +200 +100 +0 +1 +2 +8 +12 +4 +16 +# threads2 (46.4%) +3 (18.6%) +4 (10.2%) +5 +(6.6%) +6 (4.2%) +7 (2.2%) +8 (2.0%) +9 (1.2%) +10+ (8.6%)(a) Text Classification +(b) Text Recognition +(c) End-to-End Inference +Figure 4. The impact of using prun in PaddleOCR. +Figure 5. Total (end-to-end) inference latency of PaddleOCR +with a varying number of threads. Also shown the latency of +Text Classification (Cls) and Text Recognition (Rec) phases +shown that the majority of computation cycles in Transform- +ers is spent on (scalable) matrix multiplication operations, +yet up to one third of the cycles is spent elsewhere (i.e., less +scalable operations) [11]. +It is well-known that one way to improve the inference +performance (specifically, throughput) of Transformers is +through input batching [3, 15, 30]. This strategy works well, +however, when the inputs have the same length. Otherwise, +one has either give up on batching, or pad inputs to the same +length. The latter results in wasted computation cycles, since +special padding tokens are treated exactly as input tokens by +the architecture and dismissed at the end of the computation. +This situation presents an ideal case for applying the +Divide-and-Conquer Principle. Instead of padding the in- +puts of various lengths up to the longest input in the batch, +we can run inference on those inputs (as they are, without +padding) using the prun API, and let the runtime decide +how many cores should be used to process each of the in- +puts. We modify the Transformer benchmark built into the +OnnxRuntime [25] to implement this strategy. +To evaluate the effectiveness of the approach described +above, we set up an experiment where we generate 𝑋 in- +puts of a length chosen uniformly and randomly out of the +range [16, 512]. We then compare the pad-batch version in +which all 𝑋 inputs are padded to the longest length in the +given batch with the prun version in which the inference +is invoked with prun on all inputs in the batch. We show +results with the highly popular BERT model [10] (specifi- +cally, “bert-based-uncased”). We have also experimented +with other Transformer-based models (such as “bert-large- +uncased” or “roberta-base”) measuring similar qualitative +results. +We note that this experiment includes inherent amount +of randomness — a batch of small sentences is as likely to +be chosen as a batch of long sentences. In an attempt to +reduce the anticpated high varaince of the results, we opted +to repeat the experiment 1000 times, and so for each 𝑋, each +data point is an average of 1000 results. Figure 6 presents the +throughput results with batches of various sizes (i.e., 𝑋 varies +from 2 to 8), with error bars depicting the standard deviation +of the reported mean. Even though prun outperforms the +pad-batch variant across all batch sizes, the variance in the +measured results remains exceptionally high. +As a result, we setup two additional experiments in a +more controlled way likely to produce more stable results. In +the first, we simply preset the lengths of various sequences +in each batch. For instance, a batch denoted as “16-64-256” +includes three sentences, one is 16, another is 64 and yet +another is 256 tokens long. We show the results of this exper- +iment in Figure 7. Here, the prun version easily outperforms +the pad-batch variant, which has to pad all sequences to the +longest sequence in a batch. As one might expect, the benefit +from using prun increases with the number of sentences in +a batch, as this variant eliminates all the redundant work +associated with padding. +In the second experiment, we use a batch of 1 long sen- +tence (256 tokens long) and 𝑋 short sequences of 16 tokens +each, where we vary 𝑋 between 0 and 15. We show the +throughput results of this experiment in Figure 7, along with +a curve depicting the number of threads allocated by prun +for the long sequence in the batch. +There are several interesting observations that can be +made here. First, when 𝑋=0, i.e., the batch contains only one +long sentence, both variants employ all available cores to pro- +cess that batch, producing similar result. This shows that the +7 + +140 +base +120 +prun-1 +prun-def +100 +Latency (ms) +prun-eg +80 +60 +40 +20 +0 +2 +3 +4 +5 +6 +7 +8 +9 +10+ +# detected boxes900 +base +800 +prun-1 +700 +prun-def +(ms) +600 +prun-eg +500 +atency +400 +Lhhhhhhh +300 +200 +100 +0 +2 +3 +4 +5 +6 +8 +9 +10+ +# detected boxes1200 +base +1000 +prun-1 +prun-def +(ms) +800 +prun-eg +Latency ( +600 +400 +bhhhhhh +200 +0 +2 +3 +4 +5 +6 +7 +8 +9 +10+ +# detected boxes600 +base (Total) +base (Rec) +base (Cls) +500 +prun (Total) +prun (Rec) +prun (Cls) +Latency (ms) +400 +300 +200 +100 +0 +1 +2 +4 +8 +12 +16 +# threadsFigure 6. Throughput of inferencing BERT on batches of +sequences of sizes chosen randomly from the range [16, 512] +Figure 7. Throughput of inferencing BERT on batches of +sequences of various preset sizes +overhead of using prun when the input has only one chunk is +negligible. Second, the throughput of the pad-batch version +grows, but modestly with the increase in the number of short +sequences. This is because, as stated above, a larger batch of +(padded) sequences helps to achieve better throughput with +Transformers. At the same time, the throughput growth with +prun is much more dramatic up to 3 short sequences in a +batch and then it declines, but stays well above that achieved +with pad-batch. Both phenomena can be explained with the +fact that inferencing a sequence of 256 tokens takes about the +same time with 16 threads as it takes with 13. Thus, adding +a few short sequences into the batch, each allocated with +just 1 thread (as they have small relative weight), has negli- +gible impact on the latency, but improves throughput. With +more short sequences in a batch, less threads are allocated +for the long sequence (as can be seen in Figure 7) and its +inference latency grows. This causes the overall throughput +to decrease. +4.3 +Batching of Homogeneous Inputs +Our last example follows directly from the discussion in +Section 2 on the lack of scalability in ML models. As al- +ready mentioned, while Transformers models heavily use +Figure 8. Throughput of inferencing BERT on a batch con- +taining one long sentence of 256 tokens and 𝑋 short se- +quence with 16 tokens each, where 𝑋 varies 0 to 15. In +addition, we show how many threads are dedicated to the +inference of the one long sentence in the batch in the prun +variant +scalable matrix multiplication operations, they also employ +less scalable operations. The impact of the latter grows with +the increase in the number of cores. Therefore, one may +benefit form the Divide-and-Conquer Principle applied to +Transformers even when the batch includes inputs of the same +length. +As a concrete example, consider a batch of two inputs. +Instead of using all available cores to process the batch, we +will use half the cores for each input. Intuitively, the less +scalable operators create less relative overhead when less +cores are used and the input sequence is shorter (i.e., contains +half the tokens compared to the entire batch). +Figure 9 demonstrates this effect with batches of inputs of +equal lengths. In addition to the pad-batch variant (which +we simply call batch here, as no padding is required) and +prun, we include a no-batch variant, which runs inference +on each sequence in a given batch one at a time. Note that +we include the latter to simply demonstrate the benefits of +batching in general, confirming previous findings [3, 15, 30]. +Each set of bars in Figure 9 corresponds to a batch of 4 +sentences with the given length (from 64 tokens to 512). +Overall, the prun version yields a more modest (yet non- +trivial) speedup over batch compared to the case of non- +homogeneous inputs in Section 4.2. This is expected, since +in this case the room for improvement (over batch) does not +include wasted computation related to padding. +5 +Related Work +As mentioned in the Introduction, the major focus of the +ML community has been on improving the accuracy and +training performance of proposed models, while efficient +inferencing and serving of those models receives relatively +less attention. Yet, there have been some notable exceptions +of work focused specifically on inference performance, and +8 + +60 +16 +pad-batch +# +14 + threads f +Throughput (queries/s) +50 +prun +prun +12 +40 +10 +for +30 +8 +long +6 +20 + sequence +4 +10 +2 +e +0 +0 +3 +5 +0 +2 +9 +12 +15 +# of short sequences in a batch35 +pad-batch A +30 +Throughput (queries/s) +prun +25 +20 +15. +10 +5 +0 +2 +4 +6 +8 +# sequences in a batch50 +pad-batch +45 +prun +40 +35 +30 +25 +20 +15 +10 +5 +0 +16-256 +16-32-64-128-256 +16-64-256 +sequence lengths in a batchFigure 9. Throughput of inferencing BERT with batches of +4 sequences of equal size +we survey the most relevant results hereafter. As an aside, we +note that many of the results below come from less formal +blog posts published by various companies, highlighting the +great practical importance of efficient inference. +Wang et al. [30] explore various factors that influence in- +ference performance in TensorFlow, including the choice of +a specific math library, a thread pool library, availability of +SIMD (single instruction multiple data) support, etc. They +identify data preparation as one of the causes for poor scala- +bility of small matrix multiplication operations, something +we more generally attribute to framework overhead in Sec- +tion 2. They come up with a set of guidelines one can use +to tune TensorFlow settings to achieve better performance +compared to the one achieved with settings recommended +by TensorFlow authors or Intel. +With the tremendous rise in popularity of Transformers, +several papers and blog posts focus on its inference perfor- +mance. Dice and Kogan investigate inference performance of +Transformers on CPUs [11]. Their analysis shows that most +inference computation cycles are spent in matrix multipli- +cation operations. Hence, they propose an adaptive matrix +multiplication optimization aimed at reducing the latency +of those operations and subsequently improving the overall +inference performance. Intel engineers describe an effort +to optimize inference of BERT in Apache MXNet using the +GluonNLP toolkit, where one of the ideas is to quantize +the model for better performance with lower precision [31]. +Similar quantization ideas (along with distillation, another +common method of reducing the size of a model [27]) were +employed by Roblox to speedup their deployment of BERT +on CPUs [19]. The same blog post also mentions that elimi- +nating padding of input sentences has led to better perfor- +mance (though the authors did that for batches of 1 input +only). A Microsoft team [26] describes their effort on accel- +erating BERT with OnnxRuntime through operation fusion +that helps to reduce the amount of overhead (e.g., memory +copying) in invoking each kernel individually. +A few recent papers and projects have looked into the de- +ficiency of padding of heterogenous inputs. Fang et al. [15] +propose a sequence-length-aware batch scheduler, which +aims to batch requests of a similar size, thus reducing the +cost of zero padding of all requests into one batch. It re- +quires a profiling phase during which the inference cost of +various batches is collected. Du et al. [13] propose to care- +fully redesign the GPU kernels employed by Transformers to +eliminate most redundant computation associated with zero +padding. The Effective Transformer project by ByteDance [6] +aims to dynamically remove and restore padding during dif- +ferent calculation stages. All those efforts target specifically +the inferencing Transformers on GPUs, and it is not clear +how efficient they would be on CPUs and/or with other +architectures. +Beyond Transformers, Liu at et. [20] describe NeoCPU, an +approach for optimizing CNN inference on CPUs. NeoCPU +proposes a configurable design of an efficient convolution +operation that can be tuned efficiently to popular CPUs. +This design is coupled with a scheme for obtaining the best +memory layout for data in different operations of a CNN +model, in order to minimize the overhead of transforming +the data between various individual operations. +6 +Discussion +In this paper, we have discussed various reasons for the lack +of scalability of inferencing ML models. While the reasons +vary from micro to macro-levels, the common motive is that +existing ML frameworks are geared towards high perfor- +mance training. This is expressed by the fact that kernels for +common operations are typically optimized for large batches +with long inputs, ignoring relatively small overheads in var- +ious parts of those frameworks that are immaterial to the +overall training performance. However, during inference the +batches tend to be much smaller and contain shorter inputs, +thus making those overheads more prominent. A somewhat +similar observation has been made by Aminabadi et al. [3]. +We leverage this poor scalability and describe a simple, +yet powerful approach, in which the given input is broken +into chunks and each chunk is processed in parallel, instead +of using all available resources for the entire input. As we +demonstrate with a few well-known models, this approach +improves inference scalability and ultimately can lead to +over 2x latency and throughput improvements. +This work offers several directions for future research. +First, we want to explore more dynamic thread allocation +strategies, e.g., ones that can better adjust to the cases where +the weight of a work chunk does not correlate linearly with +its size and/or where the underlying model performs best +while running with a single thread. Second, we want to find +ways to automate splitting the input into chunks that can +be processed in parallel, lowering the cost (in terms of user +code changes) of using prun even further. Finally, we want +9 + +90 +三 +no-batch +80 +batch +(queries/s) +70 +prun +60 +50 +Throughput ( +40 +30 +20 +10 +0 +64 +128 +256 +512 +sequence length in a batchto explore other use cases where the use of prun would be +beneficial, including other ML models that feature a pipeline- +based architecture (e.g., [21, 29]). +Acknowledgments +The author would like to thank Dave Dice for valuable com- +ments on an early draft of this paper. +References +[1] Ahsan Ali, Riccardo Pinciroli, Feng Yan, and Evgenia Smirni. Batch: +Machine learning inference serving on serverless platforms with adap- +tive batching. In Proceedings of the International Conference for High +Performance Computing, Networking, Storage and Analysis (SC), 2020. +[2] Gene M. Amdahl. Validity of the single processor approach to achiev- +ing large scale computing capabilities. 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In +USENIX Annual Technical Conference (ATC), 2022. +10 + diff --git a/CtE4T4oBgHgl3EQfeg0W/content/tmp_files/load_file.txt b/CtE4T4oBgHgl3EQfeg0W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1c182e960cfeb9ebc4f5b51cf1f8a6f48b5ad34e --- /dev/null +++ b/CtE4T4oBgHgl3EQfeg0W/content/tmp_files/load_file.txt @@ -0,0 +1,655 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf,len=654 +page_content='Improving Inference Performance of Machine Learning with the Divide-and-Conquer Principle Alex Kogan Oracle Labs alex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='kogan@oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='com Abstract Many popular machine learning models scale poorly when deployed on CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In this paper we explore the reasons why and propose a simple, yet effective approach based on the well-known Divide-and-Conquer Principle to tackle this problem of great practical importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Given an inference job, instead of using all available computing resources (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', CPU cores) for running it, the idea is to break the job into independent parts that can be executed in parallel, each with the number of cores according to its expected computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We implement this idea in the popular OnnxRuntime framework and evaluate its effectiveness with several use cases, including the well-known models for optical character recognition (PaddleOCR) and natural language processing (BERT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 1 Introduction We live in the era of unprecedented attention to machine learning (ML) from researchers and practitioners alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' New ML models across a variety of domains (or modalities, such as video, images and text) are proposed nearly daily, the models grow bigger and more sophisticated, and their com- ponents are continuously revised to achieve better accuracy scores on various tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' While lots of attention is given to training efficiency and prediction accuracy, seemingly less effort is focused on making sure those models perform well when deployed in practice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', during inference [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' As we demonstrate in this paper, some models scale poorly (and at times, even worse!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=') when the number of available cores in a CPU-based deployment is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Why does not the inference on CPUs scale?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' There are a variety of reasons, and we devote the entire section of this paper to look into some of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Briefly, they range from the micro-level, such as the use of non-scalable operators inside ML architectures, to macro-level, such as employing ML architectures that process input iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' To mitigate those scalability challenges, one might con- sider redesigning their ML architecture or reimplementing its non-scalable operations with a more efficient version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Such approaches, however, require either substantial ML domain specific expertise, exceptional engineering skills and famil- iarity with ML frameworks used for inference, significant investments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', to retrain a new model, with a potential risk to the accuracy metrics), or all of the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In this paper, we take a different approach and propose to leverage the poor scalability of ML models by applying the Divide-and-Conquer Principle, a well-known algorithm de- sign technique in Computer Science [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Specifically, instead of allocating all available computing resources (CPU cores) to the entire problem, we propose to divide the problem into smaller chunks1, let the framework decide how the comput- ing resources should be allocated among those chunks and then run their respective computations in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We argue that in many use cases, such a division is natural and requires only trivial changes in the user code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We also describe a sim- ple mechanism that allocates computing resources based on the expected computational intensity (or weight) of each chunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Consider, for instance, a model for solving a natural lan- guage processing (NLP) task such as tweet classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Our approach allows efficient batching of inference requests of various sizes, eliminating the need for padding (a common, but wasteful solution to deal with batches of requests of variable size) and letting the framework allocate comput- ing resources proportionally to the length of each sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We implement the aforementioned allocation mechanism in OnnxRuntime [24], a popular framework for training and inferencing ML models, and extend its inference API to al- low user code to invoke parallel inference on multiple inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We demonstrate the effectiveness of our approach with sev- eral use cases, including highly popular models for image processing (PaddleOCR [14]) and NLP tasks (BERT [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The remainder of this paper is organized as following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In Section 2 we elaborate on various reasons for why the inference (on CPUs) commonly does not scale well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Next, we describe in Section 3 the concept and implementation details of the Divide-and-Conquer Principle as it applies to inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Following that, we present in Section 4 several use cases of ML models where this principle can be applied, along with the performance evaluation of its benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We discuss related work in Section 5 and conclude the paper in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 1We note that unlike the classical Divide-and-Conquer Principle [8], we divide the problem only once, although it might be possible in some cases to divide it recursively into increasingly smaller chunks that can be executed by one thread each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='05099v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='LG] 12 Jan 2023 2 Why is Inference Slow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' There are numerous reasons for this lack of scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In this section we survey some of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='1 Not “enough” work One reason is simply because the amount of computation required by a model during inference is not “enough” for efficient parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' As noted by Aminabadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' [3], kernel implementations of various ML operations are often geared towards training, which tends to consist of sizable batches of large inputs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', sentences of 512 tokens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' During inference, however, the batches tend to be much smaller, and often include just one input (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', for real-time / interactive inference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Besides, the inputs themselves can be small, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', a tweet or chatbot interaction consisting of just a few words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Consider, for instance, highly popular Transformer-based [28] models for NLP tasks, such as BERT [10] or GPT-3 [5], which rely mostly (but not solely) on matrix multiplication prim- itives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Those primitives are known to scale well for large matrices [11, 22, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' However, when the actual input to the model during inference is short, matrix multiplications involve smaller and therefore, less amendable to efficient parallelization, matrices [11, 23, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='2 Non-Scalable Operators Another reason for poor scalability of some ML models is the use of non-scalable (and often, sequential) operators in their architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Typically, the overhead of those operators would be negligible compared to other, more scalable parts of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Yet, as the number of cores increases and fol- lowing directly from the Amdahl’s Law [2], their negative impact of non-scalable operators on the overall inference per- formance would grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Considering again the Transformer- based [28] models mentioned above, Dice and Kogan have observed that while matrix multiplication scales well, at least for long inputs, other operations such as layer normaliza- tion and softmax do not, contributing to the overall poor scalability of those models [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In this paper, we consider a vision-based model, which employs sequentially imple- mented functions for internal format conversions, which similarly cause the entire model not to scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We note that some of those cases could be considered a performance bug in the underlying ML framework, which could be fixed by reimplementing the respective operators with more efficient (and parallel) alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This, however, requires lots of engineering effort, which includes perfor- mance analysis and deep understanding of corresponding framework implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Besides, some of the ML operators, such as layer normalization [4], require careful coordination among computing threads (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', to compute variance and standard deviation of all the hidden units in a layer and then use those statistics to normalize the values of the units) and therefore do not lend themselves naturally for efficient parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='3 Framework Overhead Somewhat related to the prior point, an ML framework might add small but measurable overhead in invoking model op- erations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Most popular ML frameworks, such as PyTorch, Tensorflow or OnnxRuntime, support multiple backends for executing ML operations, targeting different hardware archi- tectures (CPU, GPU, TPU), utilizing different BLAS libraries (MKL, OpenBLAS, oneDNN, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' ), different threading infras- tructure (Intel TBB, pthreads, custom implementation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' ), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Dispatching appropriate kernel (implementation) for every operator is efficient, but is sequential and requires non-trivial amount of work, especially when the model is executed interactively [11] (the default execution mode in PyTorch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This overhead becomes substantial as the actual execution time of the kernels reduces with the increased number of cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In addition to the above, various kernels might require specific memory layout for its input parameters (tensors), and the framework would add appropriate dummy operators for input/output conversion or data preparation [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' As we demonstrate later in this paper, these operators might add substantial overhead as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='4 Model Architecture Quite often the high-level architecture of an ML model itself plays a substantial role in causing inference not to scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' For instance, some ML models, especially ones built for video and image processing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', [14, 21, 29]), are composed as a multi-phase pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The first phase of the pipeline would typically identify the points of interest in the input (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', text boxes in an image or a moving object in a video), while sub- sequent phases would process those points (iteratively or as a batch) to solve the predefined problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', identify text in the boxes or classify the moving object in the video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The inference latency of such models might grow linearly with the number of objects identified in the first phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Further- more, if even one phase of the pipeline does not scale well, the scalability of the entire pipeline is impaired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='5 Padding Batching multiple inputs and processing them at once is a well-known way of improving inference throughput [1, 3, 9, 15, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In fact, multiple serving system for machine learning models (such as TensorFlow Serving [16] or TorchServe [7]) include tunable parameters that configure how long an in- ference server can wait in order to batch as many input requests as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' However, when inputs in a batch do not have exactly the same shape, they need to be padded to be processed efficiently, since underlying kernels typically an- ticipate batches of homogeneous inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The padding leads 2 to reduced computational efficiency, since it is treated by ker- nels as the rest of the input, even though the corresponding output produced by the model is dismissed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 3 Divide-and-Conquer Principle Applied to Inference In this section, we describe the application of the Divide- and-Conquer Principle [8] to the inference of ML models at the conceptual level and as a concrete realization by imple- menting it in the OnnxRuntime framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We note that applying this principle does not directly address the reasons for poor scalability detailed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In fact, the advantage of our approach is that one does not have to identify and/or fix any scalability bottlenecks in their models to rip the benefits of its underlying idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='1 Concept The basic idea is pretty straightforward — consider a compu- tation job 𝐽, which can be broken into𝑘 independent parts, 𝑗1, 𝑗2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', 𝑗𝑘, which can be executed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Assume we have an oracle assigning relative weight 𝑤𝑖 ∈ (0, 1] correspond- ing to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', the number of required floating point operations (FLOPs) or single-thread latency of the computation job part 𝑗𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Finally, assume we have 𝐶 computing cores available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We strive to allocate to each part the number of cores relative to its weight, namely, we assign 𝑐𝑖 = 𝑚𝑎𝑥{1, ⌊𝑤𝑖 ∗𝐶⌋} cores for the part 𝑗𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This effectively means allocating 𝑐𝑖 worker threads for 𝑗𝑖 since we later create one worker thread per core (as common in ML frameworks, including in OnnxRuntime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Note that �𝑘 𝑖=1 𝑐𝑖 might be larger than C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This is obvious when the number of job parts, 𝑘, is larger than C, but it is possible even when 𝑘 ≤ 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This does not create a problem other than implying that some job parts will be run after other job parts have finished (rather than running them all in parallel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' At the same time, due to the rounding-down (floor) function intended to reduce the above possibility of oversubscription, some unallocated cores might remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' To avoid this waste of available resources, we sort all the job parts by their remaining unallocated weight, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', by 𝑤𝑖 ∗𝐶 − ⌊𝑤𝑖 ∗ 𝐶⌋, and assign one core to each part in the descending order, up until all cores are allocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The C++-like pseudo- code for the entire algorithm is given in Listing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Naturally, the idea described above raises the question of how to assign relative weight to a job part 𝑗𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In all our cases considered in Section 4, the weight is simply set proportion- ally to the size of input tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Specifically, let 𝑠𝑖 be the size of the input tensor for job part 𝑗𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We set 𝑤𝑖 to 𝑠𝑖 �𝑘 𝑖=1 𝑠𝑖 , essen- tially assuming that the amount of computation (expressed as the number of required FLOPs) grows roughly linearly with the input tensors’ size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In general, however, assigning weight can be done with the help of a profiling phase and a lightweight classification mechanism, which associates job parts of the same (or similar) shape (as the one encountered 1 vector allocate(vector inputs, int numCores) { 2 vector threadAllocation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 3 vector> threadUnallocatedWeight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 4 int numInputs = inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='size();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 5 int allocatedCores = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 6 int index = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 7 int totalSize = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 8 for (auto j_i : inputs) totalSize += j_i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='size() 9 for (auto j_i : inputs) { 10 int numThreadsToUse = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 11 if (numInputs <= numCores) { 12 int size = j_i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='size();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 13 float w_i = ((float)size) / totalSize;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 14 int numThreadsToUse = floor(w_i * numCores);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 15 // this may happen due to flooring 16 if (numThreadsToUse < 1) numThreadsToUse = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 17 unallocatedWeight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='add( 18 make_tuple(index, w_i * numCores - numThreadsToUse));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 19 } 20 threadAllocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='add(numThreadsToUse);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 21 allocatedCores += numThreadsToUse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 22 index++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 23 } 24 if (allocatedCores < numCores) { 25 // sort the vector in decreasing order by 26 // comparing the second field in each tuple 27 sort(unallocatedWeight, bySecondField);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 28 int nextToAdjust = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 29 while (allocatedCores < numCores) { 30 // fetch the first field in the `nextToAdjust` tuple 31 index = 32 unallocatedWeight[nextToAdjust % numInputs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='get(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 33 threadAllocation[index]++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 34 allocatedCores++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 35 nextToAdjust++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 36 } 37 } 38 return threadAllocation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 39 } Listing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Thread allocation algorithm during the profiling phase) to the relative weight obtained during profiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='2 Implementation Details We extend the API of the InferenceSession class of On- nxRuntime with a new prun method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This method is modeled 3 1 class TextRecognizer(object): 2 def __init__(self, args): 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 4 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='predictor = ort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='InferenceSession(args.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='file_path) 5 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='postprocess_op = build_post_process(args) 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 7 def __call__(self, img_list): 8 img_num = len(img_list) 9 for beg_img_no in range(0, img_num, batch_num): 10 end_img_no = min(img_num, beg_img_no + batch_num) 11 inputs = prepare(img_list, beg_img_no, end_img_no) 12 outputs = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='run(inputs) 13 preds = outputs[0] 14 rec_result = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='postprocess_op(preds) 15 all_results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='add(rec_result) 16 return all_results Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Original (shortened and edited for clarity) TextRecognizer class implementation from PaddleOCR 1 class TextRecognizer(object): 2 def __init__(self, args): 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 4 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='predictor = ort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='InferenceSession(args.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='file_path) 5 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='postprocess_op = build_post_process(args) 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 7 def __call__(self, img_list): 8 img_num = len(img_list) 9 for beg_img_no in range(0, img_num, batch_num): 10 end_img_no = min(img_num, beg_img_no + batch_num) 11 inputs = prepare(img_list, beg_img_no, end_img_no) 12 all_inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='append(inputs) 13 all_outputs = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='prun(all_inputs) 14 for outputs in all_outputs: 15 preds = outputs[0] 16 rec_result = self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='postprocess_op(preds) 17 all_results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='add(rec_result) 18 return all_results Listing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Modified TextRecognizer class implementation (uses prun).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Added or modified lines are in red after the existing run method used as the main entry point when running inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The main difference is that prun accepts a list of inputs (instead of just one) and returns a list of outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Internally, the implementation of prun iterates over the list of inputs, calculates their size (after validating those are tensors) and corresponding relative weight, and applies the allocation algorithm described in Listing 1 to associate the number of worker threads with each input (job part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Follow- ing that, the implementation creates one worker thread for each input, and runs them in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Each worker thread, in turn, creates a thread pool of the size calculated by the alloca- tion algorithm (the thread pool includes the worker thread it- self), and invokes the run method of the InferenceSession object with that thread pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The entire patch of the On- nxRuntime codebase to implement the prun functionality and other minor internal changes (such as having the run method to accept a thread pool as an optional argument in- stead of always using the default pool) consisted of around 200 lines of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' On the user side, the code also has to change to make use of the new prun API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Those changes, however, are quite straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Instead of invoking run for every job, a user needs to create a list of job parts and call prun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In addition, the user needs to rearrange the post-processing code to iterate over the results of prun, and apply any post-processing to each returned output (object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' As an example of what the user code changes entail, we show the original Python code (edited for brevity and clarity) of the TextRecognizer class in PaddleOCR (Listing 2) alongside the modified version that makes use of the new prun API (Listing 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 4 Use Cases Before we detail the use cases where the Divide-and-Conquer Principle is beneficial and report on our performance find- ings, we give a brief summary of our evaluation setup and methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We run all our experiments on a 16-core AMD- based VM in Oracle Cloud (aka OCI VM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='Standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='Flex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' (We also ran some experiments on a newer E4 shape, but have not noticed substantial differences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' To reduce perfor- mance variability, especially as we create separate thread pools for the variants that use prun, we use thread binding (pinning), for all the evaluated variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Every experiment was repeated 5 times, and we report the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We note that the standard deviation of all reported results, except for one specific case discussed below, was extremely low (typically, less than 1% of the mean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' For our experiments, we use the latest release versions (as of the date of writing this paper) of the corresponding software, specifically OnnxRuntime v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='1 and PaddleOCR v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='1 Sequential Pipeline Our first example of where applying the Divide-and-Conquer Principle is extremely useful is PaddleOCR [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' PaddleOCR is a lightweight OCR system, which consists of three parts: Text Detection, Text Classification (called Detection Boxes Rectify in [14]) and Text Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Each of those parts corresponds to a separate ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 4 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' PaddleOCR 3-phase pipeline (edited version of Figure 2 from [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The OCR pipeline accepts an image file and passes it first through the text detection phase whose objective is to locate text areas in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The output of this phase is a list of potential text boxes’ coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Next, the list is iterated over, and each item in that list (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', a text box) is sent to the text classification model, which decides whether the box needs to be transformed into a horizontal rectangle box before the actual text recognition takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Based on the classifier’s decision, each box is altered respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Finally, the list is iterated over again, and each item is sent to the text recognition model for inference, which recognizes the text in the given box and produces the actual character sequence based on the supplied character dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This process is depicted in Figure 1, which is a redacted version of Figure 2 from [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In our experiments with PaddleOCR, we observe that the system does not scale well with the increase in the number of available cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We demonstrate that in Figure 2 depicting inference latency as a function of available cores (which directly translates into the number of worker threads used by the runtime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' For all experiments in this section, including the one in Figure 2, we use a subset of images from the OpenImages dataset [17], selected according to a criterium described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In Figure 2, we break the total latency into time spans cor- responding to the three phases of the OCR pipeline discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' As one can notice, the average inference latency goes down from 554 ms for 1 thread to 364 ms for 4 cores and then back up to 435 ms for 16 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Interestingly, the Text Clas- sification phase shows negative scalability, where it takes 27 ms to process an image, on average, with 1 thread, but it takes 38 ms to do the same with 16 threads — a slowdown of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This shows an example of a system where, beyond a certain point, adding more threads not only does not help, but actually harms performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Discussing concrete rea- sons for the lack of scalability of these specific models is not in the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' For a curious reader, however, we note that a built-in OnnxRuntime profiling tool shows inflated execution times for the output reordering operators (which are inserted by the framework, along with the input reordering operator, to convert the memory layouts of input arguments for various kernels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We apply the Divide-and-Conquer Principle to the last two phases of the OCR pipeline, namely the Text Classifica- tion and Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' To that end, instead of invoking the corresponding models for each text box produced by Text Detection, we send all the boxes to the runtime (by invoking the prun API) and effectively let the runtime decide how many cores / worker threads to allocate each box based on its relative size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The required changes to implement this functionality in the Text Recognition phase are depicted in Listing 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' the changes to the Text Classification phase are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' For our performance evaluation, we compare the prun implementation as discussed in Section 3 (and depicted in Listing 1), which we denote as prun-def on the charts, to a few simple variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The first variant, denoted as prun-1, simply allocates one worker thread to each input in the list given to prun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The second variant, denoted as prun-eq, allocates an equal number of cores for each input (but at least one), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', sets 𝑐𝑖 = 𝑚𝑎𝑥{1, ⌊𝑘/𝐶⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Our motivation is to show that trivial solutions might also be useful in certain scenarios (as discussed below), yet they tend to underperform compared to prun-def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We note that the benefit of prun in this use case is possible only when there are at least two text boxes identified in the Text Detection phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Otherwise, the other two phases would not be used (if no text boxes detected) or the prun-def vari- ant will use the same (maximum) number of cores as the base (unmodified) version (if only one text box is detected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' As a result, the subset of images used for performance evaluation in this section includes images with at least two identified text boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The pie chart in Figure 3 shows the distribution of the actual number of boxes detected in the first phase of the OCR pipeline for the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The total number of images in the dataset was 500 – this number was chosen to keep the evaluation times reasonably short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' (We note that we also ran evaluations on a larger dataset that includes images with less than two text boxes and confirmed that the use of prun does not create any overhead in those cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=') 5 营养护发器 ODM OEM ODMOEM ODMOEM ODMOEM ODMOEM Image Text Detection Detection Boxes Rectify Text Recognition (db_mv3_slim,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='4M) (dir_cls_mv3_slim, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='5M) (crnn_mv3_slim, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='6M) OutputFigure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Inference latency of PaddleOCR with a varying number of threads, broken down by the three phases of the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In light of the discussion above, we break down the com- parison of the latency results by the number of detected boxes, as depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The latency numbers in this figure were collected with 16 cores;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' we discuss the over- all scalability trends later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We also break down the per- formance in two of the phases where we have used prun, namely Text Classification (Figure 4 (a)) and Recognition (Figure 4 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Considering the results in Figure 4, one can notice that, as expected, the benefit of prun increases with the number of detected text boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' For instance, when considering the total end-to-end latency (Figure 4 (c)), with only two boxes prun-def outperforms base by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='28x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' However, with 9 and 10+ boxes, prun-def outperforms base by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='33x and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='81x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' It is interesting to compare the performance of prun-def with other pun-based variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' As one can notice in Fig- ure 4 (a), the prun-1 variant produces the lowest latency when the number of detected boxes is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In fact, the base variant also performs better than prun-def in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We attribute this to two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' First, this specific phase of the pipeline shows negative scalability, which can be also seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Therefore, best performance is achieved when fewer threads per box is used in this phase, which is what prun-1 effectively achieves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Second, prun-def (and prun-eq) cre- ate and destroy more threads than prun-1 in those cases as they create thread pools containing more threads for each prun invocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This adds small, but non-negligible over- head given that the the execution time of this phase is short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In the future work, we intend to experiment with reusing thread pools between prun invocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' As the number of detected boxes increases, however, all prun variants allocate less threads (or even just 1) per each box, and they allocate a similar number of threads for their pools, thus closing the gap with the prun-1 variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' When the Text Recognition phase is concerned (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Fig- ure 4 (b)), however, it is apparent from Figure 2 that one Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Distribution of the number of detected text boxes in the input dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' can improve its latency by using more than one thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We note that, quantitively, this phase is also far more dominant than the Text Classification one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Here, prun-def manages to achieve best or close to best result across all counts of detected boxes, which translates to overall highly competi- tive end-to-end inference performance (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Figure 4 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In general, the results in Figure 4 call for a dynamic mechanism, which would choose the best thread allocation strategy based on the given workload and available resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Devising and experimenting with such a strategy is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Finally, we shed more light on how the scalability im- proves with the use of prun in Figure 5, where we vary the number of cores (and therefore, the total number of worker threads) available for OnnxRuntime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Once again, we include the latency of each of the two last phases of PaddleOCR (denoted as Rec for Text Recognition and Cls for Text Clas- sification) along with the end-to-end (Total) latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We include only the results of the base and prun-def variants (denoted simply as prun in Figure 5), for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Overall, one can notice similar trends to the ones discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In the base version, the Text Recognition phase does scale up to 4 threads, but then its performance suffers as the number of threads increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The prun variant avoids this performance degradation, and in fact, continues to scale up to 16 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Indeed, when considering the Text Recognition phase only, the prun variant outperforms base by more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='4x at 16 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' However, since both variants have an identical Text Detection phase, which according to Figure 2 subsumes a substantial part of the total latency, the end-to- end speedup of prun is only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='5x at 16 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='2 Batching of Heterogeneous Inputs Our next example concerns with the Transformer architec- ture [28], which revolutionized the domain of NLP when it was introduced in 2017 and has been applied to other do- mains since then (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', [12, 18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This architecture consists of a stack of layers, each composed of a self-attention block followed by a fully connected network [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Past work has 6 700 Detection Recognition 600 Classification Other 500 Latency (ms) 400 300 200 100 0 1 2 8 12 4 16 # threads2 (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='4%) 3 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='6%) 4 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='2%) 5 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='6%) 6 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='2%) 7 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='2%) 8 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='0%) 9 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='2%) 10+ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='6%)(a) Text Classification (b) Text Recognition (c) End-to-End Inference Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The impact of using prun in PaddleOCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Total (end-to-end) inference latency of PaddleOCR with a varying number of threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Also shown the latency of Text Classification (Cls) and Text Recognition (Rec) phases shown that the majority of computation cycles in Transform- ers is spent on (scalable) matrix multiplication operations, yet up to one third of the cycles is spent elsewhere (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', less scalable operations) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' It is well-known that one way to improve the inference performance (specifically, throughput) of Transformers is through input batching [3, 15, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This strategy works well, however, when the inputs have the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Otherwise, one has either give up on batching, or pad inputs to the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The latter results in wasted computation cycles, since special padding tokens are treated exactly as input tokens by the architecture and dismissed at the end of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This situation presents an ideal case for applying the Divide-and-Conquer Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Instead of padding the in- puts of various lengths up to the longest input in the batch, we can run inference on those inputs (as they are, without padding) using the prun API, and let the runtime decide how many cores should be used to process each of the in- puts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We modify the Transformer benchmark built into the OnnxRuntime [25] to implement this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' To evaluate the effectiveness of the approach described above, we set up an experiment where we generate 𝑋 in- puts of a length chosen uniformly and randomly out of the range [16, 512].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We then compare the pad-batch version in which all 𝑋 inputs are padded to the longest length in the given batch with the prun version in which the inference is invoked with prun on all inputs in the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We show results with the highly popular BERT model [10] (specifi- cally, “bert-based-uncased”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We have also experimented with other Transformer-based models (such as “bert-large- uncased” or “roberta-base”) measuring similar qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We note that this experiment includes inherent amount of randomness — a batch of small sentences is as likely to be chosen as a batch of long sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In an attempt to reduce the anticpated high varaince of the results, we opted to repeat the experiment 1000 times, and so for each 𝑋, each data point is an average of 1000 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Figure 6 presents the throughput results with batches of various sizes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', 𝑋 varies from 2 to 8), with error bars depicting the standard deviation of the reported mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Even though prun outperforms the pad-batch variant across all batch sizes, the variance in the measured results remains exceptionally high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' As a result, we setup two additional experiments in a more controlled way likely to produce more stable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In the first, we simply preset the lengths of various sequences in each batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' For instance, a batch denoted as “16-64-256” includes three sentences, one is 16, another is 64 and yet another is 256 tokens long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We show the results of this exper- iment in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Here, the prun version easily outperforms the pad-batch variant, which has to pad all sequences to the longest sequence in a batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' As one might expect, the benefit from using prun increases with the number of sentences in a batch, as this variant eliminates all the redundant work associated with padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In the second experiment, we use a batch of 1 long sen- tence (256 tokens long) and 𝑋 short sequences of 16 tokens each, where we vary 𝑋 between 0 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We show the throughput results of this experiment in Figure 7, along with a curve depicting the number of threads allocated by prun for the long sequence in the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' There are several interesting observations that can be made here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' First, when 𝑋=0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', the batch contains only one long sentence, both variants employ all available cores to pro- cess that batch, producing similar result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This shows that the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='base ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='prun-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='prun-def ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='Latency (ms) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='prun-eg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='7 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='10+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='# detected boxes600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='base (Total) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='base (Rec) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='base (Cls) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='prun (Total) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='prun (Rec) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='prun (Cls) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='Latency (ms) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='# threadsFigure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Throughput of inferencing BERT on batches of sequences of sizes chosen randomly from the range [16, 512] Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Throughput of inferencing BERT on batches of sequences of various preset sizes overhead of using prun when the input has only one chunk is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Second, the throughput of the pad-batch version grows, but modestly with the increase in the number of short sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This is because, as stated above, a larger batch of (padded) sequences helps to achieve better throughput with Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' At the same time, the throughput growth with prun is much more dramatic up to 3 short sequences in a batch and then it declines, but stays well above that achieved with pad-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Both phenomena can be explained with the fact that inferencing a sequence of 256 tokens takes about the same time with 16 threads as it takes with 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Thus, adding a few short sequences into the batch, each allocated with just 1 thread (as they have small relative weight), has negli- gible impact on the latency, but improves throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' With more short sequences in a batch, less threads are allocated for the long sequence (as can be seen in Figure 7) and its inference latency grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This causes the overall throughput to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='3 Batching of Homogeneous Inputs Our last example follows directly from the discussion in Section 2 on the lack of scalability in ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' As al- ready mentioned, while Transformers models heavily use Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Throughput of inferencing BERT on a batch con- taining one long sentence of 256 tokens and 𝑋 short se- quence with 16 tokens each, where 𝑋 varies 0 to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In addition, we show how many threads are dedicated to the inference of the one long sentence in the batch in the prun variant scalable matrix multiplication operations, they also employ less scalable operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The impact of the latter grows with the increase in the number of cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Therefore, one may benefit form the Divide-and-Conquer Principle applied to Transformers even when the batch includes inputs of the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' As a concrete example, consider a batch of two inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Instead of using all available cores to process the batch, we will use half the cores for each input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Intuitively, the less scalable operators create less relative overhead when less cores are used and the input sequence is shorter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', contains half the tokens compared to the entire batch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Figure 9 demonstrates this effect with batches of inputs of equal lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In addition to the pad-batch variant (which we simply call batch here, as no padding is required) and prun, we include a no-batch variant, which runs inference on each sequence in a given batch one at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Note that we include the latter to simply demonstrate the benefits of batching in general, confirming previous findings [3, 15, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Each set of bars in Figure 9 corresponds to a batch of 4 sentences with the given length (from 64 tokens to 512).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Overall, the prun version yields a more modest (yet non- trivial) speedup over batch compared to the case of non- homogeneous inputs in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This is expected, since in this case the room for improvement (over batch) does not include wasted computation related to padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 5 Related Work As mentioned in the Introduction, the major focus of the ML community has been on improving the accuracy and training performance of proposed models, while efficient inferencing and serving of those models receives relatively less attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Yet, there have been some notable exceptions of work focused specifically on inference performance, and 8 60 16 pad-batch # 14 threads f Throughput (queries/s) 50 prun prun 12 40 10 for 30 8 long 6 20 sequence 4 10 2 e 0 0 3 5 0 2 9 12 15 # of short sequences in a batch35 pad-batch A 30 Throughput (queries/s) prun 25 20 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 10 5 0 2 4 6 8 # sequences in a batch50 pad-batch 45 prun 40 35 30 25 20 15 10 5 0 16-256 16-32-64-128-256 16-64-256 sequence lengths in a batchFigure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Throughput of inferencing BERT with batches of 4 sequences of equal size we survey the most relevant results hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' As an aside, we note that many of the results below come from less formal blog posts published by various companies, highlighting the great practical importance of efficient inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' [30] explore various factors that influence in- ference performance in TensorFlow, including the choice of a specific math library, a thread pool library, availability of SIMD (single instruction multiple data) support, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' They identify data preparation as one of the causes for poor scala- bility of small matrix multiplication operations, something we more generally attribute to framework overhead in Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' They come up with a set of guidelines one can use to tune TensorFlow settings to achieve better performance compared to the one achieved with settings recommended by TensorFlow authors or Intel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' With the tremendous rise in popularity of Transformers, several papers and blog posts focus on its inference perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Dice and Kogan investigate inference performance of Transformers on CPUs [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Their analysis shows that most inference computation cycles are spent in matrix multipli- cation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Hence, they propose an adaptive matrix multiplication optimization aimed at reducing the latency of those operations and subsequently improving the overall inference performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Intel engineers describe an effort to optimize inference of BERT in Apache MXNet using the GluonNLP toolkit, where one of the ideas is to quantize the model for better performance with lower precision [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Similar quantization ideas (along with distillation, another common method of reducing the size of a model [27]) were employed by Roblox to speedup their deployment of BERT on CPUs [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The same blog post also mentions that elimi- nating padding of input sentences has led to better perfor- mance (though the authors did that for batches of 1 input only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' A Microsoft team [26] describes their effort on accel- erating BERT with OnnxRuntime through operation fusion that helps to reduce the amount of overhead (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', memory copying) in invoking each kernel individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' A few recent papers and projects have looked into the de- ficiency of padding of heterogenous inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' [15] propose a sequence-length-aware batch scheduler, which aims to batch requests of a similar size, thus reducing the cost of zero padding of all requests into one batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' It re- quires a profiling phase during which the inference cost of various batches is collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' [13] propose to care- fully redesign the GPU kernels employed by Transformers to eliminate most redundant computation associated with zero padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' The Effective Transformer project by ByteDance [6] aims to dynamically remove and restore padding during dif- ferent calculation stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' All those efforts target specifically the inferencing Transformers on GPUs, and it is not clear how efficient they would be on CPUs and/or with other architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Beyond Transformers, Liu at et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' [20] describe NeoCPU, an approach for optimizing CNN inference on CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' NeoCPU proposes a configurable design of an efficient convolution operation that can be tuned efficiently to popular CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This design is coupled with a scheme for obtaining the best memory layout for data in different operations of a CNN model, in order to minimize the overhead of transforming the data between various individual operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 6 Discussion In this paper, we have discussed various reasons for the lack of scalability of inferencing ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' While the reasons vary from micro to macro-levels, the common motive is that existing ML frameworks are geared towards high perfor- mance training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This is expressed by the fact that kernels for common operations are typically optimized for large batches with long inputs, ignoring relatively small overheads in var- ious parts of those frameworks that are immaterial to the overall training performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' However, during inference the batches tend to be much smaller and contain shorter inputs, thus making those overheads more prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' A somewhat similar observation has been made by Aminabadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' We leverage this poor scalability and describe a simple, yet powerful approach, in which the given input is broken into chunks and each chunk is processed in parallel, instead of using all available resources for the entire input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' As we demonstrate with a few well-known models, this approach improves inference scalability and ultimately can lead to over 2x latency and throughput improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' This work offers several directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' First, we want to explore more dynamic thread allocation strategies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', ones that can better adjust to the cases where the weight of a work chunk does not correlate linearly with its size and/or where the underlying model performs best while running with a single thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Second, we want to find ways to automate splitting the input into chunks that can be processed in parallel, lowering the cost (in terms of user code changes) of using prun even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Finally, we want 9 90 三 no-batch 80 batch (queries/s) 70 prun 60 50 Throughput ( 40 30 20 10 0 64 128 256 512 sequence length in a batchto explore other use cases where the use of prun would be beneficial, including other ML models that feature a pipeline- based architecture (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' TurboTransform- ers: an efficient GPU serving system for transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' of ACM SIGPLAN PPoPP, pages 389–402, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' [16] Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Tensorflow Serving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' https://www.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Archit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Code Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=', 18(1), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' [31] Shufan Wu, Tao Lv, Pengxin Yuan, Patric Zhao, Jason Ye, and Haibin Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Optimization for BERT Inference Performance on CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' https://medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content='com/apache-mxnet/optimization-for-bert- inference-performance-on-cpu-3bb2413d376c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' Published: 09-12-19, Accessed: 08-02-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' [32] Zhe Zhou, Xuechao Wei, Jiejing Zhang, and Guangyu Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' PetS: A Unified Framework for Parameter-Efficient Transformers Serving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' In USENIX Annual Technical Conference (ATC), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} +page_content=' 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE4T4oBgHgl3EQfeg0W/content/2301.05099v1.pdf'} diff --git a/FdAzT4oBgHgl3EQfUfww/content/tmp_files/load_file.txt b/FdAzT4oBgHgl3EQfUfww/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aba645e167867a1e4b7c116489da9cd4f34bec7f --- /dev/null +++ b/FdAzT4oBgHgl3EQfUfww/content/tmp_files/load_file.txt @@ -0,0 +1,2833 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf,len=2832 +page_content='HIGGS-COULOMB CORRESPONDENCE AND WALL-CROSSING IN ABELIAN GLSMS KONSTANTIN ALESHKIN AND CHIU-CHU MELISSA LIU Dedicated to the memory of Professor Bumsig Kim Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We compute I-functions and central charges for abelian GLSMs using virtual matrix factorizations of Favero and Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In the Calabi-Yau case we provide analytic continuation for the central charges by explicit integral formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The integrals in question are called hemisphere partition functions and we call the integral representation Higgs-Coulomb correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We then use it to prove GIT stability wall-crossing for central charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Geometry of gauged linear sigma models and A-model state spaces 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Categories of B-branes and K-theories 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The Higgs Branch 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The Coulomb Branch 32 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Convergence of multivariate hypergeometric functions 41 References 46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Introduction 2d gauged linear sigma models (GLSMs) were introduced by Witten in 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Following [FJR], the input data of a GLSM is a 5-tuple (V, G, C∗ R, W, ζ), where V is a finite dimensional complex vector space, G ⊂ GL(V ) is a reductive linear group known as the gauged group, C∗ R ∼= C∗ acts linearly on V and the action commutes with the G-action, W : V → C is a G-invariant polynomial which is quasi-homogeneous with respect to the C∗ R- action, and ζ is a G-character with the property V ss G (ζ) = V s G(ζ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=', every ζ-semistable point is ζ-stable, so that the GIT quotient stack Xζ = [V//ζG] = [V ss G (ζ)/G] is an orbifold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' smooth DM stack with trivial generic stabilizer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The space of stability conditions is ˆG ⊗Z R ∼= RdimC Z(G), where ˆG = Hom(G, C∗) is the group of G characters and Z(G) is the center of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' it is decomposed into chambers called phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The G-invariant polynomial W descends to wζ : Xζ → C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' the pair (Xζ, wζ) is a Landau-Ginzburg (LG) model, where wζ is known as the superpotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' GLSM invariants of (V, G, C∗ R, W, ζ) are, roughly speaking, virtual counts of curves in the critical locus Zζ := Crit(wζ) = [(Crit(W) ∩ V ss G (ζ)) /G] which is often assumed to be compact/proper but can be singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Different phases of a GLSM (that is GLSMs which differ only by the choice of a stability parameter) are closely related to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For example, let G = C∗ act on V = C6 by weights (1, 1, 1, 1, 1, −5) with W = pW5 where W5 = x5 1 + · · · + x5 5 is the Fermat quintic polynomial in 5 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In the CY/geometric phase ζ > 0, Xζ = KP4, Zζ = X5 := {W5 = 0} ⊂ P4 is the Fermat quintic threefold, and GLSM invariants are (up to sign) Gromov-Witten (GW) invariants of X5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In the LG phase ζ < 0, Xζ ∼= [C5/µ5], where µ5 is the group of 5-th roots of unity acts diagonally on C5, Zζ is supported at the origin, and GLSM invariants are Fan-Jarvis-Ruan-Witten (FJRW) invariants of the affine LG model ([C5/µ5], W5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Chiodo-Ruan [CR] proved genus-zero LG/CY correspondence for quintic threefolds relating GW invariants of X5 and FJRW invariants of ([C5/µ5], W5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Their proof can be summarized into two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (1) (ϵ-wall-crossing) In the CY (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' LG) phase, the Givental-style mirror theorem says the J-function which governs the genus-zero GW (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' FJRW) is related to the I-function, which can be expressed in terms of explicit hypergeometric series, by explicit change of variables known as the mirror map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (2) (ζ-wall-crossing) The I-function admits a Mellin-Barnes integral representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' I-functions in the two phases are related by analytic continuation given by deforming the contour of integration in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The interpretation of Step (1) as ϵ-wall-crossing appeared in later work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For each ϵ ∈ Q>0, Ciocan-Fontanine–Kim– Maulik [CKM] introduced ϵ-stable quasimaps to certain GIT quotient W//G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Ciocan-Fontanine and Kim [CK] 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='01266v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='AG] 3 Jan 2023 introduced Jϵ which is a generating function of invariants defined by genus-zero ϵ-stable quasimaps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Jϵ specializes to the I-function and the J-function as ϵ → 0+ and ϵ → +∞, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In the presence of a good torus action, they proved ϵ-wall-crossing which relates Jϵ to the J-function J = J∞ for any ϵ ∈ Q>0, by change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' They computed the I-function I = J0+ explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In particular, they recover the mirror theorem in the geometric phase first proved by Givental [Gi96] and Lian-Liu-Yau [LLY].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The mirror theorem in the LG phase was first proved by Chiodo-Ruan [CR] and later reproved by Ross-Ruan [RR] via ϵ-wall-crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For a general GLSM, GLSM invariants are defined by integrating against virtual cycles on moduli of ϵ-stable LG quasimaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The virtual cycle is constructed for narrow sectors by Fan-Jarvis-Ruan [FJR] via cosection localization, and for both narrow and broad sectors by Favero-Kim [FK] via matrix factorization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Favero-Kim’s construction generalizes previous constructions for affine LG models [PV16] and for convex hybrid models [CFGKS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' When ϵ > 0, the definition relies on a good lift ˜ζ which is a character of the group Γ ⊂ GL(V ) generated by G and C∗ R, such that V ss Γ (˜ζ) = V ss G (ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Such a good lift does not always exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' At ϵ = 0+, a good lift is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In this paper we focus on the ϵ → 0+ stability condition and study genus-zero GLSM invariants and ζ-wall-crossing for abelian GLSMs where G = (C∗)κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In this case, Xζ is a smooth toric DM stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let �T-be the diagonal subgroup of GL(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Using the work of Favero-Kim [FK], we define and compute K-theoretic GLSM I-function IK w which takes values in the K-theory of category of matrix factorizations on the inertial stack IXζ, and the (cohomological) GLSM I-function Iw which takes values in the GLSM state space Hw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We also define and compute K-theorectic �T-equivariant I-function IK � T of (V, G, C∗ R, 0, ζ) which takes values in K � T (IXζ), and �T-equivariant I-function I � T of (V, G, C∗ R, 0, ζ) which takes values in H � T (IXζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In Gromov-Witten theory I-functions fail to capture integral structure on cohomology [Ir09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Hosono defined an object called a central charge that sees the Gamma integral structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Integral structures are crucial for integral representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Central charges are power series which are constructed from both J-function and objects of the derived category of coherent sheaves of the target manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Hori and Romo [HR] constructed explicit analytic functions called hemisphere partition functions and conjectured that their power series expansions are equal to the central charges in appropriate cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We define the GLSM central charge of a matrix factorization B of (Xζ, wζ) as Zw(B) = ⟨Iw, Γwchw([B])⟩ where Γw is an appropriate version of Iritani’s Γ-class and [B] is the K- theory class of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We also define the �T-equivariant central charge of a �T-equivariant perfect complex B on Xζ as Z � T (B) = ⟨I � T , Γ � T ch � T ([B])⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We show that our central charges indeed have integral representations of the hemisphere partition function form (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We call this representation Higgs-Coulomb correspondence because in physics GLSM central charges can be obtained by a version of the Higgs branch localization and hemisphere partition functions are computed by the Coulomb branch localization (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' [BC]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The integral representations we obtain depend continuously on the complexified stability parameter θ = ζ + 2π√−1B and do not have any restrictions on ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In this philosophy ζ-wall-crossing follows immediately by analytic continuation in ζ (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Remarkably, matrix factorizations of the central charges in question are related by the Fourier-Mukai transform [BP10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let ζ± represent stability conditions in two adjacent chambers and B+ be a matrix factorizations in the phase corresponding to ζ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then, the analytic continuation of the central charge of B+ is a central charge of B− = FM(B+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Particular Fourier-Mukai kernel is choosen by B = Im(θ)/2π and convergence of the integral representation is equivalent to the so-called Grade Restriction Rule [HL,BFK,CIJS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1) B B+ B− π+ π− F M Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We wish to thank Daniel Halpern-Leistner, Kentaro Hori, Hiroshi Iritani, Andrei Okounkov, Tudor P˘adurariu, Renata Picciotto, Alexander Polishchuk, Che Shen, Yefeng Shen, Mark Shoemaker, and Yang Zhou for helpful communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We thank the hospitality and support of the Simons Center for Geometry and Physics (SCGP) during the program Integrability, Enumerative Geometry and Quantization (August 22-September 23, 2022) where part of the paper was completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The authors are partially supported by NSF grant DMS-1564497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Geometry of gauged linear sigma models and A-model state spaces In this paper, all the schemes and algebraic stacks are defined over Spec C, where C is the field of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Gauged linear sigma models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We start with the setup of a general gauged linear sigma model (GLSM) following [FJR,FK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Part of our formulation in Section 4 is closer to that in the more general setting in [CJR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The input data of a GLSM is a 5-tuple (V, G, C∗ R, W, ζ), where (1) (linear space) V = SpecC[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , xn+κ] ≃ Cn+κ is a finite dimensional complex vector space, where κ = dim G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (2) (gauge group) G ⊂ GL(V ) is a reductive algebraic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (3) (vector R-symmetry) C∗ R ∼= C∗ acts linearly and faithfully on V , so we may view C∗ R as a subgroup of GL(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We assume that (a) the intersection G ∩ C∗ R is finite, and (b) the C∗ R-action commutes with the G-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The finite group G ∩ C∗ R must be cyclic, generated by an element J of finite order r ∈ Z>0, given explicitly in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='2) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The surjective group homomorphism C∗ R → C∗ ω := C∗/⟨J⟩ is a degree r covering map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Γ := GC∗ R ⊂ GL(V ) be the subgroup generated by G and C∗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then we have a short exact sequence of groups: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1) 1 → G → Γ χ→ C∗ ω → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' By (b), the Γ-action on V induces a C∗ ω-action on the smooth Artin stack [V/G] in the sense of [Ro].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The R-charges are (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , qn+κ) = �2c1 r , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , 2cn+κ r � , where c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , cn+κ ∈ Z are the weights of the C∗ R-action on V ≃ Cn+κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (Note that gcd(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , cn+κ) = 1 since C∗ R acts faithfully on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=') Let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='2) J = (e2π√−1c1/r, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , e2π√−1cn+κ/r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (4) (superpotential) W : V → C is a G-invariant regular function which is a quasi-homogeneous polynomial of degree r with respect to the C∗ R-action on V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' in other words, W ∈ C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , xn+κ]G and W(tc1x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , tcn+κxn+κ) = trW(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , xn+κ), t ∈ C∗ R, (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , xn+κ) ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' It descends to a function w : [V/G] → C of degree 1 with respect to the C∗ ω-action on [V/G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (5) (stability condition) ζ ∈ Hom(G, C∗) = Hom(Gab, C∗), where Gab = G/[G, G] is the abelianization of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We view Hom(G, C∗) as an additive group and let χζ : G → C∗ denote the associated G-character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let V ss G (ζ) (respectively V s G(ζ)) be the set of semistable (respectively stable) points in V determined by the G-linearization on the trivial line bundle V × C → V given by g · (v, t) = (g · p, χζ(g)t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We assume that V s G(ζ) = V ss G (ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then the quotient stack Xζ := [V ss G (ζ)/G] is an orbifold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' a smooth Deligne-Mumford stack with trivial generic stabilizer) of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The GIT quotient Xζ := V �ζ G = V ss G (ζ)/G is the coarse moduli space of Xζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Zζ := [(Crit(W) ∩ V ss G (ζ))/G] be the critical locus of wζ := w|Xζ : Xζ → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We say ζ is in a geometric phase if Crit(W) ∩ V ss(ζ) is non-singular, which implies Zζ is an orbifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The central charge of the GLSM is (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' [FJR, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='3]) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='3) ˆc := dim V − dim G − 2ˆq = n − 2ˆq, where ˆq = 1 2 n+κ � j=1 qj = 1 r n+κ � j=1 cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' By the condition (4) above, the superpotential W is semi-invariant with respect to Γ in the sense of [PV11, Section 2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='4) W(γ · x) = χ(γ)W(x) for any γ ∈ Γ and x ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In this paper the stability condition ζ is an element in Hom(G, C∗) and corresponds to the symbol θ in [FJR,FK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The 1-dimensional torus C∗ ω in this paper corresponds to C∗ ω in [CJR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' At this point, we do not assume the critical locus Zζ is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' This allows us to include the case without superpotential as a special case where the superpotential and the R-charges are zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' W = 0 and qj = 0 for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , n + κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' note that in this special case the C∗ R-action on V is trivial, and in particular, not faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Abelian GLSMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In the rest of this paper, we consider abelian GLSMs where the gauge group G = (C∗)κ is a complex algebraic torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In this case [G, G] = {1} and Gab = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Up to an inner automorphism of GL(V ), we may assume the image of ρV : G → GL(V ) is contained in the diagonal torus �T ≃ (C∗)n+κ ⊂ GL(V ) ∼= GLn+κ(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then Xζ is an n-dimensional toric orbifold, and its coarse moduli Xζ is a semi-projective simplicial toric variety which contains T := �T/G ∼= (C∗)n as a Zariski dense open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We have a short exact sequence of abelian groups (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='5) 1 → G ρV −→ �T −→ T → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The notation in this subsection is similar to but slightly different from that in [CIJ]: G, �T, and T in this paper correspond to K, T, and Q in [CIJ, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='3], respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' κ and n in this paper correspond to r and m − r in [CIJ, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Applying Hom(C∗, −) to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='38), we obtain the following short exact sequence of cocharater lattices (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='6) 0 → L := Hom(C∗, G) −→ � N := Hom(C∗, �T) −→ N := Hom(C∗, T) → 0, where L ∼= Zκ, � N ∼= Zn+κ, and N ∼= Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Applying Hom(−, C∗) to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='38), or equivalently dualizing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='6), we obtain the following short exact sequence of character lattices: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='7) 0 → M := Hom(T, C∗) −→ � M := Hom( �T, C∗) −→ L∨ := Hom(G, C∗) → 0 The map L → � N ∼= Zn+κ is given by (D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , Dn+κ) where Di ∈ L∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The stability condition ζ is an element in Hom(G, C∗) = L∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If we choose a Z-basis {ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , ξκ} of L (which is equivalent to a choice of an isomorphism G ≃ (C∗)κ) and let {ξ∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , ξ∗ κ} be the dual Z-basis of L∨, then Di = κ � a=1 Qa i ξ∗ a for some Qa i ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given any t = �κ a=1 taξ∗ a ∈ L∨, where t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , tκ ∈ Z, let χt : G → C∗ be the corresponding G character given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='8) χt(s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , sκ) = κ � a=1 sta a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then the map G ≃ (C∗)κ → �T ∼= (C∗)n+κ is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='9) s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , sκ) ∈ G �→ � χD1(s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , χDn+κ(s) � ∈ �T, where χDi(s) = κ � a=1 sQa i a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given a lattice Λ ∼= Zr and a field F, we define ΛF := Λ ⊗Z F ∼= Fr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' in this paper, F = Q, R, or C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The map G → �T is injective iff D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , Dn+κ generate the lattice L∨ over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In Section 5 (The Coulomb Branch), we work with the weaker assumption that D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , Dn+κ span the vector space L∨ Q over Q, or equivalently, the kernel K of the group homomorphism G → �T is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then Xζ is a smooth toric DM stack with a generic stabilizer K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' it is a toric orbifold iff K is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' It is also possible to work in this generality in Section 4 (The Higgs branch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We assume K is trivial in Section 4 mainly because [FJR] and [FK] assume so, but K can be non-trivial in orbifold quasimap theory [CCK] which can be viewed as a mathematical theory of GLSM without superpotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let GR ≃ U(1)κ be the maximal compact subgroup of G ≃ (C∗)κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then LR is canonically isomorphic to the Lie algebra gR of GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The G-action on V = SpecC[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , xn+κ] restricts to a Hamiltonian GR-action on the K¨ahler manifold (V, √−1 2 �n+κ i=1 dxi ∧ d¯xi) with a moment map µ : V −→ g∨ R = L∨ R, (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , xn+κ) = 1 2 κ � a=1 Qa i |xi|2ξ∗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then Xζ = [V ss G (ζ)/G] = [µ−1(ζ)/GR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' From this perspective, the stability condition ζ is a regular value of the moment map µ, and can be an element in L∨ R ∼= Rκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Anticones and the extended stacky fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The triple (V, G, ζ), which is part of the input data (V, G, C∗ R, W, ζ) of the given abelian GLSM, determines a set Aζ of anticones and an extended stacky fan Σζ = (N, Σζ, β, Sζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We describe them in this subsection, and describe V ss G (ζ) ⊂ V in terms of anticones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We fix an isomorphism V = SpecC[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , xn+κ] ∼= Cn+κ, which determines an ordered Z-basis (�e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , �en+κ) of � N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In particular, � N = n+κ � i=1 Z�ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let vi ∈ N be the image of ei under � N → N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Define β = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , vn+κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given a subset I of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , n + κ}, let I′ = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , n + κ} \\ I be its complement, and define ∠I = { � i∈I aiDi : ai ∈ R, ai > 0} ⊂ L∨ R, σI = { � i∈I aivi : ai ∈ R, ai ≥ 0} ⊂ NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If I = ∅ is the empty set, define σ∅ = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For a fixed G-action on V , a stability condition ζ ∈ L∨ R determines the following three sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Aζ = {I ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , n + κ} : ζ ∈ ∠I}, Σζ = {σI : I′ ∈ Aζ}, Sζ = {i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , n + κ} : σ{i} /∈ Σζ} = {i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , n + κ} : {i}′ /∈ Aζ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Note that Sζ ⊂ I for any I ∈ Aζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We choose the stability condition ζ ∈ L∨ R such that the following three equivalent conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (i) For any I ∈ Aζ, {Di : i ∈ I} spans L∨ Q as a vector space over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (ii) For any I ∈ Aζ, {vi : i ∈ I′} is a set of linearly independent vectors in NQ, or equivalently, σI′ is a simplicial cone in NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (iii) Σζ is a simplicial fan in NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Elements in Σζ are called cones, while elements in Aζ are called anticones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' By (i), the cardinality |I| of any anticone I ∈ Aζ is greater or equal to κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Σζ(d) be the set of d-dimensional cones in Σζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then σ ∈ Σζ(d) iff σ = σI where |I| = d and I′ ∈ Aζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Amin ζ = {I ∈ Aζ : |I| = κ} = {I ∈ Aζ : σI′ ∈ Σζ(n)} be the set of minimal anti-cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The irrelevant ideal of ζ is the ideal Bζ in C[x] := C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , xn+κ] generated by {xI := � i∈I xi : I ∈ Aζ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Zζ = Z(Bζ) be the closed subvariety of V = SpecC[x] defined by the irrelevant ideal Bζ ⊂ C[x], and let Uζ = V \\Zζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If ζ ∈ L∨ is a G character, then Uζ = V ss G (ζ), and Zζ = V un G (ζ) is the set of unstable points defined by ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For any I ∈ Aζ, define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='10) VI = V \\ Z(xI) = {(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , xn+κ) ∈ V : xi ̸= 0 if i ∈ I} = (C∗)I × CI′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then Uζ = � I∈Aζ VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Note that if I, J ∈ Aζ and I ⊂ J then VJ ⊂ VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Therefore, Uζ = � I∈Amin ζ VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We now give an alternative description of Zζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If I ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , n + κ}, and I /∈ Aζ, or equivalently σI′ /∈ Σζ, define ZI = CI × {0}I′ = {(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , xn+κ) ∈ V : xi = 0 if i ∈ I′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then Zζ = � I /∈Aζ ZI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Define Cζ := � I∈Aζ ∠I = � I∈Amin ζ ∠I ⊂ L∨ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The open cone Cζ is called the extended ample cone in [CIJ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' It is a chamber in the space of stability conditions: if ζ′ ∈ Cζ then Σζ′ = Σζ and Uζ′ = Uζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We recall the following facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 5 (1) The quotient stack Xζ = [Uζ/G] is the smooth toric Deligne-Mumford (DM) stack defined by the stacky fan (N, Σζ, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' See Borisov-Chen- Smith [BCS] for definition of toric Deligne-Mumford stacks in terms of stacky fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (2) The coarse moduli space of Xζ is the categorical (and geometric) quotient Xζ = Uζ/G which is the toric variety defined by the simplicial fan Σζ ⊂ NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' See [Fu93, CLS] for an introduction of toric varieties, and in particular the definition of general normal toric varieties in terms of fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (3) If ζ ∈ L∨ is a G-character then Xζ = [V ss G (ζ)/G] is the GIT quotient stack, and Xζ = V ss G (ζ)/G = V �ζ G is the GIT quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (4) The triple (V, G, ζ) determines a particular presentation of Xζ as a quotient stack [Uζ/G] and an extended stacky fan Σζ = (N, Σζ, β, Sζ), a notion introduced by Jiang [Ji08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Closed toric substacks and their generic stabilizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The �T-divisor �Dj = {xj = 0} ⊂ V = SpecC[x] restricts to a �T-divisor �Dj∩Uζ ⊂ Uζ which descends to a T-divisor Dj = [( �Dj∩Uζ)/G] in the toric stack Xζ = [Uζ/G] and a T-divisor Dj in the toric variety Xζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Note that Dj and Dj are empty if j ∈ Sζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given any σ ∈ Σζ(d), where 0 ≤ d ≤ n, we have σ = σI′ for some I ∈ Aζ with |I| = κ + n − d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let V(σ) = � i∈I′ Di ⊂ Xζ, V (σ) = � i∈I′ Di ⊂ Xζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then V(σ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' V (σ)) is an (n − d)-dimensional closed toric substack (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' subvariety) of Xζ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Xζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The generic stabilizer of the toric stack V(σ) is the finite group Gσ = � i∈I Ker(χDi) ⊂ G where χDi : G → C∗ is defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If τ, σ ∈ Σζ and τ ⊂ σ then V(τ) ⊃ V(σ), so Gτ ⊂ Gσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In particular, V({0}) = Xζ and G{0} is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If I ∈ Amin ζ , then σI′ ∈ Σζ(n) and pI := V(σI′) ≃ [•/GσI′ ] = BGσI′ is the unique T-fixed point in XI := [VI/G] ≃ �� (C∗)I × CI′� /G � ≃ [CI′/GσI′ ] ≃ [Cn/GσI′ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Here • = SpecC is a point, BGσI′ is the classifying space of GσI′ , and VI is defined by Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Line bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let U T j := OXζ(−Dj) ∈ PicT (Xζ), uT j := −(c1)T (U T j ) ∈ H2 T (Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then uT j is the T-equivariant Poincar´e dual of Dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Note that uT j = 0 if j ∈ Sζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The T-equivariant Chern character of U T j is chT (U T j ) = e−uT j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The T-equivariant line bundles U T j generate KT (Xζ) as an algebra over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Any group homomorphism A → T induces a map [Xζ/A] → [Xζ/T] = [V ss G (ζ)/ �T] and ring homomorphisms KT (Xζ) → KA(Xζ), φ∗ : H∗ T (Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) → H∗ A(Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Uj ∈ Pic(Xζ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' uj ∈ H2(Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q)) be the image of U T j ∈ PicT (Xζ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' uT j ∈ H2 T (Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q)) under the surjective group homomorphism PicT (Xζ) → Pic(Xζ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' H2 T (Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) → H2(Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q)) induced by the group homomorphism {1} → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then c1(Uj) = −uj and ch(Uj) = e−uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let U � T j ∈ Pic � T (Xζ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' u � T j ∈ H2 � T (Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q)) be the image of U T j ∈ PicT (Xζ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' uT j ∈ H2 T (Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q)) under the group homomorphism PicT (Xζ) → Pic � T (Xζ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' H2 T (Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) → H2 � T (Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q)) induced by the group homomorphism �T → T = �T/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then (c1) � T (U � T j ) = −u � T j and ch � T (U � T j ) = e−u � T j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , n + κ, let χDi : G → C∗ be defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , κ, let χξ∗ a : G → C∗ be the character associated to ξ∗ a ∈ L∨, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=', χξ∗ a(s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , sκ) = sa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let G act on Uζ × C by s · (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , xn+κ, y) = (χD1(s)x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , χDn+κxκ, χξ∗ a(s−1)y), 6 This defines a G-equivariant line bundle on Uζ, or equivalently a line bundle Pa on Xζ = [Uζ/G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let pa = −c1(Pa) ∈ H2(Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then ch(Pa) = e−pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , n + κ, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='11) Uj = κ � a=1 P Qa j a ∈ Pic(Xζ), uj = κ � a=1 Qa j pa ∈ H2(Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Λj ∈ Pic � T (•) = Pic(B �T) be the �T-equivariant line bundle over a point • defined by the �T-character �tj, and let λj = −(c1) � T (Λj) ∈ H2 � T (•;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Z) = H2(B �T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then K � T (•) = Z[Λ±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , Λ±1 n+κ], H∗(B �T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Z) = Z[λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , λn+κ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For later convenience, we introduce the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let I ∈ Amin ζ be a minimal anticone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then {Di : i ∈ I} is a Q-basis of L∨ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let {D∗,I i : i ∈ I} be the dual Q-basis of LQ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=', for any i, j ∈ I, ⟨Di, D∗,I j ⟩ = δij where ⟨ , ⟩ : L∨ Q × LQ → Q is the natural pairing between dual vector spaces over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given any I ∈ Amin ζ , the inclusion ιI : pI �→ Xζ of the torus fixed point pI induces a ring homomorphism ι∗ I : H∗ � T (Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) −→ H∗ � T (pI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) ≃ H∗(B �T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) = Q[λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , λn+κ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , n + κ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='12) ι∗ Iu � T i = � j∈I ⟨Di, D∗,I j ⟩λj − λi ∀I ∈ Amin ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In particular, if i ∈ I then ι∗ Iu � T i = 0, which is consistent with the fact that the T-fixed point pI is not contained in the divisor Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let �T × G act on Uζ × C by (�t, s) · (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , xn+κ, y) = (�t1χD1(s)x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , �tn+κχDn+κxκ, χξ∗ a(s−1)y), where �t = (�t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , �tn+κ) ∈ �T and s ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' This defines a �T × G-equivariant line bundle on Uζ, or equivalently a �T-equivariant line bundle P � T a on Xζ = [Uζ/G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let p � T a = −c1(P � T a ) ∈ H2 � T (Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then ch � T (P � T a ) = e−p � T a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For a = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , κ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='13) ι∗ Ip � T a = � j∈I ⟨ξ∗ a, D∗,I j ⟩λj ∀I ∈ Amin ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='14) U � T j = κ � a=1 (P � T a )Qa j · Λ−1 j ∈ Pic � T (Xζ), u � T j = κ � a=1 Qa j p � T a − λj ∈ H2 � T (Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Under K � T (Xζ) −→ K(Xζ), U � T j , P � T a , and Λj are mapped to Uj, Pa, and 1, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' under H2 � T (Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) → H2(Xζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q), u � T j , p � T a , and λj are mapped to uj, pa, and 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Our definitions of u � T j , U � T j , p � T a , P � T a , λj, Λj are consistent with the convention in Givental’s papers [GiV,GiVI] on permutation-equivariant quantum K-theory of toric manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The inertia stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The inertia stack of a general algebraic stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given a general algebraic stack X, the inertia stack IX of X is the fiber product IX � � X ∆ � X ∆ � ∆ � X × X where ∆ : X → X ×X is the diagonal morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' IX is an algebraic stack, and in particular a groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' An object in the groupoid IX is a pair (x, g) where x is an object in the groupoid X and k is an element in the automorphism group AutX (x) = HomX (x, x) of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Morphisms between two objects in IX are HomIX ((x1, g1), (x2, g2)) = {h ∈ HomX (x1, x2) : h ◦ g1 = g2 ◦ h}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 7 The map (x, g) �→ (x, g−1), where (x, g) is an object in IX, defines an involution inv : IX → IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The inertia stack of a quotient stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If X = [U/G] is a quotient stack, where U is a scheme and G is an algebraic group, then the inertia stack is also a quotient stack: IX = [IU/G] where IU := {(x, g) ∈ U × G | g · x = x} is a closed subscheme of U × G, and the G-action on IU is given by h · (x, g) = (h · x, hgh−1), where h ∈ G and (x, g) ∈ IU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In particular, if G is abelian then the action is given by h · (x, g) = (h · x, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The involution inv : IX → IX is induced by the G-equivariant involution IU → IU given by (x, g) �→ (x, g−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The inertial stack of the toric orbifold Xζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Xζ = [Uζ/G] be as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' To describe its inertia stack IXζ more explicitly, we introduce some definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given σ = σI ∈ Σζ, where I ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , n + κ}, define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='15) Box(σ) := � v ∈ N : v = � i∈I civi, 0 ≤ ci < 1 � and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='16) Box′(σ) := � v ∈ N : v = � i∈I civi, 0 < ci < 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='17) Box(Σζ) := � σ∈Σζ Box(σ) = � σ∈Σζ(n) Box(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' which is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For any v ∈ Box(Σζ) there exists a unique σ ∈ Σζ such that v ∈ Box′(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Therefore, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='18) Box(Σζ) = � σ∈Σζ Box′(σ), where the right hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='18) is a disjoint union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given any σ = σI ∈ Σζ, where I′ ∈ Aζ, define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='19) age(v) = � i∈I ci ∈ Q if v = � i∈I civi ∈ Box′(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Suppose that σ = σI ∈ Σζ where I′ ∈ Aζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' There is a bijection Box(σ) −→ Gσ given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='20) v = � i∈I civi �→ g(v) = (a1, · · · , an+κ) ∈ Gσ ⊂ G ⊂ �T ≃ (C∗)n+κ where ai = � e2π√−1ci, i ∈ I, 1, i ∈ I′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The map v �→ g(v) defines a bijection Box(Σζ) = � σ∈Σζ Box(σ) −→ � σ∈Σζ Gσ = {g ∈ G : (Uζ)g is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' }, where (Uζ)g = {x ∈ Uζ : g · x = x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='21) IUζ = {(x, g) ∈ Uζ × G : g · x = x} = � v∈Box(Σζ) (Uζ)g(v) × {g(v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The above union is a disjoint union of connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' IXζ = [IUζ/G] = � v∈Box(Σζ) Xζ,v where Xζ,v ≃ [(Uζ)g(v)/G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In particular, g(0) = 1 is the identity element of G and Xζ,0 ≃ Xζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' There is an involution inv : Box(Σζ) → Box(Σζ) characterized by g(inv(v)) = g(v)−1 ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The involution inv : IXζ −→ IXζ maps Xζ,v isomorphically to Xζ,inv(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Observe that age(v) + age(inv(v)) + dim Xζ,v = n = dim Xζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='22) η = (eπ√−1q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , eπ√−1qn+κ) ∈ C∗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 8 Then η2 = J = (e2π√−1q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , e2π√−1qn+κ) and W(η · x) = −W(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let invR : IXζ → IXζ be the map induced by the map IUζ → IUζ given by (x, g) �→ (η · x, g−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then invR maps Xζ,v isomorphically to Xζ,inv(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Note that in general invR ◦ invR is not the identity map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=', invR is not an involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' A-model GLSM state spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let M be a large positive number such that the real part of any critical values of W is less than M, and define w∞ ζ := (ReW)−1(M, ∞) ⊂ Xζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' As a graded vector space over Q, the rational GLSM state space of (V, G, C∗ R, W, ζ) is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='23) Hw,Q = � v∈Box(Σζ) H∗(Xζ,v, w∞ ζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q)[2 (age(v) − ˆq)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' where wζ,v = wζ �� Xζ,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In particular, Hw,Q ∼= H∗(IXζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) as a vector space over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Note that invR maps (Xζ,v, wζ,v) diffeomorphically to (Xζ,inv(v), −wζ,inv(v)) and induces an isomorphism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='24) inv∗ R : H∗(Xζ,inv(v), −w∞ ζ,inv(v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) ∼ = −→ H∗(Xζ,v, w∞ ζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' By [FK, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='3], Crit(wζ,v) ⊂ Zζ = Crit(wζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If Zζ is proper then there is a nondegenerate pairing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='25) H∗(Xζ,v, w∞ ζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) × H∗(Xζ,v, −w∞ ζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) → Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' for all v ∈ Box(Σζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='24) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='25), we obtain a nondegenerate pairing H∗(Xζ,v, w∞ ζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) × H∗(Xζ,inv(v), w∞ ζ,inv(v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) → Q for all v ∈ Box(Σζ), thus a non-degenerate pairing ( , )w : Hw,Q × Hw,Q → Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' As a graded vector space over C, the GLSM state space of (V, G, C∗ R, W, ζ) is Hw = � v∈Box(Σζ) H∗ � Xζ,v, (Ω• Xζ,v, dwζ,v) � [2(age(v) − ˆq)] where H∗ � Xζ,v, (Ω• Xζ,v, dwζ,v) � ≃ H∗(Xζ,v, w∞ ζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Therefore, Hw ≃ Hw,Q ⊗Q C as a graded vector space over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We say v ∈ Box(Σζ) is narrow if Xζ,v is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If v is narrow then wζ,v is constant and w∞ ζ,v is empty, so H∗(Xζ,v, w∞ ζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C) = H∗(Xζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In this case, we call the above vector space a narrow sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We say v is broad if v is not narrow, and call H∗(Xζ,v, w∞ ζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C) a broad sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We also consider a closely related GLSM (V, G, C∗ R, 0, ζ) obtained by replacing the superpotental W by zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' As a graded vector space over C, the GLSM state space of (V, G, C∗ R, 0, ζ) is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='26) H = � v∈Box(Σζ) H∗(Xζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C)[2 (age(v) − ˆq)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Note that inv and invR are homotopic as maps from Xζ,v to Xζ,inv(v), so inv∗ R = inv∗ : H∗(Xζ,inv(v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C) → H∗(Xζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The action of �T on V commutes with the action of its subgroup G and C∗ R, and preserves the zero superpotential 0 and the semistable locus V ss G (ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We define the �T-equivariant state space H � T of (V, G, C∗ R, 0, ζ) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' As a graded vector space over C, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='27) H � T = � v∈Box(Σζ) H∗ � T (Xζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C)[2 (age(v) − ˆq)] where each H∗ � T (Xζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C) is a module over H∗ � T (•;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C) = H∗(B �T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C) = C[λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , λn+κ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let C(λ) := C(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , λ) be the fractional field of C[λ] := C[λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , λn+κ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' There is a non-degenerate pairing H � T ⊗C[λ] C(λ) × H � T ⊗C[λ] C(λ) → C(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Categories of B-branes and K-theories Given an abelian GLSM, we consider several versions of the category of B-branes and the A-model state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' dg category category of B-branes K-theory A-model state space MF(Xζ, wζ) DMF(Xζ, wζ) ≃ DSg(Xζ,0) K(DMF(Xζ, wζ)) Hw = � v∈Box(Σζ) H∗(Xζ,v, w∞ ζ,v) Perf � T (Xζ) Db � T (Xζ) K � T (Xζ) H � T = � v∈Box(Σζ) H∗ � T (Xζ,v) Perf(Xζ) Db(Xζ) K(Xζ) H = � v∈Box(Σζ) H∗(Xζ,v) Db c(Xζ) Kc(Xζ) Hc = � v∈Box(Σζ) H∗ c (Xζ,v) In each version, the Chern character sends the K-theory class of a B-brane to an element in the A-model state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Kw = K(DMF(Xζ, wζ)) chw −→ Hw := � v∈Box(Σζ) Hw,v, Hw,v = H∗(Xζ,v, wζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' K � T = K � T (Xζ) ch � T −→ H � T = � v∈Box(Σζ) H � T ,v, H � T ,v = H∗ � T (Xζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' K = K(Xζ) ch −→ H = � v∈Box(Σζ) Hv, Hv := H∗(Xζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Kc = Kc(Xζ) chc −→ Hc = � v∈Box(Σζ) Hc,v, Hc,v = H∗ c (Xζ,v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' K � T is a commutative ring with unity and an algebra over K � T (•) = K(B �T) = Z[Λ± 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , Λ± n+κ], and K is a commutative ring with unity and an algebra over K(•) = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' There is a surjective ring homomorphism K � T → K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Kc and Kw are modules over the ring K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' There is a map Kc → K which is a morphism of K-modules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' the image Kct is an ideal in the ring K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Taking Euler characteristic defines non-degenerate pairings: K � T × K � T → Q(Λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , Λn+κ), K × Kc → Z, Kw × Kw → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Fix v ∈ Box(Σζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' H � T ,v is a commutative ring with unity and an algebra over H∗ � T (•) = H∗(B �T) = C[λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , λn+κ], and Hv is a commutative ring with unity and an algebra over H∗(•) = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' There is a surjec- tive ring homomorphism H � T ,v → Hv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Hc,v and Hw,v are modules over the ring Hv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' There is a map Hc,v → Hv which is a morphism of Hv-modules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' the image Hct,v is an ideal in the ring Hv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' non-degenerate pairings: H � T ,v × H � T ,inv(v) → C(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , λn+κ), Hv × Hc,inv(v) → C, Hw,v × Hw,inv(v) → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The Higgs Branch 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' A mathematical theory of GLSM: an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let (V, G, C∗ R, W, ζ) be the input date of a GLSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The first four components (V, G, C∗ R, W) give rise to the following diagram: V W � � [V/G] w � � □ C � [V/Γ] ˆw � � [C/C∗ ω] � BΓ = [•/Γ] Bχ � BC∗ ω = [•/C∗ ω] where the middle square is Cartesian, the upper triangle and the lower square are commutative, and the bottom right arrow Bχ : BΓ → BC∗ ω is induced by the group homomorphism χ : Γ → C∗ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' A Landau-Ginzburg (LG) quasimap to (V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' C∗ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' ζ) is a birational map from an orbicurve C to [V ss G (ζ)/Γ] which extends to a representable 10 morphism f : C → [V/Γ] of smooth Artin stacks and satisfies certain stability conditions such that the following diagram commutes: [V/Γ] π � C f � P � ωlog C � BΓ Bχ � BC∗ ω Recall that BΓ is the classifying space of principal Γ-bundles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' and [V/Γ] is the classifying space of a principal Γ-bundle P → C together with a section u : C → P ×Γ V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' where P ×Γ V → C is the rank n + κ vector bundle associated to the representation Γ → GL(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' A section u : C → P ×Γ V is equivalent to a Γ-equivariant map ˜f : P → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' More explicitly, we have the following cartesian diagram P � � � C = P/Γ π◦f � BΓ = [•/Γ] Let pr1 : P × V −→ P and pr2 : P × V → V be the projection to the first and second factors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We have a commutative diagram P × V pr1 � pr2 � V � P � (idP , ˜ f) � ˜ f � where all the arrows are Γ-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Taking the quotient of the above diagram by the Γ-action, we obtain the following commutative diagram P ×Γ V � � [V/Γ] π � C π◦f � u � f � BΓ = [•/Γ] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Twisted curves and their moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We follow the presentation of [AV02,AGV] on twisted curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' A genus- g, ℓ-pointed twisted prestable curve is a connected proper one-dimensional DM stack C together with ℓ disjoint zero-dimensional integral closed substacks z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ ⊂ C, such that (i) C is ´etale locally a nodal curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (ii) formally locally near a node, C is isomorphic to the quotient stack [Spec(C[x, y]/(xy))/µr], where η · (x, y) = (ηx, η−1y), η ∈ µr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (iii) each marking zi ⊂ C is contained in the smooth locus of C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (iv) C is a scheme away from the markings and the singular points of C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' the coarse moduli space C of C is a nodal curve of arithmetic genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let π : C → C be the coarse moduli morphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' let zi = π(zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The resulting (C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ) is a genus-g, ℓ-pointed prestable curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We say (C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ) is stable if (C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ) is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The moduli Mtw g,ℓ of genus-g, ℓ-pointed twisted prestable curves is a smooth Artin stack of dimension 3g − 3 + ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , ℓ, let Li be the line bundle on Mtw g,ℓ whose fiber over (C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ) is the T ∗ zjC, the cotangent line to the coarse curve C at zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The space of infinitesimal automorphisms of (C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ) is Ext0 OC(ΩC(z1 + · · · + zℓ), OC), while the space of infinitesimal deformations of (C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ) is Ext1 OC(ΩC(z1 + · · · + zℓ), OC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' zℓ) is stable if and only if Ext0 OC(ΩC(z1 + · · · + zℓ), OC) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Line bundles over a twisted curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let (C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ) be a genus-g, ℓ-pointed twisted prestable curve, where zi = Bµri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' A line bundle L on C defines a morphism C −→ BC∗ such that we have the following Cartesian diagram L � � □ [C/C∗] � C � [•/C] = BC∗ where [C/C∗] → BC∗ is the universal line bundle over the classifying space BC∗ of principal C∗-bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given any line bundle L over C, there exists a positive integer m such that L⊗m = π∗M for some line bundle M over the coarse moduli space C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Define deg L = 1 m deg M ∈ Q If zi is a scheme point, we define agezi(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If zi = Bµri, where ri > 1, then the restriction Lzi of L to zi is an element in Pic(Bµri) = Hom(µri, C∗) ∼= Z/riZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' There is a unique ai ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , ri − 1} such that Lzi ∼= (TziC)⊗ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Define agezi(L) = ai ri ∈ (0, 1) ∩ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' There is a unique line bundle L on the coarse moduli C such that L ≃ π∗L ⊗ OC( ℓ � i=1 aizi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' where deg � OC � ℓ � i=1 aizi �� = ℓ � i=1 agezi(L), deg(π∗L) = deg L ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' So deg L − ℓ � i=1 agezi(L) ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For i = 0, 1, hi(C, L) = hi(C, L), so (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1) χ(C, L) = χ(C, L) = deg L + 1 − g − ℓ � j=1 agezj(L) which is a special case of Kawasaki’s orbifold version of Riemann-Roch theorem [Ka79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The space of infinitesimal automorphisms of L on a fixed twisted prestable curve C is Ext0 OC(L, L) ≃ H0(C, OC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The space of infinitesimal deformations of L on a fixed twisted prestable curved C is Ext1 OC(L, L) ≃ H1(C, OC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Universal moduli of principal Γ-bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Mg,ℓ(BΓ) denote the moduli of pairs ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P), where (C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ) is a genus-g, ℓ-pointed twisted prestable curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' P is a principal Γ-bundle over C which corresponds to a representable morphism C −→ BΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then Mg,ℓ(BΓ) is a smooth Artin stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Note that Mg,ℓ(BΓ) can be identified with the Hom-stack HomM(CM, BΓ × M) where M = Mtw g,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Forgetting the principal bundle P defines a (non-representable) morphism of smooth Artin stacks πD/M : D := Mg,ℓ(BΓ) −→ M which is smooth of relative dimension dim Γ(g − 1) = (κ + 1)(g − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' So Mg,ℓ(BΓ) is a smooth Artin stack of dimension 3g − 3 + ℓ + (κ + 1)(g − 1) = (4 + κ)(g − 1) + ℓ = (dim BΓ − 3)(1 − g) + ℓ 12 where dim BΓ = − dim Γ = −κ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let πD : CD → D := Mg,ℓ(BΓ) be the universal curve, let fD : CD → BΓ be the universal map, and let PD → CD be the universal principal Γ-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We have the following cartesian diagrams: CD � πD � □ CM πM � D πD/M � M PD � � □ � CD fD � BΓ Let �L := Hom(C∗, Γ) ∼= Zκ+1 be the cocharacter lattice of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Its dual lattice �L∨ = Hom(Γ, C∗) is the character lattice of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' A principal Γ-bundle P −→ C determines a map C = P/Γ → BΓ = [•/Γ] whose degree is an element βΓ ∈ H2(BΓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) = �LQ characterized by the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For any Γ-character λ ∈ Hom(Γ, C∗) = �L∨ = H2(BΓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Z), let P ×λ C → C be the line bundle associated to the representation λ : Γ → C∗ = GL(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then � βΓ λ = deg(P ×λ C) = � [C] c1(P ×λ C) ∈ Q, where � βΓ λ denotes the natural pairing between βΓ ∈ �LQ = H2(BΓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) and λ ∈ �L∨ ⊂ �L∨ Q = H2(BΓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q), and � [C] c1(P ×λ C) denotes the natural pairing between [C] ∈ H2(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) and c1(P ×λ C) ∈ H2(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In other words, βΓ ∈ H2(BΓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) is the image of [C] ∈ H2(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) under P∗ : H2(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) −→ H2(BΓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The monodromy of P at zj = Bµrj is an element γj ∈ Γ of order rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The subset �LQ/�L ∼= (Q/Z)κ+1 ⊂ �LC/�L ∼= (C/Z)κ+1 = (C∗)κ+1 can be identified with {γ ∈ Γ : ord(γ) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The monodromies (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , γℓ) ∈ (�LQ/�L)ℓ and the degree βΓ ∈ �LQ satisfy the following compatibility condition: ℓ � j=1 γj = βΓ + �L ∈ �LQ/�L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Mg,⃗γ(BΓ, βΓ) ⊂ Mg,ℓ(BΓ) be the open and closed substack of pairs ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P) with degree βΓ ∈ �LQ and monodromies ⃗γ = (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , γℓ) ∈ (�LQ/�L)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then Mg,ℓ(BΓ) = � ⃗γ=(γ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=',γℓ)∈(�LQ/�L)ℓ βΓ∈�LQ, �ℓ i=1 γi=βΓ+�L Mg,⃗γ(BΓ, βΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Universal moduli of Γ-structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Polishchuk-Vaintrob introduced Γ-structures [PV16] which is an alter- native formulation for W-structures in [FJR11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given an object (C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ) in Mg,ℓ(•), a Γ-structure on (C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ) is a pair (P, ρ) where ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P) is an object in Mg,ℓ(BΓ) and ρ : P ×χ C ∼ = −→ ωlog C is an isomorphism of line bundles on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Bg,ℓ be the moduli space of triples ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P, ρ), where (C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ) is an object in Mtw g,ℓ(•) and (P, ρ) is a Γ-structure on (C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We have a commutative diagram B = Bg,ℓ πB/D� πB/M � D := Mg,ℓ(BΓ) πD/M � M = Mtw g,ℓ where πB/D : B −→ D is given by forgetting ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The map πB/M : B → M is smooth of relative dimension dim G(g − 1) = κ(g − 1), so Bg,ℓ is a smooth Artin stack of dimension 3g − 3 + ℓ + κ(g − 1) = (3 + κ)(g − 1) + ℓ = (dim BG − 3)(1 − g) + ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 13 The map (ρV , ρR) : G × C∗ R ∼= (C∗)κ+1 −→ Γ ∼= (C∗)κ+1 (h, t) �→ ht is a surjective group homomorphism and a covering map of degree r = ord(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' It induces (ρV , ρR)∗ : H2(BG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Z) × H2(BC∗ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Z) = L × � Z[P1] � −→ H2(BΓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Z) = �L which is an inclusion of lattices of finite index r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (We recall that BC∗ R = P∞, and we let [P1] ∈ H2(P∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Z) be the class of P1 ⊂ P∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=') Therefore, we obtain an isomorphism (ρV , ρR)∗ : H2(BG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) × H2(BC∗ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) = LQ × � Q[P1] � ∼ = −→ H2(BΓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) = �LQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let (βG, βR) = (ρV , ρR)−1 ∗ (βΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then βR = 2g − 2 + ℓ r [P1] ∈ Q[P1] = H2(BC∗ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) = H2(P∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Bg,ℓ(βG) ⊂ Bg,ℓ be the open and closed substack of triples ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P, ρ) with degree βΓ = � βG, 2g − 2 + ℓ r [P1] � ∈ LQ × � Q[P1] � ∼= H2(BΓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then Bg,ℓ = � βG∈LQ Bg,ℓ(βG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For each βG, let Bg,⃗γ(βG) ⊂ Bg,ℓ(βG) be the open and closed substack of triples ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P, ρ) with mon- odromies ⃗γ = (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , γℓ) ∈ (LQ/L)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then Bg,ℓ(βG) = � ⃗γ=(γ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=',γℓ)∈(LQ/L)ℓ �ℓ i=1 γi=βG+L Bg,⃗γ(βG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Moduli of sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In this subsection we fix g, ℓ, βG and ⃗γ = (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , γℓ) and let B = Bg,⃗γ(βG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Following [BF97, Section 1], we introduce the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given a coherent sheaf F of OX-modules on an algebraic stack X, let C(F) := SpecX(SymF∨) be the abelian cone associated to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In particular, when F is locally free, C(F) = tot(F) is the vector bundle associated to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let PB → CB be the universal principal Γ-bundle over the universal curve πB : CB → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Recall that Γ is a subgroup of the diagonal torus �T ⊂ GL(V ), so VB := PB ×Γ V = n+κ � i=1 Li,B where each Li,B is a line bundle over CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Consider the moduli of sections Bg,⃗γ(V, βG) := C (πB∗VB) which parametrizes 4-tuples ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P, ρ, u) where the triple ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P, ρ) is an object in B(•) and u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , un+κ) ∈ H0(C, P ×Γ V ) = H0(C, n+κ � j=1 Lj) where uj ∈ H0(C, Lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Bg,ℓ(V, βG) := C � πBg,ℓ(βG)∗VBg,ℓ(βG) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then Bg,ℓ(V, βG) = � ⃗γ=(γ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=',γℓ)∈(LQ/L)ℓ �ℓ i=1 γi=βG+L Bg,⃗γ(V, βG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given a fixed triple ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P, ρ), the space of infinitesimal deformations of the section uj of the line bundle Lj is H0(C, Lj) and the space of obstructions to deforming uj is H1(C, Lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We now compute χ(C, Lj) = h0(C, Lj) − h1(C, Lj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The line bundle Lj is of degree ⟨Dj, βG⟩ + qj 2 (2g − 2 + ℓ) 14 and has monodromy e2π√−1ageγi(Dj) around zi, where ageγi(Dj) ∈ Q ∩ [0, 1) is the unique representative in [0, 1) of the pairing ⟨Dj, βG⟩ ∈ Q/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Note that ageγi(Dj) = ageziLj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We have ⟨Dj, βG⟩ + qj 2 (2g − 2 + ℓ) − ℓ � i=1 ageγi(Dj) ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' By Kawasaki’s orbifold version of the Riemann-Roch theorem [Ka79], χ(C, Lj) = ⟨Dj, βG⟩ + qj 2 (2g − 2 + ℓ) + (1 − g) − ℓ � i=1 ageγi(Dj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='2) (c1)G(V ) = n+κ � j=1 Dj ∈ L∨, ageγ(V ) = n+κ � j=1 ageγi(Dj) ∈ Q, and recall that ˆq = 1 2 �n+κ j=1 qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then χ(C, P ×Γ V ) = n+κ � j=1 χ(C, Lj) = ⟨(c1)G(V ), βG⟩ + (n + κ − 2ˆq)(1 − g) − ℓ � i=1 � ageγi(V ) − ˆq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' There is a map πS/B : S = Bg,⃗γ(V, βG) −→ B = Bg,⃗γ(βG) given by ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P, ρ, u) �→ ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P, ρ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='e, forgetting the section u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The map πS/B is virtually smooth: there is a relative perfect obstruction theory ES/B, where E∨ S/B = π∗ S/BRπB∗VB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The relative virtual dimension of πS/B is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='3) dvir S/B = ⟨(c1)G(V ), βG⟩ + (n + κ − 2ˆq)(1 − g) − ℓ � i=1 � ageγi(V ) − ˆq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Therefore, the moduli of sections S is a possibly singular, but virtually smooth Artin stack;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' it is equipped with a perfect obstruction theory ES of virtual dimension dvir S = dvir S/B + dim B = ⟨(c1)G(V ), βG⟩ + (n + κ − 2ˆq)(1 − g) − ℓ � i=1 � ageγi(V ) − ˆq � + (κ + 3)(g − 1) + ℓ which can be rewritten as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='4) dvir S = ⟨(c1)G(V ), βG⟩ + (ˆc − 3)(1 − g) + ℓ − ℓ � i=1 � ageγi(V ) − ˆq � where ˆc = n − 2ˆq is the central charge of the GLSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Note that if γ ∈ LQ/L corresponds to v ∈ Box(Σζ) then ageγ(V ) = age(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The definition of the central charge ˆc and the degree shift 2 (age(v) − ˆq) in the definition of the A-model state spaces are motivated by the formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='4) of the virtual dimension, which is consistent with [FJR, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let PS → CS be the universal principal Γ-bundle over the universal curve πS : CS → S, and let uS : CS → VS := PS ×Γ V be the universal section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We have the following commutative diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' VS � � □ VB � CS � πS � uS � □ CB � πB � □ CM πM � S πS/B � B πB/M � M 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Landau-Ginzburg quasimaps and their moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1 (prestable LG quasimaps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' A prestable genus-g, ℓ-pointed, degree βG Landau-Ginzburg (LG) quasimap to the 5-tuple X = (V, G, C∗ R, W, ζ) is a 4-tuple Q = ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P, ρ, u), which is an object in Bg,ℓ(V, βG)(•) such that the base locus B(Q) := u−1(P ×Γ V us G (ζ)) ⊂ C of Q is purely zero-dimensional and is disjoint from the marked points and the nodes in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Note that the �T-action preserves V ss G (ζ) and V us G (ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In particular, Γ acts on V us G (ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let LGpre g,ℓ (X, βG) be the moduli of prestable genus-g, ℓ-pointed, degree βG LG quasimaps to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' It is an open substack of Bg,ℓ(V, βG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' There are evaluation maps evi : LGpre g,ℓ (X, βG) → IXζ = � v∈Box(Σ) Xζ,v, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given ⃗v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , vℓ) ∈ Box(Σζ)ℓ, let LGg,⃗v(X, βG) := ℓ� i=1 ev−1 i (Xζ,vi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then LGg,⃗v(X, βG) is an open substack of Bg,⃗γ=(γ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=',γℓ)(V, βG), where γi ∈ LQ/L corresponds to vi ∈ Box(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Therefore, the virtual dimension of LGg,⃗v(X, βG) is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='5) ⟨(c1)G(V ), βG⟩ + (ˆc − 3)(1 − g) + ℓ − ℓ � i=1 (age(v) − ˆq) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='3 (good lift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' ˜ζ ∈ �L∨ is a lift of ζ ∈ L∨ if ζ is the image of ˜ζ under �L∨ = Hom(Γ, C∗) → L∨ = Hom(G, C∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' ˜ζ is a good lift of ζ if V ss Γ (˜ζ) = V ss G (ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For any ˜ζ ∈ Hom(Γ, C∗) = �L∨, let χ˜ζ : Γ → C∗ denote the corresponding Γ-character, let L˜ζ ∈ PicΓ(V ) denote the Γ-equivariant line bundle on V determined by χ˜ζ χ ˜ζ1+˜ζ2 = χ ˜ζ1χ ˜ζ2, L˜ζ1+˜ζ2 = L˜ζ1 ⊗ L˜ζ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' A section s ∈ H0(V, L˜ζ)Γ defines a Γ-equivariant map V → C which induces a morphism s : P ×Γ V → P ×χ˜ ζ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given any u ∈ H0(C, P ×Γ V ), let u∗s := s◦u ∈ H0(C, P ×χζ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The following definition is [FJR, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='10] (which is essentially [CKM, Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1]) in slightly different notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='4 (length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Q = ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P, ρ, u) be a prestable LG quasimap to X = (V, G, C∗ R, W, ζ), let ˜ζ ∈ �L∨ be a good lift of ζ ∈ L∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The length of a point y in C with respect to Q and ˜ζ is defined to be (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='6) ℓy(Q, ˜ζ) := min �ordy(u∗s) m ��� s ∈ H0(V, Lm˜ζ = L⊗m ˜ζ )Γ, m ∈ Z>0 � where ordy(u∗s) is the order of vanishing of the section u∗s ∈ H0(C, P ×χm˜ ζ C) at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='5 (ϵ-stable LG quasimaps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let ˜ζ ∈ Hom(Γ, C∗) be a good lift of ζ ∈ Hom(G, C∗), and let ϵ be a positive rational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' A prestable LG quasimap Q = ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P, ρ, u) is ϵ-stable with respect to ˜ζ if (1) ωlog C ⊗ (P טζ C)ϵ ∈ Pic(C) ⊗Z Q is an ample Q line bundle on C, and (2) ϵℓy(Q, ˜ζ) ≤ 1 for every y ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let Cv (respectively Cv) be the connected component of the normalization of C (respectively C) associated to a vertex v in the dual graph of the coarse moduli C of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let gv be the genus of Cv and let ℓv be the number of points on Cv mapped to a marked point or a node under the normalization map, and let βΓ(v) ∈ H2(BΓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) = �LQ be the degree of Cv → C P→ BΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Condition (1) in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='5 is equivalent to the following condition: 2gv − 2 + ℓv + ϵ � βΓ(v) ˜ζ > 0 for all vertex v in the dual graph of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 16 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let m ∈ Z>0, ϵ ∈ Q>0, and ˜ζ ∈ �L∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then Q is ϵ-stable with respect to ˜ζ iff it is (ϵ/m)-stable with respect to m˜ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' see [CKM, Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='4] for the analogous statement in quasimap theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given ν ∈ �L∨ Q, choose m ∈ Z>0 such that mν ∈ �L∨ and define Q to be ν-stable if it is (1/m)-stable with respect to mν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' the definition is independent of the choice of m and agrees with [FK, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let LGpre g,ℓ (X, βG) (respectively LGϵ,˜ζ g,ℓ(X, βG)) be the moduli of genus-g, ℓ-pointed, degree βG prestable (respec- tively ϵ-stable with respect to ˜ζ) quasimaps to X := (V, G, C∗ R, W, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' More generally, let Y be a Γ-invariant closed subscheme of V such that Y ∩ V s G(ζ) = Y ∩ V ss G (ζ) is non-empty (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Y = Crit(W)), and let Y = (Y, G, C∗ R, W, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let LGpre g,ℓ (Y, βG) be the closed substack of LGpre g,ℓ (X, βG) parametrizing Q = ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P, ρ, u) such that u : C → [Y/Γ] ⊂ [V/Γ], and define LGϵ,˜ζ g,ℓ(Y, βG) similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Fan-Jarvis-Ruan proved the following result: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='8 ( [FJR]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' LGϵ,˜ζ g,ℓ(Y, βG) is a separated Deligne-Mumford stack of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' It is proper over SpecC if [(Y ∩ V ss G (ζ))/G] is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given ⃗v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , vℓ) ∈ Box(Σζ)ℓ, let LGϵ,˜ζ g,⃗v(X, βG) := LGϵ,˜ζ g,ℓ(X, βG) ∩ LGg,⃗v(X, βG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then LGϵ,˜ζ g,ℓ(X, βG) = � ⃗v∈Box(Σζ)ℓ LGϵ,˜ζ g,⃗v(X, βG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='8 and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='9 below, we fix g,⃗v, ϵ, βG, and let X = LGϵ,˜ζ g,⃗v(X, βG), Z = LGϵ,˜ζ g,⃗v(Z, βG) where Z = (Crit(W), G, C∗ R, W, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='8, if Zζ = [(Crit(W) ∩ V ss G (ζ)) /G] = Crit(wζ) is proper, then Z is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Cosection localized virtual cycle and cosection localized virtual structure sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Recall that v ∈ Box(Σζ) is narrow if Xζ,v is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We assume v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , vℓ are narrow in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Under this assumption, Fan-Jarvis-Ruan [FJR] constructed a cosection δ : ObX → OX whose zero locus is Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Applying [KL13], they obtain a cosection localized virtual cycle [X]vir loc ∈ A∗(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) such that ι∗[X]vir loc = [X]vir ∈ A∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) where ι : Z → X is the inclusion, and [X]vir is the Behrend-Fantechi virtual fundamental class [BF97] defined by the perfect obstruction theory described in previous subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' When Z is proper, [X]vir loc can be used to define (cohomological) GLSM invariants in the narrow sector1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The construction of cosection localized virtual cycle in the narrow sector in [FJR] can be viewed as generalization of Chang-Li-Li’s construction of Witten’s top Chern class via cosection localization [CLL].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We now consider the following particularly nice case, as in [BF97, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Situation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' X is smooth of dimension r0 and ObX is locally free of rank r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In Situation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='9, ObX := tot(ObX) is a vector bundle over X of rank r1, called the obstruction bundle, and the virtual dimension is r = r0 − r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let δ∨ : OX → Ob∨ X be the dual of the cosection, which is a section of Ob∨ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' [X]vir = cr1(ObX) ∩ [X] = (−1)r1cr1(Ob∨ X) ∩ [X] ∈ Ar(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='7) [X]vir loc = (−1)r1cr1(Ob∨ X, δ∨) ∩ [X] ∈ Ar(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='8) where [X] ∈ Ar0(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) is the fundamental class of the smooth DM stack X, cr1(Ob∨ X) ∩ − : Ak(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) → Ak−r1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) is the top Chern class, and cr1(Ob∨ X, δ∨) ∩ − : Ak(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) → Ak−r1(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) is the localized top Chern class [Fu98, Chapter 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 1In [FJR, Section 6], GLSM correlators are defined for compact type insertions [FJR, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='4] which are more general than narrow insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' See [Sh] for subtleties of defining GLSM invariants involving compact type insertions which are not narrow, as well as an alternative construction of genus-zero compact type GLSM invariants under additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 17 Let K0(X) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' K0(X)) denote the Grothendieck group generated by coherent sheaves (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' locally free sheaves) on X with relations [F] = [F ′] + [F ′′] whenever there is a short exact sequence 0 → F ′ → F → F ′′ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Applying [KL18], one obtains a cosection localized virtual structure sheaf Ovir X,loc ∈ K0(Z) such that ι∗Ovir X,loc = Ovir X ∈ K0(X) where Ovir X is the virtual structure sheaf defined in [BF97] (see [BF97, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='4]) and [Lee].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' When Z is proper, Ovir loc an be used to define K-theoretic GLSM invariants in the narrow sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In Situation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='9, Ovir X = r1 � i=0 (−1)i ∧i Ob∨ X = (−1)r1 det(Ob∨ X) r1 � i=0 (−1)i ∧i ObX ∈ K0(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Note that td(ObX)ch(Ovir X ) = cr1(ObX), so [X]vir = td(ObX)ch(Ovir X ) ∩ [X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The section δ∨ : OX → Ob∨ defines a Koszul complex (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='9) K(δ∨) := Symr1 � ObX iδ∨ → OX � = � 0 → ∧r1ObX iδ∨ → ∧r1−1ObX → · · · → ∧1ObX iδ∨ → OX → 0 � which is exact on X − Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , ek are local sections of ObX then iδ∨(e1 ∧ · · · ∧ ek) = k � i=1 (−1)i−1⟨δ∨, ei⟩e1 ∧ · · · ∧ ei−1 ∧ ei+1 ∧ · · · ∧ ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Note that td(Ob∨ X)chX Z (K(δ∨)) = cr1(Ob∨ X, δ∨), where chX Z (K(δ∨))∩ : A∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) → A∗(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) is the localized Chern character [Fu98, Chapter 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Virtual factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Favero-Kim [FK] construct GLSM invariants for general choice of stability in both narrow and broad sectors via matrix factorization, generalizing constructions in [PV16,CFGKS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In this subsection we briefly describe the construction (in slightly different notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The Artin stack C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let ⃗γ = (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , γℓ), where γi ∈ LQ/L corresponds to vi ∈ Box(Σζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let B := Bg,⃗γ(βG) be the universal moduli of Γ-structures defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let PB → CB be the universal principal Γ-bundle over the universal curve πB : CB → B, let VB = PB ×Γ V , and let C(πB∗VB) be the moduli of sections, which is an abelian cone over B, as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then X is an open substack of C(πB∗VB), and the image of X under the projection C(πB∗VB) → B (forgetting the section) is contained in a finite type open substack B◦ ⊂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Therefore, X is an open substack of C := C (πB◦∗VB◦) = B◦ ×B C (πB∗VB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , ℓ, there are evaluations maps (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='10) evC i : C −→ Xvi := [V g(vi)/G] which restricts to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='11) evi : X −→ Xζ,vi = �� V g(vi) ∩ V ss G (ζ) � /G � ⊂ Xvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The smooth Artin stack A and the smooth DM stack U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Over the finite type smooth Artin stack B◦, RπB◦∗VB◦ admits a global resolution RπB◦∗VB◦ = [A dA −→ B] where A, B are locally free sheaves of OB◦-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let πA/B◦ : A := tot(A) → B◦ be the projection, let tA ∈ Γ(A, π∗ A/B◦A) be the tautological section, and let βA := � π∗ A/B◦dA � tA ∈ Γ(A, π∗ A/B◦B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The zero locus of tA is the zero section in A = tot(A), and the zero locus of βA is C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' There exists an open substack U ⊂ A such that U is a DM stack of finite type and the following diagram is a Cartesian square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' X = Z(βU) ιX � jX � U jU � C = Z(βA) ιC � A In the above Cartesian diagram, the two vertical arrows are open embeddings, the two horizontal arrows are closed embeddings, and βU = j∗ UβA ∈ Γ(U, BU), where BU := j∗ Uπ∗ A/B◦B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The virtual dimension r of X is r = dim B + rankA − rankB = dim U − rankB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We have [X]vir = crankB (BU, βU) ∩ [U], where [U] ∈ Ar+rankB(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) is the fundamental class of the smooth DM stack U, crankB (BU, βU) ∩ − : Ar+rankB(U) → Ar(X) is the localized top Chern class, and [X]vir ∈ Ar(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Q) is the Behrend-Fantechi virtual fundamental class of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Evaluation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , ℓ, the evaluation map evC i : C → Xvi extends to evA i : A → Xvi which restricts to evU i : U → Xζ,vi, so that we have the following commutative diagram X ιX � � U evU i � evU i � � Xζ,vi � C ιC � A evA i � Xvi where evU i ◦ ιX = evi and evA i ◦ ιC = evC i , and all the vertical arrows are open embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let ⃗v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , vℓ), X⃗v := ℓ � i=1 Xvi, Xζ,⃗v := ℓ � i=1 Xζ,vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' A and evA i are chosen such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='12) evA := ℓ � i=1 evA i : A → X⃗v is a surjective smooth map between smooth Artin stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' More explicitly, given any object ξ = ((C, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , zℓ), P, ρ) in B◦(•), which corresponds to a morphism SpecC → B◦, let Aξ := SpecC ×B◦ A be the fiber of A = tot(A) over ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We have the following linear maps between complex vector spaces: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='13) H0(C, P ×Γ V ) ιC,ξ −→ Aξ evU ξ −→ ℓ � i=1 H0(zi, (P ×Γ V )|zi) = ℓ � i=1 V g(vi) where ιC,ξ is injective and evU ξ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' As a consequence, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='14) evU := ℓ � i=1 evU i : U → Xζ,⃗v is a smooth map between smooth DM stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Superpotentials and matrix factorizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let wvi : Xvi = [V g(vi)/G] → C denote the restriction of w : [V/G] → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Define a superpotential wA on A: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='15) wA := ℓ � i=1 (evA i )∗wvi ∈ Γ(A, OA) which restricts to a superpotential on U: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='16) wU := j∗ UwA = ℓ � i=1 (evU i )∗wζ,vi ∈ Γ(U, OU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The sum of residues of a meromorphic 1-form on a curve is zero, so ι∗ CwA = 0, ι∗ XwU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' When the GLSM X is a convex hypbrid model, it is shown in [CFGKS] that (a) U can be chosen to be separated over SpecC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (b) There exists a cosection α∨ A : π∗ A/B◦B → OA, or equivalently a section αA : OA → π∗ A/B◦B∨, such that ⟨αA, βA⟩ = −wA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (c) Let αU := j∗ UαA ∈ Γ(U, B∨ U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then Z = Z(αU) ∩ Z(βU) ⊂ X = Z(βU) ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='14, we will provide explicit construction of U and α∨ A satisfying (a)-(c) for the genus-zero one-pointed moduli spaces used to define the GLSM I-functions for all abelian GLSMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' When αA exists, one obtains a Koszul matrix factorization {αA, βA} of (A, −wA) defined by {αA, βA} = � � i Λ2iπ∗ A/B◦B∨ ∂ � � i Λ2i+1π∗ A/B◦B∨ ∂ � � , where ∂ = iβA + αA ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then KU := {αU, βU} = j∗ U{αA, βA} is a Koszul matrix factorization of (U, −wU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' It is called the fundamental factorization in [PV16, CFGKS] and called the virtual factorization in [FK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Localized Chern character and the virtual fundamental class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let U be a smooth DM stack over C and let w : U → C be a regular function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In [FK, Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='4], Favero-Kim define the Atiyah class and the localized Chern character of a matrix factorization for (U, w), following the construction of Kim-Polishchuk [KP22] when U is a smooth scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Applying the definition to KU, one obtains a localized Chern character chU Z KU ∈ Heven Z (U, (Ω• U, −dwU)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The virtual fundamental class of X is defined to be [U]vir w := � ℓ � i=1 ri � tdchU Z KU ∈ Heven Z (U, (Ω• U, −dwU)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' where ri = ord(g(vi)), or equivalently zi = Bµri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Effective classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given X = (V, G, C∗ R, W, ζ), and g, ℓ ∈ Z≥0, define Keff(X)g,ℓ := {βG ∈ LQ : LGpre g,ℓ (X, βG) is nonempty}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We now give a more explicit description of Keff(X)g,ℓ when ℓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let ((C, z), P, ρ, u) be an object in LGpre g,1(X, βG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then u(z) ∈ Xζ = � I∈Amin ζ XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If u(z) ∈ XI then ui(x) ̸= 0 for i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We observe that ui(z) ̸= 0 ⇒ deg Li ≥ 0 and agez(L) = 0 ⇔ deg Li ∈ Z≥0 20 since deg Li − agez(L) ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Therefore, Keff(X)g,1 ⊂ � I∈Amin ζ Keff I (X)g,1, where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='17) Keff I (X)g,1 = {βG ∈ LQ : ⟨Di, βG⟩ + qi 2 (2g − 1) ∈ Z≥0 for all i ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Indeed, it is not hard to see that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='18) Keff(X)g,1 = � I∈Amin ζ Keff I (X)g,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let {D∗I i : i ∈ I} be the Q-basis of LQ ∼= Qκ dual to the Q-basis {Di : i ∈ I} of L∨ Q: for any i, j ∈ I, ⟨Di, D∗I j ⟩ = δij Then Keff I (Xζ, wζ)g,1 = � � i∈I (mi − qi 2 (2g − 1))D∗I i : mi ∈ Z≥0 � 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Stacky loop spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In orbifold quasimap theory [CKK], the Jϵ-function is defined via C∗ localization on genus-zero ϵ-stable quasimap graph spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' When the gauge group G = (C∗)κ is an algebraic torus, the small I-function I = J0+|t=0 can be computed completely explicitly by C∗ localization on stacky loop spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Similarly, we may define the small Jϵ-function of a GLSM via C∗ localization on genus-zero ϵ-stable LG quasimap graph spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' When the gauge group G = (C∗)κ is an algebraic torus, the small I-function I = J0+|t=0 of the GLSM can be computed by C∗ localization on the LG version of stacky loop spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In this paper, we define and compute K-theoretic and cohomological I-functions via C∗ localization on stacky loop spaces defined in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Classical version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given any subset I of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , n + κ}, define VI = {(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , xκ+n) ∈ V = Cn+κ | xi ̸= 0 if i ∈ I} = CI′ × (C∗)I where I′ = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , n + κ} \\ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then V ss G (ζ) = � I∈Amin ζ VI, and Xζ = [V ss G (ζ)/G] = � I∈Amin ζ XI where XI = [VI/G] ⊂ Xζ is an open toric substack which is an affine toric orbifold defined by the n-dimensional cone σI′ spanned by {vi : i ∈ I′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' XI contains a unique torus fixed (possibly stacky) point pI = �� {0}I′ × (C∗)I� � G � = �� {0} ¯I × {1}I� � GσI′ � ∼= BGσI′ where GσI′ ⊂ G is the stabilizer of the point {0}I′ × {1}I ∈ V (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let NσI′ = � i∈I Zvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then GσI′ is a finite abelian group, and GσI′ ∼= N/NσI′ , Hom(GσI′ , C∗) ∼= L∨�� � i∈I ZDi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Quantum version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We introduce the following convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given any rational number m/a, where m ∈ Z, a ∈ Z>0, and m, a are coprime, we define Hi(P1, OP1(m a )) := Hi(P[a, 1], OP[a,1](m)), i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Recall that the total space of OP1[a,1](m) is [ � (C2 − {0}) × C � /C∗], where C∗ acts by weights (a, 1, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given any β ∈ Keff(Xζ, wζ)0,1, we let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='19) Vβ = n+κ � i=1 Vβ,i, where Vβ,i = H0 � P1, OP1(⟨Di, β⟩ − qi 2 ) � , and let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='20) Wβ = n+κ � i=1 Wβ,i, where Wβ,i = H1 � P1, OP1(⟨Di, β⟩ − qi 2 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let χDi : G → C∗, where 1 ≤ i ≤ n + κ, be defined as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let G act on Vβ,i and Wβ,i by g · u = χDi(g)u where g ∈ G and u ∈ Vβ,i or Wβ,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='10 (stacky loop space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We define the degree β stacky loop space by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='21) Xζ,β := [V ss β (ζ)/G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The stacky loop space in the above definition is the analogue of the stacky loop space in orbifold quasimap theory [CCK, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='2], which can be viewed as the orbifold version of Givental’s toric map space [Gi98, Section 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We have V ss β (ζ) = � I∈Amin ζ Vβ,I where Vβ,I = {(u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , un+κ) ∈ Vβ | ui ̸= 0 if i ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='11 (obstruction bundle and obstruction sheaf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We define the degree β obstruction bundle by Obζ,β = [ � V ss β (ζ) × Wβ � /G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The degree β obstruction sheaf Obζ,β is the locally free sheaf of OXζ,β-modules on Xζ,β associated to the vector bundle Obζ,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The obstruction bundle Obζ,β is a toric vector bundle over the smooth toric DM stack Xζ,β, and Obζ,β = Spec � SymOb∨ ζ,β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let TXζ,β be the tangent sheaf of Xζ,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The two-term complex (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='22) � TXζ,β 0 −→ Obζ,β � is a perfect tangent-obstruction complex [LT98] on Xβ,ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Taking the dual of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='22), we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='23) � Ob∨ ζ,β 0 −→ ΩXζ,β � which is a perfect obstruction theory [BF97] on Xζ,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In particular, the tangent-obstruction complex (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='22) and the perfect obstruction theory are objects in D(Xζ,β), the derived category of coherent sheaves on Xζ,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The virtual tangent bundle is T vir Xζ,β = TXζ,β − ObXζ,β ∈ K(Xζ,β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let TXζ,β/BG be the relative tangent bundle of the smooth map Xζ,β → BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We have the following short exact sequence of vector bundles on Xζ,β: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='24) 0 → Xζ,β × LC → TXζ,β/BG = [ � V ss β (ζ) × Vβ � /G] → TXζ,β → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given I ∈ Amin ζ , σI′ is an n-dimensional cone in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Define Keff I (X, ζ)0,1 −→ Box(σI′) = � � i∈I′ aivi : ai ∈ [0, 1) ∩ Q � β �→ v(β) := � i∈I′ {−⟨Di, β⟩ + qi 2 }vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 22 The big torus �T = (C∗)n+κ acts on Vβ by (�t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , �tn+κ) · (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , un+κ) = (�t1u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , �tn+κun+κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' This induces an action of T = �T/G ∼= (C∗)n (the flavor torus) on Xζ,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The tangent sheaf TXζ,β and the obstruction sheaf Obζ,β are T-equivariant locally free sheaves Xζ,β, so the the perfect obstruction theory is T-equivariant and is an object in DT (Xζ,β), the derived category of T-equivariant coherent sheaves on Xζ,β, and T vir Xζ,β ∈ KT (Xζ,β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Define V ◦ β,I = {(u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , un+κ) ∈ Vβ | ui(1, 0) ̸= 0 if i ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then V ◦ β,I is a Zariski open dense subset of Vβ,I, and V ss β (ζ)◦ := � I∈Amin ζ V ◦ β,I is a Zariski dense open subset of V ss β (ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Define X◦ ζ,β := [V ss β (ζ)◦/G] which is the open substack of Xζ,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (Our notation X◦ ζ,β is motivated by Okounkov’s notation in [Ok20], in which QM◦ denotes the open subtack of QM where ∞ = [1, 0] is not a base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=') Given I ∈ Amin ζ , σI′ is a top-dimensional (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' n-dimensional) cone in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Recall that g(v) ∈ GI is the image of v = � i∈I′ aivi ∈ Box(σI′) under the bijection Box(σI′) → GσI′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The g(v)-fixed subspace of V is V g(v) = {x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , xn+κ) ∈ V | xi = 0 if i ∈ I′ and ai /∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The connected component Xζ,v of the inertia stack IXζ associated to v is Xζ,v = �� V g(v) ∩ V ss(ζ) � /G � which is an open dense substack of the Artin stack Xv = [V g(v)/G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' There is an evaluation map ev∞ : Xζ,β −→ Xv(β) [u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , un+κ] �→ [u1(1, 0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , un+κ(1, 0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then ev−1 ∞ (Xζ,v(β)) = X◦ ζ,β 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Torus actions and C∗ q fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' In orbifold quasimap theory, the I-function is obtained by torus localization on the stacky loop space, using the C∗-action on P1, the coarse moduli space of P[a, 1] where a is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We denote this C∗ by C∗ q since it corresponds to C× q in [Ok20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' For �T-equivariant parameters, we use notation similar to that in [CIJ] and [GiV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' KC∗q(•) = K(BC∗ q) = Z[q±1], K � T (•) = K(B �T) = Z[Λ±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , Λ±1 n+κ], Let z = c1(q) ∈ H2 C∗ q(•;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Z), λj = −c1(Λj) ∈ H2 � T (•;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then H∗ C∗q(•;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Z) = H∗(BC∗ q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Z) = Z[z], H∗ � T (•;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Z) = H∗(B �T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Z) = Z[λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , λn+κ], � M = n+κ � j=1 Zλj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let deg(x) = a, deg(y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then C[x, y] = ∞ � m=0 C[x, y]m where C[x, y]m denote the degree m part of the graded ring C[x, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If m ∈ Z≥0 then H0(P[a, 1], OP1[a,1](m)) = C[x, y]m = ⌊ m a ⌋ � k=0 Cxkym−ka, H1(P[a, 1], OP1[a,1](m)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let C∗ q act on P[a, 1] by q · [x, y] = [qx, y] = [x, q−1/ay], and on C[x, y] by q · x = x, q · y = q−1/ay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Given any number r ∈ Q, let ⌊r⌋ be the unique integer such that ⌊r⌋ ≤ r < ⌊r⌋ + 1, and let {r} := r − ⌊r⌋ ∈ [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' As an 23 element in KC∗q(•), H0(P[a, 1], OP1[a,1](m)) − H1(P[a, 1], OP1[a,1](m)) = q−{ m a } 1 − q−1 + q− m a 1 − q = ∞ � k=0 q−{ m a }−k − ∞ � k=0 q− m a −1−k = � � � � � � � � � � � � � � � � � ⌊ m a ⌋ � k=0 q−{ m a }−k, m ≥ 0 0, −a ≤ m ≤ −1 − −⌊ m a ⌋−1 � k=1 q−{ m a }+k, m ≤ −a − 1 For j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' , n + κ, we define (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='25) dj(β) := ⟨Dj, β⟩ − qj 2 ∈ Q Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='26) V j β = � � � � � � � ⌊dj(β)⌋ � k=0 Λjq−{dj(β)}−k, dj(β) ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 0, dj(β) < 0 , W j β = � � � � � � � 0, dj(β) ≥ −1, −⌊dj(β)⌋−1 � k=1 Λjq−{dj(β)}+k, dj(β) < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (V j β )C∗ q = � C, dj(β) ∈ Z≥0, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Write V = n+κ � j=1 Vj = V + β ⊕ V − β ⊕ V ⊥ β where Vj = SpecC[xj], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='27) V + β := � j∈[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='.n+κ] dj(β)∈Z≥0 Vj, V − β := � j∈[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='.n+κ] dj(β)∈Z<0 Vj, V ⊥ β := � j∈[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='.n+κ] dj(β)/∈Z Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then V + β = V C∗ q β , V + β ⊕ V − β = V g(v(β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We define Fβ := �� V + β ∩ V ss G (ζ) �� G � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then Fβ is a closed substack of Xζ,v(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' ev∞ : X◦ ζ,β → Xζ,v(β) restricts to an isomorphism ev∞ : � X◦ ζ,β �C∗ q → Fβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' We identify � X◦ ζ,β �C∗ q with Fβ under the above isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If I ∈ Amin ζ and β ∈ Keff(Xζ, wζ) then the torus fixed point pI is contained in Fβ if and only of β ∈ Keff I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Virtual tangent and normal bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Since C∗ q acts trivially on Fβ, it acts linearly on the fibers of any C∗ q-equivariant vector bundle on Fβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' If V is a C∗ q-equivariant vector bundle over Fβ then V = � d∈Z Vd = V f ⊕ V m, where Vd is the subbundle on which C∗ q acts by weight d, and V m = � d̸=0 Vd (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' V f = V0) is the moving (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' fixed) part of V under the C∗ q-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let T 1 β := TXζ,β �� Fβ , T 2 β := Obζ,β|Fβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Then T 1,f β = TFβ is the tangent bundle of Fβ, and T 1,m β = NFβ/X◦ ζ,β is the normal bundle of Fβ in Xζ,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' 24 The virtual tangent bundle of Fβ is T 1,f β − T 2,f β = TFβ − 0 = TFβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Therefore, [Fβ]vir = [Fβ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' The virtual normal bundle of Fβ is defined to be (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='28) N vir β := T 1,m β − T 2,m β ∈ KC∗q× � T (Fβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Let ιβ→v(β) : Fβ → Xζ,v(β), ιv : Xζ,v → Xζ, ιβ = ιv(β) ◦ ιβ→v(β) : Fβ → Xζ be inclusion maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='29) N vir β = ι∗ β→v(β) � N vir β where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content='30) � N vir β = n+κ � j=1 ι∗ v(β)(U � T j )−1� ∞ � k=0 q−dj(β)+k − ∞ � k=0 q{−dj(β)}+k� + � dj(β)∈Z<0 ι∗ v(β)(U � T j )−1 ∈ KC∗q× � T � Xζ,v(β) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAzT4oBgHgl3EQfUfww/content/2301.01266v1.pdf'} +page_content=' T 1,m β = � dj(β)≥0 ι∗ β(U � T j )−1� � k∈Z 0≤k ck. When ω < ck, +∂qωE +k,q becomes imaginary. This is related to the fact that the out-of-plane wavevector q becomes imaginary and the +corresponding modes are evanescent in the out-of-plane direction. Thus, it is not possible to emit radiation at ω < ck +and the evanescent modes remain guided in the plane. +Similarly, ˜ΓAB +k +(ω) denotes the Fourier transform of Eq. (S7). This takes the form +˜ΓAB +k +(ω) = 1 +ℏ +� +dqπ +� +κA +k,qκB +k,qδ(ω − ωE +k,q) +� ++ i +ℏp.v. +� +dq +κA +k,qκB +k,q +ω − ωE +k,q +, +(S13) +where p.v. denotes the Cauchy principal value. The real part is then +Re[˜ΓAB +k +(ω)] = 1 +ℏπκA +k,qpκB +k,qpρk(ω). +(S14) +Note that, the real and imaginary parts of the functions ˜Γk(ω) are related to each other via Kramers-Kronig relations +which is due to the causality in Eq. (S6). +Unitary input-output relation +Using Eqs. (S9) and (S11), the input-output relation (S5) can be expressed as +ˆeO +k,qp = S(k, ω)ˆeI +k,qp +(S15) +with +S(k, ω) = 1 − i2πρk(ω) +ℏ +� +i,j +[M(k, ω)]−1 +ij κi +k,qpκj +k,qp. +(S16) + +3 +Importantly, the transformation (S15) is unitary since one can check that |S(k, ω)| = 1, and hence, the output +operators obey the same Bose commutation relations as the input operators. +We note that the presence of a common photonic bath with a given density of state ρk in the present theory is +important for the unitarity of the transformation (S15). In particular, in the different situation where the emitter and +cavity photon part interact with independent matter and photon baths, the resulting input-output transformation +cannot be unitary in general since it is not equivalent to have an input from the matter or from the photon bath when +these have different dispersion relations. This subtlety seems to have been missed in Ref. [28]. +Power spectrum +The system power spectrum is defined as the Fourier transform of the auto-correlation function +I(k, ω) = +� +dτ +� +dt⟨ˆc† +k(t)ˆck(t + τ) + ˆx† +k(t)ˆxk(t + τ)⟩eiωτ +(S17) += ⟨ ˆC† +k(ω) ˆCk(ω) + ˆ +X † +k(ω) ˆ +Xk(ω)⟩. +(S18) +Using Eq. (S10), it can be related to the input operators as +I(k, ω) = ⟨ +� ˆFC† +k (ω), ˆFX† +k (ω) +� +I(k, ω) +� ˆFC +k (ω) +ˆFX +k (ω) +� +⟩, +(S19) +where the matrix I(k, ω) is given by +I(k, ω) = +� +M(k, ω)−1�† M(k, ω)−1. +(S20) +The spectrum I(k, ω) can be calculated analytically and is of the form +I(k, ω) = +� +|M12|2 + |M22|2� +⟨ ˆFC† +k +ˆFC +k ⟩ + +� +|M21|2 + |M11|2� +⟨ ˆFX† +k +ˆFX +k ⟩ − 2 Re [(M11 + M22)M∗ +12] ⟨ ˆFC† +k +ˆFX +k ⟩ +| det[M(k, ω)]|2 +. (S21) +Here, we have shortened the notation by removing the k, ω dependence in the numerator, and we have used the +relation ⟨ ˆFC† +k +ˆFX +k ⟩ = ⟨ ˆFX† +k +ˆFC +k ⟩ since from Eq. (S11), one has +⟨ ˆFA† +k (ω) ˆFB +k (ω)⟩ = 4π2ρ2 +k(ω)κA +k,qpκB +k,qp⟨ˆeI† +k,qpˆeI +k,qp⟩. +(S22) +Memoryless approximation +The set of evolution equations in (S6) shows that the system operators ˆck, ˆxk at time t are affected by all their past +at times t′ < t. This influence is encoded in the functions ΓAB +k +(τ) defined in Eq. (S7). As mentioned in the main +text, one can simplify these equations by using a memoryless approximation. One can motivate this approximation +by inspecting the functions ΓAB +k +(τ) +ΓAB +k +(τ) = θ(τ)1 +ℏ +� ∞ +0 +dq κA +k,qκB +k,qe− i +ℏ ϵE +k,qτ = θ(τ)1 +ℏ +� ∞ +0 +dω ρk(ω)κA +k,qpκB +k,qpe−iωτ. +(S23) +Since the relevant frequencies are large (these are in the vicinity of the emitter frequency ωX +0 ), we extend the integral +toward negative frequencies. Neglecting the frequency dependence of κA +k,qp ≃ κA +k and assuming ρk(ω) ≃ ρk(ωX +0 ) one +has +� ∞ +−∞ +dω ρk(ω)κA +k,qpκB +k,qpe−iωτ ≃ +� ∞ +−∞ +dω ρk(ωX +0 )κA +k κB +k e−iωτ = ρk(ωX +0 )κA +k κB +k δ(τ), +(S24) +and thus one can approximate +ΓAB +k +(τ) ≃ 1 +ℏρk(ωX +0 )κA +k κB +k δ(τ) ≡ γAB +k +δ(τ). +(S25) + +4 +In this approximation Eqs. (S6) reduce to +iℏ∂tˆck = +� +ϵC +k − iγC +k +� +ˆck + +� +gR − iγCX +k +� +ˆxk + ˆF C +k , +(S26a) +iℏ∂tˆxk = +� +ϵX +k − iγX +k +� +ˆxk + +� +gR − iγXC +k +� +ˆck + ˆF X +k , +(S26b) +with the decay parameters +γC +k = ρk(ωX +0 )κC +k κC +k /ℏ, +γX +k = ρk(ωX +0 )κX +k κX +k /ℏ, +and γCX +k += γXC +k += +� +γC +k γX +k . +(S27) +This leads to a simplification of the matrix (6) into (8) in the main text. +Applying this approximation to the average of the product of force operators (S22) gives +⟨ ˆFA† +k (ω) ˆFB +k (ω)⟩ ≃ 4π2ℏρk(ωX +0 ) +� +γA +k γB +k ⟨ˆeI† +k,qpˆeI +k,qp⟩, +(S28) +and the power spectrum can be expressed as +I(k, ω) ∝ +� +|˜g|2 + |ℏω − zX +k |2� +γC +k + +� +|˜g|2 + |ℏω − zX +k |2� +γX +k + 2 Re +� +(2ℏω − zC +k − zX +k )˜g∗� � +γC +k γX +k +| det[M(k, ω)]|2 +ρk(ωX +0 )⟨ˆeI† +k,qpˆeI +k,qp⟩. +(S29) +Neglecting the k dependence of ρk and using nk(ω) ∝ ⟨ˆeI† +k,qpˆeI +k,qp⟩, one obtains the power spectrum given in Eq. (10) +of the main text. For example, taking a constant nk(ω) = const. mimics a broadband input field, i.e., an illumination +with white light. +Field amplitude dynamics +Assuming that the environment is initially in its vacuum and using the memoryless approximation, one can calculate +analytically the evolution of the system field amplitudes. Taking the average of the evolution equations in (S26), and +using the notation introduced in the main text one has: +iℏ∂t⟨ˆck⟩ = zC +k ⟨ˆck⟩ + ˜g⟨ˆxk⟩, +(S30a) +iℏ∂t⟨ˆxk⟩ = zX +k ⟨ˆxk⟩ + ˜g⟨ˆck⟩. +(S30b) +The solutions are of the form: +⟨ˆck⟩(t) = +1 +ℏωU +k −ℏωL +k {e−iωU +k t � +⟨ˆxk⟩0˜g + ⟨ˆck⟩0(ℏωU +k − zX +k ) +� +− e−iωL +k t � +⟨ˆxk⟩0˜g − ⟨ˆck⟩0(ℏωU +k − zC +k ) +� +}, +(S31a) +⟨ˆxk⟩(t) = +1 +ℏωU +k −ℏωL +k {e−iωU +k t � +⟨ˆck⟩0˜g + ⟨ˆxk⟩0(ℏωU +k − zC +k ) +� +− e−iωL +k t � +⟨ˆck⟩0˜g − ⟨ˆxk⟩0(ℏωU +k − zX +k ) +� +}. +(S31b) +Using the initial conditions ⟨ˆck⟩0 = 0 and ⟨ˆxk⟩0 = 1, one can obtain Eq. (12) of the main text for δ = δBiC and k = 0. +It is interesting to note that the model used to fit the experimental Rabi oscillations reported in Ref. [56] contains +a phenomenological “upper polariton decay/dephasing” term that seems to act as the present dissipative coupling +(Im[˜g]) in the evolution equations for the field amplitudes (S30). This suggests that the mechanism introduced in our +work could also have played a role in this time-resolved experiment. + diff --git a/KtE0T4oBgHgl3EQfSQDq/content/tmp_files/load_file.txt b/KtE0T4oBgHgl3EQfSQDq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aed85e8ec343ac7105846b08abb20dc60b18a489 --- /dev/null +++ b/KtE0T4oBgHgl3EQfSQDq/content/tmp_files/load_file.txt @@ -0,0 +1,790 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf,len=789 +page_content='Effective dissipative light-matter coupling in nonideal cavities Olivier Bleu, Kenneth Choo, Jesper Levinsen, and Meera M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Parish School of Physics and Astronomy, Monash University, Victoria 3800, Australia and ARC Centre of Excellence in Future Low-Energy Electronics Technologies, Monash University, Victoria 3800, Australia We consider the scenario of an emitter embedded in a nonideal cavity, accounting for the possibility that the emitter and cavity photons interact with a common photonic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Using an input- output approach to describe the open system, we demonstrate that this situation gives rise to an effective dissipative coupling between the emitter and the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' The underlying mechanism is independent of the nature of the emitter and exists even at zero temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' hence our results are potentially relevant for a variety of experimental platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We show that the effective dissipative coupling can lead to physical effects that do not occur in closed light-matter coupled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In particular, when the radiative decay rates exceed the conventional Rabi coupling, we can have the phenomenon of level attraction between the emitter and cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Our model thus provides a possible explanation for the level attraction observed in recent photoluminescence measurements in semiconductor microcavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Finally, we demonstrate that hybrid light-matter exceptional points and bound states in the continuum can be realized within this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In quantum optics, the spontaneous emission of radia- tion is understood as arising from the coupling between an emitter and the vacuum of the electromagnetic field in its surroundings [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' An important consequence of this observation is that the spontaneous emission is not an intrinsic property of the emitter, but can be controlled (enhanced or inhibited) by modifying its electromagnetic environment [2–5], for instance, by placing the emitter in an optical cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' This fact is at the heart of the research field of cavity quantum electrodynamics [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' A standard scenario considered in the literature is that of an emitter coupled to a single mode of an ideal cavity, which has been described using two minimal models: the Jaynes-Cummings model [9] where the emitter is a two- level system, and the coupled-oscillator model [10] which considers the emitter as a bosonic mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Both models are equivalent in the single-excitation limit, in which case the eigenstates of the light-matter coupled system consist of superpositions between the emitter and bare cavity photon states — the so-called polariton states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In practice, however, the cavity cannot be perfect and thus the light-matter-coupled system can be affected by the outside electromagnetic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In this con- text, there are two effects: the damping of the cavity, which has been extensively studied theoretically for the Jaynes-Cummings model [11–14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' and the possibility for the emitter itself to emit radiation outside of the cav- ity [15–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Such dissipative effects are generic and have been similarly modelled for a range of scenarios beyond the original case of an atomic emitter, such as semicon- ductor microcavities with either two-level systems [19–22] or bosonic modes [23–25] as emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' However, to our knowledge the decay channels for the emitter and the cavity have always been assumed to be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In this Letter, we consider the situation in which the emitter and the cavity photon interact with a common photonic environment, a scenario which is readily realized in a nonideal cavity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In principle, this configu- Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (a) Sketch of the hybrid light-matter system under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' It consists of a semiconductor layer embed- ded in a planar optical cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' One of the mirrors is not perfect, which allows the emitter and the cavity photon to interact with the common photonic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Panel (b) illustrates the analogy with two coupled oscillators immersed in a common medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' ration only requires a mirror that is not perfect: it is independent of the nature of the emitter and exists even when the environment is at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Therefore, our findings are of potential relevance for a variety of ex- perimental platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' To be concrete, we consider the system to be a planar semiconductor microcavity that hosts cavity photons and excitons [26], and use an input- output approach [27, 28] to describe the open quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We demonstrate that this situation gives rise to an effective dissipative coupling between the cavity mode and the exciton which can lead to level attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In particular, this effect provides a plausible and sim- ple explanation for recent photoluminescence measure- ments in planar semiconductor microcavities which re- ported anomalous dispersion relations [29, 30] or earlier reports of level attraction in quantum dot cavities [31– 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We also note that similar level attraction phenomena have been recently reported in cavity magnonic systems [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Finally, our results allow us to highlight that bound states in the continuum (BiC) and exceptional points (EP) can arise in the present model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content='02221v1 [quant-ph] 5 Jan 2023 个十 photonic gR gR environment 个十 c2 Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' — We employ a system-environment decom- position, as is customary for the description of open quantum systems, and hence we start with a total Hamil- tonian of the form ˆH = ˆHS + ˆHE + ˆHSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Here, ˆHS and ˆHE correspond to the system of interest and the environment, respectively, while ˆHSE describes the system-environment coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' To be concrete, we con- sider the system illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' It consists of two-dimensional excitons and cavity photons described by the Hamiltonian ˆHS = � k � ϵC k ˆc† kˆck + ϵX k ˆx† kˆxk + gR � ˆx† kˆck + ˆc† kˆxk �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (1) Here, ˆck (ˆc† k) and ˆxk (ˆx† k) are bosonic annihilation (cre- ation) operators of cavity photons and quantum-well ex- citons, respectively, with in-plane momentum ℏk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' The ki- netic energies at low momenta are ϵC k = ϵ0+ℏ2k2/2mC+δ and ϵX k = ϵ0+ℏ2k2/2mX, where k ≡ |k| and mC (mX) is the cavity photon (exciton) mass, while δ is the photon- exciton detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We explicitly include the exciton tran- sition energy ϵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' gR is the Rabi coupling, and we assume that the rotating wave approximation holds (ϵ0 ≫ gR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We consider the environment to be the electromagnetic field outside of the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' To describe this photonic bath, we use the following Hamiltonian ˆHE = � dq � k ϵE k,qˆe† kqˆekq, (2) where ˆekq (ˆe† kq) are annihilation (creation) operators for photons outside the cavity with in- and out-of-plane wavevectors k and q, respectively, and ϵE k,q = ℏc � k2 + q2 is the corresponding photon kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Since we consider the situation where both the emitter and the cavity photons interact with the photonic envi- ronment, we describe the system-environment interaction with a Hamiltonian of the form ˆHSE = ˆHXE + ˆHCE with ˆHXE = � dq � k � κX k,qˆe† kqˆxk + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' � , (3a) ˆHCE = � dq � k � κC k,qˆe† kqˆck + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (3b) ˆHXE encodes the fact that the probability of an exciton to recombine by emitting a photon directly to the envi- ronment is nonzero [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In particular, this can become the dominant process for the radiative recombination of the emitter when the cavity mode and emitter frequen- cies are not resonant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=', the opposite of the Purcell effect [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' ˆHCE describes the cavity-environment cou- pling that exists even in the absence of the semiconduc- tor layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Crucially, the form of ˆHSE allows for interfer- ence effects to take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' This is an essential difference between the present model and those which consider in- dependent environments for the emitter and the cavity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' [15–25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' For each in-plane wavevector, the scenario we consider is analogous to the problem of two coupled oscillators immersed in a common medium illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Equations for the system operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' — Starting from the Heisenberg equations with the total Hamiltonian, one can obtain evolution equations for the system operators using an input-output approach [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' These read iℏ∂tˆck = � ˆck, ˆHS � − i � ∞ −∞ dt′ΓCC k (t − t′)ˆck(t′) − i � ∞ −∞ dt′ΓCX k (t − t′)ˆxk(t′) + ˆF C k (t), (4a) iℏ∂tˆxk = � ˆxk, ˆHS � − i � ∞ −∞ dt′ΓXX k (t − t′)ˆxk(t′) − i � ∞ −∞ dt′ΓXC k (t − t′)ˆck(t′) + ˆF X k (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (4b) Here ˆF X,C k (t) encode Langevin-like force operators [37] and ΓAB k (τ) = ΓBA k (τ) = θ(τ) 1 ℏ � dq κA k,qκB k,qe− i ℏ ϵE k,qτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' All of the corresponding terms originate from the interac- tion with the common photonic environment (3) and are not captured in ˆHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Importantly, we see that ΓCX k (τ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (4) induces an additional coupling between the excitons and cavity photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Taking the Fourier transform of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (4), one gets the matrix equation M(k, ω) � ˆCk(ω) ˆ Xk(ω) � = � ˆFC k (ω) ˆFX k (ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (5) Here, ˆ Xk(ω), ˆCk(ω), ˆFk(ω) denote the Fourier transforms of the operators ˆxk(t), ˆck(t), ˆFk(t) respectively, and M(k, ω) = �ℏω − ϵC k + i˜ΓCC k (ω) −gR + i˜ΓCX k (ω) −gR + i˜ΓXC k (ω) ℏω − ϵX k + i˜ΓXX k (ω) � , (6) with ˜ΓAB k (ω) the Fourier transform of ΓAB k (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (5) one can obtain the power spectrum I(k, ω) ∝ ⟨ ˆC† k(ω) ˆCk(ω) + ˆ X † k(ω) ˆ Xk(ω)⟩, (7) which encodes the emission spectrum of the system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=', from the cavity and the semiconductor layer) for a given 3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Anomalous dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' The color plot represents the power spectrum I(k, ω) (color scale in arbitrary units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' The dashed white lines correspond to the real parts of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (9) and the dashed-black lines represent the bare exciton and cav- ity photon kinetic energies ϵX k and ϵC k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (a) δ/γC = 3, (b) δ/γC = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' The other parameters used for both panels are: nk(ω) = 1, γX/γC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content='8, gR/γX = 0, mC/mX = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' input boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In the absence of the envi- ronment, the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (5) is zero and the power spectrum vanishes, as it should.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We note that we have not used any approximation at this stage and thus the above derivation is formally exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Memoryless approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' — The set of evolution equations in (4) implies that the system operators ˆck, ˆxk at time t are affected by their past at times t′ < t through the functions ΓAB k (τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' One can neglect such memory effects if one approximates these functions as ΓAB k (τ) ≃ γAB k δ(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' This is equivalent to assuming that their Fourier transforms ˜ΓAB k (ω) are real and indepen- dent of ω [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Using this memoryless approximation, the matrix (6) simplifies to M(k, ω) = �ℏω − zC k −˜g −˜g ℏω − zX k � , (8) where we have introduced zC,X k = ϵC,X k − iγC,X k , and ˜g = gR − i � γC k γX k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Solving det[M] = 0, one obtains the complex eigenvalues ℏωL,U k = 1 2 � zC k + zX k ± � (zC k − zX k )2 + 4˜g2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (9) The power spectrum (7) can be calculated analytically and reads I(k, ω) ∝ Ak(ω)γC k + Bk(ω)γX k + Dk(ω) � γX k γC k |ℏω − ℏωL k |2|ℏω − ℏωU k |2 nk(ω), (10) where nk(ω) is proportional to the photon distribution in the input field [37] and we have defined Ak(ω) = |˜g|2 + |ℏω − zX k |2, Bk(ω) = |˜g|2 + |ℏω − zC k |2, Dk(ω) = 2 Re � (2ℏω − zC k − zX k )˜g∗� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We emphasise that, within our theory, γX k denotes the exciton radiative decay rate into the common photonic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In practice, there might be other sources of excitonic broadening such as emission into other pho- tonic modes or dephasing induced by the solid-state en- vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' This can be incorporated by adding a phe- nomenological dephasing/decay rate γ∗ in the definition of zX k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' To shorten the notation in the discussion be- low, we introduce the k−dependent photon-exciton en- ergy and linewidth detunings δϵk = Re[zC k − zX k ], and δγk = − Im[zC k − zX k ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Effective level attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' — In the limit of negligi- ble Rabi coupling, the argument of the square root can give rise to level attraction when δϵk ̸= 0 and γC k ∼ γX k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In particular, this effect provides a plausible and rela- tively simple explanation for recent puzzling photolumi- nescence measurements in semiconductor cavities which reported anomalous level attraction [29–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In the op- posite limit of strong Rabi coupling, gR ≫ γC k , γX k , δϵk, the argument of the square root in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (9) gives rise to level repulsion and to the conventional lower and up- per polariton modes [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Finally, we note that when γX k /γC k → 0, one recovers the case where the only radia- tive decay channel is from the cavity mode as considered in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In order to illustrate the phenomenon of level attrac- tion and to highlight how this can, in turn, lead to anomalous dispersion relations in planar cavities, we have plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' 2 two examples of power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' For the purposes of this illustration, we have used decay param- eters independent of the in-plane wavevector γC,X and a vanishing Rabi coupling gR/γC,X ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' The dispersion shown in panel (a) resembles the observation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' [29], which reported an anomalous inverted parabolic behavior of the lower line around k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' The dispersion displayed in panel (b) resembles that reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' [30], in which the inverted parabolic behavior of the lower line was ob- served at k ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We note that the power spectrum cal- culated here is not strictly equivalent to an experimental photoluminescence spectrum, as in such measurements some relaxation and partial thermalization take place which tends to favor the occupation of the lower energy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' It is worth noticing that the single mode version of the model in the regime gR/γC,X ≃ 0 could also poten- tially explain the level attraction reported in photolu- minescence measurements performed on semiconductor quantum-dot cavities under low pumping [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We emphasize that the present mechanism is solely due to the common photonic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Thus, it does not 1 1 1 6 (b 1 a 1 1 1 1 1 1 4 1 (03 2 0 1 10-2 10 10 100 100 2 2 2 4 0 0 4 mo m4 rely on material or temperature-dependent properties of solid-state emitters, nor on the pumping procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' EPs and BiCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' — Aside from the connection with recent experiments, the present model embeds additional interesting special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' First, we note that exceptional points [39–41] can arise when the square root in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (9) vanishes [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' This occurs when the following conditions for the photon-exciton detunings are both satisfied δEP ϵk = ±2 � γC k γX k , δEP γk = ∓2gR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (11) We note that the above condition would remain valid in the presence of an additional exciton broadening once it is incorporated into δγk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' and that when γX k = 0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (11) corresponds to the condition for the so-called weak to strong coupling crossover [19, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' EPs at such a crossover have been reported recently in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' [44] which relied on a polarization dependent Rabi splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In our case, the Rabi coupling is constant and we have an in- plane isotropy such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (11) can give rise to rings of EPs which, to leading order in mC/mX ≪ 1, occur at ℏkEP = � 2mC(δEP ϵk − δ) [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Another interesting configuration appears when δϵk = δBiC k ≡ gRδγk/ � γC k γX k , in which case the imaginary part of ωL k in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (9) vanishes exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' As a consequence, the corresponding modes remain undamped [46], which corresponds to the realisation of bound states in the con- tinuum [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' As for the EPs above, in a planar cavity, the condition δϵk = δBiC ϵk can give rise to a ring of BiCs occuring at ℏkBiC = � 2mC(δBiC k − δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' To our knowl- edge, the possibility to achieve such a ring is a novelty of the present isotropic system with respect to previous BiC realisations in other platforms [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' 3(a), we have plotted the power spectrum at k = 0 for different values of δ/δBiC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We can see that as the detuning approaches δBiC the width of the lower energy peak decreases and its amplitude increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' At exactly δ = δBiC, the width vanishes and the power spectrum (10) diverges as (ω − ωL 0 )−2 [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' To illus- trate the impact of the BiC condition on the dynamics, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' 3(b), we have plotted the amplitude of the exciton field as a function of time for the corresponding detun- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In this figure, we have used the solution of the memoryless equations of motion [37] with ⟨ˆx0⟩0 = 1 and the environment and cavity mode in their vacuum state as initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We can observe damped oscillations at short time, while at long time, we see that the closer δ is to δBiC, the lower is the decaying slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' At δ = δBiC, Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Vicinity of a bound state in the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (a) Power spectra for different detunings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (b) Dynamics of |⟨ˆx0⟩|2 with initial conditions ⟨ˆx0⟩0 = 1, ⟨ˆc0⟩0 = 0 and vacuum en- vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' The solid colored lines correspond to the same detunings used in panel (a) and the dashed black line corre- sponds to δ = δBiC given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (12b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' The other parameters used on both panels are gR/γC = 3, γX/γC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content='3, k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' the time dependent amplitudes can be expressed as |⟨ˆc0⟩(t)|2 = 1 + e− 2γ ℏ t − 2e− γ ℏ t cos[ ˜Ω ℏ t] γ2 γCγX, (12a) |⟨ˆx0⟩(t)|2 = γC γX + γX γC e− 2γ ℏ t + 2e− γ ℏ t cos[ ˜Ω ℏ t] γ2 γCγX, (12b) with γ = γC +γX and ˜Ω = gRγ/ � γCγX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We can clearly see that the amplitudes remain finite as t → ∞ when γC, γX ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In other words, the probabilities to find the emitter or the cavity excited do not vanish in the long time limit, which contrasts with the conventional behav- ior of damped vacuum Rabi oscillations in the absence of a common photonic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' — We have demonstrated that the presence of a common photonic environment can lead to an effective dissipative coupling between light and mat- ter in a nonideal cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Using a memoryless approxima- tion, we have obtained analytical results for the power spectrum and the complex eigenenergies of this open sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' This allowed us to highlight a potential connection with recent experiments that have reported anomalous level attraction and to provide simple conditions under which rings of EPs and BiCs are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Our results open intriguing perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' The inclusion of nonlin- earities, which are system-dependent, could unveil novel regimes to generate photon antibunching in cavity sys- tems [23, 51–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Furthermore, to our knowledge, the possibility for the emitters and the cavity modes to in- teract with a common photonic environment is not ac- counted for in usual laser theories [54], and it might be interesting to investigate whether it could affect some of their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In this context, one expects that the hy- brid light-matter BiC condition could favour low thresh- old lasing as evidenced with photonic BiCs [48, 49, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' 5 We gratefully acknowledge Dabrowka Bieganska and Maciej Pieczarka for fruitful exchanges that brought the problem of anomalous dispersions to our attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We also thank Matthias Wurdack, Elena Ostrovskaya, and Eliezer Estrecho for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We acknowledge support from the Australian Research Council Centre of Excellence in Future Low-Energy Electronics Technolo- gies (CE170100039).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' JL and MMP are also supported through Australian Research Council Future Fellowships FT160100244 and FT200100619, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' KC ac- knowledges support from an Australian Government Re- search Training Program (RTP) Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' [1] It is interesting to remember that the concept of spon- taneous emission predates the formal quantization of the electromagnetic field as well as the introduction of the word photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Purcell, Spontaneous Emission Probabilities at Ra- dio Frequencies, Physical Review 69, 681 (1946).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' [3] P.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Pfeiffer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Ballarini, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Nguyen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Gerace, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Sanvitto, Polariton Bose–Einstein condensate from a bound state in the continuum, Nature 605, 447 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' 1 SUPPLEMENTAL MATERIAL O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Bleu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Choo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Levinsen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Parish School of Physics and Astronomy, Monash University, Victoria 3800, Australia and ARC Centre of Excellence in Future Low-Energy Electronics Technologies, Monash University, Victoria 3800, Australia Details on the derivation using the input-output formalism Here, we provide some details on the derivation using the input-output formalism [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We start with the Heisenberg equations using the total Hamiltonian ˆH = ˆHS + ˆHE + ˆHSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We obtain the following evolution equations for the system operators: iℏ∂tˆck = � ˆck, ˆHS � + � dq κC k,qˆekq, (S1a) iℏ∂tˆxk = � ˆxk, ˆHS � + � dq κX k,qˆekq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S1b) In the same way, one obtains the evolution equations for the bath operators iℏ∂tˆekq = ϵE k,qˆekq + κC k,qˆck + κX k,qˆxk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S2) The formal solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S2) is of the form: ˆekq(t) = e− i ℏ ϵE k,q(t−t0)ˆekq(t0) + 1 iℏ � t t0 dt′e− i ℏ ϵE k,q(t−t′) � κC k,qˆck(t′) + κX k,qˆxk(t′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S3) Input-output relations If one introduces input and output operators defined as ˆeI kq = lim t0→−∞ e i ℏ ϵE k,qt0ˆekq(t0), (S4a) ˆeO kq = lim t→∞ e i ℏ ϵE k,qtˆekq(t), (S4b) one can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S3) to obtain the following input-output relation ˆeO kq = ˆeI kq + 1 iℏκC k,q ˆCk(ωE k,q) + 1 iℏκX k,q ˆ Xk(ωE k,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S5) Here, ˆ Ak(ω) = � dt eiωtˆak(t) corresponds to the Fourier transform of the operator ˆak(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Equations for the system operators Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S3) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S1) and taking the limit t0 → −∞, one obtains iℏ∂tˆck = � ˆck, ˆHS � − i � ∞ −∞ dt′ΓCC k (t − t′)ˆck(t′) − i � ∞ −∞ dt′ΓCX k (t − t′)ˆxk(t′) + ˆF C k (t), (S6a) iℏ∂tˆxk = � ˆxk, ˆHS � − i � ∞ −∞ dt′ΓXX k (t − t′)ˆxk(t′) − i � ∞ −∞ dt′ΓXC k (t − t′)ˆck(t′) + ˆF X k (t), (S6b) which corresponds to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (4) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Here, we have introduced the functions ΓAB k (τ) = θ(τ)1 ℏ � dq κA k,qκB k,qe− i ℏ ϵE k,qτ, (S7) 2 and the ‘force’ operators that are defined in terms of the input operators as ˆF C k (t) = � dq κC k,qe− i ℏ ϵE k,qtˆeI kq, (S8a) ˆF X k (t) = � dq κX k,qe− i ℏ ϵE k,qtˆeI kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S8b) Going to frequency space Taking the Fourier transform of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S6), one gets the matrix equation M(k, ω) � ˆCk(ω) ˆ Xk(ω) � = � ˆFC k (ω) ˆFX k (ω) � , (S9) with M(k, ω) = �ℏω − ϵC k + i˜ΓCC k (ω) −gR + i˜ΓCX k (ω) −gR + i˜ΓXC k (ω) ℏω − ϵX k + i˜ΓXX k (ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S10) The Fourier transform of the force operators are related to the input operator as ˆFC k (ω) = 2π � dq δ(ω − ωE k,q)κC k,qˆeI kq = 2πκC k,qpˆeI k,qpρk(ω), (S11a) ˆFX k (ω) = 2π � dq δ(ω − ωE k,q)κX k,qˆeI kq = 2πκX k,qpˆeI k,qpρk(ω), (S11b) where the wavevector qp is defined by the resonance condition ω = ωE k,qp and we have introduced ρk(ω) = (∂qωE k,q)−1 q=qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S12) Physically, ρk(ω) corresponds to the environment density of states which is well-defined for ω > ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' When ω < ck, ∂qωE k,q becomes imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' This is related to the fact that the out-of-plane wavevector q becomes imaginary and the corresponding modes are evanescent in the out-of-plane direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Thus, it is not possible to emit radiation at ω < ck and the evanescent modes remain guided in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Similarly, ˜ΓAB k (ω) denotes the Fourier transform of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' This takes the form ˜ΓAB k (ω) = 1 ℏ � dqπ � κA k,qκB k,qδ(ω − ωE k,q) � + i ℏp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' � dq κA k,qκB k,q ω − ωE k,q , (S13) where p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' denotes the Cauchy principal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' The real part is then Re[˜ΓAB k (ω)] = 1 ℏπκA k,qpκB k,qpρk(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S14) Note that, the real and imaginary parts of the functions ˜Γk(ω) are related to each other via Kramers-Kronig relations which is due to the causality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Unitary input-output relation Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S9) and (S11), the input-output relation (S5) can be expressed as ˆeO k,qp = S(k, ω)ˆeI k,qp (S15) with S(k, ω) = 1 − i2πρk(ω) ℏ � i,j [M(k, ω)]−1 ij κi k,qpκj k,qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S16) 3 Importantly, the transformation (S15) is unitary since one can check that |S(k, ω)| = 1, and hence, the output operators obey the same Bose commutation relations as the input operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' We note that the presence of a common photonic bath with a given density of state ρk in the present theory is important for the unitarity of the transformation (S15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' In particular, in the different situation where the emitter and cavity photon part interact with independent matter and photon baths, the resulting input-output transformation cannot be unitary in general since it is not equivalent to have an input from the matter or from the photon bath when these have different dispersion relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' This subtlety seems to have been missed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Power spectrum The system power spectrum is defined as the Fourier transform of the auto-correlation function I(k, ω) = � dτ � dt⟨ˆc† k(t)ˆck(t + τ) + ˆx† k(t)ˆxk(t + τ)⟩eiωτ (S17) = ⟨ ˆC† k(ω) ˆCk(ω) + ˆ X † k(ω) ˆ Xk(ω)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S18) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S10), it can be related to the input operators as I(k, ω) = ⟨ � ˆFC† k (ω), ˆFX† k (ω) � I(k, ω) � ˆFC k (ω) ˆFX k (ω) � ⟩, (S19) where the matrix I(k, ω) is given by I(k, ω) = � M(k, ω)−1�† M(k, ω)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S20) The spectrum I(k, ω) can be calculated analytically and is of the form I(k, ω) = � |M12|2 + |M22|2� ⟨ ˆFC† k ˆFC k ⟩ + � |M21|2 + |M11|2� ⟨ ˆFX† k ˆFX k ⟩ − 2 Re [(M11 + M22)M∗ 12] ⟨ ˆFC† k ˆFX k ⟩ | det[M(k, ω)]|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S21) Here, we have shortened the notation by removing the k, ω dependence in the numerator, and we have used the relation ⟨ ˆFC† k ˆFX k ⟩ = ⟨ ˆFX† k ˆFC k ⟩ since from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S11), one has ⟨ ˆFA† k (ω) ˆFB k (ω)⟩ = 4π2ρ2 k(ω)κA k,qpκB k,qp⟨ˆeI† k,qpˆeI k,qp⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S22) Memoryless approximation The set of evolution equations in (S6) shows that the system operators ˆck, ˆxk at time t are affected by all their past at times t′ < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' This influence is encoded in the functions ΓAB k (τ) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' As mentioned in the main text, one can simplify these equations by using a memoryless approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' One can motivate this approximation by inspecting the functions ΓAB k (τ) ΓAB k (τ) = θ(τ)1 ℏ � ∞ 0 dq κA k,qκB k,qe− i ℏ ϵE k,qτ = θ(τ)1 ℏ � ∞ 0 dω ρk(ω)κA k,qpκB k,qpe−iωτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S23) Since the relevant frequencies are large (these are in the vicinity of the emitter frequency ωX 0 ), we extend the integral toward negative frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Neglecting the frequency dependence of κA k,qp ≃ κA k and assuming ρk(ω) ≃ ρk(ωX 0 ) one has � ∞ −∞ dω ρk(ω)κA k,qpκB k,qpe−iωτ ≃ � ∞ −∞ dω ρk(ωX 0 )κA k κB k e−iωτ = ρk(ωX 0 )κA k κB k δ(τ), (S24) and thus one can approximate ΓAB k (τ) ≃ 1 ℏρk(ωX 0 )κA k κB k δ(τ) ≡ γAB k δ(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S25) 4 In this approximation Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S6) reduce to iℏ∂tˆck = � ϵC k − iγC k � ˆck + � gR − iγCX k � ˆxk + ˆF C k , (S26a) iℏ∂tˆxk = � ϵX k − iγX k � ˆxk + � gR − iγXC k � ˆck + ˆF X k , (S26b) with the decay parameters γC k = ρk(ωX 0 )κC k κC k /ℏ, γX k = ρk(ωX 0 )κX k κX k /ℏ, and γCX k = γXC k = � γC k γX k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S27) This leads to a simplification of the matrix (6) into (8) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Applying this approximation to the average of the product of force operators (S22) gives ⟨ ˆFA† k (ω) ˆFB k (ω)⟩ ≃ 4π2ℏρk(ωX 0 ) � γA k γB k ⟨ˆeI† k,qpˆeI k,qp⟩, (S28) and the power spectrum can be expressed as I(k, ω) ∝ � |˜g|2 + |ℏω − zX k |2� γC k + � |˜g|2 + |ℏω − zX k |2� γX k + 2 Re � (2ℏω − zC k − zX k )˜g∗� � γC k γX k | det[M(k, ω)]|2 ρk(ωX 0 )⟨ˆeI† k,qpˆeI k,qp⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S29) Neglecting the k dependence of ρk and using nk(ω) ∝ ⟨ˆeI† k,qpˆeI k,qp⟩, one obtains the power spectrum given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (10) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' For example, taking a constant nk(ω) = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' mimics a broadband input field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=', an illumination with white light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Field amplitude dynamics Assuming that the environment is initially in its vacuum and using the memoryless approximation, one can calculate analytically the evolution of the system field amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' Taking the average of the evolution equations in (S26), and using the notation introduced in the main text one has: iℏ∂t⟨ˆck⟩ = zC k ⟨ˆck⟩ + ˜g⟨ˆxk⟩, (S30a) iℏ∂t⟨ˆxk⟩ = zX k ⟨ˆxk⟩ + ˜g⟨ˆck⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S30b) The solutions are of the form: ⟨ˆck⟩(t) = 1 ℏωU k −ℏωL k {e−iωU k t � ⟨ˆxk⟩0˜g + ⟨ˆck⟩0(ℏωU k − zX k ) � − e−iωL k t � ⟨ˆxk⟩0˜g − ⟨ˆck⟩0(ℏωU k − zC k ) � }, (S31a) ⟨ˆxk⟩(t) = 1 ℏωU k −ℏωL k {e−iωU k t � ⟨ˆck⟩0˜g + ⟨ˆxk⟩0(ℏωU k − zC k ) � − e−iωL k t � ⟨ˆck⟩0˜g − ⟨ˆxk⟩0(ℏωU k − zX k ) � }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (S31b) Using the initial conditions ⟨ˆck⟩0 = 0 and ⟨ˆxk⟩0 = 1, one can obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' (12) of the main text for δ = δBiC and k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' It is interesting to note that the model used to fit the experimental Rabi oscillations reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' [56] contains a phenomenological “upper polariton decay/dephasing” term that seems to act as the present dissipative coupling (Im[˜g]) in the evolution equations for the field amplitudes (S30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} +page_content=' This suggests that the mechanism introduced in our work could also have played a role in this time-resolved experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE0T4oBgHgl3EQfSQDq/content/2301.02221v1.pdf'} diff --git a/LNAyT4oBgHgl3EQfTve8/vector_store/index.pkl b/LNAyT4oBgHgl3EQfTve8/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..dc172e5619f234339df4de71dce509f32604441a --- /dev/null +++ b/LNAyT4oBgHgl3EQfTve8/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c97f5e28a037c40784ffcc2d183fedf7a11c46a4a701f5f138dd7833a532ff57 +size 293241 diff --git a/LNE0T4oBgHgl3EQfzwJL/content/tmp_files/2301.02676v1.pdf.txt b/LNE0T4oBgHgl3EQfzwJL/content/tmp_files/2301.02676v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cc576b1ccd515abdcae58d69ecf060298fedd070 --- /dev/null +++ b/LNE0T4oBgHgl3EQfzwJL/content/tmp_files/2301.02676v1.pdf.txt @@ -0,0 +1,1041 @@ +Ionizing radiation escape enabled by galaxy merger in +reionization-era analog galaxy +Alexandra Le Reste*1, John M. Cannon2, Matthew J. Hayes1, John L. Inoue2, Amanda A. +Kepley3, Jens Melinder1, Veronica Menacho1, Angela Adamo1, Arjan Bik1, Timmy +Ejdetjärn1, Gyula I. G. Józsa4,5, Göran Östlin1, Sarah H. Taft2,6 +1. The Oskar Klein Centre, Department of Astronomy, Stockholm University, AlbaNova, SE- +10691 Stockholm, Sweden. 2. Department of Physics and Astronomy, Macalester College, +1600 Grand Avenue, Saint Paul, MN 55105, USA. 3. National Radio Astronomy +Observatory, 520 Edgemont Road, Charlottesville, VA 22903-2475, USA 4. Max-Planck- +Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany 5. Department of +Physics and Electronics, Rhodes University, P.O. Box 94, Makhanda, 6140, South Africa 6. +Minnesota Institute for Astrophysics, School of Physics & Astronomy, University of +Minnesota, 116 Church St. SE, Minneapolis, MN 55455, USA. + +Around 400 million years after the big bang, ultraviolet emission (Lyman Continuum, +LyC) from star-forming galaxies drove the reionization of the Universe. How this +radiation escapes the cold neutral gas (HI) of galaxies with sufficiently little absorption +to reionize the intergalactic medium is poorly understood. HI has never been mapped in +confirmed LyC-emitters, leaving major uncertainties on how LyC photons escape +galaxies and ionize the intergalactic medium. We imaged the 21cm HI emission of +nearby reionization-era analog galaxy Haro 11 to identify how ionizing radiation +escapes the neutral interstellar medium. We find that merger-driven interactions have +tidally displaced up to 82% of the neutral gas from the ultraviolet emission production +sites in the galaxy, allowing the escape of ionizing radiation to the intergalactic medium. +Increased galaxy interactions in the early Universe predicted by cosmological models +could contribute significantly to the reionization of the Universe. +The Universe underwent a major phase change in which almost all intergalactic hydrogen +was ionized about 13 billion years ago1. This reionization was driven by strong ultraviolet +(UV) emission from primeval star-forming galaxies2,3,4. LyC emission is absorbed by cold +neutral hydrogen gas in galaxies, hindering the escape of ionizing radiation. Observations of +galaxies at this epoch lack resolution, thus nearby analogs of early galaxies have been used to +understand the detailed physical processes responsible for reionization5,6,7,8,9. UV absorption +line measurements suggest that the main property driving LyC escape is a low covering +fraction of the neutral gas10,11, with LyC photons escaping through ionized channels within +the interstellar medium. However, measurements in the UV do not characterize the +interstellar medium in the full physical volume covered by dense neutral gas in galaxies. The +21cm line of Hydrogen is the only direct tracer of HI gas that can probe the entire extent of +the material inhibiting LyC escape. HI has never been mapped in confirmed LyC-emitters, +leaving major uncertainties on how LyC photons escape galaxies and ionize the intergalactic +medium12,13,14,15. Furthermore, even with upcoming state-of-the-art facilities, it will be +impossible to observe resolved HI in emission in galaxies at the epoch of reionization16. +Thus, observing HI in nearby galaxies with detected ionizing LyC emission is key to +revealing the gas removal and ionization mechanisms likely at play during the Epoch of +Reionization. + + +Figure 1: Neutral gas in Haro 11. Panel A: Colour-composite image of Haro 11 with 21cm +contours. High resolution (10”) MeerKAT 21cm emission is shown in blue with blue +contours overlaid, MUSE Hα emission is shown in red and HST optical stellar light in white. +The MeerKAT emission contours are shown with levels {1,2,3,4,5,6,7}× 10!"cm-2 +corresponding to the {5,10,15,20,25,30,35}× 𝜎 levels, with lower contours shown in darker +shades of blue. MeerKAT 21cm absorption contours are overlaid in dashed black lines, with +levels {-0.3,-0.2,-0.1} Jy/beam.km/s displayed. The MeerKAT synthesized beam is shown by +a white ellipse in the lower left corner of the image. Panel B: Stellar light in the HST image, +with star-forming knots identified by crosses. In all images, North is up and East is to the left. +Panel C: Low angular resolution (47”) MeerKAT 21cm spectrum integrated over the +detected 21cm, shown in black. The GBT spectrum is shown in gray for comparison. Panel +D: MeerKAT 21cm high angular resolution (10”) emission (blue) and absorption (black) +spectra. The VLA absorption spectrum is shown with gray dots for comparison. Vertical +dashed lines show the velocity centroid range of ionized gas around the star-forming knots, +where the sources emitting the absorbed radio continuum radiation are located. + +B +Knot B +Knot C +-33°32'20" +A +Knot A +40" +DEC (J2000) +33'00" +20" +40" +oh36m58s +56s +54s +52s +RA (J2000) +C +MeerKAT,lowres +Emission, high res +D +GBT +Absorption, high res +VLA absorption +Flux density [mJy] +2 +L +-1 +5800 +6000 +6200 +6400 +6600 +5800 +5900 +6000 +6100 +6200 +6300 +6400 +6500 +6600 +v [km/s] +v [km/s]Haro 11 is the closest (D=93Mpc)17 confirmed LyC-emitting galaxy and one of only three +known LyC-leaking galaxies that can be observed and resolved in 21cm with radio +interferometers18,19,20. The UV photons in this star-forming galaxy are produced in three +distinct regions21,22,23 labeled Knot A, B and C in Fig. 1. Most indirect studies agree on Knot +C as the likely producer of the bulk of LyC photons that were detected24,25,26 due to its Lyα +properties and gas covering fraction derived using lines of metals assumed to be mixed with +HI. Knot B and Knot A both produce large quantities of ionizing photons23,27,28, however +these photons are likely to escape at angles outside of the line of sight26,28,29. +The neutral gas distribution that can allow for the anisotropic escape of ionizing radiation out +of the knots has not been measured. The molecular gas distribution has been measured30, but +molecular gas is not a significant source of LyC opacity. Previous single dish observations of +Haro 11 detected the 21cm in emission13. However, the distribution of the gas has remained +unknown due to insufficient sensitivity of interferometric observations with only unresolved +absorption detections12,15. We observed the galaxy in 21cm with the MeerKAT telescope +which provides increased surface brightness sensitivity as compared to any HI interferometer +previously available. + +Results +The MeerKAT 21cm HI image and spectra are presented in Fig. 1 with the optical emission +of Haro 11 from MUSE28 tracing ionized gas and optical stellar light from HST31. We detect +and resolve HI in emission that was previously detected by the Green Bank Telescope +(GBT)13 and detect the unresolved absorption component detected by the NSF's Karl G. +Jansky Very Large Array (VLA)12,15. + +Neutral gas content and structure around LyC production regions +The 21cm integrated flux map and spectrum extracted around the star-forming knots which +produce the LyC photons is shown in Fig. 2. The prominent 21cm absorption component +indicates the presence of neutral gas in front of the optical body of the galaxy. The HI +absorption corresponds to a gas mass MHI,abs=3.30 ± 2.41 × 10# M⊙. The MeerKAT +observation does not resolve the 21cm absorption nor the radio continuum emission absorbed +by the neutral gas, but previous absorption line studies have shown that the interstellar +medium in front of the UV photon production sites is porous25,26. This porosity enables the +escape of LyC photons from their immediate neutral gas environment. Using archival VLA +data, we find radio continuum emission across all star-forming knots (extended data Fig. 3), +with an especially strong source co-spatial with star-forming Knot B (extended data Fig. 4, +5). X-Ray observations of the galaxy found two Ultra Luminous X-ray regions, respectively +overlapping with Knot C and Knot B29,32. This could indicate the presence of a low +luminosity Active Galactic Nucleus (AGN) in the X-Ray region co-spatial with Knot B. The +radio spectral index of the radio continuum source in Knot B is consistent with that of star- +forming regions33 (extended data Fig. 5). We cannot conclude on the presence or absence of +an AGN with the radio continuum observations. At the location of the 21cm absorption, there +is residual 21cm emission. Since it is not seen in absorption, the emitting 21cm gas at this +location must be located behind the optical body of the galaxy where the radio continuum +sources are located. The 21cm emission component co-spatial with the absorption +corresponds to a flux 0.052±0.053 Jy.km/s. + + + +Figure 2: Neutral gas around the LyC production sites. Left panel: MeerKAT High +angular resolution 21cm integrated flux density map. The dashed white line shows the limits +of the absorption component and the aperture used to extract the spectrum shown in the right +panel. The black crosses indicate the position of the star-forming knots. Right panel: +Spectrum of Haro 11 at the location of the absorption component, shown in black. The blue +line shows the Gaussian fit to the emission component seen at this location. Dashed black +lines show the velocity centroid range of the ionized gas around the star forming knots. +We compare the 21cm absorption feature velocity to that derived from the Hα centroid in +each knot and find that the neutral gas seen in absorption is blueshifted with maximum +velocity vmax = 145±42 km/s compared to the ionized gas. The peak of the 21cm emission +is redshifted by 71±30 km/s. The blueshifted 21cm absorption indicates that the neutral gas +in front of the star-forming knots is moving towards us, and the redshifted 21cm emission +shows the neutral gas behind the star-forming knots is moving away from us. This suggests +that the neutral gas around the optical body of the galaxy is outflowing. Previous studies have +already noted outflows of ionized gas in the galaxy23,26,28,34, and the presence of a multi-phase +outflow with partially neutral gas traced by metal absorption lines23. The MeerKAT +observation shows that the dense cold neutral gas located around the optical body of the +galaxy is also likely outflowing. +Large-scale neutral gas distribution +We then investigate the structure of the neutral gas emission around the galaxy. The bulk of +the HI emission is offset from the main body of the galaxy, in an elongated structure that is +~40 kpc long when projected on the plane of the sky. The HI emission components in +individual velocity channels are found to be connected spatially and spectrally (Fig 6, 7, 8). +The morpho-kinematics of the gas are consistent with the HI emission structure being a tidal +tail from a merger. The merger state of Haro 11 had already been found by studies of the +ionized gas kinematics of the galaxy35,36, but the impact of the merger on the neutral gas +distribution was unknown. The total 21cm emission flux is SHI,em=0.391 ± 0.042 Jy.km/s. +This yields a new mass estimate for the HI in emission MHI,em=7.99 ± 0.85 × 10# M⊙, a +value higher than that of 5.7 ± 0.8 × 10# M⊙ previously calculated with the single dish +measurement13. Only 13% of the gas mass associated to HI emission is co-spatial with the +star-forming knots. Taking into account the absorbing gas mass MHI,abs=3.30 ± +2.41 × 10# M⊙, the total atomic hydrogen gas mass of Haro 11 is MHI=1.1 ± +0.3 × 10% M⊙. While this value is higher than previous estimates, it remains smaller than +the ionized gas mass28 MHα = 1.8 × 10% M⊙ and the stellar mass35 of the galaxy M∗ = + +1 +-33°33'00" +Flux density [mJy] +DEC (J2000) +15" +2 +30" +-3 +oh36m54s +53s +52s +515 +5900 +6000 +6100 +6200 +6300 +6400 +6500 +6600 +RA (J2000) +v [km/s]1.6'".) +*!.+ × 10+" M⊙. Importantly, up to 82% of the total neutral gas mass in the galaxy is +offset from the ionizing emission production regions due to merger-driven interactions. +Discussion +The HI morphology and kinematics of Haro 11 demonstrates that pathways for LyC escape +exist from the interstellar to the circumgalactic medium. Additionally, these observations +provide a potential link between dwarf galaxy mergers and the detection of LyC emission +from galaxies. Galaxy mergers in low mass systems could play several roles in facilitating the +escape of LyC emission from galaxies. First, mergers create multiple star formation bursts37 +during the timescale of the interaction by repeatedly compressing the gas at the center of the +galaxy. These bursts create numerous massive stars, which produce the bulk of LyC emission +in galaxies; thus, mergers increase the intrinsic LyC photon production. Second, starbursts +are also responsible for the intense feedback that creates ionized channels enabling the escape +of LyC photons from their immediate environment38,39. Finally, merger interactions tidally +displace the material inhibiting LyC escape from the center of the galaxy. By creating regions +depleted of HI on large scales, galaxy mergers facilitate the anisotropic escape of LyC +photons out of the interstellar medium and into the intergalactic medium. + +In the local Universe, two other LyC-emitting galaxies have been detected that are close +enough to be imaged with interferometers: Tol 1247-23219 and Mrk 5420. Both of these +galaxies show prominent merger morphologies in the optical, however their neutral gas +content is significantly different, with Mrk 54 having a high HI mass40 MHI=1.6 ± +0.2 × 10+" M⊙, while Tol 1247-232 has an upper limit14 MHI<1 × 10% M⊙. Given the +morphology of the galaxies, it is likely that a fraction of their HI gas has been removed by +tidal interactions from the lines of sight where LyC is emitted. This would facilitate the +ionization by the starburst and explain how LyC emission can escape from environments with +very different interstellar medium properties. + +While they are not directly detected, many galaxies are considered to be LyC candidates due +to their Lyα line profiles or their high [OIII]/[OII] ratios, which indirectly trace LyC +escape7,9,41,42. Among these candidates, many display signs of ongoing merger events that +would facilitate LyC escape. Green pea galaxies are a class of objects considered excellent +analogs of high redshift LyC-emitting galaxies5,43. Single dish 21cm line measurements of +green peas galaxies have suggested that galaxies with high [OIII]/[OII] ratios are less likely +to be detected in 21cm, potentially indicating that galaxies with low HI mass are more likely +to leak LyC radiation44. However, about a fifth of the sample of green pea galaxies studied in +21cm have neutral gas and galaxy properties indicative of either recent gas accretion or the +presence of a gas-rich companion. Recently, HI imaging of the Green Pea galaxy J0213+0056 +has shown that a merger could explain Lyα escape in the galaxy, and could potentially lead to +LyC leaking45. + +The galaxy merger rate is difficult to measure in the early Universe, however a few +observational studies have found indications of an increase in merger rate at redshift 4 and up +to 646,47. From the simulation perspective, galaxy mergers also seem to be a promising +process facilitating the escape of ionizing LyC radiation during the epoch of reionization. +Indeed, hydrodynamical cosmological simulations predict an increase of mergers with +increasing redshift48. Semi-analytic galaxy formation models also show that galaxies found in +dense environments reside in larger ionized regions49, with galaxies having neighbors being +more likely to show Lyα emission at z=8. However, the environment of galaxies leaking LyC +emission has not yet been studied in a systematic way. While mergers are likely not the only + +mechanism responsible for LyC escape from galaxies, they are an effective process that +reunites several of the conditions required for ionizing photons to escape to the Intergalactic +medium. Their contribution to reionization and the impact of galaxy environment on LyC +escape should be assessed from the point of view of simulations and observations alike. + +References +1. Barkana, R., The First Stars in the Universe and Cosmic Reionization. Science 313, 931- +934 (2006). +2. Robertson, B., Ellis, R., Dunlop, J. et al. Early star-forming galaxies and the reionization +of the Universe. Nature 468, 49-55 (2010). +3. Kulkarni, G., Worseck, G., Hennawi, J. F., Evolution of the AGN UV luminosity function +from redshift 7.5. Monthly Notices of the Royal Astronomical Society 488, 1035-1065 (2019). +4. Dayal, P., Volonteri, M., Choudhury, T. R. et al. Reionization with galaxies and active +galactic nuclei. Monthly Notices of the Royal Astronomical Society 495, 3065-3078 (2020). +5. Izotov, Y., Orlitová, I., Schaerer, D., et al. Eight per cent leakage of Lyman continuum +photons from a compact, star-forming dwarf galaxy. Nature 529, 178-180 (2016). +6. Senchyna, P., Stark, D. 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The Source of Leaking Ionizing +Photons from Haro11: Clues from HST/COS Spectroscopy of Knots A, B, and C. The +Astrophysical Journal 912, 155 (2021) +27. Keenan, R. P., Oey, M. S., Jaskot, A. E. and James, B. L. Haro 11: Where is the Lyman +Continuum Source? The Astrophysical Journal 848 12 (2017). +28. Menacho V., Östlin, G., Bik. A. et al. The impact of stellar feedback from velocity- +dependent ionized gas maps - a MUSE view of Haro 11. Monthly Notices of the Royal +Astronomical Society 487, 3183-3198 (2019). +29. Gross, A. C., Prestwich, A., Kaaret, P. Resolving the ultraluminous X-ray sources in the +Lyα emitting galaxy Haro 11. Monthly Notices of the Royal Astronomical Society 505, 610- +627 (2021). +30. Gao, Y., Gu, Q., Shi, Y., et al. The molecular gas resolved by ALMA in the low- +metallicity merging dwarf galaxy Haro 11. Astronomy & Astrophysics 661, A136 (2022). +31. Östlin, G., Hayes, M., Kunth, D., et al. The Lyman Alpha Morphology of Local Starburst +Galaxies: Release of Calibrated Images. The Astronomical Journal 138, 923-940 (2009). +32. Prestwich, A. H., Jackson, F., Kaaret, P. et al. Ultra-luminous X-Ray Sources in HARO +11 and the Role of X-Ray Binaries in Feedback in Lyα Emitting Galaxies. The Astrophysical +Journal 812, 166 (2015). +33. Klein, U., Lisenfeld, U., and Verley, S. Radio synchrotron spectra of star-forming +galaxies. Astronomy & Astrophysics 611, A55 (2018). +34. Menacho, V., Östlin, G., Bik, A., et al. Ionized gas properties of the extreme starburst +galaxy Haro 11 - temperature and metal abundance discrepancies. Monthly Notices of the +Royal Astronomical Society 506, 1777-1800 (2021). +35. Östlin, G., Amram, P., Bergvall, N. et al. Dynamics of blue compact galaxies, as revealed +by their Hα velocity fields. Astronomy & Astrophysics 374, 800-823 (2001). +36. Östlin, G., Marquart, T., Cumming, R. J. et al. Kinematics of Haro 11: The miniature +Antennae. Astronomy & Astrophysics 583, A55 (2015) +37. Lahén, N., Naab, T., Johansson, P. H., et al. The GRIFFIN Project—Formation of Star +Clusters with Individual Massive Stars in a Simulated Dwarf Galaxy Starburst.The +Astrophysical Journal 891, 2 (2020). +38. Borthakur, S., Heckman, T. M., Leitherer, C., and Overzier, R. A local clue to the +reionization of the universe. Science 346, 216-219 (2014). + +39. Clarke, C. and Oey, M. S. Galactic porosity and a star formation threshold for the escape +of ionizing radiation from galaxies. Monthly Notices of the Royal Astronomical Society 337, +1299-1308 (2002). +40. Haynes, M. P., Giovanelli, R., Kent, B. R., et al. The Arecibo Legacy Fast ALFA Survey: +The ALFALFA Extragalactic H I Source Catalog. The Astrophysical Journal 861, 49 (2018). +41. Verhamme, A., Orlitová, I., Schaerer, D. and Hayes, M. Using Lyman-α to detect +galaxies that leak Lyman continuum. Astronomy & Astrophysics 578, A7 (2015). +42. Flury, S., Jaskot, A. E., Ferguson, H. C., et al. The Low-redshift Lyman Continuum +Survey. II. New Insights into LyC Diagnostics. The Astrophysical Journal 930, 126 (2022). +43. Cardamone, C., Schawinski, K., Sarzi, M., et al. Galaxy Zoo Green Peas: discovery of a +class of compact extremely star-forming galaxies. Monthly Notices of the Royal +Astronomical Society 399, 1191-1205 (2009). +44. Kanekar, N., Ghosh, T., Rhoads, J., et al. The Atomic Gas Mass of Green Pea Galaxies. +The Astrophysical Journal Letters 913, L15 (2021). +45. Purkayastha, S., Kanekar, N., Chengalur, J. N., et al. A Green Pea Starburst Arising from +a Galaxy-Galaxy Merger. The Astrophysical Journal Letters 933, L11 (2022). +46. Tasca, L. A. M., Le Fèvre, O., López-Sanjuan, C., et al. Evidence for major mergers of +galaxies at 2 ≲ z < 4 in the VVDS and VUDS surveys. Astronomy & Astrophysics 565, A10 +(2014). +47. Duncan, K., Conselice, C. J., Mundy, C., et al. Observational Constraints on the Merger +History of Galaxies since z ≈ 6: Probabilistic Galaxy Pair Counts in the CANDELS Fields. +The Astrophysical Journal 876, 110 (2019). +48. Rodriguez-Gomez, V., Genel, S., Vogelsberger, M. et al. The merger rate of galaxies in +the Illustris simulation: a comparison with observations and semi-empirical models. Monthly +Notices of the Royal Astronomical Society 449, 49-64 (2015). +49. Qin, Y., Wyithe, J. S. B., Oesch, P. A., et al. Dark-ages reionization and galaxy formation +simulation XX. The Lyα IGM transmission properties and environment of bright galaxies +during the epoch of reionization. Monthly Notices of the Royal Astronomical Society 510, +3858-3866 (2022). + + + +Methods + +Throughout this paper we assume a Hubble constant H0 = 67.4 ± 0.5 km/s/Mpc and a matter +density1 Ωm = 0.315. Using the redshift2 of Haro 11 (z=0.0206), we derive a luminosity +distance of 93 Mpc. Literature values mentioned in this paper have been corrected for this +value. + +MeerKAT observations and data reduction +Haro 11 was observed with the MeerKAT interferometer for program SCI-20210212-AL-01 +(P.I. Le Reste). The 8.8 hours of on-source integration time were split between two observing +sessions (2021 Feb 20 and 2021 Aug 14). The correlator was configured in the C856M32k +mode, wherein an 856 MHz wide bandpass is separated into 32,000 channels. J0408-6545 +and J1939-6342 were used for primary flux and bandpass calibrations for the Feb 20 and the +Aug 14 data, respectively; J0025-2602 was used as secondary calibrator in both observing +sessions. To enable study of the HI spectral line, a narrow (20 MHz) bandpass, centered on +the recessional velocity of Haro 11 derived from optical emission lines2, was extracted from +the full dataset. The 26.123 kHz channel width produces a native velocity resolution of 5.5 +km/s/ch. Data reduction and calibration followed standard prescriptions in the CASA 5.6 +environment3. Imaging was performed with CASA 6.4 using the AUTO-MULTITHRESH +algorithm4 within the CASA TCLEAN task at three different angular resolutions by using +tapering in the uv-plane. The velocity resolution was set to 10 km/s. The masks generated by +AUTO-MULTITHRESH were examined by hand, and deep cleaning (to the level of 0.5 σ) +was performed inside regions that contain real flux. The final datacubes have beam sizes and +rms noise values as follows: 47.3” ×45.7” and σ = 0.18 mJy/beam (hereafter, “low +resolution”, produced using a 2.5kλ uv-taper); 24.6” ×21.7” and σ = 0.16 mJy/beam +(hereafter, “medium resolution”, produced using a 7k uv-taper); 11.2” ×9.7” and σ = 0.17 +mJy/beam (hereafter, “high resolution”, produced with no uv-taper). + +21cm detection +To detect 21cm signal in the cubes, we used the dedicated 21cm detection software Source +FInding Algorithm5 (SoFIA, version 1.3.3). We used the Smooth+Clip algorithm with 4.5σ +threshold using the Median absolute Deviation rms mode in order to include HI absorption in +the mask. We used spatial smoothing with Gaussian kernels and spectral smoothing boxcar +kernels (with either no smoothing or 3-channel wide (~30km/s) smoothing). For each of the +datasets, we applied kernels with no smoothing, 10” smoothing (average beam size of the +high resolution dataset), 23” smoothing (average beam size of the intermediate resolution +dataset) and 46” smoothing (average beam size of the low resolution dataset). A reliability +threshold is often used to increase completeness and reliability of the detections, however +since this method relies on statistics on negative pixels and our cube presents prominent +absorption, we did not use it. Eye inspection of the masked cubes confirmed that the 21cm +detection parameters enabled the recovery of all the HI emission and absorption. + +HI Mass estimation +To estimate the gas mass associated with the 21cm emission, we use the high angular +resolution cube which best resolves the absorption in the cube, that allows us to separate the +emission and absorption components. To estimate the gas mass associated with 21cm +emission-only, we mask the absorption component from the cube. We fit a double Gaussian +function to the emission profile using least-square fitting weighted by errors. We integrate +over the fitted profile and find a flux value SHI,em=0.391 ± 0.042 Jy. km/s, taking into + +account a conservative flux calibrator error of 10%, the pixel variance in the cube and the +uncertainty on fitting parameters. To determine the emitting gas mass, we assume that the gas +is optically thin and use MHI = 2.36 × 10, D2[Mpc] SHI[Jy.km/s]. We find an HI gas mass +MHI,em=7.99 ± 0.85 × 10# M⊙ for the 21cm emission component. + +Estimating the gas mass associated with 21cm absorption yields more uncertainties, due to +the unknown size of the absorbing component, spin temperature and covering fraction. +The column density NHI of the absorbing gas can be written as a function of the spin +temperature Ts and the optical depth 𝜏(𝑣) as NHI[cm-2] = 1.823 × +10+# Ts[K] ∫ 𝜏(𝑣) d𝑣 + +. +[km/s]. The optical depth can be expressed as a function of the +observed change in flux density due to the absorption ΔS, the continuum flux density Sc and +the covering fraction of the absorbing gas 𝑓 as 𝜏(𝑣) = − ln(1 + ΔS(𝑣)/(𝑓 Sc)). Finally, the +absorbing gas mass calculation requires an assumption on the volume covered by the gas. We +express it as a function of the area covered by the gas 𝐴/01, the column density, the mass of +the Hydrogen atom 𝑚2 and assume the same volume filling factor of 𝑓3/! as in previous +21cm absorption study6, such that MHI,abs = NHI,abs 𝐴/01 𝑚2 𝑓3/!. +The radio continuum images and previous VLA absorption study indicate that the radio +emission is produced by an unresolved source, and the distribution of the gas within the area +is unknown. Thus, we use the beam size (7.27” ×10.07”) of the previous VLA absorption +observation which reached a better resolution, as an upper limit to the area covered by the +gas. Observations and modeling6,7 of the ISM in Haro 11 suggests that the spin temperature is +found within the range 91-200 K. We use values from the measurement of UV absorption +lines of metals which are a proxy for the neutral gas content of galaxies8 to constrain the +covering fraction between f=0.55 and f=0.95. We find that the mass in absorption ranges +from 8.9 × 105 M⊙ to 5.7 × 10# M⊙. Adding the mass seen in emission to the midpoint of +the absorption mass, we find a total HI gas mass MHI=1.1 ± 0.3 × 10% M⊙. + +HI distribution at the location of the absorption feature +We extract a spectrum from the non-masked cube at the location of the absorption component +to assess the content and kinematic structure of the neutral gas at this location. This also +corresponds to the location of the optical body of the galaxy and of the star-forming knots +producing LyC emission. The spectrum and aperture are shown on Fig. 2. We find that 21cm +emission is seen at the location of the absorption feature. We have shown the velocity range +of the sources emitting the radio continuum emission that is being absorbed, and find that the +peak of the 21cm emission is redshifted by 71±30 km/s, depending on the location +considered for the radio continuum source. HI gas that is located in front of the source +emitting the radio continuum is absorbed, while gas located in the background is emitting +21cm. The respective redshift of the emission and blueshift of the absorption indicate that we +are looking at a neutral gas outflow around the optical body of the galaxy. We fit the +emission component with a Gaussian using least-square fitting weighted by errors and +integrate the Gaussian profile. We find a flux of 0.052±0.053 Jy.km/s in the emission +component cospatial with the absorption, which corresponds to 13% of the 21cm emission +flux of the galaxy. Most of the HI emission is thus found in the tidal tail, away from the star- +forming knots. + +HI 21cm morpho-kinematics +Channel maps of the 21cm of Haro 11 are shown for the high-resolution, intermediate- +resolution and low-resolution cubes in Fig. 6, 7 and 8. The emission is connected (both + +spatially and spectrally) and is offset from the center of the galaxy, indicating that it traces a +tidal tail. + +Archival VLA continuum data reduction +The VLA continuum data was taken by the National Radio Astronomy Observatory’s Karl G. +Jansky Very Large Array (VLA) as part of project VLA/15B-197 (PI: Kepley). The +observing frequencies and parameters are given in Table 1. For all three bands, the strategy +was to observe a flux/bandpass calibrator once during each scheduling block and then +to periodically observe a phase calibrator. Additionally for the X and Ka-band data, a +pointing scan was run at the beginning of a scheduling block. The data were calibrated using +VLA pipeline version Pipeline-Cycle3-R1-B with CASA version 4.3.1 r32491 and imaged in +CASA 6.2.1-7, using multi-term, multi-frequency synthesis9 with nterms=2 to account for the +spectral curvature of the data and Briggs robust weighting of 0.5. In addition, the S-band data +used w-projection to correct for the curvature of the sky over the larger field of view. We +used self-calibration to improve the image dynamic range for all three bands. All three +datasets had a phase self-cal performed. The Ka-band data had an additional scan-length +amplitude self-cal. The final images were primary beam corrected accounting for the +variation of the primary beam as a function of frequency across the band. + +Haro 11 radio continuum emission +We present the VLA radio continuum images of Haro 11 in Fig. 3. The radio continuum flux +density of Haro 11 was calculated in the VLA archival images at 3GHz, 9.8 GHz and 33 GHz +using the lowest contour corresponding to the 3σ level. This was supplemented by values +taken from the literature6,10. The flux density values for Haro 11 are presented in Table 2. We +also extract the flux density in Knot B only in the VLA archival images. To do so, we fit a +2D Gaussian using the CASA task IMFIT, fixing the value of the peak and allowing for a +zero-point offset. The flux density values of Knot B only are presented in Table 3. The error +values are calculated using a conservative flux calibrator error of 10 %, which drives the +uncertainty. We show the extraction regions used for the full galaxy and Knot B on Fig. 5, +along with the radio continuum SED. The flux values agree with Knot B being the location +of the highest radio continuum emission region, but the continuum emission in Knot B is +unresolved, even with Ka-band imaging. We compare the location of the Knot B radio +continuum emission with the optical image of Haro 11 in Fig. 4. The radio spectral energy +distributions (SED) of Haro 11 and the individual components is presented on Fig. 5. + +We fit the radio SED of Haro 11 using a combination of thermal and non-thermal radiation: +S(𝜈) = 𝑆678 + 𝑆78 = 𝑐+ 𝜈'9 + 𝑐! 𝜈'".+. We use a least-square fitting approach weighted by +errors and find the following parameters: 𝑐+ = 29.9 ± 4.5 mJy, α =0.6 ± 0.27, 𝑐! = 2.8 ± +5.3 mJy. The non-thermal spectral slope is consistent with radio emission from star-forming +regions11. We fit the SED of Knot B with non-thermal radiation only, given the number of +points is insufficient for a combined fit, and will lead to overfitting. We find the following +parameters minimize the chi-square: 𝑐+ = 26.3 ± 5.1 mJy, α =0.5 ±0.1 mJy. The flatter +spectral index of Knot B is consistent within errors with that of the galaxy. + +Methods references +1. Planck Collaboration, Planck 2018 results. VI. Cosmological parameters. Astronomy & +Astrophysics 641, A6 (2020). +2. Bergvall, N., Masegosa, J., Östlin, G. and Cernicharo, J. LWS spectroscopy of the +luminous blue compact galaxy Haro 11. Astronomy & Astrophysics 359, 41-50 (2000). + +3. McMullin, J.P., Waters, B., Schiebel, D., et al. CASA Architecture and Applications. +Astronomical Data Analysis Software and Systems XVI ASP Conference Series 376, 127 +(2007). +4. Kepley, A., Tsutsumi, T., Brogan, C. et al. Auto-multithresh: A General Purpose +Automasking Algorithm. Publications of the Astronomical Society of the Pacific 132, 024595 +(2020). +5. Serra, P., Westmeier, T., Giese, N., et al. SOFIA: a flexible source finder for 3D spectral +line data. Monthly Notices of the Royal Astronomical Society 448, 1922-1929 (2015). +6. MacHattie, J., Irwin, J., Madden, S. et al. Detection of H I absorption in the dwarf galaxy +Haro 11. Monthly Notices of the Royal Astronomical Society: Letters 438, L66-L70 (2014). +7. Cormier, D., Lebouteiller, V., Madden, S. C., et al. The nature of the interstellar medium +of the starburst low-metallicity galaxy Haro 11: a multi-phase model of the infrared emission. +Astronomy & Astrophysics 548, A20 (2012). +8. Östlin, G., Rivera-Thorsen, T. E., Menacho, V. et al. The Source of Leaking Ionizing +Photons from Haro11: Clues from HST/COS Spectroscopy of Knots A, B, and C. The +Astrophysical Journal 912, 155 (2021). +9. Rau, U, and Cronwell, T .J. A multi-scale multi-frequency deconvolution algorithm for +synthesis imaging in radio interferometry. Astronomy & Astrophysics 532, A71 (2011). +10. Schmitt, H. R., Calzetti, D., Armus, L. et al. Multiwavelength Star Formation Indicators: +Observations. The Astrophysical Journal Supplement Series 164, 52-80 (2006). +11. Klein, U., Lisenfeld, U., and Verley, S. Radio synchrotron spectra of star-forming +galaxies. Astronomy & Astrophysics 611, A55 (2018). + +Acknowledgments +ALR thanks Christian Binggeli, Alba Covelo Paz and Mohammad Javad Shahhoseini for +their contributions to the initial 21cm interferometric observing proposal for Haro 11. +JMC and JLI are supported by NSF/AST-2009894. JMC and SHT acknowledge support from +Macalester College. +MJH is fellow of the Knut & Alice Wallenberg foundation. +AA acknowledges financial support from the Swedish Research Council (VR) under grant +2021-05559. +GÖ acknowledges financial support from the Swedish Research Council (VR) and the +Swedish National Space Agency (SNSA). +The MeerKAT telescope is operated by the South African Radio Astronomy Observatory, +which is a facility of the National Research Foundation, an agency of the Department of +Science and Innovation. +The National Radio Astronomy Observatory is a facility of the National Science Foundation +operated under cooperative agreement by Associated Universities Inc. +This study uses observations made with ESO Telescopes at the La Silla Paranal Observatory +under programme IDs 094.B-0944(A) and 096.B-0923(A). + +Author contributions +ALR designed the MeerKAT 21cm observing proposal of Haro 11, analyzed and interpreted +the data. JMC assisted with technical preparation of the 21cm observing proposal and +reduced and imaged the MeerKAT 21cm data. MJH made major contributions to the +scientific justification of the observing proposal and to the interpretation of the data. JLI +reduced the MeerKAT 21cm data. AAK reduced the VLA continuum data and contributed to +the radio continuum data analysis. JM contributed the HST image of Haro 11. VM reduced +the MUSE data and produced the ionized gas maps. AA, AB and GÖ contributed to the +scientific justification of the 21cm observing proposal and interpretation of the data. TE + +designed hydrodynamical simulations of the Haro 11 merger which were used for +interpretation of the 21cm data. GIGJ provided major contributions during the technical +preparation of the 21cm observing proposal. The 21cm observing proposal justification was +partly based on results obtained by SHT with the VLA. All authors contributed to the writing +of the manuscript. + +Data Availability +The datasets generated and analysed in the current study will be made publicly available in +the SARAO DataCite repository. + +Competing interest declaration +The authors declare they have no competing financial interests. + + + +Extended data + + +Figure 3: VLA continuum images. From left to right: S-band (3 GHz) continuum image, +X-band (9.8 GHz) continuum image, Ka-band (33 GHz) continuum image. Intensities are +displayed with a logarithmic stretch to highlight the full distribution of the radio continuum +emission. The white contours displayed correspond to the {3,5,10,20,40,80,160} levels. The +synthesized beam is represented by a white ellipse in the bottom left corner. + + +Figure 4: Radio continuum and optical stellar emission comparison. HST f435W filter +image with VLA continuum emission contours overlaid in pink for the S-band (3GHz), in +orange for the X-band (9.8GHz), and in yellow for the Ka-band (33GHz). The contours for +each band are also presented on Figure 5, and correspond to the Gaussian aperture used to +extract the flux of Knot B. Black crosses indicate the position of the three star-forming knots, +the red cross indicates the position of the 21cm absorption centroid. + +S-band (3GHz) +X-band (9.8GHz) +Ka-band (33GHz) +-33°33'05" +10" +(J2000) +15" +DEC +20" +25" +30" +oh36m53.5s +53.0s +52.5s +52.0s +51.5s 0h36m53.5s +53.0s +52.5s +52.0s +51.5s 0h36m53.5s +53.0s +52.5s +52.0s +51.5s +RA (J2000)-33°33'12" +15" +DEC (J2000) +18" +21" +VLA 3GHz +VLA 9.8GHz +VLA 33GHz +X +SF knots +X +2lcm abs +24" +0h36m53.0s +52.5s +52.0s +RA (J2000) + +Figure 5: Radio spectral energy distribution (SED) and VLA continuum images. The +left panels show the SED for Haro 11 (top) and star-forming knot B only (bottom). For Haro +11, we fit the data using a combination of thermal and non-thermal radiation. The combined +fit is shown in black, the thermal and non-thermal components are shown in gray solid and +dashed lines respectively. For Knot B, we fit the data using the non-thermal emission only +(gray dashed line), since we do not have enough data points to make a combined fit. The right +panels show the VLA archival images with contours used for flux extraction overlaid. + + +S-band (3GHz) +Haroll +non thermal fit +thermal fit +Combined fit +Flux density [mly] +101 +X-band (9.8GHz) +100 +100 +101 + [GHz] +Knot B +non thermal fit +Flux density [mJy] +101 +Ka-band (33GHz) +100 +100 +101 +v [GHz] +Figure 6: MeerKAT high angular resolution 21cm channel map. Levels displayed +correspond to the -10σ, -3σ, 3σ and 5σ levels, respectively shown in black, gray, light blue +and dark blue. The position of Knot B is indicated by a black cross. The two other knots are +extremely close to Knot B, they have not been represented here to help readability. + +-33°32'00" +10 kpc +6028 km/s +6038 km/s +6048 km/s +6058Km/s +6069 km/s +30" +.00. +x +x +x +x +30" +.00 +-33°32'00" +6079 km/s +6089 km/s +6099 km/s +6109 km/s +6120 km/s +30" +.00. +30" +34'00" +-33°32'00" +6130 km/s +6140 km/s +6150 km/s +6160 km/s +6171 km/s +30" +.00. +? +- +0 +(J2000) +30" +34'00" +33°32°00 +DEC +6181 km/s +6191 km/s +6201 km/s +6211 km/s +6222 km/s +30" +0 +.00, +D +? +?. +30" +34'00" +-33°32'00" +6232 km/s +6242 km/s +6252 km/s +6262 km/s +6273 km/s +30" +0Q +.00. +30", +34'00" +6283 km/s +6293 km/s +6303 km/s +6313 km/s +6323 km/s +30" +33'00" +x +30 +34'00* +oh36m58s 56s +54s +52s0h36m58 56s +54s +52*0h36m58* 56s +54s +52s0h36m58* 56s +54s +52*0h36m58* 56s +5 4s +52§ +RA(J2000) + +Figure 7: MeerKAT intermediate angular resolution 21cm channel map. Levels +displayed correspond to the -10σ, -3σ, 3σ and 5σ levels, respectively shown in black, gray, +light blue and dark blue. The position of Knot B is indicated by a black cross. The two other +knots are extremely close to Knot B, they have not been represented here to help readability. + + +-33°32'00" +10 kpc +6028km/s +6038km/s +6048 km/s +6058km/s +6069km/s +30", +.00. +x +X +x +XO +30" +x +34'00" +-33°32'00" +6079-km/s +6089 km/s +6099 km/s +6109km/s +6120km/s +30" +.00.E +? +30", +34'00" +-33°32'00" +6130km/s +6140-km/s +6150km/s +6160km/s +6171km/s +30" +33'00" +(J2000) +30" +34'00" +0 +33°32'00 +6181km/s +6191km/s +6201km/s +6211km/s +6222_km/s +DEC +30" +D +.00. +30" +C +34'00" +0 +6232km/s +6242km/s +6252km/s +6262km/s +6273km/s +30" +0 +33'00* +30" +.00, +-33°32'00" +6283kfm/s +6293km/s +6303km/s +6313km/s +6323km/s +30" +0 +.00. ++ +30" +.00 +0 +0h37m00'36m57s +54s +510h37m0036m57s +545 +510h37m0036m57s +545 +510h37m0036m57s +545 +510h37mo036m57s +54s +51 +RA(J2000) +Figure 8: MeerKAT low angular resolution 21cm channel map. Levels displayed +correspond to the -10σ, -3σ, 3σ and 5σ levels, respectively shown in black, gray, light blue +and dark blue. The position of Knot B is indicated by a black cross. The two other knots are +extremely close to Knot B, they have not been represented here to help readability. + + +Band +S-band +X-band +Ka-band +Configuration +B +CnB +DnC +Central Frequency (GHz) +3 +9.8 +33.0 +Bandwidth (GHz) +1.75 +4.0 +7.9 +Channel width (MHz) +1 +1 +1 +Number of channels/spectral +window +128 +128 +128 + +10 kpc +6028 km/s +6038 km/s +6048 km/s +6058 km/s +6069 km/s +33*32' +EE +x +34' - +6079 km/s +6089 km/s +6099 km/s +6109 km/s +6120 km/s +33°32' +.EE +x +34' - +0 +0 +6130 km/s +6140 km/s +6150 km/s +6160 km/s +6171 km/s +33°32', +33' +(J2000) +34' +EC +6181 km/s +6191 km/s +6201 km/s +6211 km/s +6222 km/s +33°32' +EE +34' +Q +6232 km/s +6242 km/s +6252 km/s +6262 km/s +6273 km/s +33°32' +34' - +6283 km/s +6293 km/s +6303 km/s +6313 km/s +6323 km/s +33*32' +X +34' +oh37m0036m56s525 +48s oh37mo036m56s 525 +48s oh37mo036m56s52s +48s 0h37m0036m56*52s +48s oh37m0036m56s52s +RA(J2000)Number of spectral windows 16 +32 +64 +Flux/BP calibrator +3C48 +3C48 +3C48 +Phase calibrator +J0024- +4202-S +J0012- +3954 +J0012-3954 +Phase calibrator cadence +14min +14min +7min +Time on Source (hr) +0.26 +0.24 +3.1 +Observing Dates +2016-05- +31 +2016-05- +05 +2016-01-10, 2016-01-13, 2016- +01-23, 2016-01-23/24 +Beam +5.3" x +1.65" +3.22" x +1.51" +2.31" x 1.35" +Image noise (microJy/beam) +15.6 +9.1 +6.7 + +Table 1: Very Large Array radio continuum observation parameters. + +[GHz] +1.4 +1.48 +3 +4.89 +8.46 +9.8 +33 +S[mJy] 27.0±0.2 27.2±0.9 18.4±1.8 15.1±0.5 10.1±0.2 9.6±1.0 6.4±0.6 +Table 2: Radio continuum flux density values. + +[GHz] +3 +9.8 +33 +S[mJy] 16.5±1.7 8.0±0.8 5.1±0.5 +Table 3: Radio continuum flux density values in Knot B. + + + + + diff --git a/LNE0T4oBgHgl3EQfzwJL/content/tmp_files/load_file.txt b/LNE0T4oBgHgl3EQfzwJL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c5eeacea4ff774edd89b11648da497adb940a0cb --- /dev/null +++ b/LNE0T4oBgHgl3EQfzwJL/content/tmp_files/load_file.txt @@ -0,0 +1,961 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf,len=960 +page_content='Ionizing radiation escape enabled by galaxy merger in reionization-era analog galaxy Alexandra Le Reste*1, John M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Cannon2, Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Hayes1, John L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Inoue2, Amanda A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Kepley3, Jens Melinder1, Veronica Menacho1, Angela Adamo1, Arjan Bik1, Timmy Ejdetjärn1, Gyula I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Józsa4,5, Göran Östlin1, Sarah H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Taft2,6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The Oskar Klein Centre, Department of Astronomy, Stockholm University, AlbaNova, SE- 10691 Stockholm, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Department of Physics and Astronomy, Macalester College, 1600 Grand Avenue, Saint Paul, MN 55105, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903-2475, USA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Max-Planck- Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Department of Physics and Electronics, Rhodes University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Box 94, Makhanda, 6140, South Africa 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Minnesota Institute for Astrophysics, School of Physics & Astronomy, University of Minnesota, 116 Church St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' SE, Minneapolis, MN 55455, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Around 400 million years after the big bang, ultraviolet emission (Lyman Continuum, LyC) from star-forming galaxies drove the reionization of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' How this radiation escapes the cold neutral gas (HI) of galaxies with sufficiently little absorption to reionize the intergalactic medium is poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' HI has never been mapped in confirmed LyC-emitters, leaving major uncertainties on how LyC photons escape galaxies and ionize the intergalactic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We imaged the 21cm HI emission of nearby reionization-era analog galaxy Haro 11 to identify how ionizing radiation escapes the neutral interstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We find that merger-driven interactions have tidally displaced up to 82% of the neutral gas from the ultraviolet emission production sites in the galaxy, allowing the escape of ionizing radiation to the intergalactic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Increased galaxy interactions in the early Universe predicted by cosmological models could contribute significantly to the reionization of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The Universe underwent a major phase change in which almost all intergalactic hydrogen was ionized about 13 billion years ago1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' This reionization was driven by strong ultraviolet (UV) emission from primeval star-forming galaxies2,3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' LyC emission is absorbed by cold neutral hydrogen gas in galaxies, hindering the escape of ionizing radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Observations of galaxies at this epoch lack resolution, thus nearby analogs of early galaxies have been used to understand the detailed physical processes responsible for reionization5,6,7,8,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' UV absorption line measurements suggest that the main property driving LyC escape is a low covering fraction of the neutral gas10,11, with LyC photons escaping through ionized channels within the interstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' However, measurements in the UV do not characterize the interstellar medium in the full physical volume covered by dense neutral gas in galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The 21cm line of Hydrogen is the only direct tracer of HI gas that can probe the entire extent of the material inhibiting LyC escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' HI has never been mapped in confirmed LyC-emitters, leaving major uncertainties on how LyC photons escape galaxies and ionize the intergalactic medium12,13,14,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Furthermore, even with upcoming state-of-the-art facilities, it will be impossible to observe resolved HI in emission in galaxies at the epoch of reionization16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Thus, observing HI in nearby galaxies with detected ionizing LyC emission is key to revealing the gas removal and ionization mechanisms likely at play during the Epoch of Reionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Figure 1: Neutral gas in Haro 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Panel A: Colour-composite image of Haro 11 with 21cm contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' High resolution (10”) MeerKAT 21cm emission is shown in blue with blue contours overlaid, MUSE Hα emission is shown in red and HST optical stellar light in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The MeerKAT emission contours are shown with levels {1,2,3,4,5,6,7}× 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' "cm-2 corresponding to the {5,10,15,20,25,30,35}× 𝜎 levels, with lower contours shown in darker shades of blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' MeerKAT 21cm absorption contours are overlaid in dashed black lines, with levels {-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='3,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='2,-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='1} Jy/beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='km/s displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The MeerKAT synthesized beam is shown by a white ellipse in the lower left corner of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Panel B: Stellar light in the HST image, with star-forming knots identified by crosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' In all images, North is up and East is to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Panel C: Low angular resolution (47”) MeerKAT 21cm spectrum integrated over the detected 21cm, shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The GBT spectrum is shown in gray for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Panel D: MeerKAT 21cm high angular resolution (10”) emission (blue) and absorption (black) spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The VLA absorption spectrum is shown with gray dots for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Vertical dashed lines show the velocity centroid range of ionized gas around the star-forming knots, where the sources emitting the absorbed radio continuum radiation are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' B Knot B Knot C 33°32\'20" A Knot A 40" DEC (J2000) 33\'00" 20" 40" oh36m58s 56s 54s 52s RA (J2000) C MeerKAT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='lowres Emission,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' high res D GBT Absorption,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' high res VLA absorption Flux density [mJy] 2 L 1 5800 6000 6200 6400 6600 5800 5900 6000 6100 6200 6300 6400 6500 6600 v [km/s] v [km/s]Haro 11 is the closest (D=93Mpc)17 confirmed LyC-emitting galaxy and one of only three known LyC-leaking galaxies that can be observed and resolved in 21cm with radio interferometers18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The UV photons in this star-forming galaxy are produced in three distinct regions21,22,23 labeled Knot A, B and C in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Most indirect studies agree on Knot C as the likely producer of the bulk of LyC photons that were detected24,25,26 due to its Lyα properties and gas covering fraction derived using lines of metals assumed to be mixed with HI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Knot B and Knot A both produce large quantities of ionizing photons23,27,28, however these photons are likely to escape at angles outside of the line of sight26,28,29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The neutral gas distribution that can allow for the anisotropic escape of ionizing radiation out of the knots has not been measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The molecular gas distribution has been measured30, but molecular gas is not a significant source of LyC opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Previous single dish observations of Haro 11 detected the 21cm in emission13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' However, the distribution of the gas has remained unknown due to insufficient sensitivity of interferometric observations with only unresolved absorption detections12,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We observed the galaxy in 21cm with the MeerKAT telescope which provides increased surface brightness sensitivity as compared to any HI interferometer previously available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Results The MeerKAT 21cm HI image and spectra are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 1 with the optical emission of Haro 11 from MUSE28 tracing ionized gas and optical stellar light from HST31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=" We detect and resolve HI in emission that was previously detected by the Green Bank Telescope (GBT)13 and detect the unresolved absorption component detected by the NSF's Karl G." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Jansky Very Large Array (VLA)12,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Neutral gas content and structure around LyC production regions The 21cm integrated flux map and spectrum extracted around the star-forming knots which produce the LyC photons is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The prominent 21cm absorption component indicates the presence of neutral gas in front of the optical body of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The HI absorption corresponds to a gas mass MHI,abs=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='30 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='41 × 10# M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The MeerKAT observation does not resolve the 21cm absorption nor the radio continuum emission absorbed by the neutral gas, but previous absorption line studies have shown that the interstellar medium in front of the UV photon production sites is porous25,26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' This porosity enables the escape of LyC photons from their immediate neutral gas environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Using archival VLA data, we find radio continuum emission across all star-forming knots (extended data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 3), with an especially strong source co-spatial with star-forming Knot B (extended data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' X-Ray observations of the galaxy found two Ultra Luminous X-ray regions, respectively overlapping with Knot C and Knot B29,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' This could indicate the presence of a low luminosity Active Galactic Nucleus (AGN) in the X-Ray region co-spatial with Knot B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The radio spectral index of the radio continuum source in Knot B is consistent with that of star- forming regions33 (extended data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We cannot conclude on the presence or absence of an AGN with the radio continuum observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' At the location of the 21cm absorption, there is residual 21cm emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Since it is not seen in absorption, the emitting 21cm gas at this location must be located behind the optical body of the galaxy where the radio continuum sources are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The 21cm emission component co-spatial with the absorption corresponds to a flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='052±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='053 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Figure 2: Neutral gas around the LyC production sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Left panel: MeerKAT High angular resolution 21cm integrated flux density map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The dashed white line shows the limits of the absorption component and the aperture used to extract the spectrum shown in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The black crosses indicate the position of the star-forming knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Right panel: Spectrum of Haro 11 at the location of the absorption component, shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The blue line shows the Gaussian fit to the emission component seen at this location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Dashed black lines show the velocity centroid range of the ionized gas around the star forming knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We compare the 21cm absorption feature velocity to that derived from the Hα centroid in each knot and find that the neutral gas seen in absorption is blueshifted with maximum velocity vmax = 145±42 km/s compared to the ionized gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The peak of the 21cm emission is redshifted by 71±30 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The blueshifted 21cm absorption indicates that the neutral gas in front of the star-forming knots is moving towards us, and the redshifted 21cm emission shows the neutral gas behind the star-forming knots is moving away from us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' This suggests that the neutral gas around the optical body of the galaxy is outflowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Previous studies have already noted outflows of ionized gas in the galaxy23,26,28,34, and the presence of a multi-phase outflow with partially neutral gas traced by metal absorption lines23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The MeerKAT observation shows that the dense cold neutral gas located around the optical body of the galaxy is also likely outflowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Large-scale neutral gas distribution We then investigate the structure of the neutral gas emission around the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The bulk of the HI emission is offset from the main body of the galaxy, in an elongated structure that is ~40 kpc long when projected on the plane of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The HI emission components in individual velocity channels are found to be connected spatially and spectrally (Fig 6, 7, 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The morpho-kinematics of the gas are consistent with the HI emission structure being a tidal tail from a merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The merger state of Haro 11 had already been found by studies of the ionized gas kinematics of the galaxy35,36, but the impact of the merger on the neutral gas distribution was unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The total 21cm emission flux is SHI,em=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='391 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='042 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' This yields a new mass estimate for the HI in emission MHI,em=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='85 × 10# M⊙, a value higher than that of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8 × 10# M⊙ previously calculated with the single dish measurement13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Only 13% of the gas mass associated to HI emission is co-spatial with the star-forming knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Taking into account the absorbing gas mass MHI,abs=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='30 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='41 × 10# M⊙, the total atomic hydrogen gas mass of Haro 11 is MHI=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='3 × 10% M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' While this value is higher than previous estimates, it remains smaller than the ionized gas mass28 MHα = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8 × 10% M⊙ and the stellar mass35 of the galaxy M∗ = 1 33°33\'00" Flux density [mJy] DEC (J2000) 15" 2 30" 3 oh36m54s 53s 52s 515 5900 6000 6100 6200 6300 6400 6500 6600 RA (J2000) v [km/s]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='6\'".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=') !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='.+ × 10+" M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Importantly, up to 82% of the total neutral gas mass in the galaxy is offset from the ionizing emission production regions due to merger-driven interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Discussion The HI morphology and kinematics of Haro 11 demonstrates that pathways for LyC escape exist from the interstellar to the circumgalactic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Additionally, these observations provide a potential link between dwarf galaxy mergers and the detection of LyC emission from galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Galaxy mergers in low mass systems could play several roles in facilitating the escape of LyC emission from galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' First, mergers create multiple star formation bursts37 during the timescale of the interaction by repeatedly compressing the gas at the center of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' These bursts create numerous massive stars, which produce the bulk of LyC emission in galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' thus, mergers increase the intrinsic LyC photon production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Second, starbursts are also responsible for the intense feedback that creates ionized channels enabling the escape of LyC photons from their immediate environment38,39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Finally, merger interactions tidally displace the material inhibiting LyC escape from the center of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' By creating regions depleted of HI on large scales, galaxy mergers facilitate the anisotropic escape of LyC photons out of the interstellar medium and into the intergalactic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' In the local Universe, two other LyC-emitting galaxies have been detected that are close enough to be imaged with interferometers: Tol 1247-23219 and Mrk 5420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Both of these galaxies show prominent merger morphologies in the optical, however their neutral gas content is significantly different, with Mrk 54 having a high HI mass40 MHI=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='2 × 10+" M⊙, while Tol 1247-232 has an upper limit14 MHI<1 × 10% M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Given the morphology of the galaxies, it is likely that a fraction of their HI gas has been removed by tidal interactions from the lines of sight where LyC is emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' This would facilitate the ionization by the starburst and explain how LyC emission can escape from environments with very different interstellar medium properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' While they are not directly detected, many galaxies are considered to be LyC candidates due to their Lyα line profiles or their high [OIII]/[OII] ratios, which indirectly trace LyC escape7,9,41,42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Among these candidates, many display signs of ongoing merger events that would facilitate LyC escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Green pea galaxies are a class of objects considered excellent analogs of high redshift LyC-emitting galaxies5,43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Single dish 21cm line measurements of green peas galaxies have suggested that galaxies with high [OIII]/[OII] ratios are less likely to be detected in 21cm, potentially indicating that galaxies with low HI mass are more likely to leak LyC radiation44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' However, about a fifth of the sample of green pea galaxies studied in 21cm have neutral gas and galaxy properties indicative of either recent gas accretion or the presence of a gas-rich companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Recently, HI imaging of the Green Pea galaxy J0213+0056 has shown that a merger could explain Lyα escape in the galaxy, and could potentially lead to LyC leaking45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The galaxy merger rate is difficult to measure in the early Universe, however a few observational studies have found indications of an increase in merger rate at redshift 4 and up to 646,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' From the simulation perspective, galaxy mergers also seem to be a promising process facilitating the escape of ionizing LyC radiation during the epoch of reionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Indeed, hydrodynamical cosmological simulations predict an increase of mergers with increasing redshift48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Semi-analytic galaxy formation models also show that galaxies found in dense environments reside in larger ionized regions49, with galaxies having neighbors being more likely to show Lyα emission at z=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' However, the environment of galaxies leaking LyC emission has not yet been studied in a systematic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' While mergers are likely not the only mechanism responsible for LyC escape from galaxies, they are an effective process that reunites several of the conditions required for ionizing photons to escape to the Intergalactic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Their contribution to reionization and the impact of galaxy environment on LyC escape should be assessed from the point of view of simulations and observations alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Barkana, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=', The First Stars in the Universe and Cosmic Reionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Science 313, 931- 934 (2006).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The Astrophysical Journal Letters 933, L11 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Tasca, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=', Le Fèvre, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=', López-Sanjuan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Evidence for major mergers of galaxies at 2 ≲ z < 4 in the VVDS and VUDS surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Astronomy & Astrophysics 565, A10 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Duncan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=', Conselice, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=', Mundy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Observational Constraints on the Merger History of Galaxies since z ≈ 6: Probabilistic Galaxy Pair Counts in the CANDELS Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The Astrophysical Journal 876, 110 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Rodriguez-Gomez, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=', Genel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=', Vogelsberger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The merger rate of galaxies in the Illustris simulation: a comparison with observations and semi-empirical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Monthly Notices of the Royal Astronomical Society 449, 49-64 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Qin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=', Wyithe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=', Oesch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Dark-ages reionization and galaxy formation simulation XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The Lyα IGM transmission properties and environment of bright galaxies during the epoch of reionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Monthly Notices of the Royal Astronomical Society 510, 3858-3866 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Methods Throughout this paper we assume a Hubble constant H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5 km/s/Mpc and a matter density1 Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Using the redshift2 of Haro 11 (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0206), we derive a luminosity distance of 93 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Literature values mentioned in this paper have been corrected for this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' MeerKAT observations and data reduction Haro 11 was observed with the MeerKAT interferometer for program SCI-20210212-AL-01 (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Le Reste).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8 hours of on-source integration time were split between two observing sessions (2021 Feb 20 and 2021 Aug 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The correlator was configured in the C856M32k mode, wherein an 856 MHz wide bandpass is separated into 32,000 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' J0408-6545 and J1939-6342 were used for primary flux and bandpass calibrations for the Feb 20 and the Aug 14 data, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' J0025-2602 was used as secondary calibrator in both observing sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' To enable study of the HI spectral line, a narrow (20 MHz) bandpass, centered on the recessional velocity of Haro 11 derived from optical emission lines2, was extracted from the full dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='123 kHz channel width produces a native velocity resolution of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5 km/s/ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Data reduction and calibration followed standard prescriptions in the CASA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='6 environment3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Imaging was performed with CASA 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='4 using the AUTO-MULTITHRESH algorithm4 within the CASA TCLEAN task at three different angular resolutions by using tapering in the uv-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The velocity resolution was set to 10 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The masks generated by AUTO-MULTITHRESH were examined by hand, and deep cleaning (to the level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5 σ) was performed inside regions that contain real flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The final datacubes have beam sizes and rms noise values as follows: 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='3” ×45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='7” and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='18 mJy/beam (hereafter, “low resolution”, produced using a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5kλ uv-taper);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='6” ×21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='7” and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='16 mJy/beam (hereafter, “medium resolution”, produced using a 7k uv-taper);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='2” ×9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='7” and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='17 mJy/beam (hereafter, “high resolution”, produced with no uv-taper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 21cm detection To detect 21cm signal in the cubes, we used the dedicated 21cm detection software Source FInding Algorithm5 (SoFIA, version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We used the Smooth+Clip algorithm with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5σ threshold using the Median absolute Deviation rms mode in order to include HI absorption in the mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We used spatial smoothing with Gaussian kernels and spectral smoothing boxcar kernels (with either no smoothing or 3-channel wide (~30km/s) smoothing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' For each of the datasets, we applied kernels with no smoothing, 10” smoothing (average beam size of the high resolution dataset), 23” smoothing (average beam size of the intermediate resolution dataset) and 46” smoothing (average beam size of the low resolution dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' A reliability threshold is often used to increase completeness and reliability of the detections, however since this method relies on statistics on negative pixels and our cube presents prominent absorption, we did not use it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Eye inspection of the masked cubes confirmed that the 21cm detection parameters enabled the recovery of all the HI emission and absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' HI Mass estimation To estimate the gas mass associated with the 21cm emission, we use the high angular resolution cube which best resolves the absorption in the cube, that allows us to separate the emission and absorption components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' To estimate the gas mass associated with 21cm emission-only, we mask the absorption component from the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We fit a double Gaussian function to the emission profile using least-square fitting weighted by errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We integrate over the fitted profile and find a flux value SHI,em=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='391 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='042 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' km/s, taking into account a conservative flux calibrator error of 10%, the pixel variance in the cube and the uncertainty on fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' To determine the emitting gas mass, we assume that the gas is optically thin and use MHI = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='36 × 10, D2[Mpc] SHI[Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='km/s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We find an HI gas mass MHI,em=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='85 × 10# M⊙ for the 21cm emission component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Estimating the gas mass associated with 21cm absorption yields more uncertainties, due to the unknown size of the absorbing component, spin temperature and covering fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The column density NHI of the absorbing gas can be written as a function of the spin temperature Ts and the optical depth 𝜏(𝑣) as NHI[cm-2] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='823 × 10+# Ts[K] ∫ 𝜏(𝑣) d𝑣 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' [km/s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The optical depth can be expressed as a function of the observed change in flux density due to the absorption ΔS, the continuum flux density Sc and the covering fraction of the absorbing gas 𝑓 as 𝜏(𝑣) = − ln(1 + ΔS(𝑣)/(𝑓 Sc)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Finally, the absorbing gas mass calculation requires an assumption on the volume covered by the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We express it as a function of the area covered by the gas 𝐴/01, the column density, the mass of the Hydrogen atom 𝑚2 and assume the same volume filling factor of 𝑓3/!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' as in previous 21cm absorption study6, such that MHI,abs = NHI,abs 𝐴/01 𝑚2 𝑓3/!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='. The radio continuum images and previous VLA absorption study indicate that the radio emission is produced by an unresolved source, and the distribution of the gas within the area is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Thus, we use the beam size (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='27” ×10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='07”) of the previous VLA absorption observation which reached a better resolution, as an upper limit to the area covered by the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Observations and modeling6,7 of the ISM in Haro 11 suggests that the spin temperature is found within the range 91-200 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We use values from the measurement of UV absorption lines of metals which are a proxy for the neutral gas content of galaxies8 to constrain the covering fraction between f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='55 and f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We find that the mass in absorption ranges from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='9 × 105 M⊙ to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='7 × 10# M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Adding the mass seen in emission to the midpoint of the absorption mass, we find a total HI gas mass MHI=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='3 × 10% M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' HI distribution at the location of the absorption feature We extract a spectrum from the non-masked cube at the location of the absorption component to assess the content and kinematic structure of the neutral gas at this location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' This also corresponds to the location of the optical body of the galaxy and of the star-forming knots producing LyC emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The spectrum and aperture are shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We find that 21cm emission is seen at the location of the absorption feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We have shown the velocity range of the sources emitting the radio continuum emission that is being absorbed, and find that the peak of the 21cm emission is redshifted by 71±30 km/s, depending on the location considered for the radio continuum source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' HI gas that is located in front of the source emitting the radio continuum is absorbed, while gas located in the background is emitting 21cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The respective redshift of the emission and blueshift of the absorption indicate that we are looking at a neutral gas outflow around the optical body of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We fit the emission component with a Gaussian using least-square fitting weighted by errors and integrate the Gaussian profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We find a flux of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='052±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='053 Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='km/s in the emission component cospatial with the absorption, which corresponds to 13% of the 21cm emission flux of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Most of the HI emission is thus found in the tidal tail, away from the star- forming knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' HI 21cm morpho-kinematics Channel maps of the 21cm of Haro 11 are shown for the high-resolution, intermediate- resolution and low-resolution cubes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 6, 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The emission is connected (both spatially and spectrally) and is offset from the center of the galaxy, indicating that it traces a tidal tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Archival VLA continuum data reduction The VLA continuum data was taken by the National Radio Astronomy Observatory’s Karl G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Jansky Very Large Array (VLA) as part of project VLA/15B-197 (PI: Kepley).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The observing frequencies and parameters are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' For all three bands, the strategy was to observe a flux/bandpass calibrator once during each scheduling block and then to periodically observe a phase calibrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Additionally for the X and Ka-band data, a pointing scan was run at the beginning of a scheduling block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The data were calibrated using VLA pipeline version Pipeline-Cycle3-R1-B with CASA version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='1 r32491 and imaged in CASA 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='1-7, using multi-term, multi-frequency synthesis9 with nterms=2 to account for the spectral curvature of the data and Briggs robust weighting of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' In addition, the S-band data used w-projection to correct for the curvature of the sky over the larger field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We used self-calibration to improve the image dynamic range for all three bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' All three datasets had a phase self-cal performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The Ka-band data had an additional scan-length amplitude self-cal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The final images were primary beam corrected accounting for the variation of the primary beam as a function of frequency across the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Haro 11 radio continuum emission We present the VLA radio continuum images of Haro 11 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The radio continuum flux density of Haro 11 was calculated in the VLA archival images at 3GHz, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8 GHz and 33 GHz using the lowest contour corresponding to the 3σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' This was supplemented by values taken from the literature6,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The flux density values for Haro 11 are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We also extract the flux density in Knot B only in the VLA archival images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' To do so, we fit a 2D Gaussian using the CASA task IMFIT, fixing the value of the peak and allowing for a zero-point offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The flux density values of Knot B only are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The error values are calculated using a conservative flux calibrator error of 10 %, which drives the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We show the extraction regions used for the full galaxy and Knot B on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 5, along with the radio continuum SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The flux values agree with Knot B being the location of the highest radio continuum emission region, but the continuum emission in Knot B is unresolved, even with Ka-band imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We compare the location of the Knot B radio continuum emission with the optical image of Haro 11 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The radio spectral energy distributions (SED) of Haro 11 and the individual components is presented on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=" We fit the radio SED of Haro 11 using a combination of thermal and non-thermal radiation: S(𝜈) = 𝑆678 + 𝑆78 = 𝑐+ 𝜈'9 + 𝑐!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 𝜈\'".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We use a least-square fitting approach weighted by errors and find the following parameters: 𝑐+ = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='9 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5 mJy, α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='27, 𝑐!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='3 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The non-thermal spectral slope is consistent with radio emission from star-forming regions11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We fit the SED of Knot B with non-thermal radiation only, given the number of points is insufficient for a combined fit, and will lead to overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' We find the following parameters minimize the chi-square: 𝑐+ = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='3 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='1 mJy, α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='1 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The flatter spectral index of Knot B is consistent within errors with that of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Methods references 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Planck Collaboration, Planck 2018 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Cosmological parameters.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Klein, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=', Lisenfeld, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=', and Verley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Radio synchrotron spectra of star-forming galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Astronomy & Astrophysics 611, A55 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Acknowledgments ALR thanks Christian Binggeli, Alba Covelo Paz and Mohammad Javad Shahhoseini for their contributions to the initial 21cm interferometric observing proposal for Haro 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' JMC and JLI are supported by NSF/AST-2009894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' JMC and SHT acknowledge support from Macalester College.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' MJH is fellow of the Knut & Alice Wallenberg foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' AA acknowledges financial support from the Swedish Research Council (VR) under grant 2021-05559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' GÖ acknowledges financial support from the Swedish Research Council (VR) and the Swedish National Space Agency (SNSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The MeerKAT telescope is operated by the South African Radio Astronomy Observatory, which is a facility of the National Research Foundation, an agency of the Department of Science and Innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' This study uses observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='B-0944(A) and 096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='B-0923(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Author contributions ALR designed the MeerKAT 21cm observing proposal of Haro 11, analyzed and interpreted the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' JMC assisted with technical preparation of the 21cm observing proposal and reduced and imaged the MeerKAT 21cm data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' MJH made major contributions to the scientific justification of the observing proposal and to the interpretation of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' JLI reduced the MeerKAT 21cm data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' AAK reduced the VLA continuum data and contributed to the radio continuum data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' JM contributed the HST image of Haro 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' VM reduced the MUSE data and produced the ionized gas maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' AA, AB and GÖ contributed to the scientific justification of the 21cm observing proposal and interpretation of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' TE designed hydrodynamical simulations of the Haro 11 merger which were used for interpretation of the 21cm data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' GIGJ provided major contributions during the technical preparation of the 21cm observing proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The 21cm observing proposal justification was partly based on results obtained by SHT with the VLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' All authors contributed to the writing of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Data Availability The datasets generated and analysed in the current study will be made publicly available in the SARAO DataCite repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Competing interest declaration The authors declare they have no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Extended data Figure 3: VLA continuum images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' From left to right: S-band (3 GHz) continuum image, X-band (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8 GHz) continuum image, Ka-band (33 GHz) continuum image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Intensities are displayed with a logarithmic stretch to highlight the full distribution of the radio continuum emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The white contours displayed correspond to the {3,5,10,20,40,80,160} levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The synthesized beam is represented by a white ellipse in the bottom left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Figure 4: Radio continuum and optical stellar emission comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' HST f435W filter image with VLA continuum emission contours overlaid in pink for the S-band (3GHz), in orange for the X-band (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8GHz), and in yellow for the Ka-band (33GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The contours for each band are also presented on Figure 5, and correspond to the Gaussian aperture used to extract the flux of Knot B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Black crosses indicate the position of the three star-forming knots, the red cross indicates the position of the 21cm absorption centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' S-band (3GHz) X-band (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8GHz) Ka-band (33GHz) 33°33\'05" 10" (J2000) 15" DEC 20" 25" 30" oh36m53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5s 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0s 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5s 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0s 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5s 0h36m53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5s 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0s 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5s 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0s 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5s 0h36m53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5s 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0s 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5s 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0s 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5s RA (J2000)-33°33\'12" 15" DEC (J2000) 18" 21" VLA 3GHz VLA 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8GHz VLA 33GHz X SF knots X 2lcm abs 24" 0h36m53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0s 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5s 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0s RA (J2000) Figure 5: Radio spectral energy distribution (SED) and VLA continuum images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The left panels show the SED for Haro 11 (top) and star-forming knot B only (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' For Haro 11, we fit the data using a combination of thermal and non-thermal radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The combined fit is shown in black, the thermal and non-thermal components are shown in gray solid and dashed lines respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' For Knot B, we fit the data using the non-thermal emission only (gray dashed line), since we do not have enough data points to make a combined fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The right panels show the VLA archival images with contours used for flux extraction overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' S-band (3GHz) Haroll non thermal fit thermal fit Combined fit Flux density [mly] 101 X-band (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8GHz) 100 100 101 [GHz] Knot B non thermal fit Flux density [mJy] 101 Ka-band (33GHz) 100 100 101 v [GHz] Figure 6: MeerKAT high angular resolution 21cm channel map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Levels displayed correspond to the -10σ, -3σ, 3σ and 5σ levels, respectively shown in black, gray, light blue and dark blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The position of Knot B is indicated by a black cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The two other knots are extremely close to Knot B, they have not been represented here to help readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 33°32\'00" 10 kpc 6028 km/s 6038 km/s 6048 km/s 6058Km/s 6069 km/s 30" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' x x x x 30" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='00 33°32\'00" 6079 km/s 6089 km/s 6099 km/s 6109 km/s 6120 km/s 30" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 30" 34\'00" 33°32\'00" 6130 km/s 6140 km/s 6150 km/s 6160 km/s 6171 km/s 30" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 0 (J2000) 30" 34\'00" 33°32°00 DEC 6181 km/s 6191 km/s 6201 km/s 6211 km/s 6222 km/s 30" 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='00, D ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='. 30" 34\'00" 33°32\'00" 6232 km/s 6242 km/s 6252 km/s 6262 km/s 6273 km/s 30" 0Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 30", 34\'00" 6283 km/s 6293 km/s 6303 km/s 6313 km/s 6323 km/s 30" 33\'00" x 30 34\'00* oh36m58s 56s 54s 52s0h36m58 56s 54s 52*0h36m58* 56s 54s 52s0h36m58* 56s 54s 52*0h36m58* 56s 5 4s 52§ RA(J2000) Figure 7: MeerKAT intermediate angular resolution 21cm channel map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Levels displayed correspond to the -10σ, -3σ, 3σ and 5σ levels, respectively shown in black, gray, light blue and dark blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The position of Knot B is indicated by a black cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The two other knots are extremely close to Knot B, they have not been represented here to help readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 33°32\'00" 10 kpc 6028km/s 6038km/s 6048 km/s 6058km/s 6069km/s 30", .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' x X x XO 30" x 34\'00" 33°32\'00" 6079-km/s 6089 km/s 6099 km/s 6109km/s 6120km/s 30" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='E ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 30", 34\'00" 33°32\'00" 6130km/s 6140-km/s 6150km/s 6160km/s 6171km/s 30" 33\'00" (J2000) 30" 34\'00" 0 33°32\'00 6181km/s 6191km/s 6201km/s 6211km/s 6222_km/s DEC 30" D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' 30" C 34\'00" 0 6232km/s 6242km/s 6252km/s 6262km/s 6273km/s 30" 0 33\'00* 30" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='00, 33°32\'00" 6283kfm/s 6293km/s 6303km/s 6313km/s 6323km/s 30" 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' + 30" .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content="00 0 0h37m00'36m57s 54s 510h37m0036m57s 545 510h37m0036m57s 545 510h37m0036m57s 545 510h37mo036m57s 54s 51 RA(J2000) Figure 8: MeerKAT low angular resolution 21cm channel map." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Levels displayed correspond to the -10σ, -3σ, 3σ and 5σ levels, respectively shown in black, gray, light blue and dark blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The position of Knot B is indicated by a black cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' The two other knots are extremely close to Knot B, they have not been represented here to help readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' Band S-band X-band Ka-band Configuration B CnB DnC Central Frequency (GHz) 3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0 Bandwidth (GHz) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content="9 Channel width (MHz) 1 1 1 Number of channels/spectral window 128 128 128 10 kpc 6028 km/s 6038 km/s 6048 km/s 6058 km/s 6069 km/s 33*32' EE x 34' - 6079 km/s 6089 km/s 6099 km/s 6109 km/s 6120 km/s 33°32' ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content="EE x 34' - 0 0 6130 km/s 6140 km/s 6150 km/s 6160 km/s 6171 km/s 33°32'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content="33' " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='(J2000) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content="34' " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='EC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='6181 km/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='6191 km/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='6201 km/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='6211 km/s ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='7min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='Time on Source (hr) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='1 Observing Dates 2016-05- 31 2016-05- 05 2016-01-10, 2016-01-13, 2016- 01-23, 2016-01-23/24 Beam 5.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='7 Table 1: Very Large Array radio continuum observation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' [GHz] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='48 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='89 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='46 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8 33 S[mJy] 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='6 Table 2: Radio continuum flux density values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content=' [GHz] 3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8 33 S[mJy] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} +page_content='5 Table 3: Radio continuum flux density values in Knot B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfzwJL/content/2301.02676v1.pdf'} diff --git a/MNE2T4oBgHgl3EQfqQhz/content/tmp_files/2301.04037v1.pdf.txt b/MNE2T4oBgHgl3EQfqQhz/content/tmp_files/2301.04037v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d76ce43564136d92c34c06647852af1658d42090 --- /dev/null +++ b/MNE2T4oBgHgl3EQfqQhz/content/tmp_files/2301.04037v1.pdf.txt @@ -0,0 +1,2047 @@ +ROBUSfT: Robust Real-Time Shape-from-Template, a C++ Library +Mohammadreza Shetab-Bushehri, Miguel Aranda, Youcef Mezouar, Adrien Bartoli, Erol ¨Ozg¨ur +Abstract— Tracking the 3D shape of a deforming object +using only monocular 2D vision is a challenging problem. +This is because one should (i) infer the 3D shape from a +2D image, which is a severely underconstrained problem, and +(ii) implement the whole solution pipeline in real-time. The +pipeline typically requires feature detection and matching, +mismatch filtering, 3D shape inference and feature tracking +algorithms. We propose ROBUSfT, a conventional pipeline based +on a template containing the object’s rest shape, texturemap +and deformation law. ROBUSfT is ready-to-use, wide-baseline, +capable of handling large deformations, fast up to 30 fps, free of +training, and robust against partial occlusions and discontinuity +in video frames. It outperforms the state-of-the-art methods +in challenging datasets. ROBUSfT is implemented as a publicly +available C + + library and we provide a tutorial on how to +use it in https : //github.com/mrshetab/ROBUSfT. +Keywords: +monocular Non-rigid reconstruction, mis- +match removal, SfT, validation procedure. +I. INTRODUCTION +Problem and challenges. Tracking the 3D shape of a de- +forming object has important applications in augmented +reality [1], [2], computer-assisted surgery [3]–[7] and +robotics [8]–[10]. However, the existing solutions are im- +practical. This is because of the following challenges: (C1) +real-time implementability and (C2) robustness. Challenge +C1 is hard to achieve because the solution usually involves +a computationally demanding multi-step pipeline. Challenge +C2 is hard to maintain because of noises, occlusions, invisi- +ble object, large deformations and fast motions. Furthermore, +in numerous applications of augmented reality, computer- +assisted surgery and robotics, a 2D camera is the de facto +sensor owing to its light weight, small size, and low cost. +The camera’s perspective projection introduces an additional +challenge, (C3) recoverability of shape’s depth from a 2D im- +age. Challenge C3 becomes extremely difficult for deforming +objects. +Shape-from-Template. Different priors and constraints have +been proposed to resolve challenge C3. The most common +ones are the object’s 3D rest shape, texturemap, deformation +law and the camera intrinsics. These form the ingredients +for a variety of methods. Among these methods, we are par- +ticularly interested in Shape-from-Template (SfT). SfT has +MR. +Shetab-Bushehri, +Y. +Mezouar, +A. +Bartoli, +and +E. +¨Ozg¨ur +are +with +the +CNRS, +Clermont +Auvergne +INP, +Institut +Pascal, +Universit´e +Clermont +Auvergne, +F-63000 +Clermont-Ferrand, +France +(e-mail: +m.r.shetab@gmail.com; +youcef.mezouar@sigma-clermont.fr; +adrien.bartoli@gmail.com; erolozgur@gmail.com). M. Aranda is with +Instituto de Investigaci´on en Ingenier´ıa de Arag´on (I3A), Universidad de +Zaragoza, E-50018 Zaragoza, Spain (e-mail: miguel.aranda@unizar.es). +been well studied for isometrically deforming objects [11]– +[13] and has been shown to uniquely resolve the depth of +each object point [14]. It uses a template formed by the +abovementioned priors. SfT’s input is a single image of a +deformed object, and its output is the object’s 3D shape +seen in the image. We adopt a conventional SfT pipeline +shown in Figure 1 to solve the 3D shape tracking problem of +deforming objects. The pipeline involves keypoint extraction +and matching, mismatch filtering, warping and 3D shape in- +ference steps, respectively. We successfully made it real-time +and robust by integrating seamlessly both novel and state-of- +the-art algorithms at different steps. We next overview the +strengths and weaknesses of current SfT methods. +State-of-the-art SfT methods. SfT can be broken down into +two main parts: registration and 3D shape inference. Fol- +lowing this, we categorize existing SfT methods into two +groups: (G1) shape inference methods and (G2) integrated +methods. G1 methods only cover the 3D shape inference +part [10]–[12], [14]–[18]. In contrast, G2 methods cover +both the registration and 3D shape inference parts [6], [19]– +[23]. We also overview Deep Neural Network (DNN) based +SfT methods, as the third group (G3), which have been +recently introduced. G3 methods cover both the registration +and 3D shape inference parts [24]–[28]. The majority of +G1 methods are wide-baseline. However, they barely run in +real-time. Furthermore, a complete solution with registration +shall be even slower. The majority of G2 methods require an +initialization close to the solution. This makes them short- +baseline. Subsequently they often fail against occlusions, fast +motions and large deformations. Once failed, they need to be +reinitialized. G3 methods are wide-baseline and run in real- +time. However, they are object-specific. They require a huge +amount of training data and proper computational resources +for each new object. These make them difficult to consider as +a general and ready-to-use solution. We therefore conclude +that there does not exist an SfT method that is complete, +real-time, robust and easily applicable to new objects. +Contributions. We list our contributions in three parts. +a) Contribution to SfT: We propose ROBUSfT, a com- +plete real-time robust SfT pipeline for monocular 3D shape +tracking of isometrically deforming thin-shell objects with +matchable appearance. It can track up to 30 fps using 640 × +480 images on off-the-shelf standard consumer hardware. +It does not require initialization and implements tracking- +by-detection. It is wide-baseline and robust to occlusions, +invisible object, large deformations and fast motions. It does +not require training. It is thus directly applicable in many +arXiv:2301.04037v1 [cs.CV] 10 Jan 2023 + +Fig. 1: Overview of ROBUSfT. +industrial and research contexts. ROBUSfT outperforms the +state-of-the-art methods in challenging datasets. +b) Contribution to mismatch removal: We introduce +myNeighbor, a novel mismatch removal algorithm. It han- +dles deforming scenes and a large percentage of mismatches. +It is lightning fast, reaching 200 fps. +c) Contribution to experimental validation: We design +a novel type of validation procedure, called Fake Realistic +Experiment (FREX). It allows us to automatically generate +semi-synthetic datasets with ground-truth. This eases the +quantitative evaluation of 2D and 3D shape tracking algo- +rithms for deforming objects to a great extent. +Paper structure. Section 2 reviews previous work. Section +3 explains ROBUSfT. Section 4 presents FREX. Section 5 +describes myNeighbor, conducts a series of experiments +and evaluates the results of myNeighbor in comparison to +previous work. Section 6 validates ROBUSfT through FREX +and real data experiments, and compares the results with +previous work. Finally, Section 7 concludes and suggests +future work. + +Offline +Online +Object in rest shape +Image acquisition →I +Step 1-2. Keypoint matching +Step 1-1. Keypoint extraction +and description +Texturemap and +keypoints (P) +Shape mesh (M.) +CAMELEON STRES! +Step 2. Mismatch removal +par +inference +Step 3. Warping computation +Shape mesh aligned +Points with known 3D coordinates +to texturemap (M) +(optional) +Loop +Default pose +Step 4. 3D shape inferenceII. PREVIOUS WORK +We review the methods for monocular shape inference of +isometrically deforming objects, following the above three +categories, namely, (G1) shape inference methods, (G2) +integrated methods, and (G3) DNN-based SfT methods. For +each category, we describe the assumptions, main character- +istics, and limitations. We finally compare ROBUSfT to these +methods. +A. (G1) Shape inference methods +These methods cover the 3D shape inference part. They +assume that the registration between the template and the +image was previously computed. For instance, they typically +use keypoint matches between the template and the image, +with generic mismatch removal methods [16], [29], [29]– +[31]. In fact, very few methods in this category could form +a complete SfT pipeline by adding an existing registration +solution [1], [16]. Three general groups are found in existing +3D shape inference methods; (i) methods using a convex re- +laxation of isometry called inextensibility [11], [12], [17], (ii) +methods using local differential geometry [14]–[16], and (iii) +methods minimizing a global non-convex cost function [10], +[17], [18]. The methods in (iii) are the most precise ones +but also computationally expensive and require initialization. +The first two groups of methods are often used to provide +an initial guess for the third group. +In the first group, Salzmann et al. [12] suggested a closed- +form solution to non-rigid 3D surface registration by solving +a set of quadratic equations accounting for inextensibility. +Later, they replaced equality constraints with inequality and +thus sharp deformations could be better recovered [11]. +Brunet et al. [17] formulated two shape inference methods +based on point-wise and continuous surface models as Sec- +ond Order Cone Programs (SOCP). In the second group, +Bartoli et al. [14] showed that in addition to keypoint 2D co- +ordinates in the image, their first-order differential structure +can be used to estimate the depth. Instead of calculating the +warp globally, which is time-consuming, Famouri et al. [16] +estimated the depth locally for each match pair with respect +to both local texture and neighboring matches. In each frame, +the most recognizable matches were selected based on offline +training. The execution speed of their algorithm is claimed to +be up to 14 fps only for the 3D shape inference. In the third +group, Brunet et al. [17] proposed a refining isometric SfT +method by reformulating the isometric constraint and solving +as a non-convex optimization problem. The method required +a reasonably accurate 3D shape of the deforming surface +as the initializing guess. ¨Ozg¨ur and Bartoli [18], developed +Particle-SfT, which handles isometric and non-isometric de- +formations. A particle system is guided by deformation and +reprojection constraints which are applied consecutively to +the particle mesh. Similar to [17], this algorithm needs an +initial guess for the 3D position of the particles, however, +for [18], sensitivity to this initial guess is very low. The closer +the guess to the true 3D shape, the faster the convergence. +Aranda et al. [10] improved this algorithm in terms of +execution speed and occlusion-resistance and used that in +real-time shape servoing of isometrically deforming objects. +They used the 3D shape estimated in one frame as the initial +guess for the next frame and thus improved the convergence +speed of the algorithm to a great extent. They showed that +their algorithm can track a paper sheet covered with markers +and being manipulated by a robotic arm. To this end, they +only needed to track a handful of markers. Knowing the +3D coordinates of several mesh points also has a significant +effect on the convergence speed of the algorithm. The last +step of ROBUSfT uses the same method to infer the 3D shape, +as explained in Section III. +B. (G2) Integrated methods +These methods handle registration and 3D shape inference +at the same time. They minimize a non-convex cost function +in order to align the 3D inferred shape with image features. +These features can be local [20], [21] or at the pixel-level [6], +[22]. +Ostlund et al. [20] and later Ngo et al. [21] used the Lapla- +cian formulation to reduce the problem size by introducing +control points on the surface of the deforming object. The +process of removing mismatches was performed iteratively +during optimization by projecting the 3D estimated shape on +the image and disregarding the correspondences with higher +reprojection errors. Using this procedure, they could reach +up to 10 fps using 640×480 input images and restricting the +maximum number of template and image keypoints to 500 +and 2000, respectively. +As for pixel-level alignment, Collins and Bartoli [22] +introduced a real-time SfT algorithm which could handle +large deformations and occlusions and reaches up to 21 fps. +They combined extracted matches with physical deformation +priors to perform shape inference. Collins et al. [6] later +extended this algorithm and used it for tracking organs in +laparoscopic videos. For achieving better performance, they +also exploited organ boundaries as a tracking constraint. +These methods are fast and can handle large deformation. +Their main drawback, however, is to be short-baseline. In +case of tracking failure, they should be re-initialized precisely +with a wide-baseline method. This restrict their usage to +video streams. +C. (G3) DNN-based methods +DNN-based SfT methods have been introduced in the +recent years, which coincides with the tendency to use +deep learning to solve many computer vision problems. +These methods are wide-baseline, fast, and cover both the +registration and shape inference steps [24]–[28]. We group +these methods based on their type of output, which may be +sparse or dense. The methods of the first group represent +the SfT solution as the 3D coordinates of a regular mesh +with a predefined size [24]–[26]. The usage of these methods +is limited to thin-shell objects with rectangular shapes. The +second group of methods gives a pixel-level depthmap as +output [27], [28]. They also apply a post-processing step +based on the As-rigid-as-possible (ARAP) model [32] to +the resulting depthmap. This step recovers the whole object, + +Category +Method +Registration +Real-time +Wide-baseline +General +geometry +Needless of +training for +new objects +Public access +code +G1 +Salzmann et al. [12] +× +NA +✓ +✓ +✓ +× +Brunet et al. [17] +× +× +✓ +✓ +✓ +✓ +Bartoli et al. [14] +× +NA +✓ +✓ +✓ +✓ +Ozgur et Bartoli [18] +× +× +✓ +✓ +✓ +× +Famouri et al. [16] +× +✓ +✓ +✓ +✓ +✓ +Aranda et al. [10] +× +✓ +✓ +✓ +✓ +× +G2 +Ostlund et al. [20] +✓ +✓ +× +✓ +✓ +× +Ngo et al. [21] +✓ +✓ +× +✓ +✓ +× +Collins and Bartoli [22] +✓ +✓ +× +✓ +✓ +× +Collins et al. [6] +✓ +✓ +× +✓ +✓ +× +G3 +Pumarola et al. [24] +✓ +× +✓ +× +× +× +Golyanik et al. [25] +✓ +✓ +✓ +× +× +× +Fuentes-Jimenez et al. [27] +✓ +✓ +✓ +✓ +× +× +Shimada et al. [26] +✓ +✓ +✓ +× +× +× +Fuentes-Jimenez et al. [28] +✓ +✓ +✓ +× +✓ +× +ROBUSfT +✓ +✓ +✓ +✓ +✓ +✓ +TABLE I: Comparison of the state-of-the-art SfT methods and ROBUSfT. +including the occluded parts, as a mesh. The method in [27] +reconstructs the shape of the object with different geometries +and texturemaps that the network is trained for. In [28], +however, the proposed method can be applied to objects with +new texturemaps unseen to the network. The geometry of +the objects is, nevertheless, limited to flat paper-like shapes. +All the aforementioned methods in this category are object- +specific. This means that they merely work for the object that +they were trained for. An exception is [28], as it works for +unseen texturemaps but the applicability is still limited to flat +rectangular objects. On the other hand, in order to use the +DNN-based methods for a new object, the network should +be fine-tuned for it. This demands proper computational +resources and potentially a huge amount of training data, +which are challenging to collect for deformable objects. +D. Positioning ROBUSfT compared to previous work +Existing methods all have one or several limitations, +including not covering the whole pipeline, not being wide- +baseline, being limited to specific texture or geometry, +requiring fine-tuning for a new object, being slow, and +lacking public code access. This information is summarized +in Table I. In contrast, ROBUSfT covers the whole pipeline +and due to the fast execution can be used to develop real-time +shape tracking applications. It can be instantly used for each +deforming object without training. Only a template contain- +ing information regarding the object’s geometry, appearance, +and deformation law as well as intrinsic parameters of the +monocular camera is necessary, but this need is common to +all existing and future SfT methods, by definition. In the next +section, we describe ROBUSfT and all its steps. +III. ROBUSfT +A. Overview of the pipeline +The overview of our pipeline is presented in Figure 1. The +pipeline is divided into an offline and an online sections. +The offline section deals with the template. The online +section includes four main steps: keypoint extraction and +matching, mismatch removal, warp estimation, and 3D shape +inference. The images coming directly from the camera +are used as the inputs for the first step. In this step, the +keypoints are extracted and matched with the ones that +were previously extracted from the template’s texturemap. +Then, the mismatches are detected and removed using our +new mismatch removal algorithm myNeighbor. The list of +estimated correct matches is then transferred to the next step +where a warp is estimated between the template’s texturemap +and the image. This warp transfers the template’s registered +mesh to the image space, which is finally used as input for +the 3D shape inference algorithm. This process is repeated +for each image, the analysis of each image being performed +independently in a tracking-by-detection manner. +In the following, both the offline and online sections of the +pipeline are described in detail. Afterwards, an implementa- +tion permitting a fast execution of the pipeline is given. +B. Offline section: creating a template +We create a template for the surface of the deforming +object that we want to track. We call this surface the tracking +surface. The template of the tracking surface consists of the +following elements: +• MT : the triangular mesh covering the tracking surface +at rest shape. +• P: the texturemap of the tracking surface. +• M: the alignment of MT to P. +The first step in creating the template is to generate the +3D model of the tracking surface. The 3D model is in fact +the textured 3D geometry of the tracking surface in real +dimensions in rest shape. We form MT by triangulating this +3D geometry. The resolution of MT should be high enough +to be well aligned to the shape of the tracking surface. The +next step is to take an image from the 3D model of the +tracking surface while it is positioned perpendicular to the +camera’s optical axis in a simple texture-less background. +In this image, P is formed by the projection of the texture +of the tracking surface and M by the projection of MT . For +simple rectangular thin-shell objects like a piece of paper, the +whole process is straightforward. For other objects, including + +thin-shell objects with arbitrary shape, such as a shoe sole, +and also volumetric objects, 3D reconstruction software like +Agisoft Photoscan [33] can be used. +Next, we extract keypoints on P. These keypoints will be +matched with the ones that will be extracted from the input +image in the online section. We use SIFT [34] for extracting +keypoints but any other feature descriptor could be swapped +in. As the final step, we initialize the pose of MT in 3D +space. This initial pose can be arbitrarily chosen as it will +be used only once by Step 4 of the online section of the +pipeline for the first input image. It will then be replaced by +the inferred 3D shape in the next images. +In order to use the ROBUSfT C++ library, first, an ob- +ject of the class ROBUSfT should be created. The whole +process of forming the template for this object is handled +by the member function build template(). This function +possesses parameters for creating templates for rectangular +and non-rectangular thin-shell objects as well as the tracking +surface of volumetric objects. Regarding thin-shell objects, +the process of forming the template is automatic by just +receiving a handful of inputs from the user. For the tracking +surface of volumetric objects, however, MT , M, and P +should be prepared by the user and imported into the library. +C. Online section: shape tracking +Step 1: keypoint extraction and matching. The first step of +the online section of the pipeline is to extract keypoints in +the input image I. To do so, we use the PopSift library [35], +which is a GPU implementation of the SIFT algorithm. +We then match these keypoints with the ones that were +previously extracted from P by comparing descriptors, using +winner-takes-all and Lowe’s ratio test. Inevitably, a number +of mismatches will be formed between P and I. The +mismatch points in I can be located on the surface of the +deforming object or even in the background. This is shown as +red lines in the Matching step of Figure 1. These mismatches +will be eliminated in Step 2 thanks to myNeighbor which +can cope with a large percentage of mismatches. As a result, +in this step, the images coming from the camera can be +used directly without pretraining on either the image for +segmenting the object from the background, or the matches +for preselection of the most reliable ones. In the library, the +member function extract keypoints GPU() handles the +keypoint extraction in I. Then, the member function match() +performs matching. +Step 2: mismatch removal. To remove the possible mis- +matches introduced in Step 1, a new mismatch removal +algorithm, myNeighbor, was developed. The main principle +used in this algorithm is the preservation of the neighborhood +structure of correct matches on a deforming object. In other +words, if all of the matches were correct, by deforming +the object, the neighbor matches of each match should be +preserved. On the contrary, mismatches lead to differences +in the neighboring matches of each matched point in I +in comparison to P. This was used as a key indication +to detect and remove mismatches. The whole process of +myNeighbor is explained in Section V. In the library, the +Fig. 2: Implementation of ROBUSfT on the CPU and GPU. +A pure CPU implementation is also available. +member function mismatch removal algorithm() handles +the mismatch removal process. The output is a list of +estimated correct matches. +Step 3: warp estimation. We use the estimated correct +matches to estimate a warp W between P and I. We then use +W to transfer M to I and form � +M. The mesh points in � +M +will be used as sightline constraints in the 3D shape inference +algorithm in Step 4. The precision of warping depends on the +number of matches, their correctness, and their distribution +all over P. Warp W can be estimated in the most precise way +if all the matches are correct between P and I. However, +due to the smoothing nature of the warping algorithms, the +transferring process can cope with a small percentage of +mistakenly selected mismatches. It should be noted that W +cannot be extremely precise in areas without matches. As a +result, in these areas, the shape of � +M might not be aligned +well to the shape of the deforming object in I. This is worse +when the matchless area is located near the boundaries of +P as the alignment cannot be guided by the surrounding +matches. Hence, in order to use just well-aligned transferred +mesh points of � +M as the input for the 3D shape inference +step, an assessment is performed over all of the mesh points +and only the qualified ones are passed to Step 4. For this, +we check M cell-by-cell. Only the mesh vertices for cells +containing at least one correct match will be qualified as + +OFFLINE +Create an Object of class ROBUSfT +ROBUSfT softObject; +Setting the parameters +Template'stexturemap +Template's dimension and mesh resolution +Deformation model's parameters +Build the template +Mesh of the template (mesh points coordinates, triangulation) +Keypoints on the template's texturemap +Isometric deformation law +softobject.build_template(); +CPULoop +GPU Loop +Matching +softobject.match(); +Capture image +Mismatch removal +capture_image(); +softobject.mismatch_removal(); +Warping +Extracting keypoints of the image +softobject.calculate_warp(); +softObject.Extract_keypoints_GPU(); +Shape inference +softObject.shapelnference();Fig. 3: Flowchart of FREX. +salient mesh points. The indices of these mesh points and +their coordinates in � +M are passed to Step 4. The other mesh +points are disregarded. +Representing and estimating W can be done with two +well-known types of warp, the Thin-Plate Spline (TPS) [36] +and the Bicubic B-Spline (BBS) warps [37], which we both +tested. The former is based on radial basis functions while the +latter is formulated on the tensor-product. Having the same +number of matches as input, the TPS warp proved to be +more precise than the BBS warp; nevertheless, its execution +time rises exponentially with increasing number of matches. +The execution time, however, remains almost constant for +the BBS warp regardless of the number of matches. Thus, +considering the criterion of fast execution of the code, the +BBS warp was chosen as the warp function in this step and +also in the mismatch removal step discussed in Section V. In +the library, the process of warp estimation is performed by +the function warp() that calls two functions BBS Function() +and BBS Evaluation(). The former estimates the warp W +while the latter uses W to transfer M and form � +M. The +process of selecting the salient mesh points is done by the +member function set sightlines(). +Step 4: 3D shape inference. We use Particle-SfT [18] as +improved for tracking in [10]. In this algorithm, a particle +system is defined from the points and edges in MT . Then, +the sightline and deformation constraints are applied con- +secutively on the particles until they converge to a stable +3D shape. As described in [10], in order to increase the +convergence speed of the algorithm, the stable 3D shape +for an image is used as initial guess for the next image. +It should be noted that Particle-SfT can work even without a +close initial guess. If the object is invisible in one or several +images, the last inferred 3D shape can be used as the initial +guess for the upcoming frame containing the object. This +results in a slightly longer computation time in that image. +For the next upcoming images the normal computation time +is resumed. This capability brings about two of the major +advantages of our pipeline, which are being wide-baseline +and robust to video discontinuities. In the library, the whole +process of shape inference is handled by the member function +shapeInference(). +As mentioned in [10], one of the optional input data +that can significantly improve the convergence of Particle- +SfT is the existence of 3D known coordinates of one or +several particles. This is shown in Figure 1. The known 3D +coordinates can be fixed in space, or can move on a certain +trajectory. The latter happens when the deforming object is +manipulated by tools with known poses in 3D space like +robotic end-effectors. +D. Implementation +In order to optimize the implementation of ROBUSfT, it +was coded in C++ in two parallel loops: one on the GPU, and +one on the CPU. The GPU loop handles keypoint extraction +in the images. These keypoints are transferred to the CPU +loop where the rest of the steps of the pipeline are taken. A +pure CPU implementation is also available. This is shown +in Figure 2. Any arbitrary resolution can be considered +for the captured images, nevertheless, we obtained the best +performance by using 640 × 480 images. The code runs on +a Dell laptop with an Intel Core i7 2.60 GHz CPU and a +Quadro T1000 GPU. + +图图 +Start +A4 Aruco template +Arbitrary texturemap +Form 2D +Estimate warp +correspondences +An image from the video +Warp texturemap +Other alterations +showing the printed A4 +to image space +(background, lighting, etc.) +Aruco template being +deformed +3D coordinates of +Aruco markers +3D Ground Truth +2D Ground Truth +Quantitative evaluation +of algorithmsIV. FAKE REALISTIC EXPERIMENT (FREX) +We introduce a novel experimental protocol, which we +used for evaluating myNeighbor and ROBUSfT in compar- +ison to the state-of-the-art methods. A single execution of +this protocol provides a large collection of scenes of an +isometrically deforming object in various conditions, with +known 2D and 3D ground truth. This collection can be +used to evaluate, compare, train, and validate new algorithms +regarding isometrically deforming objects such as mismatch +removal, 2D image registration, and isometric 3D shape +inference. In contrast to other artificially generated scenes of +an isometrically deforming surface, the generated images in +our protocol are the result of real object deformations. Being +formed of successive images with continuous deformation, +it can also be used for algorithms which exploit feature and +shape tracking. In addition, object occlusion and invisibility +can be easily simulated, by dropping frames or pasting an +occluder. +The protocol flowchart is shown in Figure 3. First, we form +the Aruco template by randomly distributing a set of Aruco +markers all over a blank image. We then print the Aruco +template on a standard A4 paper. These markers should be +big enough to be recognizable by the user’s camera in the de- +sired distance. In order to improve recognition, there should +be white space between the markers on the paper. In our +experiments, we used 100 markers with a width of 1.4 cm. +The OpenCV library was used to identify the markers. These +markers were recognizable by a 720p RGB camera from an +approximate distance of 0.6m. The next step is to deform +the printed Aruco template in front of the camera. In each +frame, the 2D and 3D coordinates of the markers’ centers +are estimated. Because each marker has its own unique id, +they can be used as correspondences between the Aruco +template and each image of the video. We exploit the 2D +coordinates of these recognized correspondences to estimate +a warp with which we can transfer an arbitrary texturemap +to the video image space. This is done firstly by resizing the +arbitrary texture to the size of the Aruco template. In order +to keep the aspect ratio of the arbitrary texturemap, white +margins can also be added before resizing. Then, an inverse +warping process with bilinear interpolation is used to transfer +the pixel color information from the arbitrary texturemap to +their corresponding pixels in the video images. The whole +procedure results in a scene with the arbitrary texturemap +being deformed exactly on top of the Aruco template. It is +also possible to add further modifications; for instance, one +can transfer the arbitrary texturemap to another scene with +any different background. Besides, as in [38], an artificial +lighting can also be added to form different variations of the +scene. +For evaluating algorithms, one can use the 2D and 3D +ground truth estimated in each frame of the video. Regarding +the 2D ground truth, the estimated warp can be used to +identify the 2D corresponding point of each pixel of the +arbitrary texturemap in the image. As for the 3D ground +truth, one can exploit the 3D estimated coordinates of the +Aruco markers in each frame which can be achieved using +the OpenCV library. +V. myNeighbor +We describe myNeighbor, our novel mismatch removal +algorithm. It works based on two main principles: +• Having an image of a textured surface and another +image of that surface undergoing a deformation, to +estimate a sufficiently accurate transfer function be- +tween them with which one can judge the correctness of +matches, there is no need to remove all the mismatches +from the list of matches. Instead, a set of correct +matches would be sufficient to estimate the transfer +function. +• This set of correct matches can be extracted considering +that in reality, under a deformation, the neighborhood +structure among the points on a deforming surface is +preserved. +We show that by using these two principles, the mis- +matches can be detected and removed in a fast and efficient +way. The proposed algorithm is illustrated in Figure 4. It +consists of three steps. First, a set of matches which are +highly probable to be correct are selected. This selection is +done by forming two triangulations using match points, one +in P and one in I, and then choosing matches with high +similarity in the list of their neighbors. Second, a small per- +centage of possible mismatches among the selected matches +are identified and removed. This is done by transferring the +selected match points from P to I and then removing those +with large distances from their correspondences in I. Third, +we transfer all the match points from P to I using a warp +estimated based on the clean set of selected matches from the +second step. The distance between the transferred template +match points and their correspondences in I is used as the +criterion to distinguish estimated mismatches from estimated +correct matches. +In order to analyze the performance of myNeighbor and +calibrate the parameters in the different steps, we used +synthetic data experiments. In the following section, we +describe the design of these experiments. Afterwards, we +describe in detail the different steps of myNeighbor. +A. Synthetic data experiments for calibrating parameters +These experiments are conducted by synthetically forming +two images of a mesh MT and a series of matches between +the two images. The first image shows MT in its flat rest +shape with all its keypoints on it. We call this image IF . +In IF , the keypoints can be considered as the extracted +keypoints from P and the 2D mesh is equivalent to M. The +second image simulates I and shows MT having undergone +a random 3D deformation. We call this deformed mesh MG. +The keypoints in this image can be positioned in their correct +locations on the mesh (correct matches) or being displaced +in the image area (mismatches). +We consider MT as a regular triangular mesh with 10 × 6 +points in 3D space. In order to deform MT , we use the +same method as in [10]. This is done by applying two 3D + +Fig. 4: Flowchart of myNeighbor. +deformations containing random translations and rotations +to two mesh cells at both sides of MT . The deformation +is calculated in an iterative process based on position-based +dynamics [39], [40]. As for generating keypoints, we first +randomly place keypoints in the inner area of M in IF . +In order to create the matches between IF and I, we then +transfer the keypoints from IF to I using a three-step +process: calculating barycentric coordinates of the keypoints +in M, transferring the keypoints to the 3D deformed mesh +using the barycentric coordinates and the new 3D mesh +points of the deformed MT , and eventually projecting the +transferred keypoints to I. To generate mismatches, an arbi- +trary percentage of the transferred keypoints were corrupted +by randomly distributing them all over the area of I. Two +samples of the generated images for 100 and 1000 matches +each with 30% mismatches can be observed in the two first +columns of Figure 5. +B. Methodology +The algorithm myNeighbor is applied on Nm matches +denoted as Cp ↔ Cq between P and I, with: +Cp = {p1, ..., pNm}, pi = (xi, yi) +(1) +Cq = {q1, ..., qNm}, qi = (ui, vi) +(2) +A pair (pi, qi) of points with the same index forms a match +pi ↔ qi. We define the set of correct matches Sin as the +collection of matches pi ↔ qi where pi and qi point to +the same location on the deforming surface in P and I. +On the contrary, when the pointing locations of the match +points are different, they are categorized as mismatches Sout. +The goal of myNeighbor is to form and remove the subsets +Op ⊂ Cp and Oq ⊂ Cq which have the largest possible +number of matches belonging to Sout and smallest possible +number of matches belonging to Sin. We explain the steps +of our algorithm to fulfill this goal. +1) Step I – Neighbor-based correct match selection: We +select subsets Cps ⊂ Cp and Cqs ⊂ Cq which are highly +probable to form correct matches. We start by defining WG +as the groundtruth warp between P and I that can transfer +all the match points Cp from P to their correct locations in +I. With this definition, we have the set of correct matches +Sin as: +Sin = {(pi, qi) | i ∈ R}, +(3) +where: +R = {i | ∥WG(pi) − qi∥ < ϵ}, +(4) +where ϵ is a very small positive number. Warp WG is an un- +known composition of isometric deformation and perspective +projection mappings. The isometric deformation mapping +preserves the geodesic distances among the points and their +topological structure on the object’s surface. However, with +the addition of perspective projection mappings, only the +topological structure of points remains preserved in visible +areas. This implies that by applying WG, the neighborhood +structure among the points on the object in P and I + +Set of matches between P and I as Cp Cg +and M (or any 2D mesh that covers P ) +Step 1: Selecting matches that are highly probable to be correct +Generate two triangulations: Tp using Cp in P, and Tg using +Cqin I +Form neighbor lists Qp and Qg for all the matches in P and I +Calculate the parameter MF for all the matches and select +Estimate warp Wi using Cp, ←→ Cas and use it to transfer +M to I to obtain Mi +Step 2: Removing probable mismatches from the selected matches +Estimate Ca, by transferring Cp, to Iusing barycentric +coordinates of match points ps with respect to M and the +transferred mesh M1 +Calculate Euclidean distance d2(i) between each Qs; ←→ qs +Remove possible mismatches from the list of Cps Ca,by +the MAD criterion +Estimate warp W2 using the modified list of matches Cps ←→ ( +and use it to transfer M to I to obtain M2 +Step 3: Removing all the mismatches from the list of all the matches +Estimate Ca by transferring all match points C, from P to I +using barycentric coordinates of p with respect to M and +the transferred mesh M2 +Calculate Euclidean distance dg(i) between each Qi qi +Form mismatch sets Op and Og by selecting matches with +dg(i) > d3TH +Set of correct matches between P and I as (Cp - Op) (Cq - Oa)Fig. 5: Two sample results of the steps for synthetic data experiments. The first row is an experiment with 100 matches +and a mismatch percentage of 30%. The second row is an experiment with 1000 matches and a mismatch percentage of +30%. The first and second columns represent IF and I with correct matches in green and mismatches in red. The third +column is the result of Step I. The wrongly chosen mismatches are shown in red. The fourth column is the result of Step +II. The mismatches along with a small percentage of correct matches are removed. The fifth column is the separation of the +estimated correct matches and the estimated mismatches from Step III. The transferred meshes � +M1, � +M2, and � +M3 are shown +in orange, yellow, and cyan for the three steps. +should be preserved. We exploit this characteristic of WG +to estimate �R as the set of indices of highly probable correct +matches Cps ↔ Cqs. To do so, first, we form two Delaunay +triangulations, Tp = D(Cp) in P, and Tq = D(Cq) in I. +Then, for each match i, we calculate two sets of first-order +neighbors Qp(i) and Qq(i) in P and I, respectively. We then +define the Mismatch Factor (MF) criterion for match i as: +MF(i) = |Qp(i) ∪ Qq(i) − Qp(i) ∩ Qq(i)| +|Qp(i) ∪ Qq(i)| +× 100 +(5) +For each match, MF represents the difference in the neigh- +bor points between P and I as a percentage. Ideally, we +expect that for all the matches MF = 0, which implies that +there is no difference in the neighbors of each match during +a deformation. However, in practice, there are two reasons +which rather put MF values in a range from 0 to 100: the +presence of mismatches and variations in triangulation. The +presence of mismatches can affect the value of MF in two +ways. First, when the match point i in I is a mismatch +and thus located in a wrong location. And second, when +the match point i in I is a correct match but one, several, +or all of its neighbors are mismatches. Both of these cases +result in different neighbors in I in comparison to P. As +for the two triangulations, it should be noted that even in +the absence of mismatches, the neighborhood structures in +Tp and Tq do not necessarily coincide. This is because of +surface deformation, change in viewpoint, and occlusions. +Calculating MF for all the matches, we can have a fair +estimation regarding the state of the matches. The lower +values of MF(i) indicate that the match i is surrounded by +similar matches in P and I and has a higher probability to be +placed in its correct location and thus be a correct match. On +the contrary, the higher values of MF(i) can stem from the +wrong location of the match i in comparison to its neighbors +which strengthens the possibility of it being a mismatch. The +basic idea in this step is to form Cps ↔ Cqs by selecting +pairs of highly probable correct matches ps ↔ qs. This is +done by choosing the matches with lower values of MF. +We examined the validity of this reasoning by evaluating +three different synthetic data experiments, each with 1000 +matches and different rates of correct matches (30%, 60%, +and 90%). Figure 6 shows the histogram of MF for each +case. We observe that the dispersion of MF spans a wider +range as the value of the correct match rate grows. For higher +numbers of correct matches, there are more similarities in +the neighbor lists of each match and, consequently, MF de- + +Mismatch removal algorithm steps +Template +Image +Step I +Step II +Step III +。 Correct matches + Correctly selected correct matches + Correctly kept correct matches +。 Final estimated correct matches +。 Mismatches +o Mistakenly selected mismatches +• Mistakenly kept mismatches +。 Final estimated mismatches + Mesh (flat in the template and deformed in the image) +- Mesh M; transformed by the warp Wi - Mesh M transformed by the warp W2 - Mesh M; transformed by the warp W3 +n,=32% AoS=93.75% +n,=25% AoS=100% +TPR=1 +FPR=0 +n,=25.2% AoS=99.2% +n,=23% AoS=100% +TPR=1 +FPR=0Fig. 6: Histogram of MF values for three sample synthetic +data experiments with 1000 matches and 30%, 60% and 90% +of correct matches. +creases. Furthermore, regardless of the values of the correct +match rate, the majority of the mismatches are accumulated +in the top bins of the graphs that correspond to higher values +of MF. This is shown in more detail for the case with the +correct match percentage of 30% by expanding the last two +bins of the graph in Figure 6.a. This validates our prior +reasoning that by selecting the matches with MF below +a certain threshold MFth, we can have a set of matches +which are highly probable to be correct. To quantify the +appropriateness of this selection, we define two criteria, +based on the following two quantities. The first quantity is +ns, which is the percentage of the selected matches compared +to the total number of matches: +ns = |Cs| +Nm +× 100, +(6) +where Cs = {(pi, qi) | i ∈ �R} is the set of selected matches. +The second quantity is AoS, which is the Accuracy of +Selection, defined as: +AoS = |Cs ∩ Sin| +|Cs| +× 100. +(7) +Our goal is to choose the value of MFth in the way that we +have both of these criteria to be as high as possible, which +means selecting a high percentage of matches with high +accuracy. However, practically, these two criteria work in +reverse. By choosing a higher value for MFth, more matches +are selected (higher ns) but with less accuracy (lower AoS) +and vice versa. In order to choose the proper value for MFth, +we analyzed the behavior of these two criteria for a series +of synthetic data experiments. We consider three scenarios +for these experiments based on the number of matches, i.e., +Dense, Moderate, and Sparse with in turn 1000, 200, and 50 +total number of matches. The experiments were done in a +wide range of correct match percentages (10% to 100%) for +each scenario. Two different values of the criterion MFth +were studied; mean and 0.9 × mean where mean is the +mean of all MF values in each experiment. The results are +presented in Figure 7.a and 7.b. Each point in the graph is +the average result of 1000 trials. The first point that should +be noted here is that, generally, the proposed match selection +method in this step is more reliable as the number of total +matches grows. This can be deduced by comparing the higher +values of AoS in the Dense case with the ones in the +Moderate and Sparse cases. As for choosing MFth, it should +be noted that setting MFth = 0.9 × mean leads to higher +values of AoS in comparison to the case with MFth = +mean. Nevertheless, as shown in Figure 7.a, this sacrifices +a high percentage of matches by dropping ns significantly, +which is undesirable. Hence, in this step, we choose mean +as the value of MFth and form �R as the set of indices of +probable correct matches. While this choice implies a higher +number of selected mismatches (lower AoS), we note that +these mismatches can be removed in Step II. +As the final operation in this step, we estimate the warp +W1 between P and I using the selected matches Cps ↔ +Cqs. We then exploit this warp to transfer M to I. We call +this new mesh � +M1. As can be seen in the third column of +Figure 5, the mesh � +M1 (shown in orange) may not be totally +faithful to the deformation of MG in I, which is due to the +inaccuracies in the calculation of the warp W1. This stems +from two main reasons; the existence of mismatches in our +selection (shown as red dots), and the insufficient number of +correct matches in some areas. In the next step, we exploit +the transferred mesh � +M1 to remove the possible remaining +mismatches from the selected matches. +2) Step II – Removing mismatches from the list of selected +matches: We remove the possible mismatches from the +selected matches Cps ↔ Cqs. We first form the set Cˆqs by +transferring Cps to I. This is done by finding the barycentric +coordinates of each selected match psi ∈ Cps with respect +to M and applying them on the transferred 2D mesh � +M1 +from Step I. We then use the following decision criterion to +identify and remove possible mismatches one by one from + +Correct Match Percentage 30% +90-100 +80-90 +70-80 +98-100 +60-70 +50-60 +40-50 +z6-06 +30-40 +20-30 +80-82 +10-20 + Correct Matches +0-10 +0 +200 +400 +600 +800 +Mismatches +0 +200 +400 +600 +800 +1000 +Number of Matches +(a) +Correct Match Percentage 60% +90-100 +80-90 +70-80 +60-70 +50-60 +MF +40-50 +30-40 +20-30 +Correct Matches +10-20 +0-10 +Mismatches +0 +200 +400 +600 +800 +1000 +Number of Matches +(b) +Correct Match Percentage 90% +90-100 +80-90 +70-80 +60-70 +% +50-60 +40-50 +30-40 +20-30 +Correct Matches +10-20 +0-10 +Mismatches +0 +200 +400 +600 +800 +1000 +Number of Matches +(c)Fig. 7: Results of applying the first two steps of the algorithm myNeighbor in synthetic data experiments in three different +scenarios; Dense (1000 matches), Moderate (200 matches), and Sparse (50 matches). Each curve is the average result of +1000 trials. The first row gives ns and AoS from Step I for two different values of MFT H. The second row gives the +results of Step II in comparison to the results of Step I with MFth = mean(MF). +Fig. 8: ROC curves resulting from the algorithm myNeighbor in synthetic data experiments in three scenarios; Dense (1000 +matches), Moderate (200 matches), and Sparse (50 matches). Each point is the average result of 1000 trials calculated with +a specific value of d3th. +the selected matches Cps ↔ Cqs: +���d2(i) − median +� +{d2(j)} +���� ⩾ 2.5 MAD, +(8) +where d2(i) = ∥ˆqsi − qsi∥ with i ∈ �R. MAD (Median of +Absolute Deviations from Median) is calculated as: +MAD = k median +�����d2(i) − median +� +{d2(j)} +���� +�� +, +(9) +where k = 1.4826 is a constant number. The values of d2 +are relatively larger for mismatches in comparison to correct +matches. This stems from two reasons. First, the small +percentage of mismatches compared to the great majority of +correct matches coming from Step I and thus lesser influence +of mismatches in the estimation of warp W1. Second, the +inconsistent location of mismatches in P and I. The decision +criterion in equation (8) is chosen due to the distribution type +of d2, with the presence of just a small percentage of large +values among the majority of small values. Figure 7.c and d +illustrate the result of this step. As can be seen, unlike the +previous strategy of choosing a smaller MFth, this method +results in improvement of AoS without losing a considerable +percentage of selected matches. This can be clearly observed +by comparing ns in Figures 7.a and c. +As the last operation in this step, warp W2 is calculated + +60 +100 +90 +50 +80 +40 +70 +(%) "u +AoS(%) +60 +50 +Dense, +40 +Dense, +MF.= mean(MF) +20 +Moderate, MF. +MF=mean(MF) +r= mean(MF) +30 +Sparse, +Sparse, +10 +Dense, +MF = 0.9.mean(MF) +20 +Dense, +MF.= 0.9 mean(MF) +- +Moderate, MF= 0.9.mean(MF) +10 +Moderate, MF= 0.9 mean(MF) +Sparse, +MF= 0.9 mean(MF) +Sparse, +MF.= 0.9.mean(MF) +0 +0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Correct Match Percentage (%) +Correct Match Percentage (%) +(a) +(b) +60 +100 +90 +50 +80 +40 +70 +n,(%) +AoS(%) +60 +30 +50 +Dense, +Step I +40 +Dense, +Step I +20 +Moderate, Step I +Moderate, Step I +Sparse, +Step I +30 +Sparse, +Step I +Dense, +Step II +20 +Dense, +Step II +10 +Moderate, Step II +Moderate, Step II +Sparse, +Step II +10 +Sparse, +Step II +0 +0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Correct Match Percentage (%) +Correct Match Percentage (%) +(c) +(d)ROCCurvefor scenarioDense +ROC Curve for scenario Moderate +ROC Curve for scenario Sparse +6'0 +60 +60 +True Positive Rate +0.8 +True Positive Rate +80 + 0.7 +0.7 +0.8 +0.7 +0.6 +0.5 +0.6 +0.5 +Correct Match Percentage =90% +0.4 ++Correct Match Percentage = 90% +0.4 ++ Corre ct Match Percentage = 90% +0.3 +Correct Match Percentage = 70% +0.3 +Corre ct Match Percentage = 70% +0.2 +Correct Match Percentage = 50% +0.2 + Corre ct Match Pe rce ntage = 50% +0.2 +0.1 +0.1 ++Correct Match Percentage = 30% +0.1 +Correct Match Percentage = 30% +0 +0 +0 +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 +False Positive Rate +False Positive Rate +False Positive Rate +(a) +(b) +(c)Fig. 9: Performance evaluation of our mismatch removal method myNeighbor in comparison to the state-of-the-art methods +using the FREX protocol. The first row shows the Aruco template and three selected images (14, 47, 60) of the deformation +of the printed Aruco template. The following rows show five datasets of generated scenes with the texturemap in the first +column, three generated images corresponding to the first row in the next columns, and the ROC curves of the mismatch +removal algorithms in the last column. For each of the images � +M3 from myNeighbor is overlaid. +Method +Average run-time (s) +myNeighbor +0.0139 +Tran et al. [31] +0.0206 +Pizarro et al. [29] +1.8925 +Famouri et al. [16] +0.0171 +TABLE II: Comparison of the average run-time of the +mismatch removal algorithms for processing all the images +of all the datasets. +using the purified selected matches Cps ↔ Cqs. This warp +is then used to transfer M to the image space and form +� +M2. The result of removing possible mismatches in this step +along with the transferred mesh � +M2 are shown in the fourth +column of Figure 5. As can be observed, in comparison to +� +M1, � +M2 has a better compliance to MG. +3) Step III – Extracting mismatches from the list of all the +matches: In this step, we exploit the transferred mesh � +M2 +to extract the mismatches Op ↔ Oq from the total matches +Cp ↔ Cq. The process is similar to Step II except that this +time all of the matches are checked. We first transfer the +template match points Cp to the image space and form the +set Cˆq. This is done by calculating barycentric coordinates of +all the match points Cp with respect to M and applying them +on the new transferred mesh � +M2. We define the following +decision criterion to detect and remove mismatches: +d3(i) = ∥ˆqi − qi∥ ⩾ d3th +(10) +Unlike Step II where we used the MAD criterion to remove +just a small rate of mismatches, this time we use a constant + +图 +图图 +图 +图 + - Tran et al. [34] +固 +图图 +四 +巴 +Pizarro et al. [23] +国 +Famouri et al. [15] +T +3 +myNeighbor +0.9 +TPR +0.8 +0.7 +0 +0.1 +0.2 +0.3 +FPR +1 +0.9 +TPR +0.8 +0.7 +0.1 +0.3 +0.9 +TPR +0.8 +0.7 +0.1 +0.2 +0.3 +FPR +0.9 +TPR +0.8 +0.7 +0 +0.1 +0.2 +0.3 +FPR +0.9 +PR +0.8 +0.7 +0 +0.1 +0.3 Fig. 10: Applying myNeighbor on four real cases: a cushion, a Spiderman poster, a shoe sole, and an elastic shirt. The +first column shows the texturemaps. The second column shows Step I. All the matches are shown in this column while the +selected matches in Step I are shown in green. These selected matches are transferred to column three that shows Step II. +In this column, those matches which are chosen as possible mismatches are shown in red. The last column is the distinction +between the estimated correct matches (in green) and the estimated mismatches (in red) in Step III. The meshes � +M1, � +M2, +and � +M3 are overlaid to illustrate the computed warps. +threshold d3th. This is due to the higher percentage of +mismatches compared to Step II. In order to make this +distinction method more robust, we consider d3th as the +multiplication of a sample length ls and a constant coefficient +αs. The sample length ls is a measure of the size of the +object in the image in pixels and is calculated as the average +distance between all the mesh points in the transferred mesh +� +M2. To choose a proper value for the constant coefficient αs, +a series of synthetic data experiments with the same three +scenarios as before (Dense, Moderate, and Sparse) and four +different correct match rates was performed. The results are +presented as ROC (Receiver Operating Characteristic) curves +in Figure 8.a-c. Each point represents the average TPR (True +Positive Rate) versus the average FPR (False Positive Rate) +computed in 1000 trials using a specific value of αs in the +range of [0, 1]. TPR is calculated as the number of selected +true mismatches over the number of all true mismatches, +and FPR is calculated as the number of true correct matches +mistakenly selected as mismatches over the number of all +true correct matches. Ideally, all the mismatches should +be discarded (TPR=100%) without discarding any correct +matches (FPR=0%). Hence, the most favorable αs in a single +ROC curve is the one that results in the maximum possible +TPR leaving the FPR below a reasonable value. We choose + +Mismatch Removal Algorithm Steps +Texture-map P +Step I +Step II +Step III + Selected Matches +。 Kept Matches from Step I +。 Estimated Correct Matches +。 Unselected Matches +。 Removed Matches from Step I +。 Estimated Mismatches +- Mesh M, Estimated by Warp W +- Mesh M, Estimated by Warp W, + Mesh M, Estimated by Warp W +DEL +CHAMELEON STRIKES!Fig. 11: Comparing the accuracy of the 3D shape inference methods with Particle-SfT with three datasets obtained by FREX. +The 3D shape inference methods are Brunet et al. [17], Chhatkuli et al. [41], Bartoli et al. [14], Ostlund et al. [20], and +Salzmann et al. [19]. +αs = 0.15 which keeps TPR above 90% while FPR remains +below 10% for most of the cases. The last column of Figure 5 +illustrates the estimated correct matches (in green) and the +estimated mismatches (in red) for each case. We also use the +estimated correct matches to estimate warp W3 and transfer +M to I and form � +M3 (shown in cyan). As can be seen, +there is a high compliance between � +M3 and MG. It should +be noted that estimating W3 and � +M3 is not necessary in +myNeighbor and we merely estimate them just to visually +present the effectiveness of the algorithm in removing the +mismatches. However, considering myNeighbor as a step +in ROBUSfT, due to the fact that the final estimated correct +matches are passed from this step to Step 3 of ROBUSfT +which is warping, W3 and � +M3 can also represent W and � +M +in the warping step, respectively. +C. Mismatch removal results +In +this +section, +we +demonstrate +the +efficiency +of +myNeighbor by evaluating its performance through various +tests. We first compare the results of the algorithm with +the state-of-the-art algorithms in the literature by testing +them through FREX. The experiment includes 60 frames of +continuous deformation of the Aruco template in front of +the camera. Five datasets were generated in this experiment +each with an arbitrary texture with a challenging pattern. +Three different types of backgrounds were also considered +for these five cases, specifically two original backgrounds, +two white backgrounds, and a background with a pattern +similar to one of the texturemaps. We apply all the mismatch +removal algorithms on all datasets. For each dataset, the +corresponding arbitrary texture was used as the texturemap +for the mismatch removal algorithms. The matches between +the texturemap and each image of the dataset are extracted +using SIFT. The results are presented in Figure 9. The first +row illustrates the Aruco template and also three selected +original images of its deformation in front of the camera. +The lower rows represent the five datasets generated by FREX. +Each row shows the arbitrary texture of the dataset in the first +column, the three selected generated images, and eventually +the resulting ROC curves for all the mismatch removal +algorithms on the dataset. In the ROC curves, for a certain +algorithm and a certain dataset, each point is the average +value of TPR and FPR over all 60 images of that dataset +using a specific value for the threshold used in the algorithm. +As can be seen, in all cases, our algorithm outperforms the +other algorithms. In order to show the performance of our +algorithm visually, for each dataset, we overlaid � +M3 for the +three selected frames. As can be observed, the transferred +meshes are visually well-aligned to the 2D deformed shape +of the object. In some cases, a small number of irregularities +can be observed in certain areas (for example in the Matrix +poster). This is because of the presence of a small number + +50 +50 +Salzmann et al. +Ostlund et al. +Bartoli et al. +Salzmann et al. +Ostlund et al. +Bartoli et al. +Chhatkuli et al. +Brunet et al. +Particle SfT +Chhatkuli et al. +Brunet et al. +Particle SfT +40 +Average 3D Error (mm) +40 +30 +30 +20 +20 +10 +10 +0 +0 +0 +10 +20 +30 +40 +50 +60 +0 +10 +20 +30 +40 +50 +60 +Frame Number +Frame Number +50 +Salzmann et al. +Ostlund et al. +Bartoli et al. +Chhatkuli et al. +Brunet et al. +Particle SfT +30 +20 +10 +0 +10 +20 +30 +40 +50 +60 +Frame NumberFig. 12: Comparison between ROBUSfT and the methods +presented by Famouri et al. [16] and Ngo et al [21] on +the public dataset provided in [38]. (a) Mean absolute 3D +error between the inferred shape and the groundtruth. (b) +Execution time in milliseconds. +of mismatches in our list of estimated correct matches and +the lack of matches in those areas. As for comparing the +execution speed of different mismatch removal algorithms, +the process run-times for all the frames of all datasets +were averaged and tabulated in Table II. It shows that our +algorithm is faster than the others. It should be however noted +that our algorithm is implemented in C++ while the others +are in Matlab. +After validating the efficiency of myNeighbor in com- +parison to the state-of-the-art algorithms in the literature, +we evaluate its performance in real cases. To this end, +we applied our algorithm to four real deforming objects +as shown in Figure 10. We chose these cases in such a +way that each one is challenging in a special way. The +cases include a cushion with non-smooth surface and severe +deformation, a Spiderman poster deformed in a scene with +background covered with almost the same posters, a shoe +sole with an almost repetitive texture, and a shirt with +elastic deformation. The texturemaps are shown in the first +column of Figure 10. The second to fourth columns show the +results of Step I to Step III of myNeighbor. In each step, +the alignment of the corresponding transferred mesh to the +2D shape of the deforming object can be considered as an +indication of the correctness and abundance of the estimated +correct matches. Like in the synthetic data experiments, this +alignment improves progressively in different steps of our +algorithm. One point that should be noted here is that the +shirt (the last case in Figure 10) is elastic. We exert a non- +isometric deformation on it by pulling from both sides, and +myNeighbor still works. This is due to the fact that we did +not make any assumption regarding isometry. In fact, the only +assumption that we made is the preservation of neighborhood +structure in the deforming object. As a result, myNeighbor +also works with non-isometric deformations which preserve +neighborhood structure. +VI. EXPERIMENTAL RESULTS +We evaluate the performance of ROBUSfT on different +deforming objects in various conditions. We divide this +section into two main parts; first, comparing the results with +the state-of-the-art methods and then evaluating ROBUSfT in +several other challenging cases. +A. Comparison to the state-of-the-art methods +We compare ROBUSfT with the state-of-the-art methods +through two different tests. The first test is conducted among +the shape inference methods (G1). The second test is carried +out among the integrated methods (G2). +We use FREX to conduct the first test. To this end, the +same 60 images of deforming Aruco marker paper sheet are +used. We create three different datasets using three arbitrary +texturemaps and apply a white background to all the scenes. +The arbitrary texturemaps include a painting, the Joker +poster, and a paper sheet filled with basic geometric shapes. +These images are shown in Figure 11. In each dataset, we +compare the result of the last two steps of ROBUSfT (warp +estimation and 3D shape inference) with five other shape +inference methods from Brunet et al. [17], Chhatkuli et +al. [41], Bartoli et al. [14], Ostlund et al. [20], and Salzmann +et al. [19]. A similar comparison was made in [18] on another +dataset. However, in [18], a random 3D shape was used as the +initial guess for Particle-SfT algorithm in each image of the +video; in contrast, we use the 3D inferred shape of the object +in each image as the initial guess for the next image. In each +dataset, the matches between P and each image are extracted +using SIFT. We then separate the correct matches and use +them as the input for all the methods. If required by a shape +inference method, a BBS warp is estimated based on these +correct matches and used as the input to that shape inference +method. The results for all three datasets are presented in +Figure 11 as the average 3D error between the 3D inferred +shapes and the ground truth. As can be observed, Particle- +SfT provides the lowest value of 3D error in comparison +to the other methods. This is more apparent in the datasets +with lower number of matches. In the last dataset, there are +several discontinuities in the 3D error graph of state-of-the- +art methods. This is due to the failure of shape inference in +those images of the video by those methods. Particle-SfT, +however, succeeds to infer the 3D shape of the object in all +of the images with a reasonable error. + +20 +Famouriet al. +Ngo et al. +15 +ROBUSfT +10 +50 +100 +150 +200 +FrameNumber +(a) +10000 +ExecutionTime(ms) +1000 +Famouri et al. +Ngo et al. +ROBUSfT +100 +10 +50 +100 +150 +200 +Frame Number +(b)Fig. 13: Evaluating ROBUSfT in three real data experiments; a Spiderman poster, a chopping mat, and a t-shirt. The +texturemaps of the templates are shown in the first column. For each case, four images are shown. Below each frame, +the reconstructed 3D shape of the deforming object with the estimated 3D coordinates of the estimated correct matches +(red particles) as well as their ground truth (green particles) are shown. The 2D projections of the 3D inferred shapes are +also overlaid on the image. For each image, the median Euclidean distance between the estimated 3D coordinates of the +estimated correct matches and their ground truth is given below the reconstructed shape. + +12号 +ELCHAMELEON STRIKES! +5.93mm +3.69mm +6.12mm +5.14mm +4.12mm +5.65mm +5.56mm +4.88mm +RKSCOT +5.35mm +11.45mm +5.37mm +8.03mmFig. 14: Evaluating ROBUSfT in a real data experiment with +two robotic arms; soft constraints are applied to bind the +constrained mesh points to the grippers. Each row shows +three images: the original camera view, the projection of +the 3D reconstructed mesh on the camera view, and the 3D +reconstructed mesh with the robots in the RViz environment. +For the second test, we ran ROBUSfT on the public dataset +provided in [38]. The dataset includes the 2D correspon- +dences as well as 3D Kinect data of 193 consecutive images +of a deforming paper. The paper is planar and no occlusion +appears in the series of images. We compared our results with +the results of Famouri et al. [16] and Ngo et al. [21] which +were presented in their papers. This is shown in Figure 12- +a and Figure 12-b. As can be observed, ROBUSfT is both +faster and more precise. It should be noted that ROBUSfT +used directly images as the input and covered the whole +process from extracting keypoints to 3D shape inference. In +contrast, the other two algorithms used the already available +correspondences in the dataset. Another relevant point is +that in this test we use a serial CPU-GPU architecture +instead of a parallel one. This is done to make sure that the +captured image that we analyze and the ground truth that +we compare to are for the same image. This consequently +reduces the execution speed of our code compared to the +parallel architecture. In the next series of tests we use the +parallel architecture. +B. Evaluation of ROBUSfT +We first evaluate the efficiency of ROBUSfT in three real +cases. These cases are shown in Figure 13. The tested objects +are a Spiderman poster, a chopping mat, and a t-shirt. In each +case, the object is deformed in front of a 3D camera with +which we capture both RGB image and the depth of each +point on the object. We use the measured depth as ground +truth for evaluating the reconstructed 3D shape. We use the +Intel RealSense D435 depth camera and built-in libraries +for aligning the depth map to the RGB image. For each +case, four images of the experiment are shown in Figure 13. +In the first case, we set the resolution of the camera to +640 × 480. In the second and third cases, we increased it +to 1280 × 720 due to the insufficient number of detected +keypoints using the previous resolution. Below each image, +the reconstructed 3D shape of the deforming object along +with the 3D coordinates of the estimated correct matches (red +particles) as well as their ground truth (green particles) are +shown. The 3D coordinates of the estimated correct matches +are estimated by calculating their barycentric coordinates +in P with respect to M and applying these coordinates +on the 3D reconstructed mesh of the object. The number +written below each frame is the median distance between +the reconstructed 3D coordinates of the estimated correct +matches and their ground truth. The median is chosen due +to the probable existence of mismatches among the list of +estimated correct matches. In 3D space, the ground truth of +these mismatches can be located in the background and not +on the object itself. This significantly increases the 3D shape +error. Using the median gives a better estimate of the 3D +shape error considering the existence of this small percentage +of mismatches with large 3D errors. +As can be observed, the pipeline succeeds to infer the +3D shape of the object in all of the cases. This success +is more visible in the second and third cases due to the +relative scarcity of keypoints and existence of repetition in +their patterns. Regarding the Spiderman poster case, it should +be noted that there are self-occlusions in the first and third +illustrated images. In these images, the 3D shape of the +object in the occluded areas is estimated by the deformation +constraints implemented in Particle-SfT. These constraints +preserve the geodesic distance between each pair of mesh +points as its initial value in MT . Regarding the runtime, using +the parallel architecture and 640 × 480 captured frames as +the input (as in the Spiderman poster case), the execution +speed reaches 30 fps. +The last experiment is a practical use case with robots. +The experiment aims at highlighting the advantage of using +known 3D coordinates in ROBUSfT. As mentioned in Step +4 and shown in Figure 14, these known coordinates are an +optional input to the last step of ROBUSfT. Their usage can +increase the robustness of the tracking process. The setup +of this experiment is the same as in [42], where we applied +ROBUSfT in a robotic case, specifically, controlling the shape +of deformable objects. The setup consists of two robotic +arms grasping and manipulating the Spiderman poster from +both sides and a top camera facing the manipulation area. +The 3D positions of the two robotic grippers are known +in camera coordinates thanks to the known pose of each +gripper in the robots’ coordinate frames and also the external +calibration between the robots and the camera. For each +gripper, we consider the closest mesh point to the gripper +as a constrained mesh point. These mesh points should be +bound to their corresponding gripper and move with it. As +described in [42], this binding is performed using a soft +constraint. In this soft constraint, for each gripper, a sphere +with a small radius centered at the gripper’s 3D position +is considered. Then, in each iteration of Particle-SfT, if the +corresponding mesh point is outside this sphere, it will be +absorbed to the closest point on the sphere surface. This +soft constraint has two main advantages over rigidly binding +the constrained mesh points to the grippers: first, they let + +the position-based dynamic equations in Particle-SfT that +preserve the distances between the mesh points be applied +on the constrained mesh points, which leads to a smoother +reconstructed shape. Second, by applying soft constraints, +we can cope with small possible errors in robot-camera +calibration. In fact, a wrong robot-camera calibration leads +to a wrong transfer of the grippers’ 3D coordinates to the +camera coordinate frame which eventually results in wrong +coordinates of the constrained mesh points. By using the +soft constraint and considering a sphere rather than a rigid +bind, we give a certain degree of flexibility to the constrained +mesh points to move in close proximity to the gripper’s +coordinates. This can compensate for slightly inaccurate +coordinates of the grippers. +VII. CONCLUSION +We have proposed ROBUSfT, a new pipeline that can effec- +tively reconstruct the 3D shape of an isometrically deforming +object using a monocular 2D camera. The proposed pipeline +addresses the well-known challenges in this area. These +challenges include ambiguities in inferring the 3D shape of +the deforming object from a single 2D image and real-time +implementation. We have introduced myNeighbor, a novel +mismatch removal algorithm for deforming objects, which +works based on the preservation of the neighborhood struc- +ture of matches. We validated the efficiency of myNeighbor +in comparison to the state-of-the-art algorithms in numerous +experiments. In order to compare ROBUSfT and myNeighbor +with the state-of-the-art methods in the literature, we have +presented a novel type of experimental protocol called FREX +(Fake Realistic Experiment). This protocol is executed once, +but it provides a large number of resulting scenes of an +isometrically deforming object in various conditions with 2D +and 3D ground truth. This collection can be used to evaluate, +compare, and validate algorithms regarding isometrically de- +forming objects. In addition, the provided 2D and 3D ground +truth may be used for training learning-based algorithms. In +contrast to other artificially made scenes of an isometrically +deforming surface, the generated images in our protocol are +the result of real isometric deformations. +Possible directions for future work include (i) exploiting +the silhouette of the object in the image for improving 3D +shape inference in challenging cases such as weakly-textured +objects, (ii) extending ROBUSfT to volumetric objects and +(iii) adding self-occlusion reasoning. +ACKNOWLEDGMENTS +This work was supported by project SOFTMANBOT, +which received funding from the European Union’s Hori- +zon 2020 research and innovation programme under grant +agreement No 869855. +REFERENCES +[1] J. Pilet, V. Lepetit, and P. Fua, “Fast non-rigid surface detection, regis- +tration and realistic augmentation,” International Journal of Computer +Vision, vol. 76, no. 2, pp. 109–122, 2008. +[2] N. Haouchine, J. Dequidt, M.-O. Berger, and S. 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Bartoli, “Stable template-based +isometric 3D reconstruction in all imaging conditions by linear least- +squares,” in Proceedings of the IEEE Conference on Computer Vision +and Pattern Recognition, pp. 708–715, 2014. +[42] M. Shetab-Bushehri, M. Aranda, Y. Mezouar, and E. ¨Ozg¨ur, “As-rigid- +as-possible shape servoing,” IEEE Robotics and Automation Letters, +vol. 7, no. 2, pp. 3898–3905, 2022. + diff --git a/MNE2T4oBgHgl3EQfqQhz/content/tmp_files/load_file.txt b/MNE2T4oBgHgl3EQfqQhz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..00c8554a606fb8e57576d8c4975fc916fffa7759 --- /dev/null +++ b/MNE2T4oBgHgl3EQfqQhz/content/tmp_files/load_file.txt @@ -0,0 +1,1231 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf,len=1230 +page_content='ROBUSfT: Robust Real-Time Shape-from-Template, a C++ Library Mohammadreza Shetab-Bushehri, Miguel Aranda, Youcef Mezouar, Adrien Bartoli, Erol ¨Ozg¨ur Abstract— Tracking the 3D shape of a deforming object using only monocular 2D vision is a challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is because one should (i) infer the 3D shape from a 2D image, which is a severely underconstrained problem, and (ii) implement the whole solution pipeline in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The pipeline typically requires feature detection and matching, mismatch filtering, 3D shape inference and feature tracking algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We propose ROBUSfT, a conventional pipeline based on a template containing the object’s rest shape, texturemap and deformation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' ROBUSfT is ready-to-use, wide-baseline, capable of handling large deformations, fast up to 30 fps, free of training, and robust against partial occlusions and discontinuity in video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It outperforms the state-of-the-art methods in challenging datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' ROBUSfT is implemented as a publicly available C + + library and we provide a tutorial on how to use it in https : //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='com/mrshetab/ROBUSfT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Keywords: monocular Non-rigid reconstruction, mis- match removal, SfT, validation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' INTRODUCTION Problem and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Tracking the 3D shape of a de- forming object has important applications in augmented reality [1], [2], computer-assisted surgery [3]–[7] and robotics [8]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' However, the existing solutions are im- practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is because of the following challenges: (C1) real-time implementability and (C2) robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Challenge C1 is hard to achieve because the solution usually involves a computationally demanding multi-step pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Challenge C2 is hard to maintain because of noises, occlusions, invisi- ble object, large deformations and fast motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Furthermore, in numerous applications of augmented reality, computer- assisted surgery and robotics, a 2D camera is the de facto sensor owing to its light weight, small size, and low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The camera’s perspective projection introduces an additional challenge, (C3) recoverability of shape’s depth from a 2D im- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Challenge C3 becomes extremely difficult for deforming objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Shape-from-Template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Different priors and constraints have been proposed to resolve challenge C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The most common ones are the object’s 3D rest shape, texturemap, deformation law and the camera intrinsics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These form the ingredients for a variety of methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Among these methods, we are par- ticularly interested in Shape-from-Template (SfT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' SfT has MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Shetab-Bushehri, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Mezouar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Bartoli, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' ¨Ozg¨ur are with the CNRS, Clermont Auvergne INP, Institut Pascal, Universit´e Clermont Auvergne, F-63000 Clermont-Ferrand, France (e-mail: m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='shetab@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' youcef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='mezouar@sigma-clermont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='fr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' adrien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='bartoli@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' erolozgur@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Aranda is with Instituto de Investigaci´on en Ingenier´ıa de Arag´on (I3A), Universidad de Zaragoza, E-50018 Zaragoza, Spain (e-mail: miguel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='aranda@unizar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='es).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' been well studied for isometrically deforming objects [11]– [13] and has been shown to uniquely resolve the depth of each object point [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It uses a template formed by the abovementioned priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' SfT’s input is a single image of a deformed object, and its output is the object’s 3D shape seen in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We adopt a conventional SfT pipeline shown in Figure 1 to solve the 3D shape tracking problem of deforming objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The pipeline involves keypoint extraction and matching, mismatch filtering, warping and 3D shape in- ference steps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We successfully made it real-time and robust by integrating seamlessly both novel and state-of- the-art algorithms at different steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We next overview the strengths and weaknesses of current SfT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' State-of-the-art SfT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' SfT can be broken down into two main parts: registration and 3D shape inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Fol- lowing this, we categorize existing SfT methods into two groups: (G1) shape inference methods and (G2) integrated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' G1 methods only cover the 3D shape inference part [10]–[12], [14]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In contrast, G2 methods cover both the registration and 3D shape inference parts [6], [19]– [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We also overview Deep Neural Network (DNN) based SfT methods, as the third group (G3), which have been recently introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' G3 methods cover both the registration and 3D shape inference parts [24]–[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The majority of G1 methods are wide-baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' However, they barely run in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Furthermore, a complete solution with registration shall be even slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The majority of G2 methods require an initialization close to the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This makes them short- baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Subsequently they often fail against occlusions, fast motions and large deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Once failed, they need to be reinitialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' G3 methods are wide-baseline and run in real- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' However, they are object-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' They require a huge amount of training data and proper computational resources for each new object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These make them difficult to consider as a general and ready-to-use solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We therefore conclude that there does not exist an SfT method that is complete, real-time, robust and easily applicable to new objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We list our contributions in three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' a) Contribution to SfT: We propose ROBUSfT, a com- plete real-time robust SfT pipeline for monocular 3D shape tracking of isometrically deforming thin-shell objects with matchable appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It can track up to 30 fps using 640 × 480 images on off-the-shelf standard consumer hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It does not require initialization and implements tracking- by-detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It is wide-baseline and robust to occlusions, invisible object, large deformations and fast motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It does not require training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It is thus directly applicable in many arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='04037v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='CV] 10 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 1: Overview of ROBUSfT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' industrial and research contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' ROBUSfT outperforms the state-of-the-art methods in challenging datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' b) Contribution to mismatch removal: We introduce myNeighbor, a novel mismatch removal algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It han- dles deforming scenes and a large percentage of mismatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It is lightning fast, reaching 200 fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' c) Contribution to experimental validation: We design a novel type of validation procedure, called Fake Realistic Experiment (FREX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It allows us to automatically generate semi-synthetic datasets with ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This eases the quantitative evaluation of 2D and 3D shape tracking algo- rithms for deforming objects to a great extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Paper structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Section 2 reviews previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Section 3 explains ROBUSfT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Section 4 presents FREX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Section 5 describes myNeighbor, conducts a series of experiments and evaluates the results of myNeighbor in comparison to previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Section 6 validates ROBUSfT through FREX and real data experiments, and compares the results with previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Finally, Section 7 concludes and suggests future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Offline Online Object in rest shape Image acquisition →I Step 1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Keypoint matching Step 1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Keypoint extraction and description Texturemap and keypoints (P) Shape mesh (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=') CAMELEON STRES!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Mismatch removal par inference Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Warping computation Shape mesh aligned Points with known 3D coordinates to texturemap (M) (optional) Loop Default pose Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 3D shape inferenceII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' PREVIOUS WORK We review the methods for monocular shape inference of isometrically deforming objects, following the above three categories, namely, (G1) shape inference methods, (G2) integrated methods, and (G3) DNN-based SfT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For each category, we describe the assumptions, main character- istics, and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We finally compare ROBUSfT to these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' (G1) Shape inference methods These methods cover the 3D shape inference part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' They assume that the registration between the template and the image was previously computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For instance, they typically use keypoint matches between the template and the image, with generic mismatch removal methods [16], [29], [29]– [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In fact, very few methods in this category could form a complete SfT pipeline by adding an existing registration solution [1], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Three general groups are found in existing 3D shape inference methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' (i) methods using a convex re- laxation of isometry called inextensibility [11], [12], [17], (ii) methods using local differential geometry [14]–[16], and (iii) methods minimizing a global non-convex cost function [10], [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The methods in (iii) are the most precise ones but also computationally expensive and require initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The first two groups of methods are often used to provide an initial guess for the third group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the first group, Salzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [12] suggested a closed- form solution to non-rigid 3D surface registration by solving a set of quadratic equations accounting for inextensibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Later, they replaced equality constraints with inequality and thus sharp deformations could be better recovered [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Brunet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [17] formulated two shape inference methods based on point-wise and continuous surface models as Sec- ond Order Cone Programs (SOCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the second group, Bartoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [14] showed that in addition to keypoint 2D co- ordinates in the image, their first-order differential structure can be used to estimate the depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Instead of calculating the warp globally, which is time-consuming, Famouri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [16] estimated the depth locally for each match pair with respect to both local texture and neighboring matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In each frame, the most recognizable matches were selected based on offline training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The execution speed of their algorithm is claimed to be up to 14 fps only for the 3D shape inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the third group, Brunet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [17] proposed a refining isometric SfT method by reformulating the isometric constraint and solving as a non-convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The method required a reasonably accurate 3D shape of the deforming surface as the initializing guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' ¨Ozg¨ur and Bartoli [18], developed Particle-SfT, which handles isometric and non-isometric de- formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' A particle system is guided by deformation and reprojection constraints which are applied consecutively to the particle mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Similar to [17], this algorithm needs an initial guess for the 3D position of the particles, however, for [18], sensitivity to this initial guess is very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The closer the guess to the true 3D shape, the faster the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Aranda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [10] improved this algorithm in terms of execution speed and occlusion-resistance and used that in real-time shape servoing of isometrically deforming objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' They used the 3D shape estimated in one frame as the initial guess for the next frame and thus improved the convergence speed of the algorithm to a great extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' They showed that their algorithm can track a paper sheet covered with markers and being manipulated by a robotic arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' To this end, they only needed to track a handful of markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Knowing the 3D coordinates of several mesh points also has a significant effect on the convergence speed of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The last step of ROBUSfT uses the same method to infer the 3D shape, as explained in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' (G2) Integrated methods These methods handle registration and 3D shape inference at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' They minimize a non-convex cost function in order to align the 3D inferred shape with image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These features can be local [20], [21] or at the pixel-level [6], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Ostlund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [20] and later Ngo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [21] used the Lapla- cian formulation to reduce the problem size by introducing control points on the surface of the deforming object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The process of removing mismatches was performed iteratively during optimization by projecting the 3D estimated shape on the image and disregarding the correspondences with higher reprojection errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Using this procedure, they could reach up to 10 fps using 640×480 input images and restricting the maximum number of template and image keypoints to 500 and 2000, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As for pixel-level alignment, Collins and Bartoli [22] introduced a real-time SfT algorithm which could handle large deformations and occlusions and reaches up to 21 fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' They combined extracted matches with physical deformation priors to perform shape inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [6] later extended this algorithm and used it for tracking organs in laparoscopic videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For achieving better performance, they also exploited organ boundaries as a tracking constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These methods are fast and can handle large deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Their main drawback, however, is to be short-baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In case of tracking failure, they should be re-initialized precisely with a wide-baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This restrict their usage to video streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' (G3) DNN-based methods DNN-based SfT methods have been introduced in the recent years, which coincides with the tendency to use deep learning to solve many computer vision problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These methods are wide-baseline, fast, and cover both the registration and shape inference steps [24]–[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We group these methods based on their type of output, which may be sparse or dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The methods of the first group represent the SfT solution as the 3D coordinates of a regular mesh with a predefined size [24]–[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The usage of these methods is limited to thin-shell objects with rectangular shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The second group of methods gives a pixel-level depthmap as output [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' They also apply a post-processing step based on the As-rigid-as-possible (ARAP) model [32] to the resulting depthmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This step recovers the whole object, Category Method Registration Real-time Wide-baseline General geometry Needless of training for new objects Public access code G1 Salzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [12] × NA ✓ ✓ ✓ × Brunet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [17] × × ✓ ✓ ✓ ✓ Bartoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [14] × NA ✓ ✓ ✓ ✓ Ozgur et Bartoli [18] × × ✓ ✓ ✓ × Famouri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [16] × ✓ ✓ ✓ ✓ ✓ Aranda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [10] × ✓ ✓ ✓ ✓ × G2 Ostlund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [20] ✓ ✓ × ✓ ✓ × Ngo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [21] ✓ ✓ × ✓ ✓ × Collins and Bartoli [22] ✓ ✓ × ✓ ✓ × Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [6] ✓ ✓ × ✓ ✓ × G3 Pumarola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [24] ✓ × ✓ × × × Golyanik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [25] ✓ ✓ ✓ × × × Fuentes-Jimenez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [27] ✓ ✓ ✓ ✓ × × Shimada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [26] ✓ ✓ ✓ × × × Fuentes-Jimenez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [28] ✓ ✓ ✓ × ✓ × ROBUSfT ✓ ✓ ✓ ✓ ✓ ✓ TABLE I: Comparison of the state-of-the-art SfT methods and ROBUSfT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' including the occluded parts, as a mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The method in [27] reconstructs the shape of the object with different geometries and texturemaps that the network is trained for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In [28], however, the proposed method can be applied to objects with new texturemaps unseen to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The geometry of the objects is, nevertheless, limited to flat paper-like shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' All the aforementioned methods in this category are object- specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This means that they merely work for the object that they were trained for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' An exception is [28], as it works for unseen texturemaps but the applicability is still limited to flat rectangular objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' On the other hand, in order to use the DNN-based methods for a new object, the network should be fine-tuned for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This demands proper computational resources and potentially a huge amount of training data, which are challenging to collect for deformable objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Positioning ROBUSfT compared to previous work Existing methods all have one or several limitations, including not covering the whole pipeline, not being wide- baseline, being limited to specific texture or geometry, requiring fine-tuning for a new object, being slow, and lacking public code access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This information is summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In contrast, ROBUSfT covers the whole pipeline and due to the fast execution can be used to develop real-time shape tracking applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It can be instantly used for each deforming object without training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Only a template contain- ing information regarding the object’s geometry, appearance, and deformation law as well as intrinsic parameters of the monocular camera is necessary, but this need is common to all existing and future SfT methods, by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the next section, we describe ROBUSfT and all its steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' ROBUSfT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Overview of the pipeline The overview of our pipeline is presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The pipeline is divided into an offline and an online sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The offline section deals with the template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The online section includes four main steps: keypoint extraction and matching, mismatch removal, warp estimation, and 3D shape inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The images coming directly from the camera are used as the inputs for the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In this step, the keypoints are extracted and matched with the ones that were previously extracted from the template’s texturemap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Then, the mismatches are detected and removed using our new mismatch removal algorithm myNeighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The list of estimated correct matches is then transferred to the next step where a warp is estimated between the template’s texturemap and the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This warp transfers the template’s registered mesh to the image space, which is finally used as input for the 3D shape inference algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This process is repeated for each image, the analysis of each image being performed independently in a tracking-by-detection manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the following, both the offline and online sections of the pipeline are described in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Afterwards, an implementa- tion permitting a fast execution of the pipeline is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Offline section: creating a template We create a template for the surface of the deforming object that we want to track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We call this surface the tracking surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The template of the tracking surface consists of the following elements: MT : the triangular mesh covering the tracking surface at rest shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' P: the texturemap of the tracking surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' M: the alignment of MT to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The first step in creating the template is to generate the 3D model of the tracking surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The 3D model is in fact the textured 3D geometry of the tracking surface in real dimensions in rest shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We form MT by triangulating this 3D geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The resolution of MT should be high enough to be well aligned to the shape of the tracking surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The next step is to take an image from the 3D model of the tracking surface while it is positioned perpendicular to the camera’s optical axis in a simple texture-less background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In this image, P is formed by the projection of the texture of the tracking surface and M by the projection of MT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For simple rectangular thin-shell objects like a piece of paper, the whole process is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For other objects, including thin-shell objects with arbitrary shape, such as a shoe sole, and also volumetric objects, 3D reconstruction software like Agisoft Photoscan [33] can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Next, we extract keypoints on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These keypoints will be matched with the ones that will be extracted from the input image in the online section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We use SIFT [34] for extracting keypoints but any other feature descriptor could be swapped in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As the final step, we initialize the pose of MT in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This initial pose can be arbitrarily chosen as it will be used only once by Step 4 of the online section of the pipeline for the first input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It will then be replaced by the inferred 3D shape in the next images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In order to use the ROBUSfT C++ library, first, an ob- ject of the class ROBUSfT should be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The whole process of forming the template for this object is handled by the member function build template().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This function possesses parameters for creating templates for rectangular and non-rectangular thin-shell objects as well as the tracking surface of volumetric objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Regarding thin-shell objects, the process of forming the template is automatic by just receiving a handful of inputs from the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For the tracking surface of volumetric objects, however, MT , M, and P should be prepared by the user and imported into the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Online section: shape tracking Step 1: keypoint extraction and matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The first step of the online section of the pipeline is to extract keypoints in the input image I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' To do so, we use the PopSift library [35], which is a GPU implementation of the SIFT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We then match these keypoints with the ones that were previously extracted from P by comparing descriptors, using winner-takes-all and Lowe’s ratio test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Inevitably, a number of mismatches will be formed between P and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The mismatch points in I can be located on the surface of the deforming object or even in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is shown as red lines in the Matching step of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These mismatches will be eliminated in Step 2 thanks to myNeighbor which can cope with a large percentage of mismatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As a result, in this step, the images coming from the camera can be used directly without pretraining on either the image for segmenting the object from the background, or the matches for preselection of the most reliable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the library, the member function extract keypoints GPU() handles the keypoint extraction in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Then, the member function match() performs matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step 2: mismatch removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' To remove the possible mis- matches introduced in Step 1, a new mismatch removal algorithm, myNeighbor, was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The main principle used in this algorithm is the preservation of the neighborhood structure of correct matches on a deforming object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In other words, if all of the matches were correct, by deforming the object, the neighbor matches of each match should be preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' On the contrary, mismatches lead to differences in the neighboring matches of each matched point in I in comparison to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This was used as a key indication to detect and remove mismatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The whole process of myNeighbor is explained in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the library, the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 2: Implementation of ROBUSfT on the CPU and GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' A pure CPU implementation is also available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' member function mismatch removal algorithm() handles the mismatch removal process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The output is a list of estimated correct matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step 3: warp estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We use the estimated correct matches to estimate a warp W between P and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We then use W to transfer M to I and form � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The mesh points in � M will be used as sightline constraints in the 3D shape inference algorithm in Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The precision of warping depends on the number of matches, their correctness, and their distribution all over P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Warp W can be estimated in the most precise way if all the matches are correct between P and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' However, due to the smoothing nature of the warping algorithms, the transferring process can cope with a small percentage of mistakenly selected mismatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It should be noted that W cannot be extremely precise in areas without matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As a result, in these areas, the shape of � M might not be aligned well to the shape of the deforming object in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is worse when the matchless area is located near the boundaries of P as the alignment cannot be guided by the surrounding matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Hence, in order to use just well-aligned transferred mesh points of � M as the input for the 3D shape inference step, an assessment is performed over all of the mesh points and only the qualified ones are passed to Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For this, we check M cell-by-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Only the mesh vertices for cells containing at least one correct match will be qualified as OFFLINE Create an Object of class ROBUSfT ROBUSfT softObject;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=" Setting the parameters Template'stexturemap Template's dimension and mesh resolution Deformation model's parameters Build the template Mesh of the template (mesh points coordinates, triangulation) Keypoints on the template's texturemap Isometric deformation law softobject." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='build_template();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' CPULoop GPU Loop Matching softobject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='match();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Capture image Mismatch removal capture_image();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' softobject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='mismatch_removal();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Warping Extracting keypoints of the image softobject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='calculate_warp();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' softObject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Extract_keypoints_GPU();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Shape inference softObject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='shapelnference();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 3: Flowchart of FREX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' salient mesh points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The indices of these mesh points and their coordinates in � M are passed to Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The other mesh points are disregarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Representing and estimating W can be done with two well-known types of warp, the Thin-Plate Spline (TPS) [36] and the Bicubic B-Spline (BBS) warps [37], which we both tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The former is based on radial basis functions while the latter is formulated on the tensor-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Having the same number of matches as input, the TPS warp proved to be more precise than the BBS warp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' nevertheless, its execution time rises exponentially with increasing number of matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The execution time, however, remains almost constant for the BBS warp regardless of the number of matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Thus, considering the criterion of fast execution of the code, the BBS warp was chosen as the warp function in this step and also in the mismatch removal step discussed in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the library, the process of warp estimation is performed by the function warp() that calls two functions BBS Function() and BBS Evaluation().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The former estimates the warp W while the latter uses W to transfer M and form � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The process of selecting the salient mesh points is done by the member function set sightlines().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step 4: 3D shape inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We use Particle-SfT [18] as improved for tracking in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In this algorithm, a particle system is defined from the points and edges in MT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Then, the sightline and deformation constraints are applied con- secutively on the particles until they converge to a stable 3D shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As described in [10], in order to increase the convergence speed of the algorithm, the stable 3D shape for an image is used as initial guess for the next image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It should be noted that Particle-SfT can work even without a close initial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' If the object is invisible in one or several images, the last inferred 3D shape can be used as the initial guess for the upcoming frame containing the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This results in a slightly longer computation time in that image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For the next upcoming images the normal computation time is resumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This capability brings about two of the major advantages of our pipeline, which are being wide-baseline and robust to video discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the library, the whole process of shape inference is handled by the member function shapeInference().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As mentioned in [10], one of the optional input data that can significantly improve the convergence of Particle- SfT is the existence of 3D known coordinates of one or several particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The known 3D coordinates can be fixed in space, or can move on a certain trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The latter happens when the deforming object is manipulated by tools with known poses in 3D space like robotic end-effectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Implementation In order to optimize the implementation of ROBUSfT, it was coded in C++ in two parallel loops: one on the GPU, and one on the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The GPU loop handles keypoint extraction in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These keypoints are transferred to the CPU loop where the rest of the steps of the pipeline are taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' A pure CPU implementation is also available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Any arbitrary resolution can be considered for the captured images, nevertheless, we obtained the best performance by using 640 × 480 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The code runs on a Dell laptop with an Intel Core i7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='60 GHz CPU and a Quadro T1000 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 图图 Start A4 Aruco template Arbitrary texturemap Form 2D Estimate warp correspondences An image from the video Warp texturemap Other alterations showing the printed A4 to image space (background, lighting, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=') Aruco template being deformed 3D coordinates of Aruco markers 3D Ground Truth 2D Ground Truth Quantitative evaluation of algorithmsIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' FAKE REALISTIC EXPERIMENT (FREX) We introduce a novel experimental protocol, which we used for evaluating myNeighbor and ROBUSfT in compar- ison to the state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' A single execution of this protocol provides a large collection of scenes of an isometrically deforming object in various conditions, with known 2D and 3D ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This collection can be used to evaluate, compare, train, and validate new algorithms regarding isometrically deforming objects such as mismatch removal, 2D image registration, and isometric 3D shape inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In contrast to other artificially generated scenes of an isometrically deforming surface, the generated images in our protocol are the result of real object deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Being formed of successive images with continuous deformation, it can also be used for algorithms which exploit feature and shape tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In addition, object occlusion and invisibility can be easily simulated, by dropping frames or pasting an occluder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The protocol flowchart is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' First, we form the Aruco template by randomly distributing a set of Aruco markers all over a blank image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We then print the Aruco template on a standard A4 paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These markers should be big enough to be recognizable by the user’s camera in the de- sired distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In order to improve recognition, there should be white space between the markers on the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In our experiments, we used 100 markers with a width of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='4 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The OpenCV library was used to identify the markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These markers were recognizable by a 720p RGB camera from an approximate distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='6m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The next step is to deform the printed Aruco template in front of the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In each frame, the 2D and 3D coordinates of the markers’ centers are estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Because each marker has its own unique id, they can be used as correspondences between the Aruco template and each image of the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We exploit the 2D coordinates of these recognized correspondences to estimate a warp with which we can transfer an arbitrary texturemap to the video image space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is done firstly by resizing the arbitrary texture to the size of the Aruco template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In order to keep the aspect ratio of the arbitrary texturemap, white margins can also be added before resizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Then, an inverse warping process with bilinear interpolation is used to transfer the pixel color information from the arbitrary texturemap to their corresponding pixels in the video images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The whole procedure results in a scene with the arbitrary texturemap being deformed exactly on top of the Aruco template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It is also possible to add further modifications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' for instance, one can transfer the arbitrary texturemap to another scene with any different background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Besides, as in [38], an artificial lighting can also be added to form different variations of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For evaluating algorithms, one can use the 2D and 3D ground truth estimated in each frame of the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Regarding the 2D ground truth, the estimated warp can be used to identify the 2D corresponding point of each pixel of the arbitrary texturemap in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As for the 3D ground truth, one can exploit the 3D estimated coordinates of the Aruco markers in each frame which can be achieved using the OpenCV library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' myNeighbor We describe myNeighbor, our novel mismatch removal algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It works based on two main principles: Having an image of a textured surface and another image of that surface undergoing a deformation, to estimate a sufficiently accurate transfer function be- tween them with which one can judge the correctness of matches, there is no need to remove all the mismatches from the list of matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Instead, a set of correct matches would be sufficient to estimate the transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This set of correct matches can be extracted considering that in reality, under a deformation, the neighborhood structure among the points on a deforming surface is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We show that by using these two principles, the mis- matches can be detected and removed in a fast and efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The proposed algorithm is illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It consists of three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' First, a set of matches which are highly probable to be correct are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This selection is done by forming two triangulations using match points, one in P and one in I, and then choosing matches with high similarity in the list of their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Second, a small per- centage of possible mismatches among the selected matches are identified and removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is done by transferring the selected match points from P to I and then removing those with large distances from their correspondences in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Third, we transfer all the match points from P to I using a warp estimated based on the clean set of selected matches from the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The distance between the transferred template match points and their correspondences in I is used as the criterion to distinguish estimated mismatches from estimated correct matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In order to analyze the performance of myNeighbor and calibrate the parameters in the different steps, we used synthetic data experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the following section, we describe the design of these experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Afterwards, we describe in detail the different steps of myNeighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Synthetic data experiments for calibrating parameters These experiments are conducted by synthetically forming two images of a mesh MT and a series of matches between the two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The first image shows MT in its flat rest shape with all its keypoints on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We call this image IF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In IF , the keypoints can be considered as the extracted keypoints from P and the 2D mesh is equivalent to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The second image simulates I and shows MT having undergone a random 3D deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We call this deformed mesh MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The keypoints in this image can be positioned in their correct locations on the mesh (correct matches) or being displaced in the image area (mismatches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We consider MT as a regular triangular mesh with 10 × 6 points in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In order to deform MT , we use the same method as in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is done by applying two 3D Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 4: Flowchart of myNeighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' deformations containing random translations and rotations to two mesh cells at both sides of MT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The deformation is calculated in an iterative process based on position-based dynamics [39], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As for generating keypoints, we first randomly place keypoints in the inner area of M in IF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In order to create the matches between IF and I, we then transfer the keypoints from IF to I using a three-step process: calculating barycentric coordinates of the keypoints in M, transferring the keypoints to the 3D deformed mesh using the barycentric coordinates and the new 3D mesh points of the deformed MT , and eventually projecting the transferred keypoints to I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' To generate mismatches, an arbi- trary percentage of the transferred keypoints were corrupted by randomly distributing them all over the area of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Two samples of the generated images for 100 and 1000 matches each with 30% mismatches can be observed in the two first columns of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Methodology The algorithm myNeighbor is applied on Nm matches denoted as Cp ↔ Cq between P and I, with: Cp = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=', pNm}, pi = (xi, yi) (1) Cq = {q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=', qNm}, qi = (ui, vi) (2) A pair (pi, qi) of points with the same index forms a match pi ↔ qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We define the set of correct matches Sin as the collection of matches pi ↔ qi where pi and qi point to the same location on the deforming surface in P and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' On the contrary, when the pointing locations of the match points are different, they are categorized as mismatches Sout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The goal of myNeighbor is to form and remove the subsets Op ⊂ Cp and Oq ⊂ Cq which have the largest possible number of matches belonging to Sout and smallest possible number of matches belonging to Sin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We explain the steps of our algorithm to fulfill this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 1) Step I – Neighbor-based correct match selection: We select subsets Cps ⊂ Cp and Cqs ⊂ Cq which are highly probable to form correct matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We start by defining WG as the groundtruth warp between P and I that can transfer all the match points Cp from P to their correct locations in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' With this definition, we have the set of correct matches Sin as: Sin = {(pi, qi) | i ∈ R}, (3) where: R = {i | ∥WG(pi) − qi∥ < ϵ}, (4) where ϵ is a very small positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Warp WG is an un- known composition of isometric deformation and perspective projection mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The isometric deformation mapping preserves the geodesic distances among the points and their topological structure on the object’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' However, with the addition of perspective projection mappings, only the topological structure of points remains preserved in visible areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This implies that by applying WG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' the neighborhood structure among the points on the object in P and I Set of matches between P and I as Cp Cg and M (or any 2D mesh that covers P ) Step 1: Selecting matches that are highly probable to be correct Generate two triangulations: Tp using Cp in P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' and Tg using Cqin I Form neighbor lists Qp and Qg for all the matches in P and I Calculate the parameter MF for all the matches and select Estimate warp Wi using Cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' ←→ Cas and use it to transfer M to I to obtain Mi Step 2: Removing probable mismatches from the selected matches Estimate Ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' by transferring Cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' to Iusing barycentric coordinates of match points ps with respect to M and the transferred mesh M1 Calculate Euclidean distance d2(i) between each Qs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' ←→ qs Remove possible mismatches from the list of Cps Ca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='by the MAD criterion Estimate warp W2 using the modified list of matches Cps ←→ ( and use it to transfer M to I to obtain M2 Step 3: Removing all the mismatches from the list of all the matches Estimate Ca by transferring all match points C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' from P to I using barycentric coordinates of p with respect to M and the transferred mesh M2 Calculate Euclidean distance dg(i) between each Qi qi Form mismatch sets Op and Og by selecting matches with dg(i) > d3TH Set of correct matches between P and I as (Cp - Op) (Cq - Oa)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 5: Two sample results of the steps for synthetic data experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The first row is an experiment with 100 matches and a mismatch percentage of 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The second row is an experiment with 1000 matches and a mismatch percentage of 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The first and second columns represent IF and I with correct matches in green and mismatches in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The third column is the result of Step I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The wrongly chosen mismatches are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The fourth column is the result of Step II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The mismatches along with a small percentage of correct matches are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The fifth column is the separation of the estimated correct matches and the estimated mismatches from Step III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The transferred meshes � M1, � M2, and � M3 are shown in orange, yellow, and cyan for the three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' should be preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We exploit this characteristic of WG to estimate �R as the set of indices of highly probable correct matches Cps ↔ Cqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' To do so, first, we form two Delaunay triangulations, Tp = D(Cp) in P, and Tq = D(Cq) in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Then, for each match i, we calculate two sets of first-order neighbors Qp(i) and Qq(i) in P and I, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We then define the Mismatch Factor (MF) criterion for match i as: MF(i) = |Qp(i) ∪ Qq(i) − Qp(i) ∩ Qq(i)| |Qp(i) ∪ Qq(i)| × 100 (5) For each match, MF represents the difference in the neigh- bor points between P and I as a percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Ideally, we expect that for all the matches MF = 0, which implies that there is no difference in the neighbors of each match during a deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' However, in practice, there are two reasons which rather put MF values in a range from 0 to 100: the presence of mismatches and variations in triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The presence of mismatches can affect the value of MF in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' First, when the match point i in I is a mismatch and thus located in a wrong location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' And second, when the match point i in I is a correct match but one, several, or all of its neighbors are mismatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Both of these cases result in different neighbors in I in comparison to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As for the two triangulations, it should be noted that even in the absence of mismatches, the neighborhood structures in Tp and Tq do not necessarily coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is because of surface deformation, change in viewpoint, and occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Calculating MF for all the matches, we can have a fair estimation regarding the state of the matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The lower values of MF(i) indicate that the match i is surrounded by similar matches in P and I and has a higher probability to be placed in its correct location and thus be a correct match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' On the contrary, the higher values of MF(i) can stem from the wrong location of the match i in comparison to its neighbors which strengthens the possibility of it being a mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The basic idea in this step is to form Cps ↔ Cqs by selecting pairs of highly probable correct matches ps ↔ qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is done by choosing the matches with lower values of MF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We examined the validity of this reasoning by evaluating three different synthetic data experiments, each with 1000 matches and different rates of correct matches (30%, 60%, and 90%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Figure 6 shows the histogram of MF for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We observe that the dispersion of MF spans a wider range as the value of the correct match rate grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For higher numbers of correct matches, there are more similarities in the neighbor lists of each match and, consequently, MF de- Mismatch removal algorithm steps Template Image Step I Step II Step III 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Correct matches Correctly selected correct matches Correctly kept correct matches 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Final estimated correct matches 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Mismatches o Mistakenly selected mismatches Mistakenly kept mismatches 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Final estimated mismatches Mesh (flat in the template and deformed in the image) Mesh M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' transformed by the warp Wi - Mesh M transformed by the warp W2 - Mesh M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' transformed by the warp W3 n,=32% AoS=93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='75% n,=25% AoS=100% TPR=1 FPR=0 n,=25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='2% AoS=99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='2% n,=23% AoS=100% TPR=1 FPR=0Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 6: Histogram of MF values for three sample synthetic data experiments with 1000 matches and 30%, 60% and 90% of correct matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Furthermore, regardless of the values of the correct match rate, the majority of the mismatches are accumulated in the top bins of the graphs that correspond to higher values of MF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is shown in more detail for the case with the correct match percentage of 30% by expanding the last two bins of the graph in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This validates our prior reasoning that by selecting the matches with MF below a certain threshold MFth, we can have a set of matches which are highly probable to be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' To quantify the appropriateness of this selection, we define two criteria, based on the following two quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The first quantity is ns, which is the percentage of the selected matches compared to the total number of matches: ns = |Cs| Nm × 100, (6) where Cs = {(pi, qi) | i ∈ �R} is the set of selected matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The second quantity is AoS, which is the Accuracy of Selection, defined as: AoS = |Cs ∩ Sin| |Cs| × 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' (7) Our goal is to choose the value of MFth in the way that we have both of these criteria to be as high as possible, which means selecting a high percentage of matches with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' However, practically, these two criteria work in reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' By choosing a higher value for MFth, more matches are selected (higher ns) but with less accuracy (lower AoS) and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In order to choose the proper value for MFth, we analyzed the behavior of these two criteria for a series of synthetic data experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We consider three scenarios for these experiments based on the number of matches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=', Dense, Moderate, and Sparse with in turn 1000, 200, and 50 total number of matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The experiments were done in a wide range of correct match percentages (10% to 100%) for each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Two different values of the criterion MFth were studied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' mean and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='9 × mean where mean is the mean of all MF values in each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The results are presented in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='a and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Each point in the graph is the average result of 1000 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The first point that should be noted here is that, generally, the proposed match selection method in this step is more reliable as the number of total matches grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This can be deduced by comparing the higher values of AoS in the Dense case with the ones in the Moderate and Sparse cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As for choosing MFth, it should be noted that setting MFth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='9 × mean leads to higher values of AoS in comparison to the case with MFth = mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Nevertheless, as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='a, this sacrifices a high percentage of matches by dropping ns significantly, which is undesirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Hence, in this step, we choose mean as the value of MFth and form �R as the set of indices of probable correct matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' While this choice implies a higher number of selected mismatches (lower AoS), we note that these mismatches can be removed in Step II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As the final operation in this step, we estimate the warp W1 between P and I using the selected matches Cps ↔ Cqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We then exploit this warp to transfer M to I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We call this new mesh � M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As can be seen in the third column of Figure 5, the mesh � M1 (shown in orange) may not be totally faithful to the deformation of MG in I, which is due to the inaccuracies in the calculation of the warp W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This stems from two main reasons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' the existence of mismatches in our selection (shown as red dots), and the insufficient number of correct matches in some areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the next step, we exploit the transferred mesh � M1 to remove the possible remaining mismatches from the selected matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 2) Step II – Removing mismatches from the list of selected matches: We remove the possible mismatches from the selected matches Cps ↔ Cqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We first form the set Cˆqs by transferring Cps to I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is done by finding the barycentric coordinates of each selected match psi ∈ Cps with respect to M and applying them on the transferred 2D mesh � M1 from Step I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We then use the following decision criterion to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='identify and remove possible mismatches one by one from ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Correct Match Percentage 30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='90-100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='80-90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='70-80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='98-100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='60-70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='50-60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='40-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='z6-06 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='30-40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='20-30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='80-82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='10-20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Correct Matches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='0-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Mismatches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Number of Matches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Correct Match Percentage 60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='90-100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='80-90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='70-80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='60-70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='50-60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='MF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='40-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='30-40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='20-30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Correct Matches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='10-20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='0-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Mismatches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Number of Matches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Correct Match Percentage 90% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='90-100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='80-90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='70-80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='60-70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='50-60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='40-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='30-40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='20-30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Correct Matches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='10-20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='0-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Mismatches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Number of Matches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='(c)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 7: Results of applying the first two steps of the algorithm myNeighbor in synthetic data experiments in three different scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Dense (1000 matches), Moderate (200 matches), and Sparse (50 matches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Each curve is the average result of 1000 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The first row gives ns and AoS from Step I for two different values of MFT H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The second row gives the results of Step II in comparison to the results of Step I with MFth = mean(MF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 8: ROC curves resulting from the algorithm myNeighbor in synthetic data experiments in three scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Dense (1000 matches), Moderate (200 matches), and Sparse (50 matches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Each point is the average result of 1000 trials calculated with a specific value of d3th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' the selected matches Cps ↔ Cqs: ���d2(i) − median � {d2(j)} ���� ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='5 MAD, (8) where d2(i) = ∥ˆqsi − qsi∥ with i ∈ �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' MAD (Median of Absolute Deviations from Median) is calculated as: MAD = k median �����d2(i) − median � {d2(j)} ���� �� , (9) where k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='4826 is a constant number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The values of d2 are relatively larger for mismatches in comparison to correct matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This stems from two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' First, the small percentage of mismatches compared to the great majority of correct matches coming from Step I and thus lesser influence of mismatches in the estimation of warp W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Second, the inconsistent location of mismatches in P and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The decision criterion in equation (8) is chosen due to the distribution type of d2, with the presence of just a small percentage of large values among the majority of small values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='c and d illustrate the result of this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As can be seen, unlike the previous strategy of choosing a smaller MFth, this method results in improvement of AoS without losing a considerable percentage of selected matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This can be clearly observed by comparing ns in Figures 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='a and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As the last operation in this step, warp W2 is calculated 60 100 90 50 80 40 70 (%) "u AoS(%) 60 50 Dense, 40 Dense, MF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='= mean(MF) 20 Moderate, MF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' MF=mean(MF) r= mean(MF) 30 Sparse, Sparse, 10 Dense, MF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='mean(MF) 20 Dense, MF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='9 mean(MF) Moderate, MF= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='mean(MF) 10 Moderate, MF= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='9 mean(MF) Sparse, MF= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='9 mean(MF) Sparse, MF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='mean(MF) 0 0 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 Correct Match Percentage (%) Correct Match Percentage (%) (a) (b) 60 100 90 50 80 40 70 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='(%) AoS(%) 60 30 50 Dense,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step I 40 Dense,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step I 20 Moderate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step I Moderate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step I Sparse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step I 30 Sparse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step I Dense,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step II 20 Dense,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step II 10 Moderate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step II Moderate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step II Sparse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Step II 10 Sparse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=" Step II 0 0 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 Correct Match Percentage (%) Correct Match Percentage (%) (c) (d)ROCCurvefor scenarioDense ROC Curve for scenario Moderate ROC Curve for scenario Sparse 6'0 60 60 True Positive Rate 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='8 True Positive Rate 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='5 Correct Match Percentage =90% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='4 +Correct Match Percentage = 90% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='4 + Corre ct Match Percentage = 90% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='3 Correct Match Percentage = 70% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='3 Corre ct Match Percentage = 70% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='2 Correct Match Percentage = 50% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='2 Corre ct Match Pe rce ntage = 50% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='1 +Correct Match Percentage = 30% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='1 Correct Match Percentage = 30% 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='5 False Positive Rate False Positive Rate False Positive Rate (a) (b) (c)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 9: Performance evaluation of our mismatch removal method myNeighbor in comparison to the state-of-the-art methods using the FREX protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The first row shows the Aruco template and three selected images (14, 47, 60) of the deformation of the printed Aruco template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The following rows show five datasets of generated scenes with the texturemap in the first column, three generated images corresponding to the first row in the next columns, and the ROC curves of the mismatch removal algorithms in the last column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For each of the images � M3 from myNeighbor is overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Method Average run-time (s) myNeighbor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='0139 Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [31] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='0206 Pizarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [29] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='8925 Famouri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [16] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='0171 TABLE II: Comparison of the average run-time of the mismatch removal algorithms for processing all the images of all the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' using the purified selected matches Cps ↔ Cqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This warp is then used to transfer M to the image space and form � M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The result of removing possible mismatches in this step along with the transferred mesh � M2 are shown in the fourth column of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As can be observed, in comparison to � M1, � M2 has a better compliance to MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 3) Step III – Extracting mismatches from the list of all the matches: In this step, we exploit the transferred mesh � M2 to extract the mismatches Op ↔ Oq from the total matches Cp ↔ Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The process is similar to Step II except that this time all of the matches are checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We first transfer the template match points Cp to the image space and form the set Cˆq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is done by calculating barycentric coordinates of all the match points Cp with respect to M and applying them on the new transferred mesh � M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We define the following decision criterion to detect and remove mismatches: d3(i) = ∥ˆqi − qi∥ ⩾ d3th (10) Unlike Step II where we used the MAD criterion to remove just a small rate of mismatches, this time we use a constant 图 图图 图 图 Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [34] 固 图图 四 巴 Pizarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [23] 国 Famouri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [15] T 3 myNeighbor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='9 TPR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='3 FPR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='9 TPR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='3 FPR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='9 PR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 10: Applying myNeighbor on four real cases: a cushion, a Spiderman poster, a shoe sole, and an elastic shirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The first column shows the texturemaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The second column shows Step I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' All the matches are shown in this column while the selected matches in Step I are shown in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These selected matches are transferred to column three that shows Step II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In this column, those matches which are chosen as possible mismatches are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The last column is the distinction between the estimated correct matches (in green) and the estimated mismatches (in red) in Step III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The meshes � M1, � M2, and � M3 are overlaid to illustrate the computed warps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' threshold d3th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is due to the higher percentage of mismatches compared to Step II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In order to make this distinction method more robust, we consider d3th as the multiplication of a sample length ls and a constant coefficient αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The sample length ls is a measure of the size of the object in the image in pixels and is calculated as the average distance between all the mesh points in the transferred mesh � M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' To choose a proper value for the constant coefficient αs, a series of synthetic data experiments with the same three scenarios as before (Dense, Moderate, and Sparse) and four different correct match rates was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The results are presented as ROC (Receiver Operating Characteristic) curves in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='a-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Each point represents the average TPR (True Positive Rate) versus the average FPR (False Positive Rate) computed in 1000 trials using a specific value of αs in the range of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' TPR is calculated as the number of selected true mismatches over the number of all true mismatches, and FPR is calculated as the number of true correct matches mistakenly selected as mismatches over the number of all true correct matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Ideally, all the mismatches should be discarded (TPR=100%) without discarding any correct matches (FPR=0%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Hence, the most favorable αs in a single ROC curve is the one that results in the maximum possible TPR leaving the FPR below a reasonable value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We choose Mismatch Removal Algorithm Steps Texture-map P Step I Step II Step III Selected Matches 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Kept Matches from Step I 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Estimated Correct Matches 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Unselected Matches 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Removed Matches from Step I 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Estimated Mismatches Mesh M, Estimated by Warp W Mesh M, Estimated by Warp W, Mesh M, Estimated by Warp W DEL CHAMELEON STRIKES!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 11: Comparing the accuracy of the 3D shape inference methods with Particle-SfT with three datasets obtained by FREX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The 3D shape inference methods are Brunet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [17], Chhatkuli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [41], Bartoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [14], Ostlund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [20], and Salzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' αs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='15 which keeps TPR above 90% while FPR remains below 10% for most of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The last column of Figure 5 illustrates the estimated correct matches (in green) and the estimated mismatches (in red) for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We also use the estimated correct matches to estimate warp W3 and transfer M to I and form � M3 (shown in cyan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As can be seen, there is a high compliance between � M3 and MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It should be noted that estimating W3 and � M3 is not necessary in myNeighbor and we merely estimate them just to visually present the effectiveness of the algorithm in removing the mismatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' However, considering myNeighbor as a step in ROBUSfT, due to the fact that the final estimated correct matches are passed from this step to Step 3 of ROBUSfT which is warping, W3 and � M3 can also represent W and � M in the warping step, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Mismatch removal results In this section, we demonstrate the efficiency of myNeighbor by evaluating its performance through various tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We first compare the results of the algorithm with the state-of-the-art algorithms in the literature by testing them through FREX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The experiment includes 60 frames of continuous deformation of the Aruco template in front of the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Five datasets were generated in this experiment each with an arbitrary texture with a challenging pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Three different types of backgrounds were also considered for these five cases, specifically two original backgrounds, two white backgrounds, and a background with a pattern similar to one of the texturemaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We apply all the mismatch removal algorithms on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For each dataset, the corresponding arbitrary texture was used as the texturemap for the mismatch removal algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The matches between the texturemap and each image of the dataset are extracted using SIFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The results are presented in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The first row illustrates the Aruco template and also three selected original images of its deformation in front of the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The lower rows represent the five datasets generated by FREX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Each row shows the arbitrary texture of the dataset in the first column, the three selected generated images, and eventually the resulting ROC curves for all the mismatch removal algorithms on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the ROC curves, for a certain algorithm and a certain dataset, each point is the average value of TPR and FPR over all 60 images of that dataset using a specific value for the threshold used in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As can be seen, in all cases, our algorithm outperforms the other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In order to show the performance of our algorithm visually, for each dataset, we overlaid � M3 for the three selected frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As can be observed, the transferred meshes are visually well-aligned to the 2D deformed shape of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In some cases, a small number of irregularities can be observed in certain areas (for example in the Matrix poster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is because of the presence of a small number 50 50 Salzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Ostlund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Bartoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Salzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Ostlund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Bartoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Chhatkuli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Brunet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Particle SfT Chhatkuli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Brunet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Particle SfT 40 Average 3D Error (mm) 40 30 30 20 20 10 10 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Frame Number Frame Number 50 Salzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Ostlund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Bartoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Chhatkuli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Brunet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Particle SfT 30 20 10 0 10 20 30 40 50 60 Frame NumberFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 12: Comparison between ROBUSfT and the methods presented by Famouri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [16] and Ngo et al [21] on the public dataset provided in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' (a) Mean absolute 3D error between the inferred shape and the groundtruth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' (b) Execution time in milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' of mismatches in our list of estimated correct matches and the lack of matches in those areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As for comparing the execution speed of different mismatch removal algorithms, the process run-times for all the frames of all datasets were averaged and tabulated in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It shows that our algorithm is faster than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It should be however noted that our algorithm is implemented in C++ while the others are in Matlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' After validating the efficiency of myNeighbor in com- parison to the state-of-the-art algorithms in the literature, we evaluate its performance in real cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' To this end, we applied our algorithm to four real deforming objects as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We chose these cases in such a way that each one is challenging in a special way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The cases include a cushion with non-smooth surface and severe deformation, a Spiderman poster deformed in a scene with background covered with almost the same posters, a shoe sole with an almost repetitive texture, and a shirt with elastic deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The texturemaps are shown in the first column of Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The second to fourth columns show the results of Step I to Step III of myNeighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In each step, the alignment of the corresponding transferred mesh to the 2D shape of the deforming object can be considered as an indication of the correctness and abundance of the estimated correct matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Like in the synthetic data experiments, this alignment improves progressively in different steps of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' One point that should be noted here is that the shirt (the last case in Figure 10) is elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We exert a non- isometric deformation on it by pulling from both sides, and myNeighbor still works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is due to the fact that we did not make any assumption regarding isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In fact, the only assumption that we made is the preservation of neighborhood structure in the deforming object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As a result, myNeighbor also works with non-isometric deformations which preserve neighborhood structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' EXPERIMENTAL RESULTS We evaluate the performance of ROBUSfT on different deforming objects in various conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We divide this section into two main parts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' first, comparing the results with the state-of-the-art methods and then evaluating ROBUSfT in several other challenging cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Comparison to the state-of-the-art methods We compare ROBUSfT with the state-of-the-art methods through two different tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The first test is conducted among the shape inference methods (G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The second test is carried out among the integrated methods (G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We use FREX to conduct the first test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' To this end, the same 60 images of deforming Aruco marker paper sheet are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We create three different datasets using three arbitrary texturemaps and apply a white background to all the scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The arbitrary texturemaps include a painting, the Joker poster, and a paper sheet filled with basic geometric shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These images are shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In each dataset, we compare the result of the last two steps of ROBUSfT (warp estimation and 3D shape inference) with five other shape inference methods from Brunet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [17], Chhatkuli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [41], Bartoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [14], Ostlund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [20], and Salzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' A similar comparison was made in [18] on another dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' However, in [18], a random 3D shape was used as the initial guess for Particle-SfT algorithm in each image of the video;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' in contrast, we use the 3D inferred shape of the object in each image as the initial guess for the next image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In each dataset, the matches between P and each image are extracted using SIFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We then separate the correct matches and use them as the input for all the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' If required by a shape inference method, a BBS warp is estimated based on these correct matches and used as the input to that shape inference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The results for all three datasets are presented in Figure 11 as the average 3D error between the 3D inferred shapes and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As can be observed, Particle- SfT provides the lowest value of 3D error in comparison to the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is more apparent in the datasets with lower number of matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the last dataset, there are several discontinuities in the 3D error graph of state-of-the- art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is due to the failure of shape inference in those images of the video by those methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Particle-SfT, however, succeeds to infer the 3D shape of the object in all of the images with a reasonable error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 20 Famouriet al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Ngo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 15 ROBUSfT 10 50 100 150 200 FrameNumber (a) 10000 ExecutionTime(ms) 1000 Famouri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Ngo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' ROBUSfT 100 10 50 100 150 200 Frame Number (b)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 13: Evaluating ROBUSfT in three real data experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' a Spiderman poster, a chopping mat, and a t-shirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The texturemaps of the templates are shown in the first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For each case, four images are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Below each frame, the reconstructed 3D shape of the deforming object with the estimated 3D coordinates of the estimated correct matches (red particles) as well as their ground truth (green particles) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The 2D projections of the 3D inferred shapes are also overlaid on the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For each image, the median Euclidean distance between the estimated 3D coordinates of the estimated correct matches and their ground truth is given below the reconstructed shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 12号 ELCHAMELEON STRIKES!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='93mm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='69mm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='12mm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='14mm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='12mm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='65mm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='56mm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='88mm RKSCOT 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='35mm 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='45mm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='37mm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content='03mmFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' 14: Evaluating ROBUSfT in a real data experiment with two robotic arms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' soft constraints are applied to bind the constrained mesh points to the grippers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Each row shows three images: the original camera view, the projection of the 3D reconstructed mesh on the camera view, and the 3D reconstructed mesh with the robots in the RViz environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For the second test, we ran ROBUSfT on the public dataset provided in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The dataset includes the 2D correspon- dences as well as 3D Kinect data of 193 consecutive images of a deforming paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The paper is planar and no occlusion appears in the series of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We compared our results with the results of Famouri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [16] and Ngo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' [21] which were presented in their papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is shown in Figure 12- a and Figure 12-b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As can be observed, ROBUSfT is both faster and more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' It should be noted that ROBUSfT used directly images as the input and covered the whole process from extracting keypoints to 3D shape inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In contrast, the other two algorithms used the already available correspondences in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Another relevant point is that in this test we use a serial CPU-GPU architecture instead of a parallel one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This is done to make sure that the captured image that we analyze and the ground truth that we compare to are for the same image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This consequently reduces the execution speed of our code compared to the parallel architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the next series of tests we use the parallel architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Evaluation of ROBUSfT We first evaluate the efficiency of ROBUSfT in three real cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These cases are shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The tested objects are a Spiderman poster, a chopping mat, and a t-shirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In each case, the object is deformed in front of a 3D camera with which we capture both RGB image and the depth of each point on the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We use the measured depth as ground truth for evaluating the reconstructed 3D shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We use the Intel RealSense D435 depth camera and built-in libraries for aligning the depth map to the RGB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For each case, four images of the experiment are shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the first case, we set the resolution of the camera to 640 × 480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In the second and third cases, we increased it to 1280 × 720 due to the insufficient number of detected keypoints using the previous resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Below each image, the reconstructed 3D shape of the deforming object along with the 3D coordinates of the estimated correct matches (red particles) as well as their ground truth (green particles) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The 3D coordinates of the estimated correct matches are estimated by calculating their barycentric coordinates in P with respect to M and applying these coordinates on the 3D reconstructed mesh of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The number written below each frame is the median distance between the reconstructed 3D coordinates of the estimated correct matches and their ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The median is chosen due to the probable existence of mismatches among the list of estimated correct matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In 3D space, the ground truth of these mismatches can be located in the background and not on the object itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This significantly increases the 3D shape error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Using the median gives a better estimate of the 3D shape error considering the existence of this small percentage of mismatches with large 3D errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As can be observed, the pipeline succeeds to infer the 3D shape of the object in all of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This success is more visible in the second and third cases due to the relative scarcity of keypoints and existence of repetition in their patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Regarding the Spiderman poster case, it should be noted that there are self-occlusions in the first and third illustrated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In these images, the 3D shape of the object in the occluded areas is estimated by the deformation constraints implemented in Particle-SfT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These constraints preserve the geodesic distance between each pair of mesh points as its initial value in MT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Regarding the runtime, using the parallel architecture and 640 × 480 captured frames as the input (as in the Spiderman poster case), the execution speed reaches 30 fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The last experiment is a practical use case with robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The experiment aims at highlighting the advantage of using known 3D coordinates in ROBUSfT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As mentioned in Step 4 and shown in Figure 14, these known coordinates are an optional input to the last step of ROBUSfT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Their usage can increase the robustness of the tracking process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The setup of this experiment is the same as in [42], where we applied ROBUSfT in a robotic case, specifically, controlling the shape of deformable objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The setup consists of two robotic arms grasping and manipulating the Spiderman poster from both sides and a top camera facing the manipulation area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The 3D positions of the two robotic grippers are known in camera coordinates thanks to the known pose of each gripper in the robots’ coordinate frames and also the external calibration between the robots and the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' For each gripper, we consider the closest mesh point to the gripper as a constrained mesh point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These mesh points should be bound to their corresponding gripper and move with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' As described in [42], this binding is performed using a soft constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In this soft constraint, for each gripper, a sphere with a small radius centered at the gripper’s 3D position is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Then, in each iteration of Particle-SfT, if the corresponding mesh point is outside this sphere, it will be absorbed to the closest point on the sphere surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This soft constraint has two main advantages over rigidly binding the constrained mesh points to the grippers: first, they let the position-based dynamic equations in Particle-SfT that preserve the distances between the mesh points be applied on the constrained mesh points, which leads to a smoother reconstructed shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Second, by applying soft constraints, we can cope with small possible errors in robot-camera calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In fact, a wrong robot-camera calibration leads to a wrong transfer of the grippers’ 3D coordinates to the camera coordinate frame which eventually results in wrong coordinates of the constrained mesh points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' By using the soft constraint and considering a sphere rather than a rigid bind, we give a certain degree of flexibility to the constrained mesh points to move in close proximity to the gripper’s coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This can compensate for slightly inaccurate coordinates of the grippers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' CONCLUSION We have proposed ROBUSfT, a new pipeline that can effec- tively reconstruct the 3D shape of an isometrically deforming object using a monocular 2D camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' The proposed pipeline addresses the well-known challenges in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' These challenges include ambiguities in inferring the 3D shape of the deforming object from a single 2D image and real-time implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We have introduced myNeighbor, a novel mismatch removal algorithm for deforming objects, which works based on the preservation of the neighborhood struc- ture of matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' We validated the efficiency of myNeighbor in comparison to the state-of-the-art algorithms in numerous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In order to compare ROBUSfT and myNeighbor with the state-of-the-art methods in the literature, we have presented a novel type of experimental protocol called FREX (Fake Realistic Experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This protocol is executed once, but it provides a large number of resulting scenes of an isometrically deforming object in various conditions with 2D and 3D ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' This collection can be used to evaluate, compare, and validate algorithms regarding isometrically de- forming objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In addition, the provided 2D and 3D ground truth may be used for training learning-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' In contrast to other artificially made scenes of an isometrically deforming surface, the generated images in our protocol are the result of real isometric deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' Possible directions for future work include (i) exploiting the silhouette of the object in the image for improving 3D shape inference in challenging cases such as weakly-textured objects, (ii) extending ROBUSfT to volumetric objects and (iii) adding self-occlusion reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by project SOFTMANBOT, which received funding from the European Union’s Hori- zon 2020 research and innovation programme under grant agreement No 869855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfqQhz/content/2301.04037v1.pdf'} +page_content=' REFERENCES [1] J.' 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Conrada, Philipp Lorenz-Spreenc and Christof Sch¨uttea,d,1 +aZuse Institute Berlin +bPotsdam Institute for Climate Impact Research +cCenter for Adaptive Rationality, Max Planck Institute for Human Development +dInstitute of Mathematics, Freie Universit¨at Berlin +1Corresponding author: schuette@zib.de +January 2023 +Abstract +Digital communication has made the public discourse considerably more complex, and new actors and strategies have +emerged as a result of this seismic shift. +Aside from the often-studied interactions among individuals during opinion +formation, which have been facilitated on a large scale by social media platforms, the changing role of traditional media +and the emerging role of ”influencers” are not well understood, and the implications of their engagement strategies arising +from the incentive structure of the attention economy even less so. Here we propose a novel opinion dynamics model that +accounts for these different roles, namely that media and influencers change their own positions on slower time scales than +individuals, while influencers dynamically gain and lose followers. Numerical simulations show the importance of their +relative influence in creating qualitatively different opinion formation dynamics: with influencers, fragmented but short- +lived clusters emerge, which are then counteracted by more stable media positions. Mean-field approximations by partial +differential equations reproduce this dynamic. Based on the mean-field model, we study how strategies of influencers to +gain more followers can influence the overall opinion distribution. We show that moving towards extreme positions can +be a beneficial strategy for influencers to gain followers. Finally, we demonstrate that optimal control strategies allow +other influencers or media to counteract such attempts and prevent further fragmentation of the opinion landscape. Our +modelling framework contributes to better understanding the different roles and strategies in the increasingly complex +information ecosystem and their impact on public opinion formation. +1 +arXiv:2301.13661v1 [physics.soc-ph] 31 Jan 2023 + +Introduction +In the era of many-to-many communication on social media, +polarization and radicalism can increasingly be understood +as self-organized phenomena emerging from opinion dynam- +ics on social networks [3]. Despite the global use of those large +communication platforms and the resulting interactions, they +have not been thoroughly studied as effects of a dynamical +system consisting of multiple interacting components, partic- +ularly distinguishing their different roles and goals. +More specifically, social media did not replace traditional +media, but became an intermediary entity that allows peer- +to-peer communication among individuals but also traditional +media outlets to disseminate their content and interact with +their audience. In principle, they are treated like individuals +on those platforms, but have very different internal dynam- +ics (e.g., editorial processes, reputation and economic incen- +tives). +As more and more people around the world access +traditional news media through social platforms, this depen- +dency becomes increasingly pronounced [45]. +At the same +time a new role emerged in this system, namely, the influ- +encer, who is a private person with many followers on a so- +cial media network [4]. Although mostly following commercial +goals, they do turn to politics [60] and can polarize public dis- +cussions [51], as well as compete with traditional media and +with one another for audience [54]. Potentially this increased +competition could contribute to the declining trust in tradi- +tional media and polarized debates on social media around +the world [38]. We aim to capture the scenario of opinion +dynamics under the impact of influencers and media with a +generalized modeling framework designed to distinguish these +different roles, but allow them to interact with each other. +So far in most cases, opinion dynamics are mathematically +modelled by network- or agent-based models (ABMs) that +aim to reproduce how individuals change their opinions based +on the feedback of their peers in the respective social environ- +ment. The largest group of such models describes the change +of opinions based on the (dynamical) interaction network be- +tween individuals. The well-known voter model [36] in which +individuals copy the discrete opinion of a random neighbour +may serve as an example. Alternative models use continuous +opinion spaces where usually individuals’ opinions are drawn +to attracting opinions in their social network (in some cases +repelled by others). The DeGroot model [17] with individuals +being drawn to the weighted average opinion of their neigh- +bours and bounded confidence models [26, 57] where attract- +ing interactions only take place with like-minded individuals +may be the most prominent. It has been demonstrated that +these basic models and their generalizations, e.g. [27, 50, 48], +allow to describe the emergence of opinion clusters or commu- +nities, including phenomena like radicalization, social bubbles +and echo chambers [29]. +Additionally, biased assimilation models [15, 59] have been +developed based on the insight that linear weighted averag- +ing of opinions cannot really lead to increasing polarization +since people who hold strong opinions are likely to examine +information in a biased manner. Further in [5, 6, 9], multi- +dimensional opinion dynamics models have been proposed +where individuals are drawn to the neutral opinion while at +the same time reinforcing each other to more extreme opin- +ions. Opinion interactions are saturated by a non-linear in- +teraction function including tunable sensitivity to input. It +has been demonstrated that these models allow for describing +real-world political polarization [33]. +The development of these opinion dynamics models has +been taken up in the literature with a focus on understand- +ing consensus and community building, or cluster formation. +For simple linear models like the DeGroot model this can +be studied analytically [17]. +For nonlinear opinion models +like bounded confidence models this has been studied numer- +ically [26, 57, 12] or by considering the mean-field limit in +terms of partial differential equations (PDEs), providing rig- +orous theorems on cluster numbers and size, effective timescales +and bifurcations based on changing interaction parameters +[56, 21, 18]. Moreover, it is understood how to extend the the- +ory from agent-based models to mean-field limit models with +stochastic partial differential equations (SPDEs) [28, 19]. +On the background of this progress in opinion dynamics +in social networks, this article concentrates on opinion clus- +ters as coherent structures [20] meaning emerging groups of +individuals whose opinions stay similar for rather long peri- +ods of time before eventually disintegrating or merging with +other clusters. The notion of coherence implies temporal sta- +bility on timescales that are neither fast nor asymptotically +long, indicating complex dynamical behavior with multiple +characteristic timescales. We aim at understanding how, on +these different timescales, traditional media and social media +influencers may interact with individuals in order to adapt +the overall opinion distribution according to their objectives, +e.g., political agendas or economic interests. +A few first steps towards opinion models under the impact +of different types of agents have been taken: The influence +of external sources on the opinion dynamics of individuals +has been studied for example in [11, 42, 27]. These external +sources are often regarded as media or charismatic leaders and +have also been termed zealots or stubborn agents in the liter- +ature. More recently their influence has also been analysed in +network-based opinion models [7, 10] but without explicitly +taking coherence or multiple timescales into account. In [47], +the influence of a social group or clan at multiple timescales +on the opinion formation on complex networks has been ana- +lyzed. However, so far those agents are mostly considered to +have constant opinions and relations to their followers that +do not change in time. +Here, we present an opinion dynamics model with a con- +tinuous opinion space that generalizes most of the available +models (DeGroot, bounded confidence, biased assimilation,...) +and complements individual opinion dynamics with tradi- +tional media and social media influencers that follow political +agendas and economic interests, and that adapt their opinions +on different timescales than the individuals, see Figure 1. +The question of what role the media and influencers play +on the public opinion distribution became increasingly com- +2 + +(high inertia) +opinion space +(medium inertia) +Media +Individuals +Influencers +opinion space +opinion space +Figure 1: Structure of the agent-based model consisting of in- +dividuals, media and influencers. Each of the agents adapts +its opinion in the continuous opinion space by interacting with +its network neighbours. The network between individuals is +given by A(t) while the relations between individuals and +media resp. influencers are defined by B(t) resp. C(t). In +the figure, the shape of an individual indicates the medium +they read and their color indicates the influencer they fol- +low. Opinion changes happen on different time scales as de- +termined by the inertia parameters where a higher inertia +corresponds to slower opinion changes. +plex. In the attention economy, the main goal of influencers +and media is to receive more attention and increase their fol- +lowership [53], that is, to find strategies that achieve this goal +in an optimal way. This makes them themselves dynamic en- +tities that can adjust their positions or agendas in order to +achieve those goals [22]. The effects these optimal strategies +in such an incentive structure could have on the overall opin- +ion dynamics have not yet been fully understood. +Recent attempts to find control strategies of opinion dy- +namics consist of a variety of seemingly independent approaches. +On the one hand, different heuristics have been developed to +determine which agents should be targeted by an external +controller to maximize their influence [41, 43, 34]. On the +other hand, in [2, 31], it is studied how external controllers +have to act to bring the opinion distribution of agents close to +a specified target distribution, while, in [34], it is established +how to control opinion distributions by (externally) interfer- +ing with the interaction network between individuals, and, +in [1, 25], dynamic external controls for mean-field opinion +models are considered. +We will approach the design of optimal strategies in the +following way: First, we show how to construct the mean- +field model related to the proposed opinion dynamics model +in the limit of large agent numbers and with finitely many in- +fluencers and media. This mean-field model takes the form of +a system of partial differential equations for the individuals’ +opinion distribution coupled to stochastic differential equa- +tions for the opinions of media and influencers. Then, we will +demonstrate how this mean-field model can be used to study +the effect of strategies in the attention economy. In particu- +lar, we will show how to derive optimal control schemes for +counteracting agendas to influence opinion distributions. The +advantage of the mean-field model compared to the ABM is +that the computational cost is independent of the number of +agents, and that the model is deterministic (when assuming +that influencers and media are not affected by stochasticity), +which, together, means that it is considerably easier to find +insight into collective behavior or optimal strategies via the +mean-field model. +Opinion model with influencers and +media +In the following we start by defining a general opinion model +resulting from a large number of individuals adapting their +opinions through interaction with each other as well as due +to the influence of a few specific agents with particular roles, +namely traditional media and social media influencers. See +Figure 1 for the model components. We consider the situ- +ation of the early formation of opinions, which is of great +importance in the accelerating public discourse [37]. Hence, +we focus on the transient model dynamics to study the forma- +tion, splitting and merging of clusters, while the asymptotic +regime is not of interest here. In this transient regime, indi- +viduals adapt their opinions on a fast time scale, while media +and influencer agents change their opinion positions on a sig- +nificantly slower time scale. +We consider individuals i = 1, . . . , N with opinions xi(t) +lying in a continuous d-dimensional opinion space D ⊂ Rd.1 +The vector x(t) = (xi(t))N +i=1 summarizes the opinions of all +N individuals. In addition, we introduce M highly influential +agents that can be considered as media outlets with contin- +uous opinions denoted by the vector y(t) = (ym(t))M +m=1 ∈ +DM, as well as L highly influential influencers with opinions +z(t) = (zl(t))L +l=1 ∈ DL. To describe the real-world situation, +we assume that there are much fewer influencers than individ- +uals but still more influencers than media, i.e., M < L ≪ N. +When political opinions are modelled, the opinion space D +with d = 2 could span the dimensions economic left ↔ right +and libertarian ↔ authoritarian. When opinions with respect +to acting against climate change are considered, two possible +dimensions could be climate change believers ↔ deniers and +technocentric ↔ ecocentric. +Relations among individuals such as friendship or connec- +tions on social media are defined through a binary adjacency +matrix A(t) ∈ {0, 1}N×N that can depend on time t. The +resulting network determines which individuals can interact +(when there is an edge between two individuals) and which +cannot (when there is no edge). When the edges are addition- +ally weighted, such that A(t) ∈ [0, 1]N×N, or directed, then +the network can also describe the strength of a tie or the direc- +tion of social influence as in [17]. Media and influencer agents +on the other hand are assumed to only interact with those +individuals that are their followers (readers/users). In the bi- +nary adjacency matrix B(t) ∈ {0, 1}N×M of medium-follower +1We regard the opinion space D to be bounded with no flux boundary +conditions. +3 + +relations we store which individual follows which medium at +time t while the binary adjacency matrix C(t) ∈ {0, 1}L×N +defines the connections between individuals and influencers +at time t. +Matrices A, B and C represent the complex network struc- +tures of social interactions and can be derived for example +from online social media or survey data. For the sake of sim- +plicity, we will in our examples consider A and B to be con- +stant on the chosen time scale and moreover an all-to-all con- +nectivity between individuals, i.e. Aij = 1 for all (i, j). Ad- +ditionally, we assume that each individual follows exactly one +medium and one influencer, i.e., each row of B and of C con- +tains exactly one entry. Since individuals usually change in- +fluencers very dynamically, we will propose an explicit model +that defines how individuals are changing influencers in time +and thus describes the evolution of C. This simplified set- +ting allows us to better illustrate dynamical properties of the +model and distinguish them from effects arising from the so- +cial ties. Our analytical results hold in the general case. +Individuals i = 1, . . . , N adapt their opinions in time ac- +cording to the following stochastic differential equation (SDE) +dxi(t) = Fi(x, y, t)dt + σdWi(t) +(1) +where Fi defines the interaction force on individual i, σ > 0 +gives the strength of noise and Wi(t) are i.i.d. d−dimensional +Wiener processes. +The noise can, for example, model un- +known external influences, uncertainties in the communica- +tion between individuals or free will. The interaction force +on individual i is given by a weighted sum of attractive forces +from all other connected individuals j scaled by the param- +eter a > 0 as well as attractive forces from the respective +media scaled by the parameter b > 0 and from the respective +influencer scaled by the parameter c > 0 +Fi(x, y, z, t) = +a +� +j′ wij′(t) +N +� +j=1 +wij(t)(xj(t) − xi(t)) ++ +b +� +m Bim′(t) +M +� +m=1 +Bim(t) (ym(t) − xi(t)) ++ +c +� +l′ Cil′(t) +L +� +l=1 +Cil(t)(zl(t) − xi(t)). +(2) +The interaction weights between pairs of individuals i, j are +given by +wij(t) = Aij(t) φ (|xj(t) − xi(t)|) , +(3) +i.e., they depend on the network and on the opinion distance +between the pair of individuals. The weights are normalized +to ensure a unitary total contribution from all individuals. +While the weight wij is certainly zero when there is no edge +between individuals i and j, the weight depends on the opin- +ion distance between i and j when there is an edge.2 Possible +choices for the non-negative, pair function φ(|xj − xi|) are +given by: +2When an individual has an interaction weight wij = 0 to all other +other individuals, then the first term of the attraction force in (2) is +assumed to be zero. +• φ(x) = exp(−x) as in [40] that places exponentially +more weight on close-by individuals, +• φ(x) = 1[0,d](x) as in bounded confidence models [57, +26, 27] that only allows interactions with other individu- +als that are within a radius d, here 1[0,d] is the indicator +function on the set [0, d], +• φ(x) = 1 as in the DeGroot model [17] resulting in +interactions irrespective of the opinion distance between +individuals. +The first two choices imply that individuals that are already +close in opinion space excert higher social influence on each +other (homophily), while the third choice results in a weight +irrespective of the opinions of the interacting individuals. +Remark (Opinion differentiation and higher-order interac- +tions). Apart from the described opinion attraction (assimi- +lation), it is also possible to model opinion repulsion (differen- +tiation) between individuals. In [39], it is suggested to classify +pairs of individuals (i, j) either as friends that attract in their +opinions or as enemies that repel in their opinions. These +pairs (i, j) will have a positive resp. +negative edge weight +Aij(t) and thereby can turn the first term of the force (2) +from attraction to repulsion. In [29], another approach that +extends bounded confidence models is suggested: pairs of in- +dividuals can not only become closer in their opinions when +their opinion distance is smaller than the distance d, but +they can also repel from each other when they are further +apart than the distance D > d. By using the pair function +φ(x) = 1[0,d](x) − 1[D,∞)(x), this can be incorporated. But +when including differentiation, the weights wij(t) from (3) +can become negative and hence can no longer be normalized, +instead one can for example normalize the force by the num- +ber of individuals, N, or by the number of neighbours of an +individual. +Additionally, it is possible to describe higher-order inter- +actions among more than two individuals at the same time +with this model and thereby to more accurately describe group +effects such as peer pressure, see e.g. [44, 19]. +Not only individuals change their opinions, but also media +agents and influencers, however they adapt their opinions on +a much slower time scale. The resistance to rapid change is +determined by the inertia parameter Γ > 1 for media agents +and by γ > 1 for influencers. With γ < Γ media agents are +changing their opinions on an even slower time scale than in- +fluencers. In the limit when the parameters Γ, γ diverge, the +opinions of media and influencers become constant in time. +There is a lot of research [11, 27, 8, 10] that studies interac- +tions of individuals with constant agents (also termed stub- +born agents or zealots) but to our knowledge adaptive media +and influencers have not been studied so far. In particular +here media agents m = 1, . . . , M slowly adapt their opinions +according to the SDE +Γdym(t) = f(˜xm(t) − ym(t))dt + ˜σ ˜ +dW m(t), +(4) +4 + +where the force function f can be used to model nonlinear +influence effects but is set to f(x) = x subsequently, i.e., +media agents are drawn in the direction of the average opinion +of their followers +˜xm(t) = +1 +� +k Bkm(t) +N +� +i=1 +Bim(t)xi(t). +In the SDE, ˜σ > 0 gives the strength of noise on the opin- +ion dynamics and ˜Wm(t) denote i.i.d. d−dimensional Wiener +processes. Similar to media, influencers l = 1, . . . , L slowly +change their opinions in the direction of their average follow- +ership according to the SDE +γdzl(t) = g(ˆxl(t) − zl(t))dt + ˆσ ˆ +dW l(t), +(5) +where the average opinion of followers is given by +ˆxl(t) = +1 +� +k Ckl(t) +N +� +i=1 +Cil(t) xi(t). +The noise strength is given by ˆσ > 0, while ˆWl(t) denote i.i.d. +d−dimensional Wiener processes. The function g can again +be used to model nonlinear effects but is set to g(x) = x in +the following. +We have seen that influencers are similar to media agents +but are usually more numerous and adapt their opinions on a +faster time scale. To reflect that relationships on social media +are more dynamic than to traditional media outlets, we fur- +ther propose an explicit model of how individuals can switch +the influencer they are currently following. In particular, each +individual i can at any time t switch its current influencer l′ +to another influencer l with a given rate Λ→l +m (x, t) where m +is the medium of individual i and x is the opinion of i.3 The +rate could for example take the following form +Λ→l +m (x, t) = η ψ(|zl − x|) r +� +nm,l(t) +�M +m′=1 nm′,l(t) +� +(6) +with scaling parameter η > 0, pair function ψ, the link rec- +ommendation function r and nm,l denoting the fraction of +individuals that follow medium m and influencer l. By set- +ting the pair function for example to ψ(x) = exp(−x), an +individual has an exponentially higher rate to switch to an +influencer that has a similar opinion than to an influencer +with a very different opinion, i.e., there is homophily between +influencers and individuals when connections are made. On +social media platforms, link recommendation algorithms are +often used to suggest new connections to users that have the +greatest potential to become established [35, 49]. We incor- +porate link recommendation via the function r by assuming +that individuals have a higher chance of switching to an in- +fluencer with a structurally similar followership. We measure +the structural similarity of the followers of l to individual i +3Note that the change rate does not depend on the current influencer +of individual i and that an individual can therefore also change to the +influencer it is currently following without any effect. +by the ratio of followers that are connected to the same in- +fluencer (after switching) and medium as i, this proportion +is given by +nm,l +� +m′ nm′,l . We then assume that r is an increas- +ing function on [0, 1], such as for example the ReLu function +r(x) = max{0.1, −1 + 2x}. +Example 1: ABM with media and influencers +In Figure 2, we show snapshots of one realization of the ABM +with 250 individuals, 2 media and 4 influencers. +The pa- +rameters are chosen to demonstrate how opinion clusters can +form, split and merge. Initially at t = 0, individuals are ran- +domly distributed in opinion space and uniformly at random +assigned to the 2 media that are initially at y1(0) = (−1, −1) +and y2(0) = (1, 1). Individuals in each of the 4 quadrants are +assigned to a different influencer and the initial opinion of +the influencer is set to the mean opinion of its followers. All +individuals are interacting with each other, i.e., the network +A is fully-connected. +When we let the model run, the strong attraction force +to influencers (c = 4) results in individuals quickly being +attracted by their respective influencer and forming 4 clus- +ters (compare with t = 0.1). After some time, the 4 clus- +ters split further because individuals are also attracted to +their medium (compare t = 0.4), s.t. individuals now form +roughly 8 groups. Some individuals switch the influencer to +a more suitable influencer, i.e., one that is closer in opinion +space and whose majority of followers are connected to the +same medium as the individual. They then get attracted to +the new influencer (t = 1.2), until finally (t = 2) individuals +have formed 2 mixed clusters near the 2 media opinions. +In Figure 3, we show the evolution of the proportion of in- +dividuals that follow a certain influencer and medium. Around +t = 0, the proportions are roughly the same. But after t = +0.5, individuals start switching their influencer (the medium +cannot be switched), s.t. quickly the followers of each influ- +encer are dominated by individuals following the same medium. +Towards t = 2, the proportion of individuals that follow the +medium and influencer in the upper right corner (indicated as +green triangles) and the proportion of individuals that follow +the medium and influencer in the lower left corner (shown by +blue circles) dominate. +Also, Fig. 2 shows that at t = 2 most of the followers of the +medium near (−1, −1) (denoted by circles) follow the same +influencer that is also near (−1, −1) (colored in blue) while +most followers of the medium near (1, 1) (denoted by trian- +gles) also follow the influencer near (1, 1) (colored in green). +The reason for this dominance is that for individuals it is +favorable to be close to their medium and their influencer, +otherwise they might switch the influencer to a more suitable +candidate. The two less suitable influencers in the upper left +and lower right corner now very slowly move towards their few +remaining followers. On a larger time scale they will reach +the clusters. Asymptotically, also the two clusters of individ- +uals, media and influencers will approach another and merge. +For different simulation runs, the agents behave qualitatively +5 + +2 +1 +0 +1 +2 +t = 0.0 +2 +1 +0 +1 +2 +t = 0.1 +2 +1 +0 +1 +2 +t = 0.4 +2 +1 +0 +1 +2 +t = 1.2 +2 +1 +0 +1 +2 +2 +1 +0 +1 +2 +t = 2.0 +Figure 2: Realization of the ABM with media and influencers +(Example 1). There are 2 media agents marked in black, 4 +influencers indicated by large circles in 4 colours, and 250 in- +dividuals that inherit the shape of the medium they read and +the color of the influencer they currently follow. Parameters: +a = 1, b = 2, c = 4, σ = 0.5, ˜σ = 0, ˆσ = 0, Γ = 100, γ = 10, +Aij = 1 for all (i, j), φ(x) = ψ(x) = exp(−x), η = 15. +0.0 +0.5 +1.0 +1.5 +2.0 +t +0.00 +0.25 +0.50 +0.75 +1.00 +Proportion of followers +Figure 3: Stack plot showing how the proportion of individ- +uals that follow a certain influencer and medium (marked by +the symbols on the left) evolve in time in Example 1. +similar. But the individuals following the influencer initially +at (−1, 1) and (1, −1) are sometimes attracted to a different +medium than in the shown simulation. +The code for all examples is contained in the GitHub +repository www.github.com/LuzieH/SocialMediaModel. +Rich dynamical behaviour +The situation studied in Example 1 is somewhat symmet- +ric and idealistic. For other initial configurations and other +choices of parameters and pair functions φ different forms of +complex dynamical behavior emerge. This ranges from sta- +ble opinion clusters centered around ”their” influencers (for +φ(x) = ψ(x) = 1[0,d](x) and no media) to a complex interplay +of cluster formation and reformation for several influencers +with smaller and larger inertia γ, and a less symmetric con- +figuration as in Example 1. The dynamics can become even +richer, if the interaction network A is already exhibiting clus- +ters. Moreover, different initial conditions lead to different +transient dynamics. +Partial mean-field (opinion) model +For situations with many individuals but few influencers and +media, one can derive the mean-field limit by a partial dif- +ferential equation (PDE) that describes the opinion dynam- +ics of individuals in the limit of infinitely many individu- +als [56, 21, 24] but is usually already a good approximation +to the dynamics for finitely many individuals. +Since here +the number of influencers and media is still small and finite, +their dynamics are still best described by SDEs but now cou- +pled to PDEs for the evolution of the opinion distributions +of individuals. +We therefore also call the model a partial +mean-field model, compare Figure 4 for the structure of the +model. The coupled system of PDEs and SDEs is not only +computationally advantageous [28] since the computational +effort no longer scales with N 2 (due to the expensive com- +putations of pair-wise distances of individuals in the ABM) +6 + +Media +opinion space +opinion space +(high inertia) +(medium inertia) +opinion space +Influencers +Individuals +Figure 4: Structure of the partial mean-field model of dis- +tinct media agents (indicated by black circles) and influencers +(shown as colored circles) in interaction with the opinion dis- +tributions of individuals. The opinion distribution of individ- +uals that follow a certain medium m and influencer l is given +by ρm,l(x, t). +but with the number of spatial grid cells and independently +of N. Additionally, the model is conceptually easier to study +for example to find critical parameters [56, 21, 18] or to use +the partial mean-field model to control the evolution of the +opinion distribution through different influencer and media +strategies (see next section). +For the sake of simplicity, we subsequently assume a fully- +connected interaction network between individuals, such that +Aij = 1 for all pairs of individuals (i, j)4, and that each in- +dividual follows exactly one medium and one influencer at a +time and only influencers can be switched at the rate given +in (6). +Then let us define the empirical distribution of individuals +that follow medium m and influencer l at time t by the sum of +Dirac Delta distributions δ placed at the individuals’ opinions +ρ(N) +m,l (x, t) = 1 +N +� +i:Bim=1, +Cil(t)=1 +δ(x − xi(t)). +(7) +This distribution describes the stochastic opinion instances +at a given time t and integrates to +� +D +ρ(N) +m,l (x, t)dx =: n(N) +m,l (t), +the proportion of individuals that follow medium m and in- +fluencer l. +It can be shown (see Supplementary Material) that as +N → ∞, the empirical distribution ρ(N) +m,l (x, t) can be replaced +by the limiting distribution ρm,l(x, t) solving the following +4Without this assumption, one would need to derive mean-field dy- +namics for interacting agents on realistic complex networks that have a +network limit in terms of graphons [14] or graphops [32, 23], but here we +will make the standard assumption of a fully-connected network between +individuals. +PDE on the domain D +∂tρm,l(x, t) = 1 +2σ2∆ρm,l(x, t) − ∇ · (ρm,l(x, t) F(x, ym, zl, ρ)) ++ +� +l′̸=l +� +−Λ→l′ +m (x, t) ρm,l(x, t) + Λ→l +m (x, t) ρm,l′(x, t) +� +(8) +for each m = 1, . . . , M and l = 1, . . . , L, where ∇· denotes +the divergence operator and ∆ the Laplace operator on opin- +ion space. The PDE is accompanied by boundary conditions +ensuring that the number of individuals in the system is con- +served. +The changes of ρm,l are governed by three processes: (i) The +first term on the RHS of the PDE is responsible for the +stochastic diffusion of opinions. +(ii) The divergence term +models the interaction of the distribution ρm,l with the over- +all distribution of all individuals ρ = � +m,l ρm,l as well as with +the respective medium ym and the influencer zl according to +the attraction force at opinion x +F(x, ym, zl, ρ) =a +� +D ρ(x′, t)φ(|x′ − x|)(x′ − x) dx′ +� +D ρ(x′, t)φ(|x′ − x|) dx′ ++ b (ym(t) − x) + c (zl(t) − x). +(9) +(iii) The last term of the PDE is responsible for the mass +exchange between different distributions due to individuals +switching the influencer away from l′ or towards l. Note that +due to the second and third term, the PDE is non-local. +The PDEs are coupled to the SDEs of the media m = +1, . . . , M +Γdym(t) = (˜xm(t) − ym(t))dt + ˜σ ˜ +dW m(t) +(10) +where the average opinion of followers is now given by +˜xm(t) = +�L +l=1 +� +D xρm,l(x, t)dx +�L +l=1 nm,l(t) +with nm,l(t) denoting the limit of n(N) +m,l (t), as well as to the +SDE dynamics of influencers l = 1, . . . , L +γdzl(t) = (ˆxl(t) − zl(t))dt + ˆσ ˆ +dW l(t), +(11) +with +ˆxl(t) = +�M +m=1 +� +D xρm,l(x, t)dx +�M +m=1 nm,l(t) +denoting the average opinion of individuals that follow influ- +encer l. +Example 2: Comparison on ABM and partial +mean-field dynamics +In Fig. 5, we show a comparison of ABM and partial mean- +field simulations averaged over 1000 realizations, the parame- +ters are as in Example 1. To compare the ABM configurations +against the opinion distributions in the partial mean-field, +7 + +2 +1 +0 +1 +2 +t=0.0 +ABM +PDE +2 +1 +0 +1 +2 +t=0.5 +2 +1 +0 +1 +2 +t=1.0 +2 +1 +0 +1 +2 +t=1.5 +2 +1 +0 +1 +2 +2 +1 +0 +1 +2 +t=2.0 +2 +1 +0 +1 +2 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.0 +0.2 +0.4 +0.6 +0.8 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +Figure 5: Mean configuration of the ABM (left column) and +partial mean-field PDE (right column) over 1000 simulations +with influencers and media. Influencers are now marked by +grey circles, and media agents by black circles. +The opin- +ion distribution of all individuals (i.e., independent of the +media/influencer they follow) is shown with a heat plot. Pa- +rameters as in Example 1. +the opinion distributions resulting from the ABM are visu- +alized via Kernel density estimation with Gaussian kernels. +Even though the PDE is deterministic, we initialized it with +random initial distributions given by the Kernel density esti- +mation of 250 individuals placed uniformly at random in the +domain. Different realizations of these random initial condi- +tions lead to different transient dynamics and we therefore +also averaged the partial mean-field simulations. The com- +parison shows that the results of the ABM and the partial +mean-field are very consistent already for N = 250 individu- +als in the system. +Strategies in the partial mean-field model +In the previous section we have stated the partial mean-field +model as a reduced model describing the opinion dynamics of +infinitely many individuals coupled to the opinion dynamics of +finitely many media and influencers. Assuming deterministic +opinion changes of influencers and media (i.e., ˆσ = ˜σ = 0), +and a deterministic initial opinion distribution, the partial +mean-field model is deterministic and faster to solve than the +ABM, allowing us to study the effect of different (optimal) +influencer and media strategies. +2 +1 +0 +1 +2 +t = 5.0 +m nm, 5 = 0.0 +2 +1 +0 +1 +2 +t = 7.0 +m nm, 5 = 0.069 +2 +1 +0 +1 +2 +t = 8.5 +m nm, 5 = 0.137 +2 +1 +0 +1 +2 +2 +1 +0 +1 +2 +t = 10.0 +m nm, 5 = 0.177 +0.00 +0.25 +0.50 +0.75 +1.00 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +1.5 +2.0 +Figure 6: Opinion distribution ρ evolving in time (Example 3) +started from a deterministic uniform opinion distribution at +t = 0. At time t = 5, a 5th influencer (marked by a triangle) +is inserted into the system and moves at a constant pace in +the direction of the right upper corner while collecting new +followers. Parameters as in Example 1, except η = 1. +8 + +Increasing followership +In the version of our opinion dynamics model as specified +above, influencers adapt their opinions in the direction of the +average opinion of their followers. This is a very simple strat- +egy for influencers trying to keep their followers attached to +them [53]. +Other strategies to keep and even increase the +number of followers might be more fruitful. In the follow- +ing example we discuss a simple strategy for a new influencer +with initially zero followers to substantially increase its fol- +lowership. +Example 3: Strategy of an influencer to in- +crease followership +We consider the dynamics as before in Example 1 except with +η = 1. With this choice of parameters, influencers strongly +affect and attract individuals (since c > a, b) and individuals +only slowly change influencers (due to η being small). In this +way, once individuals follow a certain influencer, they will +remain with that influencer for some time. +At time t = 5 we insert an additional influencer into the +system with opinion (0, 0) and zero followers. This new influ- +encer then moves at a constant speed to the opinion (1.5, 1.5) +during the time interval [5, 10]. +In Fig. 6, we show snap- +shots of the realization. The inserted influencer l = 5 quickly +collects and attracts many followers behind. Starting with +initially zero followers, the final followershare is +2 +� +m=1 +nm,5(10.0) ≈ 0.18. +Moreover the time-averaged proportion of followers of influ- +encer l = 5 over the interval [5, 10] comes out to be approxi- +mately +1 +5 +� t=10 +t=5 +2 +� +m=1 +nm,5(t) dt ≈ 0.10. +Thus the added influencer could substantially increase its fol- +lowership by moving to an extreme opinion position. Even a +new cluster of followers has formed near the influencer. +Optimal counteraction +More generally, the partial mean-field model also allows to +apply optimal control techniques in order to derive the best +strategy for a single influencer or a medium to achieve a cer- +tain objective, such as maximizing the number of their fol- +lowers or optimally counteracting the goal of another agent. +Let ϱ(u) = (ρm,l(t))m=1,...,M,l=1,...,L, the matrix of all dif- +ferent opinion distributions on the time interval [0, T], satisfy +Eqs. (8), (10), and (11), except for the chosen controlled in- +fluencer or medium. When we are interested in the strategy +for an influencer agent l = l⋆, we let the control u determine +the behaviour of the l⋆th influencer, i.e., zl⋆ = u, when on the +other hand we are searching for a media strategy, we control +the m⋆th medium with ym⋆ = u. We can then express the +aim of influencer l⋆ to maximize its temporally aggregated +followership with the objective function +J ϱ = +� T +0 +M +� +m=1 +nm,l⋆(t) dt, +while maximizing solely the final followership means drop- +ping the integral and taking the integrand at time t = T. In +contrast, if the goal of influencer l⋆ or medium m⋆ is to coun- +teract the maximization of followership of another influencer +l′, this can be achieved by maximizing the objective +J ϱ = − +� T +0 +M +� +m=1 +nm,l′(t) dt. +When the control determines the behavior of influencer l⋆ or +medium m⋆, the control u needs to satisfy certain restric- +tions. That is, the control function u has to come from a set +of admissible controls U. On the one hand, U is restricted +by the necessary domain constraints u(t) ∈ D for almost all +t ∈ [0, T]. On the other hand, with every control u certain +costs are associated. +Control discontinuities or even chat- +tering controls could, e.g., increase the risk of the control +activities being detected, and countermeasures taken. Such +concerns can be included as a penalty term in the objective, +e.g., by +J p = α|∂tu|2 +L2(]0,T [), +where the parameter α regulates the penalty for large opinion +changes. Then, the optimal control problem has the general +form +max +u∈U J ϱ − J p. +(12) +After an appropriate control discretization with, e.g., piece- +wise polynomials, the resulting nonlinear programming prob- +lem can be solved by derivative-free methods like Nelder- +Mead or by more efficient inexact gradient descent [46] or +stochastic approximation methods. Sufficiently accurate gra- +dient evaluations can be obtained by finite differencing the +PDE/SDE forward equations for ϱ(u) or by Feynman-Kac +type gradient sampling. +Example 4: Influencer counteraction +We are interested in the optimal counteraction of influencer +l⋆ = 6 when influencer l′ = 5 moves at a constant speed +to the upper right corner (1.5, 1.5) (as in Example 3) and +thereby drastically increases its followership. +We therefore +want influencer l⋆ to move in an optimal way to satisfy +max +u∈U +� +− +� 10 +5 +2 +� +m=1 +nm,5(t) dt − α|∂tu|2 +L2(]5,10[) +� +(13) +with α = 0.05. That is, the goal is to minimize the follow- +ershare of influencer l′, while avoiding conspicuously drastic +opinion changes. +The control set U contains all functions +with u(t) ∈ D and with piecewise constant velocities that +only change at 3 chosen time points. +9 + +2 +1 +0 +1 +2 +t = 5.0 +m nm, 5 = 0.0 +2 +1 +0 +1 +2 +t = 6.0 +m nm, 5 = 0.041 +2 +1 +0 +1 +2 +t = 7.5 +m nm, 5 = 0.066 +2 +1 +0 +1 +2 +2 +1 +0 +1 +2 +t = 10.0 +m nm, 5 = 0.044 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +1 +2 +3 +Figure 7: Opinion distribution ρ evolving in time (Example 4) +started from a uniform distribution at t = 0. At time t = 5, a +5th influencer (marked by a grey triangle) is inserted into the +system and moves at a constant pace in the direction of the +right upper corner. Simultaneously, a 6th influencer (marked +by a cross) starts and moves in a optimal way to prevent +influencer l′ = 5 from collecting many followers. Parameters +as in Example 4. +The optimal strategy is shown in Fig. 7 and results in an +average followershare of influencer l′ of +1 +5 +� t=10 +t=5 +2 +� +m=1 +nm,5(t) dt ≈ 0.05, +which is significantly smaller compared to the value 0.1 that +was obtained in Example 3 without counteraction. Note also, +that the final followershare at time t = 10.0 was decreased +from 0.18 without counteraction to 0.04 with counteraction. +The snapshots in Fig. 7 show that the optimal counterstrategy +of influencer l⋆ = 6 is to steel followers of influencer l′ = 5 +by moving along; it thereby stabilizes the opinion cluster of +individuals. Note that according to the objective specified +in (13), this counteraction comes out to be more effective than +the movement in the opposite direction of influencer l′ = 5. +Example 5: Media counteraction +Next, we study the optimal counterstrategy of the media +agent m⋆ = 2 in the upper right corner when trying to mini- +mize the followershare of influencer l′ = 5, analogously to the +previous example with the objective function given by (13). +The resulting optimal strategy is shown in Fig. 8. By chang- +ing the opinion drastically from the upper right to the lower +right corner, the media agent manages to decrease the average +followershare of influencer l′ = 5 down to +1 +5 +� t=10 +t=5 +� +m +nm,5(t) dt ≈ 0.06. +Note that since the system dynamics are symmetric wrt. the +diagonal axis x = y, an equivalently good strategy would be +to move to the upper left corner. The counterstrategy of the +media agent is very different to the counteraction of an influ- +encer (given in Example 4). This is because influencers com- +pete for followers and can steal influencers from each other +while media agents can only make an opinion topic unattrac- +tive to individuals by changing the topic. +Conclusion +In this work we provided new mathematical means for the sys- +tematic study of how traditional media and influencers might +impact coherent structures of the public opinion distribution +as modelled by opinion dynamics models. +It does neither +claim to describe opinion shifts in real-world social networks +nor does it state anything about influencing the opinion of +an individual human being. It is still an idealized model and, +like most contributions to the field, can neither claim to de- +scribe human opinion formation processes realistically, nor +to be validated with observational data and controlled soci- +ological experiments. However, it may help to describe how +shifting individual perspectives and social exchange lead to +archetypal states of public opinion distribution like coherent +opinion clusters. By providing a strategy for understanding +how influencing the public opinion distribution may be done +and counteracted in an optimal way, this work may help us +to understand how to face current challenges of opinion po- +larization in complex scenarios. +In particular, our model demonstrates the impact influ- +encers and media might have on the opinion distribution by +creating stable and coherent opinion clusters that are drawn +to the positions of the respective influencer or media. +We +also see that when influencers move to extreme positions, +10 + +2 +1 +0 +1 +2 +t = 5.0 +m nm, 5 = 0.0 +2 +1 +0 +1 +2 +t = 6.0 +m nm, 5 = 0.042 +2 +1 +0 +1 +2 +t = 7.5 +m nm, 5 = 0.071 +2 +1 +0 +1 +2 +2 +1 +0 +1 +2 +t = 10.0 +m nm, 5 = 0.058 +0.00 +0.25 +0.50 +0.75 +1.00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +1.5 +2.0 +Figure 8: Opinion distribution ρ evolving in time (Example +5) started from a uniform distribution at t = 0. +At time +t = 5, a 5th influencer (marked by a triangle) is inserted into +the system and moves at a constant pace in the direction +of the right upper corner, simultaneously, the media in the +upper right corner (marked by a cross) moves in a optimal way +to prevent influencer l′ = 5 from collecting many followers. +Parameters as in Example 4. +they fragment the opinion landscape and gain more follow- +ers and more attention in the competitive scenario of so- +cial media. Hence, these results suggest that those compet- +itive goals could contribute to a polarized and fragmented +debate. However, our work also offers avenues towards solu- +tions through optimal counteracting those attempts via other +influencers (or media). These counteracting influencers can +stabilize the existing opinion landscape against extreme in- +fluencers by strengthening existing opinion clusters. Empir- +ical work needs to be done to validate those strategies and +interplays that our model is suggesting, potentially through +large-scale field experiments on social media. Our theoreti- +cal work potentially delivers a good basis for developing such +alternative strategies for influencers in the current system to +stabilize online discourse. +In this work we did not explore the full generality of the +opinion dynamics model for different interaction terms, pa- +rameters and different influencer and media strategies. In- +stead we focused in this work on the feasibility by designing +the pipeline from the inclusion of influencers and media via +the demonstration of the emergence of temporarily coherent +opinion clusters to the options for influencing the public opin- +ion by optimally chosen strategies. The results provided do +not include any analysis of the dynamical patterns such as +numbers, birth and decay of opinion clusters, that the model +may exhibit for different parameter combinations. This was +beyond the focus and scope of this study and will be left to +future investigations. +Acknowledgments +We would like to thank Jobst Heitzig +for his insights into the opinion model, Ana Djurdjevac, Ste- +fanie Winkelmann and Alberto Montefusco for their com- +ments on mean-field models, Alexander Sikorski for discus- +sions on PDE discretizations and insights into code speed- +ups, and Martin Weiser for valuable conversations on optimal +control theory. This work has been partially funded by the +Deutsche Forschungsgemeinschaft (DFG) under Germany’s +Excellence Strategy through grant EXC-2046 The Berlin Math- +ematics Research Center MATH+ (project no. 390685689). +P.L.-S. acknowledges financial support from the Volkswagen +Foundation (grant ‘Reclaiming individual autonomy and demo- +cratic discourse online: How to rebalance human and algorith- +mic decision-making’). +11 + +Supplementary Material +In the following, we will outline the derivation of the partial mean-field model for the limiting dynamics of infinitely many +individuals, while keeping the number of media and influencers fixed. +PDE Derivation +We start by deriving the mean-field PDEs for the opinion dynamics in the limit of infinitely many individuals. Related +mean-field PDEs have already been derived in the context of other opinion models [55, 21, 24]. +Assuming constant influencer-follower connections. +For the derivation, we will assume that the network between +individuals is fully-connected, i.e. that Aij = 1 for all i, j, and that each individual follows exactly one medium and one +influencer. Further in this first part, we will suppose that individuals do not change the influencer they are following, i.e. +the matrix of influencer-follower relations remains constant C(t) ≡ C. In a next step we will relax this assumption. For the +purpose of this derivation, we add (N) to the model quantities when the number of individuals is still finite. +Since there are M media and L influencers in the system, we have M × L different types of individuals depending on +which combination of a medium and an influencer a particular individual is following. Since the network between individuals +is fully-connected, individuals that follow the same influencer and medium can be considered as identical (exchangeable). +For that reason, we can switch from a description in terms of labeled individuals to a description in terms of empirical +distributions. Let us define the empirical distribution of individuals that follow medium m and influencer l by +ρ(N) +m,l (x, t) := 1 +N +� +i:B(N) +im =1, +C(N) +il +(t)=1 +δ(x − x(N) +i +(t)) +(14) +and the proportion of these individuals compared to the total number of individuals in the system, N, by +n(N) +m,l (t) := +� +D +ρ(N) +m,l (x, t)dx. +The proportion n(N) +m,l (t) is constant in time for now, since influencer-follower relations cannot be changed. Further, we denote +the total empirical distribution of individuals that follow any influencer or medium by +ρ(N)(x, t) := 1 +N +N +� +i=1 +δ(x − x(N) +i +(t)) = +M +� +m=1 +L +� +l=1 +ρ(N) +m,l (x, t) +(15) +with +� +D ρ(N)(x, t)dx = � +m,l n(N) +m,l (t) = 1. +The opinion dynamics of individuals i = 1, . . . , N are given by +dx(N) +i +(t) = Fi(x(N), y(N), z(N))dt + σdWi(t). +(16) +With the definition of the empirical distribution in (15), we can rewrite the interaction force Fi by +Fi(x(N), y(N), z(N)) = a +� +D ρ(N)(x, t)φ(|x − x(N) +i +(t)|)(x − x(N) +i +(t)) dx +� +D ρ(N)(x, t)φ(|x − x(N) +i +(t)|) dx ++ b (y(N) +m (t) − x(N) +i +(t)) + c (z(N) +l +(t) − x(N) +i +(t)) +whenever the agent i follows medium m and influencer l. Thus the SDE in (16) depends no longer on the opinions of other +individuals, but instead on the total empirical distribution ρ(N). +We are now interested in the limit N → ∞. We assume that as N grows, the number of individuals that follow a certain +medium m and a certain influencer l grows like Nn(N) +m,l (t) with n(N) +m,l (t) independent of N. It can then be expected [52, 13] +that as N → ∞, any individual that follows a certain medium m and influencer l has an i.i.d. distributed opinion ¯xm,l that +satisfies the SDE +d¯xm,l(t) = F(¯xm,l, ym, zl, ρ)dt + σdW(t) +(17) +with interaction force +F(¯xm,l, ym, zl, ρ) = a +� +D ρ(x, t)φ(|x − ¯xm,l|)(x − ¯xm,l) dx +� +D ρ(x, t)φ(|x − ¯xm,l|) dx ++ b (ym(t) − ¯xm,l) + c (zl(t) − ¯xm,l). +12 + +The opinion process ¯xm,l from (17) depends on the density ρ which can be interpreted as the limit of ρ(N) when N → ∞. +More precisely, denoting by µm,l(x, t) the probability density of an individuals’ opinion with medium m and influencer l +at time t, ρm,l(x, t) := nm,l(t)µm,l(x, t) denotes the probability density scaled by the proportion of these individuals in the +system, and ρ(x, t) = � +m,l ρm,l(x, t) denotes the probability density of any individual in the system. Stochastic processes of +the form (17) are also called McKean-Vlasov processes. Each scaled density ρm,l fulfils a McKean Vlasov-type PDE +∂tρm,l(x, t) += +1 +2σ2∆ρm,l(x, t) − ∇ · (ρm,l(x, t) F(x, ym, zl, ρ)) +(18) +and we can also interpret each PDE for a fixed m, l as describing the density of infinitely many copies of individuals that +follow medium m and influencer l. There is also a growing body of literature that considers an intermediate level of many +but not infinitely many individuals, whose empirical distribution in the opinion space can approximately be described by a +stochastic partial differential equation [16, 30, 28, 19]. +Allowing influencer-follower connections to change. +We now additionally allow individuals to switch the influencer +in time. In the ABM each individual i can change the influencer to l at the rate Λ→l +m +(N)(x, t) where m is the medium of +individual i and x its current opinion. In the PDE we want these changes from influencer l′ to l to correspond to mass +flowing from ρm,l′ to ρm,l. We will in the following derive the corresponding terms that have to be added to the PDE (18) +and are sometimes also called reaction terms. +The total number of individuals that follow medium m and influencer l at time t in the ABM is given by Y (N) +m,l (t) := +N n(N) +m,l (t) and can be considered a jump process that only changes by +1 when an individual changes its influencer to l or +by −1 when an individual changes its influencer away from l. The rate of any individual changing from (m, l′) to (m, l) is +given by the sum of the individual change rates as follows +αl′→l +m +(N)(t) = +� +i: B(N) +im =1, +C(N) +il′ (t)=1 +Λ→l +m +(N)(xi, t) = N +� +D +Λ→l +m +(N)(x, t) ρ(N) +m,l′(x, t)dx. +With this, we can write down the evolution of the jump process Y (N) +m,l (t) as +Y (N) +m,l (t) = Y (N) +m,l (0) + +� +l′̸=l +� +Pl′→l +m +�� t +0 +αl′→l +m +(N)(t′)dt′ +� +− Pl→l′ +m +�� t +0 +αl→l′ +m +(N)(t′)dt′ +�� +with unit-rate Poisson processes Pl→l′ +m +. +In the limit N → ∞, the rates become large and we can replace the Poisson processes by their mean to get the reaction +rate equation [58] written in terms of the limiting proportions nm,l(t) +dnm,l +dt +(t) = +� +l′̸=l +� +nm,l′(t)rl′→l +m +(t) − nm,l(t)rl→l′ +m +(t) +� +(19) +with the spatially-averaged rates in the limit +rl′→l +m +(t) = +1 +nm,l′(t) +� +D +Λ→l +m (x, t) ρm,l′(x, t)dx +and the limiting individual change rates +Λ→l +m (x, t) = η ψ(|zl − x|) r +� +nm,l(t) +� +m′ nm′,l(t) +� +. +The ODE (19) can be spatially extended to the following term [30, 28] +∂tρm,l(x, t) = +� +l′̸=l +−Λ→l′ +m (x, t) ρm,l(x, t) + Λ→l +m (x, t) ρm,l′(x, t), +(20) +such that the complete PDE describing opinion changes and influencer changes is given by the sum of (18) and (20). +13 + +Limiting SDE dynamics of influencers and media +The limiting dynamics of media and influencers follow by considering the limiting average position of followers of medium m +˜x(N) +m (t) += +1 +�N +k=1 B(N) +km +N +� +i=1 +B(N) +im x(N) +i +(t) += +�L +l=1 +� +D x ρ(N) +m,l (x, t)dx +�L +l=1 n(N) +m,l (t) +→ +�L +l=1 +� +D x ρm,l(x, t)dx +�L +l=1 nm,l(t) +=: ˜xm(t) +and of influencer l +ˆx(N) +l +(t) += +1 +�N +k=1 C(N) +kl (t) +N +� +i=1 +C(N) +il +(t)x(N) +i +(t) += +�M +m=1 +� +D x ρ(N) +m,l (x, t)dx +�M +m=1 n(N) +m,l (t) +→ +�M +m=1 +� +D x ρm,l(x, t)dx +�M +m=1 nm,l(t) +=: ˆxl(t). +No-flux boundary conditions +We are using boundary conditions such that the total number of individuals in the system remains constant in time, i.e., we +want +� +D +∂tρ(x, t)dx = 0 +for all t. 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Current Psychology, pages 1–17, 2020. +17 + diff --git a/MNFRT4oBgHgl3EQf2ji4/content/tmp_files/load_file.txt b/MNFRT4oBgHgl3EQf2ji4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9eea00f89294efcfaf71b94d1a91606c8a31cb4e --- /dev/null +++ b/MNFRT4oBgHgl3EQf2ji4/content/tmp_files/load_file.txt @@ -0,0 +1,788 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf,len=787 +page_content='Modelling opinion dynamics under the impact of influencer and media strategies Luzie Helfmanna,b, Nataˇsa Djurdjevac Conrada, Philipp Lorenz-Spreenc and Christof Sch¨uttea,d,1 aZuse Institute Berlin bPotsdam Institute for Climate Impact Research cCenter for Adaptive Rationality, Max Planck Institute for Human Development dInstitute of Mathematics, Freie Universit¨at Berlin 1Corresponding author: schuette@zib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='de January 2023 Abstract Digital communication has made the public discourse considerably more complex, and new actors and strategies have emerged as a result of this seismic shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Aside from the often-studied interactions among individuals during opinion formation, which have been facilitated on a large scale by social media platforms, the changing role of traditional media and the emerging role of ”influencers” are not well understood, and the implications of their engagement strategies arising from the incentive structure of the attention economy even less so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Here we propose a novel opinion dynamics model that accounts for these different roles, namely that media and influencers change their own positions on slower time scales than individuals, while influencers dynamically gain and lose followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Numerical simulations show the importance of their relative influence in creating qualitatively different opinion formation dynamics: with influencers, fragmented but short- lived clusters emerge, which are then counteracted by more stable media positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Mean-field approximations by partial differential equations reproduce this dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Based on the mean-field model, we study how strategies of influencers to gain more followers can influence the overall opinion distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We show that moving towards extreme positions can be a beneficial strategy for influencers to gain followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Finally, we demonstrate that optimal control strategies allow other influencers or media to counteract such attempts and prevent further fragmentation of the opinion landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Our modelling framework contributes to better understanding the different roles and strategies in the increasingly complex information ecosystem and their impact on public opinion formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='13661v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='soc-ph] 31 Jan 2023 Introduction In the era of many-to-many communication on social media, polarization and radicalism can increasingly be understood as self-organized phenomena emerging from opinion dynam- ics on social networks [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Despite the global use of those large communication platforms and the resulting interactions, they have not been thoroughly studied as effects of a dynamical system consisting of multiple interacting components, partic- ularly distinguishing their different roles and goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' More specifically, social media did not replace traditional media, but became an intermediary entity that allows peer- to-peer communication among individuals but also traditional media outlets to disseminate their content and interact with their audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In principle, they are treated like individuals on those platforms, but have very different internal dynam- ics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', editorial processes, reputation and economic incen- tives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' As more and more people around the world access traditional news media through social platforms, this depen- dency becomes increasingly pronounced [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' At the same time a new role emerged in this system, namely, the influ- encer, who is a private person with many followers on a so- cial media network [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Although mostly following commercial goals, they do turn to politics [60] and can polarize public dis- cussions [51], as well as compete with traditional media and with one another for audience [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Potentially this increased competition could contribute to the declining trust in tradi- tional media and polarized debates on social media around the world [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We aim to capture the scenario of opinion dynamics under the impact of influencers and media with a generalized modeling framework designed to distinguish these different roles, but allow them to interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' So far in most cases, opinion dynamics are mathematically modelled by network- or agent-based models (ABMs) that aim to reproduce how individuals change their opinions based on the feedback of their peers in the respective social environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The largest group of such models describes the change of opinions based on the (dynamical) interaction network be- tween individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The well-known voter model [36] in which individuals copy the discrete opinion of a random neighbour may serve as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Alternative models use continuous opinion spaces where usually individuals’ opinions are drawn to attracting opinions in their social network (in some cases repelled by others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The DeGroot model [17] with individuals being drawn to the weighted average opinion of their neigh- bours and bounded confidence models [26, 57] where attract- ing interactions only take place with like-minded individuals may be the most prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' It has been demonstrated that these basic models and their generalizations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' [27, 50, 48], allow to describe the emergence of opinion clusters or commu- nities, including phenomena like radicalization, social bubbles and echo chambers [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Additionally, biased assimilation models [15, 59] have been developed based on the insight that linear weighted averag- ing of opinions cannot really lead to increasing polarization since people who hold strong opinions are likely to examine information in a biased manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Further in [5, 6, 9], multi- dimensional opinion dynamics models have been proposed where individuals are drawn to the neutral opinion while at the same time reinforcing each other to more extreme opin- ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Opinion interactions are saturated by a non-linear in- teraction function including tunable sensitivity to input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' It has been demonstrated that these models allow for describing real-world political polarization [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The development of these opinion dynamics models has been taken up in the literature with a focus on understand- ing consensus and community building, or cluster formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' For simple linear models like the DeGroot model this can be studied analytically [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' For nonlinear opinion models like bounded confidence models this has been studied numer- ically [26, 57, 12] or by considering the mean-field limit in terms of partial differential equations (PDEs), providing rig- orous theorems on cluster numbers and size, effective timescales and bifurcations based on changing interaction parameters [56, 21, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Moreover, it is understood how to extend the the- ory from agent-based models to mean-field limit models with stochastic partial differential equations (SPDEs) [28, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' On the background of this progress in opinion dynamics in social networks, this article concentrates on opinion clus- ters as coherent structures [20] meaning emerging groups of individuals whose opinions stay similar for rather long peri- ods of time before eventually disintegrating or merging with other clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The notion of coherence implies temporal sta- bility on timescales that are neither fast nor asymptotically long, indicating complex dynamical behavior with multiple characteristic timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We aim at understanding how, on these different timescales, traditional media and social media influencers may interact with individuals in order to adapt the overall opinion distribution according to their objectives, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', political agendas or economic interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' A few first steps towards opinion models under the impact of different types of agents have been taken: The influence of external sources on the opinion dynamics of individuals has been studied for example in [11, 42, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' These external sources are often regarded as media or charismatic leaders and have also been termed zealots or stubborn agents in the liter- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' More recently their influence has also been analysed in network-based opinion models [7, 10] but without explicitly taking coherence or multiple timescales into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In [47], the influence of a social group or clan at multiple timescales on the opinion formation on complex networks has been ana- lyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' However, so far those agents are mostly considered to have constant opinions and relations to their followers that do not change in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Here, we present an opinion dynamics model with a con- tinuous opinion space that generalizes most of the available models (DeGroot, bounded confidence, biased assimilation,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=') and complements individual opinion dynamics with tradi- tional media and social media influencers that follow political agendas and economic interests, and that adapt their opinions on different timescales than the individuals, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The question of what role the media and influencers play on the public opinion distribution became increasingly com- 2 (high inertia) opinion space (medium inertia) Media Individuals Influencers opinion space opinion space Figure 1: Structure of the agent-based model consisting of in- dividuals, media and influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Each of the agents adapts its opinion in the continuous opinion space by interacting with its network neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The network between individuals is given by A(t) while the relations between individuals and media resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' influencers are defined by B(t) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' C(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In the figure, the shape of an individual indicates the medium they read and their color indicates the influencer they fol- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Opinion changes happen on different time scales as de- termined by the inertia parameters where a higher inertia corresponds to slower opinion changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' plex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In the attention economy, the main goal of influencers and media is to receive more attention and increase their fol- lowership [53], that is, to find strategies that achieve this goal in an optimal way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' This makes them themselves dynamic en- tities that can adjust their positions or agendas in order to achieve those goals [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The effects these optimal strategies in such an incentive structure could have on the overall opin- ion dynamics have not yet been fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Recent attempts to find control strategies of opinion dy- namics consist of a variety of seemingly independent approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' On the one hand, different heuristics have been developed to determine which agents should be targeted by an external controller to maximize their influence [41, 43, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' On the other hand, in [2, 31], it is studied how external controllers have to act to bring the opinion distribution of agents close to a specified target distribution, while, in [34], it is established how to control opinion distributions by (externally) interfer- ing with the interaction network between individuals, and, in [1, 25], dynamic external controls for mean-field opinion models are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We will approach the design of optimal strategies in the following way: First, we show how to construct the mean- field model related to the proposed opinion dynamics model in the limit of large agent numbers and with finitely many in- fluencers and media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' This mean-field model takes the form of a system of partial differential equations for the individuals’ opinion distribution coupled to stochastic differential equa- tions for the opinions of media and influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Then, we will demonstrate how this mean-field model can be used to study the effect of strategies in the attention economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In particu- lar, we will show how to derive optimal control schemes for counteracting agendas to influence opinion distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The advantage of the mean-field model compared to the ABM is that the computational cost is independent of the number of agents, and that the model is deterministic (when assuming that influencers and media are not affected by stochasticity), which, together, means that it is considerably easier to find insight into collective behavior or optimal strategies via the mean-field model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Opinion model with influencers and media In the following we start by defining a general opinion model resulting from a large number of individuals adapting their opinions through interaction with each other as well as due to the influence of a few specific agents with particular roles, namely traditional media and social media influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' See Figure 1 for the model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We consider the situ- ation of the early formation of opinions, which is of great importance in the accelerating public discourse [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Hence, we focus on the transient model dynamics to study the forma- tion, splitting and merging of clusters, while the asymptotic regime is not of interest here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In this transient regime, indi- viduals adapt their opinions on a fast time scale, while media and influencer agents change their opinion positions on a sig- nificantly slower time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We consider individuals i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' , N with opinions xi(t) lying in a continuous d-dimensional opinion space D ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='1 The vector x(t) = (xi(t))N i=1 summarizes the opinions of all N individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In addition, we introduce M highly influential agents that can be considered as media outlets with contin- uous opinions denoted by the vector y(t) = (ym(t))M m=1 ∈ DM, as well as L highly influential influencers with opinions z(t) = (zl(t))L l=1 ∈ DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' To describe the real-world situation, we assume that there are much fewer influencers than individ- uals but still more influencers than media, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', M < L ≪ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' When political opinions are modelled, the opinion space D with d = 2 could span the dimensions economic left ↔ right and libertarian ↔ authoritarian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' When opinions with respect to acting against climate change are considered, two possible dimensions could be climate change believers ↔ deniers and technocentric ↔ ecocentric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Relations among individuals such as friendship or connec- tions on social media are defined through a binary adjacency matrix A(t) ∈ {0, 1}N×N that can depend on time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The resulting network determines which individuals can interact (when there is an edge between two individuals) and which cannot (when there is no edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' When the edges are addition- ally weighted, such that A(t) ∈ [0, 1]N×N, or directed, then the network can also describe the strength of a tie or the direc- tion of social influence as in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Media and influencer agents on the other hand are assumed to only interact with those individuals that are their followers (readers/users).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In the bi- nary adjacency matrix B(t) ∈ {0, 1}N×M of medium-follower 1We regard the opinion space D to be bounded with no flux boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 3 relations we store which individual follows which medium at time t while the binary adjacency matrix C(t) ∈ {0, 1}L×N defines the connections between individuals and influencers at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Matrices A, B and C represent the complex network struc- tures of social interactions and can be derived for example from online social media or survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' For the sake of sim- plicity, we will in our examples consider A and B to be con- stant on the chosen time scale and moreover an all-to-all con- nectivity between individuals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Aij = 1 for all (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Ad- ditionally, we assume that each individual follows exactly one medium and one influencer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', each row of B and of C con- tains exactly one entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Since individuals usually change in- fluencers very dynamically, we will propose an explicit model that defines how individuals are changing influencers in time and thus describes the evolution of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' This simplified set- ting allows us to better illustrate dynamical properties of the model and distinguish them from effects arising from the so- cial ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Our analytical results hold in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Individuals i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' , N adapt their opinions in time ac- cording to the following stochastic differential equation (SDE) dxi(t) = Fi(x, y, t)dt + σdWi(t) (1) where Fi defines the interaction force on individual i, σ > 0 gives the strength of noise and Wi(t) are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' d−dimensional Wiener processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The noise can, for example, model un- known external influences, uncertainties in the communica- tion between individuals or free will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The interaction force on individual i is given by a weighted sum of attractive forces from all other connected individuals j scaled by the param- eter a > 0 as well as attractive forces from the respective media scaled by the parameter b > 0 and from the respective influencer scaled by the parameter c > 0 Fi(x, y, z, t) = a � j′ wij′(t) N � j=1 wij(t)(xj(t) − xi(t)) + b � m Bim′(t) M � m=1 Bim(t) (ym(t) − xi(t)) + c � l′ Cil′(t) L � l=1 Cil(t)(zl(t) − xi(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' (2) The interaction weights between pairs of individuals i, j are given by wij(t) = Aij(t) φ (|xj(t) − xi(t)|) , (3) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', they depend on the network and on the opinion distance between the pair of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The weights are normalized to ensure a unitary total contribution from all individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' While the weight wij is certainly zero when there is no edge between individuals i and j, the weight depends on the opin- ion distance between i and j when there is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='2 Possible choices for the non-negative, pair function φ(|xj − xi|) are given by: 2When an individual has an interaction weight wij = 0 to all other other individuals, then the first term of the attraction force in (2) is assumed to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' φ(x) = exp(−x) as in [40] that places exponentially more weight on close-by individuals, φ(x) = 1[0,d](x) as in bounded confidence models [57, 26, 27] that only allows interactions with other individu- als that are within a radius d, here 1[0,d] is the indicator function on the set [0, d], φ(x) = 1 as in the DeGroot model [17] resulting in interactions irrespective of the opinion distance between individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The first two choices imply that individuals that are already close in opinion space excert higher social influence on each other (homophily), while the third choice results in a weight irrespective of the opinions of the interacting individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Remark (Opinion differentiation and higher-order interac- tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Apart from the described opinion attraction (assimi- lation), it is also possible to model opinion repulsion (differen- tiation) between individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In [39], it is suggested to classify pairs of individuals (i, j) either as friends that attract in their opinions or as enemies that repel in their opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' These pairs (i, j) will have a positive resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' negative edge weight Aij(t) and thereby can turn the first term of the force (2) from attraction to repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In [29], another approach that extends bounded confidence models is suggested: pairs of in- dividuals can not only become closer in their opinions when their opinion distance is smaller than the distance d, but they can also repel from each other when they are further apart than the distance D > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' By using the pair function φ(x) = 1[0,d](x) − 1[D,∞)(x), this can be incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' But when including differentiation, the weights wij(t) from (3) can become negative and hence can no longer be normalized, instead one can for example normalize the force by the num- ber of individuals, N, or by the number of neighbours of an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Additionally, it is possible to describe higher-order inter- actions among more than two individuals at the same time with this model and thereby to more accurately describe group effects such as peer pressure, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' [44, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Not only individuals change their opinions, but also media agents and influencers, however they adapt their opinions on a much slower time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The resistance to rapid change is determined by the inertia parameter Γ > 1 for media agents and by γ > 1 for influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' With γ < Γ media agents are changing their opinions on an even slower time scale than in- fluencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In the limit when the parameters Γ, γ diverge, the opinions of media and influencers become constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' There is a lot of research [11, 27, 8, 10] that studies interac- tions of individuals with constant agents (also termed stub- born agents or zealots) but to our knowledge adaptive media and influencers have not been studied so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In particular here media agents m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' , M slowly adapt their opinions according to the SDE Γdym(t) = f(˜xm(t) − ym(t))dt + ˜σ ˜ dW m(t), (4) 4 where the force function f can be used to model nonlinear influence effects but is set to f(x) = x subsequently, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', media agents are drawn in the direction of the average opinion of their followers ˜xm(t) = 1 � k Bkm(t) N � i=1 Bim(t)xi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In the SDE, ˜σ > 0 gives the strength of noise on the opin- ion dynamics and ˜Wm(t) denote i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' d−dimensional Wiener processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Similar to media, influencers l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' , L slowly change their opinions in the direction of their average follow- ership according to the SDE γdzl(t) = g(ˆxl(t) − zl(t))dt + ˆσ ˆ dW l(t), (5) where the average opinion of followers is given by ˆxl(t) = 1 � k Ckl(t) N � i=1 Cil(t) xi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The noise strength is given by ˆσ > 0, while ˆWl(t) denote i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' d−dimensional Wiener processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The function g can again be used to model nonlinear effects but is set to g(x) = x in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We have seen that influencers are similar to media agents but are usually more numerous and adapt their opinions on a faster time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' To reflect that relationships on social media are more dynamic than to traditional media outlets, we fur- ther propose an explicit model of how individuals can switch the influencer they are currently following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In particular, each individual i can at any time t switch its current influencer l′ to another influencer l with a given rate Λ→l m (x, t) where m is the medium of individual i and x is the opinion of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='3 The rate could for example take the following form Λ→l m (x, t) = η ψ(|zl − x|) r � nm,l(t) �M m′=1 nm′,l(t) � (6) with scaling parameter η > 0, pair function ψ, the link rec- ommendation function r and nm,l denoting the fraction of individuals that follow medium m and influencer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' By set- ting the pair function for example to ψ(x) = exp(−x), an individual has an exponentially higher rate to switch to an influencer that has a similar opinion than to an influencer with a very different opinion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', there is homophily between influencers and individuals when connections are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' On social media platforms, link recommendation algorithms are often used to suggest new connections to users that have the greatest potential to become established [35, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We incor- porate link recommendation via the function r by assuming that individuals have a higher chance of switching to an in- fluencer with a structurally similar followership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We measure the structural similarity of the followers of l to individual i 3Note that the change rate does not depend on the current influencer of individual i and that an individual can therefore also change to the influencer it is currently following without any effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' by the ratio of followers that are connected to the same in- fluencer (after switching) and medium as i, this proportion is given by nm,l � m′ nm′,l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We then assume that r is an increas- ing function on [0, 1], such as for example the ReLu function r(x) = max{0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='1, −1 + 2x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Example 1: ABM with media and influencers In Figure 2, we show snapshots of one realization of the ABM with 250 individuals, 2 media and 4 influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The pa- rameters are chosen to demonstrate how opinion clusters can form, split and merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Initially at t = 0, individuals are ran- domly distributed in opinion space and uniformly at random assigned to the 2 media that are initially at y1(0) = (−1, −1) and y2(0) = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Individuals in each of the 4 quadrants are assigned to a different influencer and the initial opinion of the influencer is set to the mean opinion of its followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' All individuals are interacting with each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', the network A is fully-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' When we let the model run, the strong attraction force to influencers (c = 4) results in individuals quickly being attracted by their respective influencer and forming 4 clus- ters (compare with t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' After some time, the 4 clus- ters split further because individuals are also attracted to their medium (compare t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='4), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' individuals now form roughly 8 groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Some individuals switch the influencer to a more suitable influencer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', one that is closer in opinion space and whose majority of followers are connected to the same medium as the individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' They then get attracted to the new influencer (t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='2), until finally (t = 2) individuals have formed 2 mixed clusters near the 2 media opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In Figure 3, we show the evolution of the proportion of in- dividuals that follow a certain influencer and medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Around t = 0, the proportions are roughly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' But after t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5, individuals start switching their influencer (the medium cannot be switched), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' quickly the followers of each influ- encer are dominated by individuals following the same medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Towards t = 2, the proportion of individuals that follow the medium and influencer in the upper right corner (indicated as green triangles) and the proportion of individuals that follow the medium and influencer in the lower left corner (shown by blue circles) dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Also, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 2 shows that at t = 2 most of the followers of the medium near (−1, −1) (denoted by circles) follow the same influencer that is also near (−1, −1) (colored in blue) while most followers of the medium near (1, 1) (denoted by trian- gles) also follow the influencer near (1, 1) (colored in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The reason for this dominance is that for individuals it is favorable to be close to their medium and their influencer, otherwise they might switch the influencer to a more suitable candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The two less suitable influencers in the upper left and lower right corner now very slowly move towards their few remaining followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' On a larger time scale they will reach the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Asymptotically, also the two clusters of individ- uals, media and influencers will approach another and merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' For different simulation runs, the agents behave qualitatively 5 2 1 0 1 2 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 2 1 0 1 2 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='1 2 1 0 1 2 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='4 2 1 0 1 2 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='2 2 1 0 1 2 2 1 0 1 2 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 Figure 2: Realization of the ABM with media and influencers (Example 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' There are 2 media agents marked in black, 4 influencers indicated by large circles in 4 colours, and 250 in- dividuals that inherit the shape of the medium they read and the color of the influencer they currently follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Parameters: a = 1, b = 2, c = 4, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5, ˜σ = 0, ˆσ = 0, Γ = 100, γ = 10, Aij = 1 for all (i, j), φ(x) = ψ(x) = exp(−x), η = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='00 Proportion of followers Figure 3: Stack plot showing how the proportion of individ- uals that follow a certain influencer and medium (marked by the symbols on the left) evolve in time in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' But the individuals following the influencer initially at (−1, 1) and (1, −1) are sometimes attracted to a different medium than in the shown simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The code for all examples is contained in the GitHub repository www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='com/LuzieH/SocialMediaModel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Rich dynamical behaviour The situation studied in Example 1 is somewhat symmet- ric and idealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' For other initial configurations and other choices of parameters and pair functions φ different forms of complex dynamical behavior emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' This ranges from sta- ble opinion clusters centered around ”their” influencers (for φ(x) = ψ(x) = 1[0,d](x) and no media) to a complex interplay of cluster formation and reformation for several influencers with smaller and larger inertia γ, and a less symmetric con- figuration as in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The dynamics can become even richer, if the interaction network A is already exhibiting clus- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Moreover, different initial conditions lead to different transient dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Partial mean-field (opinion) model For situations with many individuals but few influencers and media, one can derive the mean-field limit by a partial dif- ferential equation (PDE) that describes the opinion dynam- ics of individuals in the limit of infinitely many individu- als [56, 21, 24] but is usually already a good approximation to the dynamics for finitely many individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Since here the number of influencers and media is still small and finite, their dynamics are still best described by SDEs but now cou- pled to PDEs for the evolution of the opinion distributions of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We therefore also call the model a partial mean-field model, compare Figure 4 for the structure of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The coupled system of PDEs and SDEs is not only ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='computationally advantageous [28] since the computational ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='effort no longer scales with N 2 (due to the expensive com- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='putations of pair-wise distances of individuals in the ABM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='Media ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='opinion space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='opinion space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='(high inertia) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='(medium inertia) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='opinion space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='Influencers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='Individuals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='Figure 4: Structure of the partial mean-field model of dis- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='tinct media agents (indicated by black circles) and influencers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='(shown as colored circles) in interaction with the opinion dis- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='tributions of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The opinion distribution of individ- uals that follow a certain medium m and influencer l is given by ρm,l(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' but with the number of spatial grid cells and independently of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Additionally, the model is conceptually easier to study for example to find critical parameters [56, 21, 18] or to use the partial mean-field model to control the evolution of the opinion distribution through different influencer and media strategies (see next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' For the sake of simplicity, we subsequently assume a fully- connected interaction network between individuals, such that Aij = 1 for all pairs of individuals (i, j)4, and that each in- dividual follows exactly one medium and one influencer at a time and only influencers can be switched at the rate given in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Then let us define the empirical distribution of individuals that follow medium m and influencer l at time t by the sum of Dirac Delta distributions δ placed at the individuals’ opinions ρ(N) m,l (x, t) = 1 N � i:Bim=1, Cil(t)=1 δ(x − xi(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' (7) This distribution describes the stochastic opinion instances at a given time t and integrates to � D ρ(N) m,l (x, t)dx =: n(N) m,l (t), the proportion of individuals that follow medium m and in- fluencer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' It can be shown (see Supplementary Material) that as N → ∞, the empirical distribution ρ(N) m,l (x, t) can be replaced by the limiting distribution ρm,l(x, t) solving the following 4Without this assumption, one would need to derive mean-field dy- namics for interacting agents on realistic complex networks that have a network limit in terms of graphons [14] or graphops [32, 23], but here we will make the standard assumption of a fully-connected network between individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' PDE on the domain D ∂tρm,l(x, t) = 1 2σ2∆ρm,l(x, t) − ∇ · (ρm,l(x, t) F(x, ym, zl, ρ)) + � l′̸=l � −Λ→l′ m (x, t) ρm,l(x, t) + Λ→l m (x, t) ρm,l′(x, t) � (8) for each m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' , M and l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' , L, where ∇· denotes the divergence operator and ∆ the Laplace operator on opin- ion space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The PDE is accompanied by boundary conditions ensuring that the number of individuals in the system is con- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The changes of ρm,l are governed by three processes: (i) The first term on the RHS of the PDE is responsible for the stochastic diffusion of opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' (ii) The divergence term models the interaction of the distribution ρm,l with the over- all distribution of all individuals ρ = � m,l ρm,l as well as with the respective medium ym and the influencer zl according to the attraction force at opinion x F(x, ym, zl, ρ) =a � D ρ(x′, t)φ(|x′ − x|)(x′ − x) dx′ � D ρ(x′, t)φ(|x′ − x|) dx′ + b (ym(t) − x) + c (zl(t) − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' (9) (iii) The last term of the PDE is responsible for the mass exchange between different distributions due to individuals switching the influencer away from l′ or towards l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Note that due to the second and third term, the PDE is non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The PDEs are coupled to the SDEs of the media m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' , M Γdym(t) = (˜xm(t) − ym(t))dt + ˜σ ˜ dW m(t) (10) where the average opinion of followers is now given by ˜xm(t) = �L l=1 � D xρm,l(x, t)dx �L l=1 nm,l(t) with nm,l(t) denoting the limit of n(N) m,l (t), as well as to the SDE dynamics of influencers l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' , L γdzl(t) = (ˆxl(t) − zl(t))dt + ˆσ ˆ dW l(t), (11) with ˆxl(t) = �M m=1 � D xρm,l(x, t)dx �M m=1 nm,l(t) denoting the average opinion of individuals that follow influ- encer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Example 2: Comparison on ABM and partial mean-field dynamics In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 5, we show a comparison of ABM and partial mean- field simulations averaged over 1000 realizations, the parame- ters are as in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' To compare the ABM configurations against the opinion distributions in the partial mean-field, 7 2 1 0 1 2 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 ABM PDE 2 1 0 1 2 t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 2 1 0 1 2 t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 2 1 0 1 2 t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 2 1 0 1 2 2 1 0 1 2 t=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 2 1 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 Figure 5: Mean configuration of the ABM (left column) and partial mean-field PDE (right column) over 1000 simulations with influencers and media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Influencers are now marked by grey circles, and media agents by black circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The opin- ion distribution of all individuals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', independent of the media/influencer they follow) is shown with a heat plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Pa- rameters as in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' the opinion distributions resulting from the ABM are visu- alized via Kernel density estimation with Gaussian kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Even though the PDE is deterministic, we initialized it with random initial distributions given by the Kernel density esti- mation of 250 individuals placed uniformly at random in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Different realizations of these random initial condi- tions lead to different transient dynamics and we therefore also averaged the partial mean-field simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The com- parison shows that the results of the ABM and the partial mean-field are very consistent already for N = 250 individu- als in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Strategies in the partial mean-field model In the previous section we have stated the partial mean-field model as a reduced model describing the opinion dynamics of infinitely many individuals coupled to the opinion dynamics of finitely many media and influencers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Assuming deterministic opinion changes of influencers and media (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', ˆσ = ˜σ = 0), and a deterministic initial opinion distribution, the partial mean-field model is deterministic and faster to solve than the ABM, allowing us to study the effect of different (optimal) influencer and media strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 2 1 0 1 2 t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 m nm, 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 2 1 0 1 2 t = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 m nm, 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='069 2 1 0 1 2 t = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 m nm, 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='137 2 1 0 1 2 2 1 0 1 2 t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 m nm, 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 Figure 6: Opinion distribution ρ evolving in time (Example 3) started from a deterministic uniform opinion distribution at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' At time t = 5, a 5th influencer (marked by a triangle) is inserted into the system and moves at a constant pace in the direction of the right upper corner while collecting new followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Parameters as in Example 1, except η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 8 Increasing followership In the version of our opinion dynamics model as specified above, influencers adapt their opinions in the direction of the average opinion of their followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' This is a very simple strat- egy for influencers trying to keep their followers attached to them [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Other strategies to keep and even increase the number of followers might be more fruitful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In the follow- ing example we discuss a simple strategy for a new influencer with initially zero followers to substantially increase its fol- lowership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Example 3: Strategy of an influencer to in- crease followership We consider the dynamics as before in Example 1 except with η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' With this choice of parameters, influencers strongly affect and attract individuals (since c > a, b) and individuals only slowly change influencers (due to η being small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In this way, once individuals follow a certain influencer, they will remain with that influencer for some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' At time t = 5 we insert an additional influencer into the system with opinion (0, 0) and zero followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' This new influ- encer then moves at a constant speed to the opinion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5) during the time interval [5, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 6, we show snap- shots of the realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The inserted influencer l = 5 quickly collects and attracts many followers behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Starting with initially zero followers, the final followershare is 2 � m=1 nm,5(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Moreover the time-averaged proportion of followers of influ- encer l = 5 over the interval [5, 10] comes out to be approxi- mately 1 5 � t=10 t=5 2 � m=1 nm,5(t) dt ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Thus the added influencer could substantially increase its fol- lowership by moving to an extreme opinion position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Even a new cluster of followers has formed near the influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Optimal counteraction More generally, the partial mean-field model also allows to apply optimal control techniques in order to derive the best strategy for a single influencer or a medium to achieve a cer- tain objective, such as maximizing the number of their fol- lowers or optimally counteracting the goal of another agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Let ϱ(u) = (ρm,l(t))m=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=',M,l=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=',L, the matrix of all dif- ferent opinion distributions on the time interval [0, T], satisfy Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' (8), (10), and (11), except for the chosen controlled in- fluencer or medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' When we are interested in the strategy for an influencer agent l = l⋆, we let the control u determine the behaviour of the l⋆th influencer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', zl⋆ = u, when on the other hand we are searching for a media strategy, we control the m⋆th medium with ym⋆ = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We can then express the aim of influencer l⋆ to maximize its temporally aggregated followership with the objective function J ϱ = � T 0 M � m=1 nm,l⋆(t) dt, while maximizing solely the final followership means drop- ping the integral and taking the integrand at time t = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In contrast, if the goal of influencer l⋆ or medium m⋆ is to coun- teract the maximization of followership of another influencer l′, this can be achieved by maximizing the objective J ϱ = − � T 0 M � m=1 nm,l′(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' When the control determines the behavior of influencer l⋆ or medium m⋆, the control u needs to satisfy certain restric- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' That is, the control function u has to come from a set of admissible controls U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' On the one hand, U is restricted by the necessary domain constraints u(t) ∈ D for almost all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' On the other hand, with every control u certain costs are associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Control discontinuities or even chat- tering controls could, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', increase the risk of the control activities being detected, and countermeasures taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Such concerns can be included as a penalty term in the objective, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', by J p = α|∂tu|2 L2(]0,T [), where the parameter α regulates the penalty for large opinion changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Then, the optimal control problem has the general form max u∈U J ϱ − J p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' (12) After an appropriate control discretization with, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', piece- wise polynomials, the resulting nonlinear programming prob- lem can be solved by derivative-free methods like Nelder- Mead or by more efficient inexact gradient descent [46] or stochastic approximation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Sufficiently accurate gra- dient evaluations can be obtained by finite differencing the PDE/SDE forward equations for ϱ(u) or by Feynman-Kac type gradient sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Example 4: Influencer counteraction We are interested in the optimal counteraction of influencer l⋆ = 6 when influencer l′ = 5 moves at a constant speed to the upper right corner (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5) (as in Example 3) and thereby drastically increases its followership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We therefore want influencer l⋆ to move in an optimal way to satisfy max u∈U � − � 10 5 2 � m=1 nm,5(t) dt − α|∂tu|2 L2(]5,10[) � (13) with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' That is, the goal is to minimize the follow- ershare of influencer l′, while avoiding conspicuously drastic opinion changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The control set U contains all functions with u(t) ∈ D and with piecewise constant velocities that only change at 3 chosen time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 9 2 1 0 1 2 t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 m nm, 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 2 1 0 1 2 t = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 m nm, 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='041 2 1 0 1 2 t = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 m nm, 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='066 2 1 0 1 2 2 1 0 1 2 t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 m nm, 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 0 1 2 3 Figure 7: Opinion distribution ρ evolving in time (Example 4) started from a uniform distribution at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' At time t = 5, a 5th influencer (marked by a grey triangle) is inserted into the system and moves at a constant pace in the direction of the right upper corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Simultaneously, a 6th influencer (marked by a cross) starts and moves in a optimal way to prevent influencer l′ = 5 from collecting many followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Parameters as in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The optimal strategy is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 7 and results in an average followershare of influencer l′ of 1 5 � t=10 t=5 2 � m=1 nm,5(t) dt ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='05, which is significantly smaller compared to the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='1 that was obtained in Example 3 without counteraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Note also, that the final followershare at time t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 was decreased from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='18 without counteraction to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='04 with counteraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The snapshots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 7 show that the optimal counterstrategy of influencer l⋆ = 6 is to steel followers of influencer l′ = 5 by moving along;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' it thereby stabilizes the opinion cluster of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Note that according to the objective specified in (13), this counteraction comes out to be more effective than the movement in the opposite direction of influencer l′ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Example 5: Media counteraction Next, we study the optimal counterstrategy of the media agent m⋆ = 2 in the upper right corner when trying to mini- mize the followershare of influencer l′ = 5, analogously to the previous example with the objective function given by (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The resulting optimal strategy is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' By chang- ing the opinion drastically from the upper right to the lower right corner, the media agent manages to decrease the average followershare of influencer l′ = 5 down to 1 5 � t=10 t=5 � m nm,5(t) dt ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Note that since the system dynamics are symmetric wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' the diagonal axis x = y, an equivalently good strategy would be to move to the upper left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The counterstrategy of the media agent is very different to the counteraction of an influ- encer (given in Example 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' This is because influencers com- pete for followers and can steal influencers from each other while media agents can only make an opinion topic unattrac- tive to individuals by changing the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Conclusion In this work we provided new mathematical means for the sys- tematic study of how traditional media and influencers might impact coherent structures of the public opinion distribution as modelled by opinion dynamics models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' It does neither claim to describe opinion shifts in real-world social networks nor does it state anything about influencing the opinion of an individual human being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' It is still an idealized model and, like most contributions to the field, can neither claim to de- scribe human opinion formation processes realistically, nor to be validated with observational data and controlled soci- ological experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' However, it may help to describe how shifting individual perspectives and social exchange lead to archetypal states of public opinion distribution like coherent opinion clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' By providing a strategy for understanding how influencing the public opinion distribution may be done and counteracted in an optimal way, this work may help us to understand how to face current challenges of opinion po- larization in complex scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In particular, our model demonstrates the impact influ- encers and media might have on the opinion distribution by creating stable and coherent opinion clusters that are drawn to the positions of the respective influencer or media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We also see that when influencers move to extreme positions, 10 2 1 0 1 2 t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 m nm, 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 2 1 0 1 2 t = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 m nm, 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='042 2 1 0 1 2 t = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 m nm, 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='071 2 1 0 1 2 2 1 0 1 2 t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 m nm, 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='0 Figure 8: Opinion distribution ρ evolving in time (Example 5) started from a uniform distribution at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' At time t = 5, a 5th influencer (marked by a triangle) is inserted into the system and moves at a constant pace in the direction of the right upper corner, simultaneously, the media in the upper right corner (marked by a cross) moves in a optimal way to prevent influencer l′ = 5 from collecting many followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Parameters as in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' they fragment the opinion landscape and gain more follow- ers and more attention in the competitive scenario of so- cial media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Hence, these results suggest that those compet- itive goals could contribute to a polarized and fragmented debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' However, our work also offers avenues towards solu- tions through optimal counteracting those attempts via other influencers (or media).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' These counteracting influencers can stabilize the existing opinion landscape against extreme in- fluencers by strengthening existing opinion clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Empir- ical work needs to be done to validate those strategies and interplays that our model is suggesting, potentially through large-scale field experiments on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Our theoreti- cal work potentially delivers a good basis for developing such alternative strategies for influencers in the current system to stabilize online discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In this work we did not explore the full generality of the opinion dynamics model for different interaction terms, pa- rameters and different influencer and media strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In- stead we focused in this work on the feasibility by designing the pipeline from the inclusion of influencers and media via the demonstration of the emergence of temporarily coherent opinion clusters to the options for influencing the public opin- ion by optimally chosen strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The results provided do not include any analysis of the dynamical patterns such as numbers, birth and decay of opinion clusters, that the model may exhibit for different parameter combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' This was beyond the focus and scope of this study and will be left to future investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Acknowledgments We would like to thank Jobst Heitzig for his insights into the opinion model, Ana Djurdjevac, Ste- fanie Winkelmann and Alberto Montefusco for their com- ments on mean-field models, Alexander Sikorski for discus- sions on PDE discretizations and insights into code speed- ups, and Martin Weiser for valuable conversations on optimal control theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' This work has been partially funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy through grant EXC-2046 The Berlin Math- ematics Research Center MATH+ (project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 390685689).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' acknowledges financial support from the Volkswagen Foundation (grant ‘Reclaiming individual autonomy and demo- cratic discourse online: How to rebalance human and algorith- mic decision-making’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 11 Supplementary Material In the following, we will outline the derivation of the partial mean-field model for the limiting dynamics of infinitely many individuals, while keeping the number of media and influencers fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' PDE Derivation We start by deriving the mean-field PDEs for the opinion dynamics in the limit of infinitely many individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Related mean-field PDEs have already been derived in the context of other opinion models [55, 21, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Assuming constant influencer-follower connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' For the derivation, we will assume that the network between individuals is fully-connected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' that Aij = 1 for all i, j, and that each individual follows exactly one medium and one influencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Further in this first part, we will suppose that individuals do not change the influencer they are following, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' the matrix of influencer-follower relations remains constant C(t) ≡ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In a next step we will relax this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' For the purpose of this derivation, we add (N) to the model quantities when the number of individuals is still finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Since there are M media and L influencers in the system, we have M × L different types of individuals depending on which combination of a medium and an influencer a particular individual is following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Since the network between individuals is fully-connected, individuals that follow the same influencer and medium can be considered as identical (exchangeable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' For that reason, we can switch from a description in terms of labeled individuals to a description in terms of empirical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Let us define the empirical distribution of individuals that follow medium m and influencer l by ρ(N) m,l (x, t) := 1 N � i:B(N) im =1, C(N) il (t)=1 δ(x − x(N) i (t)) (14) and the proportion of these individuals compared to the total number of individuals in the system, N, by n(N) m,l (t) := � D ρ(N) m,l (x, t)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The proportion n(N) m,l (t) is constant in time for now, since influencer-follower relations cannot be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Further, we denote the total empirical distribution of individuals that follow any influencer or medium by ρ(N)(x, t) := 1 N N � i=1 δ(x − x(N) i (t)) = M � m=1 L � l=1 ρ(N) m,l (x, t) (15) with � D ρ(N)(x, t)dx = � m,l n(N) m,l (t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The opinion dynamics of individuals i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' , N are given by dx(N) i (t) = Fi(x(N), y(N), z(N))dt + σdWi(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' (16) With the definition of the empirical distribution in (15), we can rewrite the interaction force Fi by Fi(x(N), y(N), z(N)) = a � D ρ(N)(x, t)φ(|x − x(N) i (t)|)(x − x(N) i (t)) dx � D ρ(N)(x, t)φ(|x − x(N) i (t)|) dx + b (y(N) m (t) − x(N) i (t)) + c (z(N) l (t) − x(N) i (t)) whenever the agent i follows medium m and influencer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Thus the SDE in (16) depends no longer on the opinions of other individuals, but instead on the total empirical distribution ρ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We are now interested in the limit N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We assume that as N grows, the number of individuals that follow a certain medium m and a certain influencer l grows like Nn(N) m,l (t) with n(N) m,l (t) independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' It can then be expected [52, 13] that as N → ∞, any individual that follows a certain medium m and influencer l has an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' distributed opinion ¯xm,l that satisfies the SDE d¯xm,l(t) = F(¯xm,l, ym, zl, ρ)dt + σdW(t) (17) with interaction force F(¯xm,l, ym, zl, ρ) = a � D ρ(x, t)φ(|x − ¯xm,l|)(x − ¯xm,l) dx � D ρ(x, t)φ(|x − ¯xm,l|) dx + b (ym(t) − ¯xm,l) + c (zl(t) − ¯xm,l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 12 The opinion process ¯xm,l from (17) depends on the density ρ which can be interpreted as the limit of ρ(N) when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' More precisely, denoting by µm,l(x, t) the probability density of an individuals’ opinion with medium m and influencer l at time t, ρm,l(x, t) := nm,l(t)µm,l(x, t) denotes the probability density scaled by the proportion of these individuals in the system, and ρ(x, t) = � m,l ρm,l(x, t) denotes the probability density of any individual in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Stochastic processes of the form (17) are also called McKean-Vlasov processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Each scaled density ρm,l fulfils a McKean Vlasov-type PDE ∂tρm,l(x, t) = 1 2σ2∆ρm,l(x, t) − ∇ · (ρm,l(x, t) F(x, ym, zl, ρ)) (18) and we can also interpret each PDE for a fixed m, l as describing the density of infinitely many copies of individuals that follow medium m and influencer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' There is also a growing body of literature that considers an intermediate level of many but not infinitely many individuals, whose empirical distribution in the opinion space can approximately be described by a stochastic partial differential equation [16, 30, 28, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' Allowing influencer-follower connections to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We now additionally allow individuals to switch the influencer in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In the ABM each individual i can change the influencer to l at the rate Λ→l m (N)(x, t) where m is the medium of individual i and x its current opinion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In the PDE we want these changes from influencer l′ to l to correspond to mass flowing from ρm,l′ to ρm,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We will in the following derive the corresponding terms that have to be added to the PDE (18) and are sometimes also called reaction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The total number of individuals that follow medium m and influencer l at time t in the ABM is given by Y (N) m,l (t) := N n(N) m,l (t) and can be considered a jump process that only changes by +1 when an individual changes its influencer to l or by −1 when an individual changes its influencer away from l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The rate of any individual changing from (m, l′) to (m, l) is given by the sum of the individual change rates as follows αl′→l m (N)(t) = � i: B(N) im =1, C(N) il′ (t)=1 Λ→l m (N)(xi, t) = N � D Λ→l m (N)(x, t) ρ(N) m,l′(x, t)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' With this, we can write down the evolution of the jump process Y (N) m,l (t) as Y (N) m,l (t) = Y (N) m,l (0) + � l′̸=l � Pl′→l m �� t 0 αl′→l m (N)(t′)dt′ � − Pl→l′ m �� t 0 αl→l′ m (N)(t′)dt′ �� with unit-rate Poisson processes Pl→l′ m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' In the limit N → ∞, the rates become large and we can replace the Poisson processes by their mean to get the reaction rate equation [58] written in terms of the limiting proportions nm,l(t) dnm,l dt (t) = � l′̸=l � nm,l′(t)rl′→l m (t) − nm,l(t)rl→l′ m (t) � (19) with the spatially-averaged rates in the limit rl′→l m (t) = 1 nm,l′(t) � D Λ→l m (x, t) ρm,l′(x, t)dx and the limiting individual change rates Λ→l m (x, t) = η ψ(|zl − x|) r � nm,l(t) � m′ nm′,l(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The ODE (19) can be spatially extended to the following term [30, 28] ∂tρm,l(x, t) = � l′̸=l −Λ→l′ m (x, t) ρm,l(x, t) + Λ→l m (x, t) ρm,l′(x, t), (20) such that the complete PDE describing opinion changes and influencer changes is given by the sum of (18) and (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' 13 Limiting SDE dynamics of influencers and media The limiting dynamics of media and influencers follow by considering the limiting average position of followers of medium m ˜x(N) m (t) = 1 �N k=1 B(N) km N � i=1 B(N) im x(N) i (t) = �L l=1 � D x ρ(N) m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='l (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' t)dx �L l=1 n(N) m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='l (t) → �L l=1 � D x ρm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='l(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' t)dx �L l=1 nm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='l(t) =: ˜xm(t) and of influencer l ˆx(N) l (t) = 1 �N k=1 C(N) kl (t) N � i=1 C(N) il (t)x(N) i (t) = �M m=1 � D x ρ(N) m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='l (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' t)dx �M m=1 n(N) m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='l (t) → �M m=1 � D x ρm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='l(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' t)dx �M m=1 nm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='l(t) =: ˆxl(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' No-flux boundary conditions We are using boundary conditions such that the total number of individuals in the system remains constant in time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=', we want � D ∂tρ(x, t)dx = 0 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' We can equivalently phrase this using the PDE equation and the divergence theorem as 0 = � D ∂tρ(x, t)dx = � m,l � D 1 2σ2∆ρm,l(x, t) − ∇ · (ρm,l(x, t)F(x, ym, zl, ρ)) dx = � m,l � dD �1 2σ2∇ρm,l(x, t) − ρm,l(x, t)F(x, ym, zl, ρ) � n dx where dD is the boundary of D, n the unit outer normal to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' The terms for influencer changes disappeared since they only shift mass between the densities ρm,l and therefore disappear when summing over m, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFRT4oBgHgl3EQf2ji4/content/2301.13661v1.pdf'} +page_content=' A sufficient condition to ensure mass conservation is therefore to ensure that the balance equation 1 2σ2∇ρm,l(x, t) · n = ρm,l(x, t)F(x, ym, zl, ρ) · n holds everywhere on the boundary of D.' metadata={'source': 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+B.T.T.Wong∗ +Abstract +The Standard Model is the paradigm of particle physics which gives an ac- +curate theory for fundamental particle interactions. However, the extension of +Standard Model with higher-order derivative is not a well-studied subject. This +paper is a follow-up work of the previous study of the generalized Abelian gauge +field theory and Yang-Mills theory under rotor mechanism of order n of higher +order derivatives, and we apply to the Standard Model of particle physics. Ro- +tor mechanism on scalar field and Dirac field is also studied. We will study the +quantization of the rotored Standard Model using path integral approach. We +also inherit the previous result from the path integral quantization of generalized +abelian gauge field and apply it to our non-abelian case. Finally, we discuss the +possibility of rotor model on taming the infinities arise from the self-energy cor- +rection of the Higgs boson in high spacetime dimension, thus provide a partial +solution and new insight to the Hierarchy problem. +1 +Introduction +The Standard Model (SM) of particle physics is a well-established theory which de- +scribes particle interaction with great precision, with quantum field theory as an under- +lying mathematical foundation. Yet, higher-order derivative SM with a great potential +to tame UV divergence is not a well studied subject. Promising model includes a re- +cent study in Lee-Wick Standard Model which stabilizes quadratic divergence [1], of +which arises from the generalization of Lee-Wick electrodynamics [2, 3]. +The study of higher order derivative quantum field theory is particular interesting +because it can eliminate ultraviolet (UV) divergences in scattering amplitudes [4, 5, +6, 7, 8]. +There are studies on higher order derivative scalar field and gauge field +theories, and these theories have contributions in quantum gravity and modified gravity +[9, 10, 11, 12, 13, 14, 15, 16]. Higher order derivative theory also shows its appearance +in string theory [17, 18, 19, 20, 21]. In our context of study, Yang-Mills theory with +higher order derivative has been studied in references [22, 23, 24]. Quantization of +higher order derivative quantum field theory using path integral approach has been +studied in [25, 26, 27, 28, 29]. +In our previous work, we have established the generalized, higher-order-derivative +Yang-Mills theory by rotor mechanism [30], which follows upon our further previous +∗CERN, u3500478@connect.hku.hk +1 +arXiv:2301.12944v1 [physics.gen-ph] 10 Jan 2023 + +work in the abelian counterpart [31]. The path integral quantization of generalized +abelian gauge field theory under rotor model is conducted in our previous work [32]. +The Yang-Mills action in D dimensional spacetime is given by [33] +SYM = −1 +2 +� +dDxTr GµνGµν = −1 +4 +� +dDx Ga +µνGµν a +(1) +where +Gµν = ∂µTν − ∂νTµ − ig[Tµ, Tν] , +(2) +for which Gµν = Ga +µνta and Tµ = T a +µta are matrices with ta the generators of SU(N) +Lie group. Using the Lie algebra [ta, tb] = if abctc, this gives the gauge field strength +as, +Ga +µν = ∂µT a +ν − ∂νT a +µ + gf abcT b +µT c +ν . +(3) +We have D = 4n+4 for a renormalizable theory with unity gauge field dimension [32]. +By carrying our integration by parts on the kinetic term and expanding the remaining +terms, equation(1) explicitly is, +SYM = +� +dDx +� +T µa ˆRµνT νa − g +2f abc(∂µT νa − ∂νT µa)T b +µT c +ν − g2 +4 f abcf adeT b +µT c +νT µdT νe +� +, +(4) +where ˆRµν = +1 +2(□ηµν − ∂µ∂ν) is defined as the projection tensor. Under the rotor +mechanism we introduced in our previous work in [31], which is the successive action +of the projection tensors on the original gauge field, +Tn µn = ˆRµnµn−1 ˆRµn−1µn−2 · · · ˆRµ3µ2 ˆRµ2µ1 ˆRµ1µ0T µ0 = +1 +2n−1P µn−1 +µn +P µn−2 +µn−1 · · · P µ2 +µ3 P µ1 +µ2 ˆRµ1µ0T µ0 , +(5) +for which +P µj−1 +µj += □δ µj−1 +µj +(6) +is defined as the propagator. This is known as the rotor transformation which generates +high-order derivative gauge fields in the action [31, 32]. The n-th order Yang-Mills +action after rotor transformation of gauge field under Lorentz gauge is [32] +S(n) +YM = −1 +4 +� +dDx Ga +n µνGµν a +n += +� +dDx +� +T µa +n ˆRµνT νa +n − g +2f abc(∂µT νa +n − ∂νT µa +n )T b +µT c +n ν − g2 +4 f abcf adeT b +n µT c +n νT µd +n T νe +n +� += +� +dDx +� 1 +4n□nT µa ˆRµν□nT νa − +1 +2 · 8ngf abc(∂µ□nT νa − ∂ν□nT µa)□nT b +µ□nT c +ν +− +g2 +4 · 16nf abcf ade□nT b +µ□nT c +ν□nT µd□nT νe +� +(7) +Therefore, under the rotor mechanism, the gauge field transforms as [30], +Tµ → Tn µ = 1 +2n□nTµ . +(8) +And the gauge field strength becomes [30] +Ga +nµν = 1 +2n∂µ□nT a +ν − 1 +2n∂ν□nT a +µ + 1 +4ngf abc□nT b +µ□nT c +ν . +(9) +2 + +The n−th ordered covariant derivative is given by [30], +Dn µ = ∂µ − i +2ng□nT c +µtc , +(10) +The equation of motion of this generalized Yang-Mills theory is [30], +Dn µGµνa +n += ∂µGµνa +n ++ 1 +2ngf abc□nT b +µGµνc +n += 0 . +(11) +The Noether’s current is [30] +Jα = +� +− 1 +4n□n ˜Gαβ k − +1 +2 · 8ngf kbc(□nT αb□nT βc − □nT αc□nT βb) +� +δ□nT k +β , +(12) +and the associated Noether’s charge is given by [30] +Q = +� +dD−1xj0 = +� +dD−1x +� +− 1 +4n□n ˜G0β k− +1 +2 · 8ngf kbc(□nT 0b□nT βc−□nT 0c□nT βb) +� +δ□nT k +β , +(13) +where ˜Gαβ k is the Maxwellian gauge field strength. It can be seen that when n = 0 +(no rotation), we get back the original Yang-Mills theory. +Under the unitary transformation by SU(N) representation, the spinor transforms +as +ψ′(x) = U(x)ψ(x) , +(14) +where U(x) = eiαa(x)ta. The covariant derivative and field operator under rotor model +transform, respectively as, +� +D′ +n µ(x) += U(x)Dn µ(x)U †(x) +G′ +n µν(x) += U(x)Gn µν(x)U †(x) . +(15) +It follows that the rotor gauge field transforms as, +□nT ′ +µ = U□nTµU † + i · 2n +g +U∂µU † . +(16) +Infinitesimally, the rotored gauge field transforms as +□nT ′a +µ = □nT a +µ + 2n +g ∂µαa + f abc□nT b +µαc +(17) +Therefore the infinitesimal change in rotored gauge field is +δ□nT a +µ = □nT ′a +µ − □nT a +µ = 2n +g (Dn µα)a . +(18) +The n-th ordered gauge field strength can be defined through the commutator of the +n-th ordered covariant derivative, +Gn µν = i +g[Dn µ, Dn ν] . +(19) +And infinitesimally it transforms as +G′ +n µν = Gn µν − f abcαbGc +n µν . +(20) +3 + +It is noted that when n = 0, this gives us back all the properties of the transformation +rules of the original Yang-Mills theory. +We will proceed the quantization of the generalized Yang-Mills theory under rotor +mechanism with Feynman path integral approach. The quantum amplitude can be +computed as an integral of all possible field configurations over the exponential of the +action [34, 35, 36]. Similarly to our previous work for the abelian case in [32], as in +the generalized model it involves the transformation of field by T µ → □nT µ, therefore +in the path integral we sum over all possible configurations of □nT µ instead of T µ, i.e. +the integration measure changes as +� +DT µ(x) → +� +D□nT µ(x) . +(21) +2 +Generalized spin-0 scalar field theory under rotor +mechanism +In this section, we will complete the study of generalized scalar field theory under +rotor mechanism, this will generate scalar fields with higher-order derivatives. First +consider the massless scalar field theory in D-dimension +S = +� +dDx 1 +2∂µφ∂µφ = +� +dDx φ +� +− □ +2 +� +φ . +(22) +According to the definition of rotor mechanism in (5), we define the rotor mechanism +as the successive operations of the operator that couples to the gauge fields. For the +gauge field case, the operator is the projection tensor ˆRµν = 1 +2(□ηµν − ∂µ∂ν). For the +scalar field case, we have the rotor operator as ˆ˜R = − □ +2 . This scalar rotor operator +can be recovered from tracing the the projection tensor, up to some scaling factor, +ˆR = ηµν ˆRµν = ˆRµ +µ = D − 1 +2 +□ . +(23) +It follows that +ˆ˜R = − +1 +D − 1 +ˆR . +(24) +The rotor mechanism on scalar field is simply +φ → +n +� +j=1 +� +− +1 +D − 1 +� +ˆRµj +µjφ = (−1)n +2n +□ · · · □□φ = (−1)n +2n +□nφ . +(25) +The generalized massless scalar field theory is therefore +S = +� +dDx 1 +2∂µ +�(−1)n +2n +□nφ +� +∂µ +�(−1)n +2n +□nφ +� += +� +dDx 1 +2∂µ +� 1 +2n□nφ +� +∂µ +� 1 +2n□nφ +� += 1 +4n +� +dDx □nφ +� +− □ +2 +� +□nφ . +(26) +Notice that when n = 0, this restores back to the original case. Therefore, we see +that both the gauge field and scalar field transforms under the rotor mechanism by +the same form, +Tµ → Tn µ = 1 +2n□nTµ +, +φ → φn = 1 +2n□nφ . +(27) +4 + +For the massive scalar field theory, the generalized action under rotor mechanism is +S = +1 +2 · 4n +� +dDx ∂µ□nφ∂µ□nφ − m2□nφ□nφ = 1 +4n +� +dDx □nφ +� +− □ + m2 +2 +� +□nφ . +(28) +The Euler-Lagrangian equation is +∂µ +∂L +∂µ□nφ = +∂L +∂□nφ . +(29) +This gives the equation of motion as +□n+1φ + m2□nφ = 0 . +(30) +Now consider the complex-scalar field theory. First consider the action with two +scalar fields, +S = +1 +2 · 4n +� +dDx +� +i=1,2 +(∂µ□nφi∂µ□nφi − m2□nφi□nφi) += 1 +4n +� +dDx +� +i=1,2 +□nφi +� +− □ + m2 +2 +� +□nφi . +(31) +Now define a n-rotored complex scalar field, +□nΦ = 1 +√ +2(□nφ1 + i□nφ2) +and +□nΦ† = 1 +√ +2(□nφ1 − i□nφ2) . +(32) +Then the action in 31 can be written as +S = 1 +4n +� +dDx ∂µ□nΦ†∂µ□nΦ − m2□nΦ†□nΦ = 1 +4n +� +dDx □nΦ†(−□n − m2)□nΦ . +(33) +The two Euler-Lagrangian equations are +∂µ +∂L +∂µ□nΦ = +∂L +∂□nΦ +and +∂µ +∂L +∂µ□nΦ† = +∂L +∂□nΦ† . +(34) +This gives the two equations of motion as follow: +□n+1Φ† + m2□nΦ† = 0 +and +□n+1Φ + m2□nΦ = 0 . +(35) +Similar to the argument in our previous paper [31], the scalar field action under rotor +mechanism is renormalizable in D = 4n+4 dimension with unity scalar field dimension. +3 +Generalized spin-1/2 Dirac field theory theory +under rotor mechanism +Next, we will investigate how the Dirac field transforms under rotor mechanism. First +we consider the massless Dirac action in D-dimensional spacetime, +S = +� +dDx ¯ψiγµ∂µψ = +� +dDx ¯ψi/∂ψ ≡ +� +dDx ¯ψaiγµ +ab∂µψb . +(36) +5 + +So in analogy to the case of gauge field and scalar field, we have the rotor operator as +the matrix ˆRab = iγµ +ab∂µ. Hence under rotor transformation, the spinor field transforms +as +ψa0 → (in)γµn +anan−1γµn−1 +an−1an−2 · · · γµ1 +a1a0∂µn∂µn−1 · · · ∂µ1ψa0 . +(37) +Now let’s see how the adjoint spinor transforms. Consider , +γ0 +a0b0ψb0 → (in)γµn +anan−1γµn−1 +an−1an−2 · · · γµ1 +a1a0∂µn∂µn−1 · · · ∂µ1γ0 +a0b0ψb0 . +(38) +Then by taking the Hermitian conjugate +(γ0 +a0b0ψb0)† → (−i)n(∂µn∂µn−1 · · · ∂µ1ψ† +b0γ0† +b0a0)γµ1† +a0a1 · · · γµn−1† +an−2an−1㵆 +an−1an +(39) +As (γ0 +a0b0ψb0)† = ψ† +b0γ0† +b0a0 and because (γ0)† = γ0, therefore ¯ψa0 = ψ† +b0γ0 +b0a0. Hence the +adjoint spinor transforms as follow: +¯ψa0 → (−i)n(∂µn∂µn−1 · · · ∂µ1 ¯ψa0)γµ1† +a0a1 · · · γµn−1† +an−2an−1㵆 +an−1an +(40) +Therefore, the generalized Dirac field theory under rotor mechanism is +S = +� +dDx(∂νn∂νn−1 · · · ∂ν1 ¯ψ)γν1†γν2† · · · γνn† i/∂ γµnγµn−1 · · · γµ1∂µn−1∂µn−2 · · · ∂µ1ψ . +(41) +Now, let’s define short-hand tensor notation by +γαβγ··· ≡ γαγβγγ · · · +and +∇αβγ··· ≡ ∂α∂β∂γ · · · . +(42) +Then the action in 41 can be formally written as +S = +� +dDx(∇νn···ν1 ¯ψ) γ†ν1···νni/∂ γµn···µ1∇µn···µ1ψ . +(43) +The Dirac action with the mass term is given by +S = +� +dDx ¯ψiγµ∂µψ − m ¯ψψ . +(44) +Under rotor mechanism, the whole action transforms as +S = +� +dDx(∇νn···ν1 ¯ψ) γ†ν1···νni/∂ γµn···µ1∇µn···µ1ψ−m(∇νn···ν1 ¯ψ) γ†ν1···νn γµn···µ1∇µn···µ1ψ . +(45) +The Euler-Lagrangian equation is given by +∂µ +∂L +∂µ(∇νn···ν1 ¯ψ) γ†ν1···νn = +∂L +∂(∇νn···ν1 ¯ψ) γ†ν1···νn . +(46) +This gives the equation of motion as +i/∂ γµn···µ1∇µn···µ1ψ − mγµn···µ1∇µn···µ1ψ = 0 . +(47) +6 + +3.1 +Generalized Quantum electrodynamics under rotor mech- +anism +Now we proceed to develop the theory of higher-order derivative quantum electrody- +namics (QED) by rotor mechanism. From [31], the general Maxwell action under rotor +mechanism is +S = − 1 +4n+1 +� +dDx □nGµν□nGµν . +(48) +From the Dirac action under rotor mechanism, we expect the interactive term can be +achieved by replacing the ordinary partial derivative into covariant derivative Dn +S = +� +dDx(∇νn···ν1 ¯ψ) γ†ν1···νni /Dn γµn···µ1∇µn···µ1ψ , +(49) +where +Dn α = ∂α + ie +2n□nTα +(50) +under Lorentz gauge. Therefore, the full QED action with interaction under rotor +mechanism is given by +SQED = +� +dDx +� +− +1 +4n+1□nGµν□nGµν + (∇νn···ν1 ¯ψ) γ†ν1···νn(iγα∂α − m) γµn···µ1∇µn···µ1ψ +− e +2n(∇νn···ν1 ¯ψ) γ†ν1···νnγα□nTαγµn···µ1∇µn···µ1ψ +� +. +(51) +4 +Path integral quantization under rotor mecha- +nism +4.1 +Generalized Yang-Mills Theory +In this section, we will study the quantization of general Yang-Mills theory by path +integral approach in detail. From now on, we take the transformed □nT µ field as the +field variable. Simply speaking, the physics is changed by T µ → □nT µ. The quantum +amplitude of the □nT µ field in the renormalizable 4n + 4 dimension is +⟨□nT µ +f (tf,xxx)|e−i ˆH(tf−ti)|□nT µ +i (ti,xxx)⟩ = +� +D□nT µ(x) exp +� +iS(n) +YM[□nT µ] +� +, +(52) +where |□nT µ +i (ti,xxx)⟩ is the field state at initial time ti and |□nT µ +i (tf,xxx)⟩ is the field +state at final time tf, and ˆH is the Hamiltonian. The sourced generating functional is +a functional of 4-(covariant) transformed vector current □nJµ(x), +Z[□nJµ(x)] = +� +D□nT µ(x) exp +� +iS(n) +YM[□nT µ] + 2i +� +d4n+4x Tr +� +□nJµ(x)□nT µ(x) +�� +. +(53) +The normalized generating functional for gauge field is +Z[□nJµ] = Z[□nJµ] +Z[0] += +� +D□nT µ(x) exp +� +iS(n) +YM[□nT µ] + 2i +� +d4n+4xTr +� +□nJµ(x)□nT µ(x) +�� +� +D□nT µ(x) exp +� +iS(n) +YM[□nT µ] +� +. +(54) +7 + +The free n-point correlation function is given by +⟨0|T□n ˆT ν1(x1)□n ˆT ν2(x2) · · · □n ˆT νn(xn)|0⟩ = 1 +in +δn +δ□nJν1(x1)δ□nJν2(x2) · · · δ□nJνn(xn)Z[□nJµ] +���� +□nJµ=0 +, +(55) +where noting that each gauge field has different Lorentz indices and T means the time +ordering operator. Then the path integral representation is +⟨0|T□n ˆT ν1(x1) · · · □n ˆT νn(xn)|0⟩ += +� +D□nT µ(x) □nT ν1(x1) · · · □nT νn(xn) exp +� +iS(n) +YM[□nT µ] +� +� +D□nT µ(x) exp +� +iS(n) +YM[□nT µ] +� +. +(56) +Since the projection tensor in the action S(n) +YM is not invertible, we need to perform +gauge fixing. We will use the Fadeev-Popov method. Recall the gauge field transforms +as (16) +□nT ′ +µ = □nT U +µ = U□nTµU † + i · 2n +g +U∂µU † . +(57) +Now we need to choose a gauge fixing functional, +G(□nT µ) = ∂µ□nT µ(x) − w(x) , +(58) +where w(x) is some arbitrary matrix (w(x) = wa(x)ta ) . Now we use the identity, +1 = +� +DUδ[G(□nT µU)] det +�δG[□nT µU(x)] +δU(y) +� +(59) +Next we insert this into (4.1), +Z[□nJµ(x)] = +� +DUD□nT µ(x) δ[G(□nT Uµ)] det +�δG[□nT Uµ(x)] +δU(y) +� +× exp +� +iS(n) +YM[□nT µ] + 2i +� +d4n+4x Tr +� +□nJµ(x)□nT µ(x) +�� +. +(60) +But since the action is gauge invariant, S(n) +YM[□nT Uµ] = S(n) +YM[□nT µ], also the integration +measure remains unchanged, D□nT µU = D□nT µ, therefore we can write +Z[□nJµ(x)] = +� +DUD□nT Uµ(x) δ[G(□nT Uµ)] det +�δG[□nT Uµ(x)] +δU(y) +� +× exp +� +iS(n) +YM[□nT Uµ] + 2i +� +d4n+4x Tr +� +□nJµ(x)□nT µ(x) +�� +. +(61) +It is remarked that the source term is not gauge invariant unless Dµ□nTµ = 0. Now +we can relabel the transformed, rotored gauge field variable to □nT Uµ = □nT ′µ, +Z[□nJµ(x)] = +� +DUD□nT ′µ(x) δ[G(□nT ′µ)] det +�δG[□nT ′µ(x)] +δU(y) +� +× exp +� +iS(n) +YM[□nT ′µ] + 2i +� +d4n+4x Tr +� +□nJµ(x)□nT µ(x) +�� +. +(62) +8 + +Now we choose some normalization N(ξ) which is dependent on the gauge fixing +parameter ξ such that +1 = N(ξ) +� +Dw(x) exp +� +− i +� +d4n+4x Tr +�w2(x) +2ξ +�� +. +(63) +Then we relate the rotored field variable □nT ′µ(x) to □nT µ(x). And inserting equa- +tion(63) to (62), now we obtain, +Z[□nJµ(x)] = +� +N(ξ) +� +DU +� � +Dw(x)D□nT µ(x) δ[G(□nT µ)] det +�δG[□nT µ(x)] +δU(y) +� +× exp +� +iS(n) +YM[□nT µ] + 2i +� +d4n+4x Tr +� +□nJµ(x)□nT µ(x) +� +− i +� +d4n+4x Tr +�w2(x) +ξ +�� +, +(64) +where N(ξ) +� +DU is some constant and can be integrated out. Now employing the +gauge condition in 58, in which the Dirac delta function picks out the term of ∂µ□nT µ(x), +we obtain, +Z[□nJµ(x)] ∝ +� +D□nT µ(x) det +�δG[□nT µ(x)] +δU(y) +� +× exp +� +iS(n) +YM[□nT µ] + 2i +� +d4n+4x Tr +� +□nJµ(x)□nT µ(x) +� +− i +ξ +� +d4n+4x Tr +� +∂µ□nT µ(x)∂µ□nT µ(x) +�� +, +(65) +where the last term in equation (65) is the gauge fixing term. We define the gauge-fixed +action as +S(n) +ξ,YM = +� +d4n+4x +� +− 1 +2Tr Gn µνGµν +n − 1 +ξ Tr (∂µ□nT µ(x)∂µ□nT µ(x)) +� +. +(66) +Now we proceed to calculate the terms in the determinant. First notice that +δ□nT U +µ = −1 +g +� +∂µα + ig +� 1 +2n□nT U +µ , α +�� += −1 +gDn µα . +(67) +For G[□nTµ(x)] = g∂µ□nTµ(x), then +δG[□nTµ(x)] = −∂µDn µα . +(68) +Since iα = δU · U −1, we have +δG[□nT µ(x)] +δU(y) += iδ4n+4(x − y)∂µDn µ · U(y). +(69) +Since the determinant of U(y) is unity, it follows that +det +�δG[□nT µ(x)] +δU(y) +� += det +� +iδ4n+4(x − y)∂µDn µ +� +. +(70) +Now we can implement the Grassmann action as follow +det(i∂µDn µ) = +� +D□nc∗D□nc exp +� +− 2i +� +d4n+4x Tr(□nc∗∂µDn µ□nc) +� +, +(71) +9 + +where □nc∗, □nc are rotored transformed Grassmann fields. Therefore, the full sourced +partition functional now gives +Z[□nJµ(x)] ∝ +� +D□nT µ(x)D□nc∗(x)D□nc(x) +× exp +� +iS(n) +YM[□nT µ] + 2i +� +d4n+4x Tr +� +□nJµ(x)□nT µ(x) +� +− i +ξ +� +d4n+4x Tr +� +∂µ□nT µ(x)∂µ□nT µ(x) +�� +− 2i +� +d4n+4x Tr +� +□nc∗(x)∂µDn µ□nc(x) +�� +. +(72) +4.2 +Generalized Scalar field theory +The quantization of generalized scalar field theory follows similarly to that of the +abelian gauge field case in our previous paper [30], where we will not repeat the steps +here. Generally speaking, the quantum amplitude follows the form of +⟨□nφf(tf,xxx)|e−i ˆH(tf−ti)|□nφi(ti,xxx)⟩ = +� +D□nφ(x) exp +� i +4n +� +d4n+4x □nφ +� +−□ + m2 +2 +� +□nφ +� +, +(73) +The sourced partition functional is then given by +Z[□nJ(x)] = +� +D□nφ(x) exp +� i +4n +� +d4n+4x □nφ +� +−□ + m2 +2 +� +□nφ+i +� +d4n+4x □nJ□nφ +� +, +(74) +The normalized generating functional for the scalar field is +Z[□nJ] = Z[□nJ] +Z[0] += +� +D□nφ(x) exp +� +i +4n +� +d4n+4x □nφ +� +− □+m2 +2 +� +□nφ + i +� +d4n+4x □nJ□nφ +� +� +D□nφ(x) exp +� +i +4n +� +d4n+4x □nφ +� +− □+m2 +2 +� +□nφ +� +(75) +The free n-point correlation function is given by +⟨0|T□nφ(x1)□nφ(x2) · · · □nφ(xn)|0⟩ = 1 +in +δn +δ□nJ(x1)δ□nJ(x2) · · · δ□nJ(xn)Z[□nJ] +���� +□nJ=0 +, +(76) +which is evaluated to be +⟨0|T□nφ(x1)□nφ(x2) · · · □nφ(xn)|0⟩ += +� +D□n(x)□nφ(x1)□nφ(x2) · · · □nφ(xn) exp +� +i +4n +� +d4n+4x □nφ +� +− □+m2 +2 +� +□nφ +� +� +D□nφ(x) exp +� +i +4n +� +d4n+4x □nφ +� +− □+m2 +2 +� +□nφ +� +. +(77) +Using the same technique in [32] similar to the case of gauge field, one finds the +Feynman propagator as +⟨0|T□nφ(x)□nφ(y)|0⟩ = +� +d4n+4p +(2π)4n+4 +i · 4n +p4n(p2 − m2)e−ip·(x−y) . +(78) +10 + +Or in the most generalized case, for the case of inhomogeneous order, we have +⟨0|T□nφ(x)□kφ(y)|0⟩ = +� +d2n+2k+4p +(2π)2n+2k+4 +i · 2n+k +p2n+2k(p2 − m2)e−ip·(x−y) . +(79) +From 78, when n = 0 which is the unrotored case this restores us back to the original +two point correlation and the Feynman propagator of the scalar fields, +⟨0|Tφ(x)φ(y)|0⟩ = +� +d4p +(2π)4 +i +(p2 − m2)e−ip·(x−y) . +(80) +4.3 +Generalized Dirac field and Electrodynamics +For the quantization of Dirac field under rotor mechanism, first we consider the quan- +tum amplitude as follow: +⟨γµn···µ1∇µn···µ1ψ(tf,xxx)|e−i ˆH(tf−ti)|∇νn···ν1 ¯ψ(ti,xxx) γ†ν1···νn⟩ += +� +D[(∇νn···ν1 ¯ψ) γ†ν1···νn]D[γµn···µ1∇µn···µ1ψ] +× exp +� +i +� +dDx (∇νn···ν1 ¯ψ) γ†ν1···νn(i/∂ − m) γµn···µ1∇µn···µ1ψ +� +. +(81) +The sourced partition functional is then given by +Z[∇βn···β1 ¯J(x) γ†β1···βn , γαn···α1∇αn···α1J(x)] += +� +D[(∇νn···ν1 ¯ψ) γ†ν1···νn]D[γµn···µ1∇µn···µ1ψ] +× exp +� +i +� +dDx (∇νn···ν1 ¯ψ) γ†ν1···νn(i/∂ − m) γµn···µ1∇µn···µ1ψ ++ i +� +dDx(∇νn···ν1 ¯ψ) γ†ν1···νnγαn···α1∇αn···α1J(x) ++ i +� +dDx∇βn···β1 ¯J(x) γ†β1···βnγµn···µ1∇µn···µ1ψ +� +(82) +And it is remarked that +iS[(∇νn···ν1 ¯ψ) γ†ν1···νn, γµn···µ1∇µn···µ1ψ] += +�� +dDxdDy(∇νn···ν1 ¯ψ(x)) γ†ν1···νniδD(x − y)(i/∂ − m)γµn···µ1∇µn···µ1ψ(y) . +(83) +Now perform fourier transform, +ψ(x) = +� +dDp +(2π)D ˜ψ(p)e−ip ˙x +and +¯ψ(x) = +� +dDp +(2π)D +¯˜ψ(p)e+ip ˙x . +(84) +11 + +Equation 83 yields +iS[(∇νn···ν1 ¯ψ) γ†ν1···νn, γµn···µ1∇µn···µ1ψ] += +�� +dDxdDy +�� +dDp +(2π)D +dDq +(2π)D (in)(−i)npνn · · · pν1 ¯˜ψ(p)γ†ν1···νniδD(x − y)(/q − m)γµn···µ1 +× qµn · · · qµ1 ˜ψ(q)e−ip·x+iq·y += +� +dDx +�� +dDp +(2π)D +dDq +(2π)D pνn · · · pν1 ¯˜ψ(p)γν1† · · · γνn†i(/q − m)γµn · · · γµ1qµn · · · qµ1 ˜ψ(q)e−i(p−q)·x += +�� +dDp +(2π)D +dDq +(2π)D (/p†)n ¯˜ψ(p)i(2π)DδD(p − q)(/q − m)/qn ˜ψ(q) += +� +dDp +(2π)D +¯˜ψ(p)(/p†)ni(/p − m)(/p)n ˜ψ(p) . +(85) +Therefore, we have the matrix element as +˜ +M(p, q) = −i(2π)Dδ(p − q)(/p†)n(/q − m)(/q)n +(86) +Therefore, the momentum green’s function for Dirac field under rotor mechanism is +˜ +M −1(p, q) = +i +(/p†)n(/q − m)(/q)n(2π)Dδ(p − q) . +(87) +Hence, the Feynman propagator in momentum space is +S(n) +F (p) = +i +(/p†)n(/p − m)(/p)n . +(88) +Now, let’s evaluate the sourceless partition functional, which is +Z[0, 0] = +� +D[(∇νn···ν1 ¯ψ) γ†ν1···νn]D[γµn···µ1∇µn···µ1ψ] +× exp +� +i +� +dDx (∇νn···ν1 ¯ψ) γ†ν1···νn(i/∂ − m) γµn···µ1∇µn···µ1ψ +� +, +(89) +which is +Z[0, 0] = +� +D[(∇νn···ν1 ¯ψ) γ†ν1···νn]D[γµn···µ1∇µn···µ1ψ] +× exp +� +i +�� +dDxdDy (∇νn···ν1 ¯ψ(x)) γ†ν1···νnδD(x − y)(i/∂ − m) γµn···µ1∇µn···µ1ψ(y) +� +, +(90) +They by using the identity from Gaussian integral, +� � +n +� +j=1 +dθ∗ +jdθj +� +exp(−θ∗TMθ) = det(M) , +(91) +we obtain +Z[0, 0] = det[δD(x − y)(i/∂ − m)] +(92) +12 + +Now, using the usual fact from Gaussian integral with sourced term, +� � +n +� +j=1 +dθ∗ +jdθj +� +exp(−θ∗TMθ + η∗Tθ + θ∗Tη) = det(M) exp(η∗TM −1η) , +(93) +and with equations (82) and (83), the sourced partition functional for the Dirac field +is evaluated to be +Z[∇βn···β1 ¯J(x) γ†β1···βn , γαn···α1∇αn···α1J(x)] += det[δD(x − y)(i/∂ − m)] exp +� �� +dDxdDy∇βn···β1 ¯J(x) γ†β1···βnM −1(x, y)γαn···α1∇αn···α1J(y) +� += Z[0, 0] exp +� �� +dDxdDy∇βn···β1 ¯J(x) γ†β1···βnM −1(x, y)γαn···α1∇αn···α1J(y) +� +(94) +The normalized generating functional for the Dirac field is +Z[∇βn···β1 ¯J(x) γ†β1···βn , γαn···α1∇αn···α1J(x)] = Z[∇βn···β1 ¯J(x) γ†β1···βn , γαn···α1∇αn···α1J(x)] +Z[0, 0] +. +(95) +The two point correlation function is given by +⟨0|T[γµn···µ1∇µn···µ1ψ(x1)][(∇νn···ν1 ¯ψ(x2)) γ†ν1···νn]|0⟩ += 1 +i2 +δ2Z[∇βn···β1 ¯J(x) γ†β1···βn , γαn···α1∇αn···α1J(x)] +δ[γµn···µ1∇µn···µ1J(x1)]δ[(∇νn···ν1 ¯J(x2)) γ†ν1···νn] +���� +γµn···µ1∇µn···µ1J(x1)=(∇νn···ν1 ¯J(x2)) γ†ν1···νn=0 +. +(96) +The variational principle follows that +δγαn···α1∇αn···α1J(x) +δγµn···µ1∇µn···µ1J(y) = δD(x − y) +(97) +and +δ∇βn···β1 ¯ψ(x) γ†β1···βn +δ∇νn···ν1 ¯ψ(y) γ†ν1···νn = δD(x − y) . +(98) +This would give +⟨0|T[γµn···µ1∇µn···µ1ψ(x1)][(∇νn···ν1 ¯ψ(x2)) γ†ν1···νn]|0⟩ = M −1(x, y) += +� +dDp +(2π)D +i +(/p†)n(/p − m)(/p)ne−ip·(x−y) . +(99) +For the inhomogeneous case, we will have, +⟨0|T[γµn···µ1∇µn···µ1ψ(x1)][(∇νk···ν1 ¯ψ(x2)) γ†ν1···νk]|0⟩ = +� +dDp +(2π)D +i +(/p†)n(/p − m)(/p)k e−ip·(x−y) . +(100) +From equation (99), when n = 0, this restores us back the fermion two-point correlation +function and Feynman propagator +⟨0|Tψ(x) ¯ψ(y)|0⟩ = +� +dDp +(2π)D +i +/p − me−ip·(x−y) +(101) +13 + +To quantize the generalized electrodynamics under rotor mechanism, we first con- +sider the sourceless functional as follow: +Z[0, 0, 0] = +� +D[(∇νn···ν1 ¯ψ) γ†ν1···νn]D[γµn···µ1∇µn···µ1ψ]D□nT µ exp +� +iSQED +� +, +(102) +where SQED is the quantum electrodynamic action in 51. The normalized generating +functional for QED is +ZQED[∇βn···β1 ¯J(x) γ†β1···βn , γαn···α1∇αn···α1J(x), □nJµ] += +exp (i +� +dDzLint[source])Z[∇βn···β1 ¯J(x) γ†β1···βn , γαn···α1∇αn···α1J(x), □nJµ] +{exp (i +� +dDzLint[source])Z[∇βn···β1 ¯J(x) γ†β1···βn , γαn···α1∇αn···α1J(x), □nJµ]}|sourced terms=0 +, +(103) +where +exp +� +i +� +d4z .Lint[source] +� += exp +� +i +� +dDz +1 +iδ∇βn···β1 ¯J(z) γ†β1···βn +1 +iδγαn···α1∇αn···α1J(y) +1 +iδ□nJµ(z) +� . +(104) +Then the full physical two-point correlation function is +⟨Ω|Tψ(x) ¯ψ(y)|Ω⟩ = 1 +i2 +δ2ZQED +δ∇βn···β1 ¯J(x) γ†β1···βnδγαn···α1∇αn···α1J(y) +���� +source terms=0 +(105) +with |Ω⟩ the physical vacuum. +4.4 +A summary of Feynman propagators under rotor mecha- +nism +Using the path integral quantization technique approach, we can obtain the Feynman +propagators under rotor mechanism ( n-rotors of □n operators) as follow: +Scalar spin-0 boson: +∆(n) +F (x − y) = +� +dDp +(2π)D +i · 4n +p4n(p2 − m2)e−ip·(x−y) . +(106) +Massless spin-1 gauge boson: +D(n) +µν (x − y) = +� +dDp +(2π)D +−i · 4ngµν +p2+4n +e−ip·(x−y) . +(107) +Massless spin-1 gauge boson in Lorentz gauge: +D(n) +µν (x − y) = +� +dDp +(2π)D +−i · 4n +p2+4n +� +gµν − (1 − ξ)pµpν +p2 +� +e−ip·(x−y) . +(108) +Massive spin-1 gauge boson: +D(n) +µν (x − y) = +� +dDp +(2π)D +−i · 4n +p4n(p2 − M 2) +� +gµν − pµpν +M 2 +� +e−ip·(x−y) . +(109) +14 + +Massive spin-1 gauge boson in Lorentz gauge +D(n) +µν (x−y) = +� +dDp +(2π)D +−i · 4n +p4n +� +gµν +p2 − M 2− +1 +p2 − M 2 +� +1−1 +ξ +�� +1 +M 2 − p2 +ξ +� +pµpν +� +e−ip·(x−y) . +(110) +Dirac spin-1/2 fermion: +SF(x − y) = +� +dDp +(2π)D +i +(/p†)n(/p − m)(/p)ne−ip·(x−y) . +(111) +Massless spin-1 gluon: +Dab (n) +µν +(x − y) = δabD(n) +µν (x − y) = +� +dDp +(2π)D +−iδab · 4n +p2+4n +� +gµν − (1 − ξ)pµpν +p2 +� +e−ip·(x−y) . +(112) +Ghost field: +Gab (n)(x − y) = +� +dDp +(2π)D +iδab +p2+4ne−ip·(x−y) . +(113) +5 +Generalized Higgs mechanism under rotor mech- +anism +The Higgs mechanism is responsible for the explanation of mass acquisition of gauge +boson through the process of spontaneous symmetry breaking, it also explains how the +fermions couple with the Higgs field to main mass [40, 41, 42, 43, 44, 45]. In addition, +it also predicts all possible Higgs interactions and decays. +Under rotor mechanism, the potential term is +V = µ2 +4n □nφ†□nφ + λ +16n(□nφ†□nφ)2 , +(114) +where µ2 < 0 and λ > 0 for spontaneous symmetry breaking. The Higgs action is +S = +� +dDx +� +− 1 +4n+1□nGµν□nGµν+ 1 +4n(Dn µ□nφ)†(Dn µ□nφ)−µ2 +4n □nφ†□nφ− λ +16n(□nφ†□nφ)2 +� +, +(115) +This Lagrangian is invariant under U(1) transformation, +(□nφ) → (□nφ)′ = eiθ(□nφ) +(116) +When expressed in terms of two real rotored scalar fields □nφ1 and □nφ2, action reads +V (□nφ1, □nφ2) = +µ2 +2 · 4n +� +(□nφ1)2 + (□nφ2)2� ++ +λ +4 · 16n +� +(□nφ1)2 + (□nφ2)2�2 . (117) +The potential has an infinite minimum when +(□nφ1)2 + (□nφ2)2 = −4n · µ2 +λ += v2 +n . +(118) +15 + +Upon the elimination of Goldstone boson by Unitary gauge and by perturbation of +the non-zero vacuum expectation value, we have +□nφ(x) = 1 +√ +2 +� +vn + □nh(x) +� +(119) +Then after spontaneously symmetry breaking, we have +S = +� +dDx +� +− +1 +4n+1□nGµν□nGµν ++ +1 +2 · 4n +� +∂µ − i +2ng□nTµ(vn + □nh) +�� +∂µ + i +2ng□nTµ(vn + □nh) +� +− +µ2 +2 · 4n(vn + □nh)2 − +λ +4 · 16n(vn + □nh)4 +� += +� +dDx +� +− +1 +4n+1□nGµν□nGµν + +1 +2 · 4ng2v2 +n□nTµ□nT µ ++ +1 +2 · 4n∂µ□nh∂µ□nh − 1 +4nλv2 +n□nh□nh ++ 1 +8ng2vn□nTµ□nT µ□nh + +1 +2 · 8ng2□nTµ□nT µ□nh□nh +− 1 +8nλvn□nh□nh□nh − +1 +4 · 16nλ□nh□nh□nh□nh +(120) +The mass term of the rotored massive gauge boson is identified as Mn = gvn, and the +mass term of the rotored Higgs boson is mn = +√ +2λvn. It is noted that when n = 0, the +action (120) will restore back to the normal Higgs action with spontaneous symmetry +breaking. From the third line in (120), this gives the interactions between the rotored +Higgs boson and rotored gauge field. We can rewrite them in terms of the metric, +1 +8ngMngµν□nT µ□nT ν□nh , +1 +2 · 8ng2gµν□nT µ□nT ν□nh□nh . +(121) +In terms of vertex diagram, we have the following feynman rules, +Figure 1: Feyman rules for rotored Higgs boson decaying into two rotored bosons (left) and +rotored scattering process (right). +From the fourth line in 120, this gives the rotored Higgs boson self interactions. +The Feynman rules are as follow: +16 + +Figure 2: Feyman rules for third order and fourth order rotored Higgs boson interactions. +6 +A discussion of Hierarchy Problem and the par- +tial solution provided by the rotor mechanism +It has been known that high-order derivative quantum field theory makes a good job +in eliminating UV divergence [4, 5, 6, 7, 8]. In this section, we will investigate how the +high-order-derivative quantum field theory by the rotor mechanism can suppress UV +divergence in the quantum loop processes. Although the standard model is renormal- +izable, i.e. the divergences in the Lagrangian can be treated by adding finite number +of counter terms, the Hierarchy Problem which involves fine tunning of the Higgs mass +remains a long-lasting problem in high-energy particle physics. Recently, reference [1] +generalizes Lee-Wick electrodynamics with high-order field derivatives to the Standard +Model and offers a solution to the Hierarchy Problem to tame UV divergences in one- +loop level. Here, we will see how high-order derivative field theory by rotor mechanism +can eliminate the UV divergence in amplitudes. +Let us have a revisit to the Hierarchy problem. For example, taking the 4-fields +self-interaction of the Higgs boson the one-particle irreducible (1 PI) function can be +easily calculated by integrating through the loop momentum k, +˜Γ(p) = −iλ2 +8 +� +d4k +(2π)4 +i +k2 − m2 +H += −iλ2 +128π2 +� +Λ2 − 2m2 +H ln +� Λ +mH +� ++ O +�m2 +H +Λ4 +� � ++ O(λ4), +(122) +which has a dominant quadratic divergence of high energy scale Λ. Hence, the renor- +malization of the bare Higgs boson’s mass for the 4-field self interaction is, +˜m2 +H = m2 +H + i˜Γ(p) = m2 +H + +λ2 +128π2 +� +Λ2 − 2m2 +H ln +� Λ +mH +� ++ O +�m2 +H +Λ2 +� � ++ O(λ4). (123) +It is noted that mH = µ above is the bare mass, while ˜mH is the physical, observed +mass-the renormalized mass. The full calculation of all quantum loop corrections from +electroweak and Yukawa interaction to the nth-order loop takes the general form [46], +˜m2 +H = m2 +H + Λ2 +16π2Cn(µ2 = m2 +H) , +(124) +where Cn is the a polynomial expansion of the bare Higgs mass scale and it is a function +of the Higgs self-coupling λ, electroweak couplings gW, g′ and the Yukawa coupling Y f +ij. +To the first-order loop calculation [46], +˜m2 +H = m2 +H + Λ2 +32π2(4λ2 + 3g′2 + 9g2 +W − 24Y f 2 +ij ) + O(λ2, g′4, g4 +W, Y f 4 +ij ) . +(125) +17 + +The physical Higgs boson mass is measured to be ˜mH ≃ 126 GeV and hence ˜m2 +H ∼ +104, GeV2, while the quantum corrections lead to the quadratic divergence of the +Planck scale Λ ∼ 1019 GeV. From equation (125) we can see that the physical Higgs +mass depends strictly on the strength of each of the coupling. The nature requires an +extremely precise fine-tuning cancellation of the couplings and the bare Higgs mass to +each nth order of quantum loop divergence by 1017 GeV, so as to obtain the low-energy, +EW-scale Standard Model we observe today [47]. This is known as the Hierarchy Prob- +lem. +The Hierarchy Problem gives a strong motivation to search for New Physics (NP) +particles with mass scales ΛNP > ΛEW such that they can cancel these loop divergences. +One important scheme is supersymmetry, which introduces new particles between +bosonic and fermonic symmetry, such that each divergent bosonic loop has a fermionic +counterpart, vice versa. Since fermonic amplitude has an extra factor of minus sign, +this can cancels the bosonic loop. +Using the rotor mechanism, we will see that the divergent loop processes are sup- +pressed and this will lead to convergent result. The generic problematic 1-loop di- +vergent process includes, for example, the self-correction of the Higgs boson by the +W-boson loop. The Feynman diagram is illustrated below. +Figure 3: Feyman diagram of self-energy correction of the Higgs boson by a W-boson loop. +The n-rotored amplitude is given by +M(n) = g2 +W(16)nM 2 +W +64n(2π)D +� +dDk gαβgδγ gβδ − kβkδ/M 2 +W +k4n(k2 − M 2 +W) +gγα − (p − k)γ(p − k)α)/M 2 +W +(p − k)4n((p − k)2 − M 2 +W) +. +(126) +By expanding, we obtain +M(n) = g2 +WM 2 +W +4n(2π)D +� � +dDk +gαβgβδgδγgγα +k4n(p − k)4n(k2 − M 2 +W)((p − k)2 − M 2 +W) +− +1 +M 2 +W +� +dDk +(p − k)α(p − k)α +k4n(p − k)4n(k2 − M 2 +W)((p − k)2 − M 2 +W) +− +1 +M 2 +W +� +dDk +kαkα +k4n(p − k)4n(k2 − M 2 +W)((p − k)2 − M 2 +W) ++ +1 +M 4 +W +� +dDk +kδ(p − k)δkα(p − k)α +k4n(p − k)4n(k2 − M 2 +W)((p − k)2 − M 2 +W) +� +. +(127) +First, we can give a rough estimate of the amplitude by the leading term, +M(n) +1 +∼ g2 +WM 2 +W +4n(2π)D +� +dDk +gαβgβδgδγgγα +k4n(p − k)4n(k2 − M 2 +W)((p − k)2 − M 2 +W) +∼ 21−DDg2 +WM 2 +W +4n√ +πDΓ +� D +2 +� +� Λ kD−1dk +k8n+4 , +(128) +18 + +..where have have used the fact that of the integration of D-dimensional solid angle is +given by +� +dΩD = +2πD/2 +Γ(D/2) with Γ(x) the Gamma function. When n = 0 and in four +spacetime dimension D = 4 which is the unrotored case in 4D spacetime (which is just +the normal case), we have logarithmic divergence +M(0) +1 +∼ g2 +WM 2 +W +2π2 +ln Λ . +(129) +Hence in the normal case, the self-correction of the Higgs boson by the W boson is +logarithmic divergent. But under rotor mechanism, by the second line of 136. we have +M(n) +1 +∼ +21−DDg2 +WM 2 +W +4n√ +πDΓ +� D +2 +� +(D − 8n − 4) +ΛD−8n−4 , +(130) +In our 4D universe, we simply have +M(n) +1 +∼ − g2 +WM 2 +W +22n+4π2n +1 +Λ8n +(131) +As Λ is a very large value, which can be up to Planck scale 1019 GeV, even for n = 1 +M(1) +1 +→ 0. For the second term, +M(n) +2 += +g2 +W +4n(2π)D +� +dDk +1 +k4n(p − k)4n−2(k2 − M 2 +W)((p − k)2 − M 2 +W) +∼ +21−Dg2 +W +4n√ +πDΓ +� D +2 +� +� Λ kD−1dk +k8n+2 += +21−Dg2 +W +4n√ +πDΓ +� D +2 +� +(D − 8n − 2) +ΛD−8n−2 . +(132) +For the unrotored case n = 0 and in D = 4, +M(0) +2 +∼ g2 +W +16π2Λ2 , +(133) +which contributes to a quadratic divergence. But under rotor mechanism, +M(n) +2 +∼ +g2 +W +2(2n+4)π2(1 − 4n)Λ2−8n . +(134) +For example for n = 1, M(1) +2 +∝ +1 +Λ6 which is convergent. For the third term, +M(n) +3 += +g2 +W +4n(2π)D +� +dDk +1 +k4n−2(p − k)4n(k2 − M 2 +W)((p − k)2 − M 2 +W) +∼ +21−Dg2 +W +4n√ +πDΓ +� D +2 +� +� Λ kD−1dk +k8n+2 += +21−Dg2 +W +4n√ +πDΓ +� D +2 +� +(D − 8n − 2) +ΛD−8n−2 , +(135) +19 + +which has the same result as the M(n) +2 +case. For the last term, +M(n) +4 +∼ +g2 +W +4n(2π)DM 2 +W +� +dDk +(k · p − k2)2 +k4n(p − k)4n(k2 − M 2 +W)((p − k)2 − M 2 +W) += +g2 +W +4n(2π)DM 2 +W +� +dDk +(k · p)2 − 2(k · p)k2 + k4 +k4n(p − k)4n(k2 − M 2 +W)((p − k)2 − M 2 +W) +∼ +g2 +W +4n(2π)DM 2 +W +� +dDk +k2p2 − 2(k · p)k2 + k4 +[k2(p − k)2]2n(k2 − M 2 +W)((p − k)2 − M 2 +W) += +g2 +W +4n(2π)DM 2 +W +� +dDk +1 +[k2(p − k)2]2n−1(k2 − M 2 +W)((p − k)2 − M 2 +W) . +(136) +From the second line to the third line, we have made the order approximation. No- +tice that (k · p)2 = kµpµkαpα = (kµpα)pµkα. Then notice that k2p2 = kµkµpαpα = +(kµpα)pαkµ. They are obviously different but they have the same order in k and p. As +here we are interested in order calculation, to simplify the analysis we make the above +approximation. Then we have +M(n) +4 +∼ +21−Dg2 +W +4n√ +πDΓ +� D +2 +� +M 2 +W +� Λ kD−1dk +k8n += +21−Dg2 +W +4n√ +πDM 2 +WΓ +� D +2 +� +(D − 8n) +ΛD−8n . +(137) +For the unrotored case n = 0 and in D = 4, +M(0) +4 +∼ g2 +W +32π2Λ4 , +(138) +which is a seriously 4-th order divergent term. But under rotor mechanism, +M(n) +4 +∼ +g2 +W +2(2n+5)π2M 2 +W(1 − 2n)Λ4−8n . +(139) +For example for n = 1, M(1) +2 +∝ +1 +Λ4 which is convergent. Therefore, the amplitude is +M(n) = M(n) +1 ++ M(n) +2 ++ M(n) +3 ++ M(n) +4 +∼ +g2 +W +4n+2π2Λ8n +�M 2 +W +n ++ +Λ2 +1 − 4n + +Λ4 +4M 2 +W(1 − 2n) +� +(140) +in D = 4 spacetime dimension. We can see that when n = 1 under rotor mechanism, +the amplitude is already convergent in the high-energy regime. Therefore, we see how +the rotor mechanism can remove infinities in loop diagrams at high energies. +Next, consider the self-correction of Higgs Boson by a fermion loop. The Feynman +diagram is as follow: +Figure 4: Feyman diagram of self-energy correction of the Higgs boson by a fermionic loop. +20 + +EASince the vertex factor is just −i mf +vn , the n-rotored amplitude is given by +M(n) = − +m2 +f +v2 +n(2π)D +� +dDk Tr +� +1 +(/k†)n(/k − mf)(/k)n +1 +(/p† − /k†)n(/p − /k − mf)(/p − /k)n +� +. +(141) +The minus sign is due to the Feynman rule for the fermion loop. +Now using the +following fact that +γµγµ = 4ID +(γµγµ)† = 4I† +D +㵆ㆠ+µ = 4ID . +(142) +Equivalently, the amplitude in (141) reads +M(n) = − +m2 +f +v2 +n(2π)D +� +dDk Tr +�(/k)n(/k + mf)(/k†)n +k4n(k2 − m2 +f) +(/p − /k)n(/p − /k + mf)(/p† − /k†)n +(p − k)4n((p − k)2 − m2 +f) +� +(143) +The rough estimation of (141) gives +M(n) ∼ − +m2 +f +v2 +n(2π)D +� +dDk Tr +� +1 +/k2+4n +� +. +(144) +For the unrotored case, n = 0 and in 4D spacetime, we have +M(n) ∼ − +m2 +f +v2(2π)D Λ2 +(145) +which has a quadratic divergence. But under rotor mechanism we have +M(n) ∼ − +m2 +f +v2 +n(2π)D Λ2−4n . +(146) +When n = 1, we can see that M(n) ∝ +1 +Λ2, which is convergent. Thus the divergence is +eliminated. +Next, we consider the self-energy correction of the Higgs boson by the 3rd order +self-interaction. The vertex factor is +i +2·8n +m2 +H +vn . The Feynman diagram is +Figure 5: Feyman diagram of self-energy correction of the Higgs boson by a 3rd-order self +interaction. +The amplitude for such process is +M(n) = +(16)nm4 +H +4 · 64n(2π)Dv2 +n +� +dDk +1 +k4n(k2 − m2 +H) +1 +(p − k)4n((p − k)2 − m2 +H) +(147) +21 + +By rough estimation, we have +M(n) ∼ +m4 +H +2D+2n+1√ +πDv2 +n +� Λ kD−1dk +k4+8n +∼ +m4 +H +2D+2n+1√ +πD(D − 8n − 4)v2 +n +ΛD−8n−4 +(148) +For the unrotored case in 4D spacetime, n = 0, we have logarithmic divergence +M(0) ∼ +m4 +H +32π2v2 +n +ln Λ +(149) +For the general case, if we consider D = 4 for our universe, we have +M(n) ∼ − +m4 +H +2D+4n+4√ +πDnv2 +n +1 +Λ8n . +(150) +For n = 1, the matrix element goes for M(n) ∝ +1 +Λ8, which drops very quickly to zero +for large Λ. Therefore, the problem of divergence vanishes in the rotor mechanism. +In general, the above method of power counting used by the above example applies +to other diagrams with higher number of loops. The generic form takes the following: +M(n) ∼ +� +· · · +� +dDk1dDk2 · · · dDkL +k4n +i · · · k4n +l k2 +i · · · k2 +l (/k† +j)n(/kj)n+1 · · · (/k† +p)n(/kp)n+1 +(151) +Let S be the superficial degree of divergence. And let L be the number of loops, NB be +the number of internal spin-1 gauge boson propagator(s), NH be the number of internal +Higgs boson propagator(s) and Nf be the number of internal fermion propagators. +Then we have +S = (power of momentum in numerator)- (power of momentum in demoninator ) += DL − (2NB + 4NBn) − (2NH + 4NHn) − (Nf + 2Nfn) += DL − (2NB + 2NH + Nf) − 2n(2NB + 2NH + Nf) += DL − (2n + 1)(2NB + 2NH + Nf) +(152) +Naively, we expect a diagram to have a divergence proportional to LambdaS . We +expect logarithmic divergence log Λ when S = 0, and no divergence when S < 0. This +takes place when the following inequality holds, +DL − (2n + 1)(2NB + 2NH + Nf) < 0 +n > +DL +(2NB + 2NH + Nf) − 1 +2 , +(153) +and also we demand n > 0. In addition, as the number of loops satisfy the following +equation, +L = I − V + 1 , +(154) +where I is the number of internal lines, V is the number of vertices, it follows that +L = NB + NH + Nf − V + 1 +(155) +22 + +Let us illustrate by an example. Consider the following 3rd-order self-energy correction +of the Higgs boson, +Figure 6: Feyman diagram of self-energy correction of the Higgs boson by a 3-loop interaction. +The Feynman amplitude can be easily written down as +M(n) = −M 3 +Wm2 +fm2 +H +2(2π)3Dv3 +n +��� +dDk dDq dDl gβαgγδgµν +× gαγ − (p − k)α(p − k)γ/M 2 +W +(p − k)4n((p − k)2 − M 2 +W) +gδµ − (p − k − q)δ(p − k − q)µ/M 2 +W +(p − k − q)4n((p − k − q)2 − m2 +W) +gνβ − kνkβ/M 2 +W +k4n(k2 − M 2 +W) +× +� +1 +q4n(q2 − m2 +H) +�2 +1 +(p − q)4n(p − q)2 − m2 +H +× Tr +� +1 +(/l +∗)n(/l − m)(/l +n) +1 +(/q∗ − /l +∗)n(/q − /l − m)(/q − /l)n +� +. +(156) +To compute the superficial degree of UV divergence, we utilize the power counting +formula in (152). First notice that NB = 3, NH = 3, Nf = 2, V = 6, then L = 3. +Hence it follows that S = 3D − 14(2n + 1) = 3D − 28n − 14. We can see that when +n = 0 for the unrotored case and D = 4 spacetime S = −2, thus this diagram is +convergent. We can also see that in D = 4 spacetime S = −2 − 28n < 0, so there is +no UV divergence. +Therefore, in this section, we have seen how infinities arise from high-energy ends +of loop diagrams are tamed by higher-order-derivatives fields under rotor mechanism. +In such way, the UV divergences of from the 1-loop self-correction of Higgs boson +propagator are eliminated. +7 +Explicit calculation of 1-loop self-correction of +Higgs boson under rotor mechanism and a dis- +cussion on the Hierarchy Problem +7.1 +Correction by the 3rd order Higgs vertex +In this section, we will demonstrate the explicit calculation of 1-loop self correction +of Higgs boson by third order Higgs vertex under rotor mechanism. We will show +that although the rotor mechanism can tame ultraviolet UV divergence at high ener- +gies, it introduces Infra-red IR divergence at low energies in 4-dimensional spacetime. +However, we will demonstrate that the rotor mechanism can remove all infinities in +23 + +higher dimension using the example of 1-loop self-correction of Higgs Boson by third +order Higgs vertex. According to equation (147), the n-th order rotored amplitude is +proportional to the following integral, +M(n) ∝ +� +dDk +1 +k4n(k2 − m2 +H) +1 +(p − k)4n((p − k)2 − m2 +H) +(157) +Define the integral as +I(n) = +� +dDk +1 +k4n(k2 − m2 +H) +1 +(p − k)4n((p − k)2 − m2 +H) +(158) +We will first demonstrate the calculation of n = 1 case, then the general n rotored +case. Using the Feynman parameter method given in [36] +1 +A1A2 · · · An += +� 1 +0 +· · · +� 1 +0 +dx1dx2 · · · dxn δ +� +n +� +i=1 +xi − 1 +� +(n − 1)! +(x1A1 + x2A2 + · · · xnAn)n . +(159) +For the n = 1 rotored case, we can rewrite the Feynman integral using equation (159) +1 +k4(p − k)4(k2 − m2 +H)((p − k)2 − m2 +H) += +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsdudvδ(x + y + r + s + u + v − 1) +× +5! +[xk2 + yk2 + r(p − k)2 + s(p − k)2 + u(k2 − m2 +H) + v((p − k)2 − m2 +H)]6 +(160) +Now let us evaluate the terms in the denominator. +xk2 + yk2 + r(p − k)2 + s(p − k)2 + u(k2 − m2 +H) + v((p − k)2 − m2 +H) += xk2 + yk2 + r(p2 − 2p · k + k2) + s(p2 − 2p · k + k2) + u(k2 − m2 +H) ++ v(p2 − 2p · k + k2 − m2 +H) += (x + y + r + s + u + v)k2 + (r + s + v)p2 − 2(r + s + v)(p · k) − (u + v)m2 +H += k2 − 2(r + s + v) + (r + s + v)p2 − (u + v)m2 +H += (k − (r + s + v)p)2 − (u + v)m2 +H , +(161) +where in the fourth line we used the fact that x + y + r + s + u + v = 1. We define +new variable l = k − (r + s + v)p and the effective mass ∆ = (u + v)m2 +H. Also it is +clearly that dDk = dDl. Therefore, now the integral reads +I(1) = +� +dDk +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsdudv +× δ(x + y + r + s + u + v − 1) +120 +[(k − (r + s + v)p)2) − (u + v)m2 +H]6 +(162) +Then we have +I(1) = +� +dDl +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsdudvδ(x + y + r + s + u + v − 1) +120 +(l2 − ∆)6 +(163) +24 + +Now we carry out Wick’s rotation. The Wick’s rotation simply amount to substitute +l0 = il0 +E and lll = lllE. Then the integral now reads +I(1) = 120i(−1)6 +� +dΩ4 +� ∞ +0 +dlE +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsdudv +× δ(x + y + r + s + u + v − 1) +l3 +E +(l2 +E + ∆)6 += 120i(2π2) +Γ(2) +� ∞ +0 +dlE +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsdudv +× δ(x + y + r + s + u + v − 1) +l3 +E +(l2 +E + ∆)6 +(164) +Next using the integral fact that +� ∞ +0 +x3 +(x2 + a)6dx = − +a + 5x2 +40(a + x2)5 +���� +∞ +0 += +1 +40a4 . +(165) +Therefore, now the integral reads +I(1) = 240iπ2 +40m8 +H +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsdudvδ(x + y + r + s + u + v − 1) +1 +(u + v)4 += 6iπ2 +m8 +H +� 1 +0 +du +� 1−u +0 +dv +� 1−u−v +0 +ds +� 1−s−u−v +0 +dr +� 1−r−s−u−v +0 +dy +1 +(u + v)4 += 6iπ2 +m8 +H +lim +u→0 +� +− 1 +4 + +1 +36u2 − 1 +4u − 1 +2 ln u +� += −3iπ2 +2m8 +H ++ infinity . +(166) +We can see that the integral diverges when u → 0, this is referred as the infra-red IR +divergence. Finally, we obtain the amplitude as +M(1) = +3i +512π2m4 +Hv2 +n ++ infinity . +(167) +Therefore, we can see that although the rotor mechanism resolves the high-energy +infinity problem, it introduces infra-red IR divergences at low energy in D = 4 space- +time. +The General n case +To calculate the amplitude of general n-rotored case, we need to use the following +Feynman integral formula given in [36] +1 +Am1 +1 Am2 +2 +· · · Amn +n += +� 1 +0 +· · · +� 1 +0 +� +n +� +i=1 +dxi +� +δ +� +n +� +i=1 +xi − 1 +� +�n +i=1 xmi−1 +i +� �n +i=1 xiAi +��n +i=1 mi +× Γ(m1 + m2 + · · · + mn) +Γ(m1)Γ(m2) · · · Γ(mn) . +(168) +25 + +The n-rotored integral is calculated to be +I(n) = +� +dDk +1 +(k2)2n(k2 − m2 +H) +1 +[(p − k)2]2n((p − k)2 − m2 +H) += +� +dDk +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) +× +x2n−1y2n−1r0s0 +[xk2 + y(p − k)2 + r(k2 − m2 +H) + s((p − k)2 − m2 +H)]4n+2 +Γ(4n + 2) +Γ(2n)Γ(2n)Γ(1)Γ(1) += +� +dDk +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +δ(x + y + r + s − 1)dxdydrds +× +x2n−1y2n−1 +[(k − (y + s)p)2 − (r + s)m2 +H]4n+2 +Γ(4n + 2) +Γ2(2n) +(169) +This time take l = k − (y + s)p and the effective mass ∆(r + s) = m2 +H. Then we have +I(n) = Γ(4n + 2) +Γ2(2n) +� +dDl +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) x2n−1y2n−1 +(l2 − ∆)4n+2 += i(−1)4n+2Γ(4n + 2) +Γ2(2n) +� ∞ +0 +dlE +� +dΩD +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) +× x2n−1y2n−1 +lD−1 +E +(l2 +E + ∆)4n+2 += 2i(−1)4n+2πD/2Γ(4n + 2) +Γ2(2n)Γ +� D +2 +� +� ∞ +0 +dlE +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) +× x2n−1y2n−1 +lD−1 +E +(l2 +E + ∆)4n+2 +(170) +Next we calculate the lE integral +� ∞ +0 +dlE +lD−1 +E +(l2 +E + ∆)4n+2 = Γ +� D +2 +� +Γ +� +4n + 2 − D +2 +� +2∆4n+2− D +2 Γ(4n + 2) +if 0 < D < 8n + 4 . +(171) +Not D is out of the range, the integral does not converge. Therefore, now the I(n) +integral is +I(n) = i(−1)4n+2πD/2Γ +� +4n + 2 − D +2 +� +Γ2(2n) +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1)x2n−1y2n−1 +∆4n+2− D +2 += i(−1)4n+2πD/2Γ +� +4n + 2 − D +2 +� +Γ2(2n)m8n+4−D +H +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) x2n−1y2n−1 +(r + s)4n+2− D +2 += i(−1)4n+2πD/2Γ +� +4n + 2 − D +2 +� +Γ2(2n)m8n+4−D +H +� 1 +0 +dr +� 1−r +0 +ds +� 1−r−s +0 +dy(1 − y − r − s)2n−1y2n−1 +(r + s)4n+2− D +2 += i(−1)4n+2πD/2Γ +� +4n + 2 − D +2 +� +Γ2(2n)m8n+4−D +H +Γ2(2n) +Γ(4n) +� 1 +0 +dr +� 1−r +0 +ds(1 − r − s)4n−1 +(r + s)4n+2− D +2 += i(−1)4n+2πD/2Γ +� +4n + 2 − D +2 +� +(−1)4nΓ(4n)m8n+4−D +H +� 1 +0 +drB +� +1 − 1 +r; 4n, 2 − D +2 +� +, +(172) +26 + +where +B +� +1 − 1 +r; 4n, 2 − D +2 +� += +� 1− 1 +r +0 +t4n−1(1 − t)1− D +2 dt +(173) +is the incomplete Beta function. Hence, we need to compute the following integral +I(n)(n, D) = iπD/2Γ +� +4n + 2 − D +2 +� +Γ(4n)m8n+4−D +H +� 1 +0 +dr +� 1− 1 +r +0 +t4n−1(1 − t)1− D +2 dt . +(174) +Let us define the integral +J(n, D) = +� 1 +0 +dr +� 1− 1 +r +0 +t4n−1(1 − t)1− D +2 dt . +(175) +First we analyse the case when n = 1/4, which is not an integer. +Although this +corresponds to an unphysical rotor number, it is worth to study this case. It is found +that, when n = 1/4, the integral is computed to be +J +�1 +4, D +� += +2 +D − 4 +� +2 +D − 2 − 1 +� +if D > 2 . +(176) +The plot is as follow: +Figure 7: The plot of J +� +1 +4, D +� +against spacetime dimension D. +When n = 1/4, so the converged amplitude takes place only when +−iM(1/4) +�1 +4, D +� += +Γ +� +3 − D +2 +� +2D+2√ +2πDm2−D +H +v2 +1 +4 +� +2 +D − 4 +� +2 +D − 2 − 1 +�� +. +(177) +The amplitude against spacetime dimension is plotted as follow: +27 + +I,d +4 +5 +4 +6 +7 +8 +9 +10 +-0.5 +1.0 +-1.5 +-2.0Figure 8: The plot of −iM +� +1 +4, D +� +against spacetime dimension D. +In particular, in our living spacetime dimension D = 4, using L-Hospital rule, +J +�1 +4, 4 +� += lim +D→4 +2 +D − 4 +� +2 +D − 2 − 1 +� += −1 . +(178) +We thus obtain a convergent amplitude for n = 1/4, D = 4 +iM(1/4) +�1 +4, 4 +� += +m2 +H +64 +√ +2π2v2 +1 +4 += +m2 +H +128π2v2 = 2.044 × 10−4 . +(179) +where we have used the fact that v2 +n = 4nv2 in equation (118) and v = 246 GeV is the +vacuum expectation value. The amplitude is inversely proportional to the square of +the Higgs boson mass. It is noted that the amplitude is ill-defined when D = 2, for +which it diverges. +Now, we investigate the physical n, D ∈ Z+ cases. We find that for each integer +n, the minimum dimension D that gives convergent J integral and amplitude iM(n) +is given by the formula +Dmin(n) = 8n + 1 . +(180) +For example, for n = 1 case, Dmin(1) = 9. That means the lowest spacetime dimension +for which the rotor mechanism to give finite result is nine. For n = 1, if D < 9, the +result will be all infinite. For example, for the n = 2 case, Dmin(2) = 17. That means +the lowest spacetime dimension for which the rotor mechanism to give finite result +is seventeen. For n = 2, if D < 17, this will be all infinite. This applies to higher +positive integer n values. And we conclude that for the Higgs boson self-correction +by 3rd order Higgs-self interaction, the minimum physical dimension for taming the +divergences (for both IR and UV) of the lowest-order rotor model (n = 1) is 9. +Now we numerically work out the pair of (n, d) which contributes the finite con- +vergent J(n, D) integral. The result is plotted in figure (9). +28 + +-iM +4 +5.5 +3.5 +4.0 +4.5 +5.0 +6.0 +-100 +-200 +-300Figure 9: The plot of J(n, D) against rotor number n and spacetime dimension D. +The full amplitude is given by +−iM(n)(n, D) = +Γ +� +4n + 2 − D +2 +� +2D+4n+2Γ(4n) +√ +πDm8n−D +H +v2J(n, D) . +(181) +In general, in the nominator, the Gamma function Γ +� +4n + 2 − D +2 +� +imposes constraints +such that the amplitude is finite. First notice that Γ(−m) is infinity for m equal to +positive integers. Therefore, such condition occurs when D = 2k and 4n + 2 − k ≤ 0, +so it takes place when k ≥ 4n + 2. Together with D ≥ 8n + 1 and D < 8n + 4, the +conditions for finite amplitude are +� +� +� +� +� +� +� +� +� +D +< 8n + 4 +D +≥ 8n + 1 +k +≱ 4n + 2 +D +̸= 2k +. +(182) +For example n = 1, we have k ≥ 6 , D ≥ 12, this will give Γ(−m), which is divergent. +So for n = 1, the required (n, D) pair for finite amplitude is (1, 9), (1, 10), (1, 11). For +example n = 2, the required (n, D) pair for finite amplitude is (2, 17), (2, 18), (2, 19). +For example, +−iM(1)(1, 9) = +Γ +� 3 +2 +� +215Γ(4)π9/2m−1 +H v2J(1, 9) = 4.931 × 10−11 GeV−1 , +(183) +which is a very small value. Therefore, under the rotor mechanism with rotor index +n = 1, at D = 9, the divergence is removed in high dimension. Thus this demonstrates +how high-order-derivative field theory under rotor mechanism can remove both UV and +IR divergences in high spacetime dimension. This shows that the Hierarchy problem +can be solved by rotor mechanism in high spacetime dimension. +29 + +2 +n +4 +6 +0.004 +0.003 +J(n,d) +0.002 +0.001 +0.000 +60 +40 +20 +d7.2 +Correction by the W boson +Next, we will study the self-energy correction of Higgs boson by massive W boson. +Using the result in (136), we have the amplitude as +M(n) = g2 +WM 2 +W +4n(2π)D +� � +dDk +D +k4n(p − k)4n(k2 − M 2 +W)((p − k)2 − M 2 +W) +− +1 +M 2 +W +� +dDk +(p − k)2 +k4n(p − k)4n(k2 − M 2 +W)((p − k)2 − M 2 +W) +− +1 +M 2 +W +� +dDk +k2 +k4n(p − k)4n(k2 − M 2 +W)((p − k)2 − M 2 +W) ++ +1 +M 4 +W +� +dDk +[k · (p − k)]2 +k4n(p − k)4n(k2 − M 2 +W)((p − k)2 − M 2 +W) +� +. +(184) +From this we define the following integrals. First from the first line, we have +I(n) +1 += +� +dDk +D +(k2)2n[(p − k)2]2n(k2 − M 2 +W)((p − k)2 − M 2 +W) . +(185) +From the second line we have +I(n) +2 += +1 +M 2 +W +� +dDk +1 +(k2)2n[(p − k)2]2n−1(k2 − M 2 +W)((p − k)2 − M 2 +W) . +(186) +From the third line we have +I(n) +3 += +1 +M 2 +W +� +dDk +1 +(k2)2n−1[(p − k)2]2n(k2 − M 2 +W)((p − k)2 − M 2 +W) . +(187) +For the last time, we use the following standard trick to convert the inner product of +k · p to squared values. The following identity is used: +a · b = 1 +2[a2 + b2 − (a − b)2] . +(188) +Therefore, +I(n) +4 += +1 +M 4 +W +� +dDk +(k · p − k2)2 +(k2)2n[(p − k)2]2n(k2 − M 2 +W)((p − k)2 − M 2 +W) += +1 +M 4 +W +� +dDk +[ 1 +2 +� +(k2 + p2) − (k − p)2� +− k2]2 +(k2)2n[(p − k)2]2n(k2 − M 2 +W)((p − k)2 − M 2 +W) += +1 +4M 4 +W +� +dDk +(p2 − k2)2 − 2(p2 − k2)(p − k)2 + (p − k)4 +(k2)2n[(p − k)2]2n(k2 − M 2 +W)((p − k)2 − M 2 +W) += +1 +4M 4 +W +� +dDk(p4 − 2p2k2 + k4) − 2(p2 − k2)(p − k)2 + (p − k)4 +(k2)2n[(p − k)2]2n(k2 − M 2 +W)((p − k)2 − M 2 +W) +(189) +For each term in the last line, we define six more integrals: +I(n) +4a = +p4 +4M 4 +W +� +dDk +1 +(k2)2n[(p − k)2]2n(k2 − M 2 +W)((p − k)2 − M 2 +W) , +(190) +I(n) +4b = −2p2 +4M 4 +W +� +dDk +1 +(k2)2n−1[(p − k)2]2n(k2 − M 2 +W)((p − k)2 − M 2 +W) , +(191) +30 + +I(n) +4c = +1 +4M 4 +W +� +dDk +1 +(k2)2n−2[(p − k)2]2n(k2 − M 2 +W)((p − k)2 − M 2 +W) , +(192) +I(n) +4d = −2p2 +4M 4 +W +� +dDk +1 +(k2)2n[(p − k)2]2n−1(k2 − M 2 +W)((p − k)2 − M 2 +W) , +(193) +I(n) +4e = +2 +4M 4 +W +� +dDk +1 +(k2)2n−1[(p − k)2]2n−1(k2 − M 2 +W)((p − k)2 − M 2 +W) , +(194) +I(n) +4f = +1 +4M 4 +W +� +dDk +1 +(k2)2n[(p − k)2]2n−2(k2 − M 2 +W)((p − k)2 − M 2 +W) , +(195) +Then now the computation becomes straight forward. The computation is similar to +that of the 3rd order Higgs vertex case, therefore we will just show the final result in +case some special issues are noted. First of all, +I(n) +1 += iDπD/2Γ +� +4n + 2 − D +2 +� +Γ(4n)m8n+4−D +W +� 1 +0 +B +� +1 − 1 +r, 4n, 2 − D +2 +� +. +(196) +if 8n + 1 ≤ D < 8n + 4. Secondly, +I(n) +2 += iπD/2Γ +� +4n + 1 − D +2 +� +Γ(4n − 1)m8n+4−D +W +� 1 +0 +B +� +1 − 1 +r, 4n − 1, 2 − D +2 +� +. +(197) +if 8n − 1 ≤ D < 8n + 2. Thirdly, +I(n) +3 += +iπD/2√πΓ +� +4n + 1 − D +2 +� +42n−1Γ(2n)Γ +� +2n − 1 +2 +� +m8n+4−D +W +� 1 +0 +B +� +1 − 1 +r, 4n − 1, 2 − D +2 +� +. +(198) +if 8n − 1 ≤ D < 8n + 2. Then +I(n) +4a = +p4 +4M 4 +W +× I(n) +1 +D = iπD/2Γ +� +4n + 2 − D +2 +� +p4 +4Γ(4n)m8n+8−D +W +� 1 +0 +B +� +1 − 1 +r, 4n, 2 − D +2 +� +. +(199) +if 8n + 1 ≤ D < 8n + 4. Then +I(n) +4b = − 2p2 +4M 2 +W +I(n) +3 += − 2iπD/2√πΓ +� +4n + 1 − D +2 +� +p2 +42nΓ(2n)Γ +� +2n − 1 +2 +� +m8n+6−D +W +� 1 +0 +B +� +1−1 +r, 4n−1, 2−D +2 +� +(200) +31 + +For the I(n) +4c +integral, the situation is more complicated. It it is worth to go through +the whole computation process. +I(n) +4c = +1 +4M 4 +W +� +dDl +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1)x2n−3y2n−1 +(l2 − ∆)4n +Γ(4n) +Γ(2n − 2)Γ(2n) += +i(−1)4nΓ(4n)(2πD/2) +4M 4 +WΓ(2n − 2)Γ(2n)Γ +� D +2 +� +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1)x2n−3y2n−1 +× +� +dlE +lD−1 +E +(l2 +E + ∆)4n += +i(−1)4nΓ(4n)(2πD/2) +4M 4 +WΓ(2n − 2)Γ(2n)Γ +� D +2 +� +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1)x2n−3y2n−1 +× 2nΓ +� D +2 +� +Γ +� +4n − D +2 +� +∆4n− D +2 Γ(4n + 1) += +i(−1)4nnπD/2Γ(4n)Γ +� +4n − D +2 +� +Γ(2n − 2)Γ(2n)Γ(4n + 1)M 8n+4−D +W +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) x2n−3y2n−1 +(r + s)4n− D +2 += +i(−1)4nnπD/2Γ(4n)Γ +� +4n − D +2 +� +Γ(2n − 2)Γ(2n)Γ(4n + 1)M 8n+4−D +W +� 1 +0 +dr +� 1−r +0 +ds +� 1−r−s +0 +dy(1 − y − r − s)2n−3y2n−1 +(r + s)4n− D +2 += +i(−1)4nnπD/2Γ(4n)Γ +� +4n − D +2 +� +Γ(2n − 2)Γ(2n)Γ(4n + 1)M 8n+4−D +W +� 1 +0 +dr +� 1−r +0 +ds(1 − r − s)4n−3 +(r + s)4n− D +2 +Γ(2n)Γ(2n − 2) +Γ(4n − 2) += i(−1)4nnπD/2Γ(4n)Γ +� +4n − D +2 +� +Γ(4n + 1)Γ(4n − 2)M 8n+4−D +W +� 1 +0 +dr +� 1−r +0 +ds(1 − r − s)4n−3 +(r + s)4n− D +2 +(201) +if 0 < D < 8. Now we evaluate the last integral. This gives +� 1−r +0 +ds(1 − r − s)4n−3 +(r + s)4n− D +2 += +Γ(4n − 2) +(1 − r)4n−2r4n− D +2 +2 ˜F1 +� +1, 4n − D +2 , 4n − 1, 1 − 1 +r +� +, +(202) +where 2 ˜F1 is the regularized generalized hypergeometric (2, 1) function. We will give +the definition of generalized hypergeometric function and regularized generalized hy- +pergeometric (p, q) function here, which is formally denoted as +pFq. It is given that +pFq(a1, · · · ap; b1, · · · bq; z) = +∞ +� +n=0 +�p +k=1(ak)(n) +�q +k=1(bk)(n) +zn +n! , +(203) +where the Pochhammer symbol for rising factorial is used, +(a)(0) = 1, · · · , (a)(n) = a(a + 1)(a + 2) · · · (a + n − 1) = +n−1 +� +l=0 +(a + l) . +(204) +The corresponding regularized generalized hypergeometric (p, q) function is defined by +p ˜Fq(a1, · · · ap; b1, · · · bq : z) = +pFq(a1, · · · ap; b1, · · · bq; z) +Γ(b1) · · · Γ(bq) +(205) +For our case, we are concerned with regularized generalized hypergeometric (2, 1) func- +tion, which is +2 ˜F1(a1, a2; b1; z) = +2F1(a1, a2; b1; z) +Γ(b1) += +1 +Γ(b1) +∞ +� +n=0 +(a1)(n)(a2)(n) +(b1)(n) +zn +n! . +(206) +32 + +Therefore, for our case in (202), we have +2 ˜F1 +� +1, 4n − D +2 , 4n − 1, 1 − 1 +r +� += +1 +Γ(4n − 1) +∞ +� +m=0 +1(m)� +4n − D +2 +�(m) +m!(4n − 1)(m) +� +1 − 1 +r +�m += +1 +Γ(4n − 1) +∞ +� +m=0 +� +4n − D +2 +�(m) +(4n − 1)(m) +� +1 − 1 +r +�m +. +(207) +Therefore, the last line of (201) yields +I(n) +4c = i(−1)4nnπD/2Γ(4n)Γ +� +4n − D +2 +� +Γ(4n + 1)Γ(4n − 1)M 8n+4−D +W +� 1 +0 +dr +1 +(1 − r)4n−2r4n− D +2 +2 ˜F1 +� +1, 4n − D +2 , 4n − 1, 1 − 1 +r +� += i(−1)4nnπD/2Γ(4n)Γ +� +4n − D +2 +� +Γ(4n + 1)Γ(4n − 1)M 8n+4−D +W +� 1 +0 +dr +1 +(1 − r)4n−2r4n− D +2 +∞ +� +m=0 +� +4n − D +2 +�(m) +(4n − 1)(m) +� +1 − 1 +r +�m +. +(208) +Thus this completes the mathematical calculation of I(n) +4c . +Then we continue to calculate the integral I(n) +4d . This is easy, +I(n) +4d = −2p2 +4M 2 +W +I(n) +2 += −iπD/2Γ +� +4n + 1 − D +2 +� +p2 +2Γ(4n − 1)m8n+6−D +W +� 1 +0 +B +� +1 − 1 +r, 4n − 1, 2 − D +2 +� +. +(209) +Next, +I(n) +4e = +i(−1)4nπD/2√πΓ(4n)Γ(4n − 2)Γ +� +4n − D +2 +� +24n−4Γ(2n − 1)Γ(4n + 1)Γ +� +2n − 1 +2 +� +M 8n+4−D +W +� 1 +0 +dr +2 ˜F1 +� +1, 4n − D +2 , 4n − 1, 1 − 1 +r +� +(1 − r)4n−2r4n− D +2 +. +(210) +Finally, +I(n) +4f = i(−1)4nΓ(4n)Γ +� +4n − D +2 +� +4Γ(4n + 1)M 8n+4−D +W +� 1 +0 +dr +2 ˜F1 +� +1, 4n − D +2 , 4n − 1, 1 − 1 +r +� +(1 − r)4n−2r4n− D +2 +. +(211) +The amplitude of the Higgs-self energy correction by W boson is hence +M(n) = g2M 2 +W +4n(2π)D +� +I(n) +1 +− I(n) +2 +− I(n) +3 ++ I(n) +4a + I(n) +4b + I(n) +4c + I(n) +4d + I(n) +4e + I(n) +4f +� +. +(212) +It is worth to analyse the properties of these integrals. We can see that unlike the case +of the self-energy correction for the Higgs-boson from the 3rd order Higgs interaction, +here the amplitude is ill-defined for the n = 1/4 case, as we have the denominator of +Γ(4n − 1) in I(n) +2 , I(n) +4c , I(n) +4d , I(n) +4e , I(n) +4f , which diverges when n = 1/4. Now, in order to +get an idea of what pairs of (n, D) give us convergent result, first we would like to +analyse the integral in (208). Define +p(n, D, r) = +1 +(1 − r)4n−2r4n− D +2 +2 ˜F1 +� +1, 4n − D +2 , 4n − 1, 1 − 1 +r +� +(213) +Then we plot p(n, D, r) with different n and D values. +33 + +Figure 10: The plot of p(n, D, r) with of n = 1 and different values of D. +Figure 11: The plot of p(n, D, r) with of n = 2 and different values of D. +34 + +— p(1, 1, r) +p(2, 1, r) +5 +p(3, 1, r) +— p(4, 1, r) +4 +— p(5, 1, r) +3 +— p(6, 1, r) +— p(7, 1, r) +- p(8, 1, r) +- p(9, 1, r) +— p(10, 1, r) +2 +3 +4 +5 p(10, 2, r) +0.07 +p(11, 2, r) +0.06 +p(12, 2, r) +0.05 + p(13, 2, r) +0.04 + p(14, 2, r) + p(15, 2, r) +0.03 + p(16, 2, r) +0.02 + p(17, 2, r) +0.01 + p(18, 2, r) +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +— p(19, 2, r)Figure 12: The plot of p(n, D, r) with of n = 3 and different values of D. +Figure 13: The plot of p(n, D, r) with of n = 4 and different values of D. +35 + +p(20, 3, r) +8.×10-8 +p(21, 3, r) +p(22, 3, r) +6.× 10-8 +p(23, 3, r) +p(24, 3, r) +4. × 10-8 + p(25, 3, r) + p(26, 3, r) +2. × 10-8 + p(27, 3, r) +p(28, 3, r) +0.5 +1.0 +1.5 +2.0 +— p(29, 3, r)— p(28, 4, r) +1.5× 10-12 +p(29, 4, r) +p(30, 4, r) +1.× 10-12 +p(31, 4, r) +- p(32,4,r) +5.× 10-13 +p(33, 4, r) + p(34, 4, r) +p(35, 4, r) +0.5 +1.0 +1.5 +2.0Figure 14: The plot of p(n, D, r) with of n = 5 and different values of D. +We study the general properties of the p(n, D, r) function and the integration of +it from 0 to 1. Let’s first study the (1, D) case in detail. For each set of (1, D), we +compute p(1, D, r) and +� 1 +0 p(1, D, r)dr. +(n, D) +p(n, D, r) +� 1 +0 p(1, D, r)dr +(1, 1) +− 2 +3r−3/2 + 2 +5r−5/2 + 4 +15 +∞ +(1, 2) +(r−1)2 +2r2 +∞ +(1, 3) +− 2 +√r + 2 +3r−3/2 + 4 +∞ +(1, 4) +1 +r + ln r − 1 +∞ +(1, 5) +2√r +�� +1 +r − 1 +�2 +1.333 +(1, 6) +r − 1 − ln r +0.5 +(1, 7) +2 +3 +√r +� +r + 2 +� +1 +r − 3 +� +0.267 +(1, 8) +1 +2(r − 1)2 +0.167 +(1, 9) +2 +15 +� +2 +� +1 +r − 5r + 3r2 +� +0.1143 +(1, 10) +1 +6(r − 1)2(2r + 1) +0.0833 +We can see that for the n = 1 case, the integral only converges when D > 4. And +in general, for higher dimension D the integral decreases. This pattern also applies for +other n. For each n, there exists a threshold dimension Dmin(n) such that below this +dimension the integral diverges, upon computation we find that +Dmin(n) = 8n − 3 . +(214) +When n = 1, the threshold is Dmin = 5, which is confirmed by the above table. And +when n = 2, Dmin = 13, and so on. From all the plots above, we can see that larger n +with larger D results in smaller values of the integral. This shows that we will obtain +smaller amplitudes with increasing (n, D) values. +36 + +.x10 +p(36, 5, r) +8. × 10-18 +p(37, 5, r) +p(38, 5, r) +6. × 10-18 +p(39, 5, r) + p(40,5,r) +4. × 10-18 + p(41, 5, r) +2. × 10-18 + p(42, 5, r) +p(43, 5, r) +0.5 +1.0 +1.5 +2.0The full amplitude is now given by +M(n) = +ig2 +W +22n+D√ +πDM 8n+2−D +W +�Γ +� +4n + 2 − D +2 +� +Γ(4n) +� +D + +p4 +4M 4 +W +� � 1 +0 +B +� +1 − 1 +r, 4n, 2 − D +2 +� +− Γ +� +4n + 1 − D +2 +�� +1 +Γ(4n − 1) +� +1 + +p2 +2M 2 +W +� ++ +√π +42nΓ(2n)Γ +� +2n − 1 +2 +� +� +4 + 2p2 +M 2 +W +�� +× +� 1 +0 +B +� +1 − 1 +r, 4n − 1, 2 − D +2 +� ++ (−1)4nΓ(4n)Γ +� +4n − D +2 +� +Γ(4n + 1) +� +n +Γ(4n − 1) + +√πΓ(4n − 2) +Γ(2n − 1)Γ +� +2n − 1 +2 +� + 1 +4 +� +× +� 1 +0 +dr +2 ˜F1 +� +1, 4n − D +2 , 4n − 1, 1 − 1 +r +� +(1 − r)4n−2r4n− D +2 +� +(215) +The full amplitude is a function of n,D and incoming momentum p, M(n) ≡ M(n)(n, D, p). +It is found that for the unrotored case (n = 0) in our D = 4 spacetime, this amplitude +diverges to infinity. This is obvious due to the term Γ(4n) appearing in the denom- +inator. This result is expected as in the Standard Model, one-loop correction of the +Higgs boson by W boson diverges. +However, as by the above results, we see that the integrals converge only in different +set of (n, D) values. We want to find the common (n, D) values for all the integrals, +which amount to solve the three inequalities: +� +� +� +� +� +� +� +� +� +8n + 1 ≤ D < 4n + 4 +8n − 1 ≤ D < 8n + 2 +0 ≤ D < 8 +D ≥ 8n − 3 +. +(216) +However, there is no solution for this set of inequality. Therefore, the amplitude in +(215) means to be diverged in any particular set of (n, D) values. +7.3 +Correction by the fermion loop +Finally, we remain to calculate the self-energy correction of Higgs Boson by fermion +loop under rotor mechanism. This is essentially to compute the amplitude calculation +in (143). We need to develop some Dirac algebra and trace identities involving 㵆, as +well as γµ and 㵆 together in the most general D spacetime dimension. First notice +that the complex conjugate of the γµ matrix is defined by +㵆 = γ0γµγ0† = γ0γµγ0 , +(217) +where we have used the fact that γ0† = γ0. Then we immediately obtain our first +identity : +Tr(㵆) = Tr(γ0γµγ0) = Tr(γ0γ0γµ) = Tr(γµ) = 0 , +(218) +where we have used the trace identity Tr(ABC) = Tr(CBA) and the fact that (γ0)2 = +I. Next, we prove the Dirac algebra for hermitian gamma matrices +{㵆, γν†} = 2ηµνID . +(219) +37 + +The proof is straight forward, +{㵆, γν†} = 㵆γν† + γν†γµ† += γ0γµγ0γ0γνγ0 + γ0γνγ0γ0γµγ0 += γ0(γµγν + γνγµ)γ0 += γ0(2ηµνID×D)γ0 += 2ηµνγ0ID×Dγ0 += 2ηµνID×D , +(220) +where we have used the fact that (γ0)2 = I. This result implies +/k/k = /k†/k† = k2ID×D +and +/k/p = /k†/p† = (k · p)ID×D . +(221) +Next we will prove the following identity: +Tr(γµγρ†) = D(2η0µη0ρ − ηµρ) . +(222) +By the definition (217), it is noticed that Tr(γµγρ†) = Tr(γµγ0γργ0). Then using the +identity of Tr(γµγνγργσ) = D(ηρσηµν − ηνσηµρ + ηµσηνρ), by putting ν and σ equal to +0, then we get +Tr(γµγρ†) = Tr(γµγ0γργ0) += D(ηρ0ηµ0 − η00ηµρ + ηµ0η0ρ) += D(2η0µη0ρ − ηµρ) , +(223) +where η00 = 1 in our diag(+ − −−) metric convention. Next we verify the following +computationally, +Tr(㵆γνγρ†) = Tr(γ0γµγ0γνγ0γργ0) = Tr(γµγν†γρ) = 0 +(224) +that the trace of a mixture of odd number of γµ and 㵆 matrices is zero. Next, another +useful identity we need to use later is +Tr(γµγν†γργσ†) = Tr(γµγ0γνγ0γργ0γσγ0) += D(8η0µη0νη0ρη0σ − 2η0ρη0σηµν − 2η0µη0σηνρ − 2η0νη0ρηµσ − 2η0µη0νηρσ ++ ηµνηρσ − ηµρηνσ + ηµσηνρ) . +(225) +For simplicity, we will consider the n = 1 case first in (143). Now we can evaluate the +nominator of (143) +(/k)(/k + mf)(/k†)(/p − /k)(/p − /k + mf)(/p† − /k†) += (k2/k† + mf/k/k†) [(p − k)2(/p† − /k†) + mf(/p − /k)(/p† − /k†)] += k2(p − k)2(k · p − k2)ID×D + mf(p − k)2/k/k†(/p† − /k†) ++ mfk2/k†(/p − /k)(/p† − /k†) + m2 +f/k/k†(/p − /k)(/p† − /k†) += k2(p − k)2(k · p − k2)ID×D + mf(p − k)2/k(k · p − k2) ++ mfk2/k†(/p/p† − /p†/k† − /k†/p† + /k/k†) + m2 +f/k/k†(/p/p† − /p†/k† − /k†/p† + /k/k†) +(226) +Now we need to take the trace of the above expression. First notice that, +Tr(/k) = kµTr(γµ) = 0 . +(227) +38 + +Then using the identity of 224, +Tr(/k†/p/q†) = kµpνqρTr(㵆γνγρ†) = 0 . +(228) +Therefore using these two results and +Tr +� +k2(p − k)2(k · p − k2)ID×D + mf(p − k)2/k(k · p − k2) ++ mfk2/k†(/p/p† − /p†/k† − /k†/p† + /k/k†) + m2 +f/k/k†(/p/p† − /p†/k† − /k†/p† + /k/k†) +� += Dk2(p − k)2(k · p − k2) + m2 +fTr(/k/k†/p/p† − /k/k†/p/k† − /k/k†/k/p†) + /k/k†/k/k†) += Dk2(p − k)2(k · p − k2) + m2 +f(kµkνpρpσ − kµkνpρkσ − kµkνkρpσ + kµkνkρkσ)Tr(γµγν†γργσ†) += Dk2(p − k)2(k · p − k2) + m2 +f(kµkνpρpσ − kµkνpρkσ − kµkνkρpσ + kµkνkρkσ) +× D(8η0µη0νη0ρη0σ − 2η0ρη0σηµν − 2η0µη0σηνρ − 2η0νη0ρηµσ − 2η0µη0νηρσ ++ ηµνηρσ − ηµρηνσ + ηµσηνρ) += Dk2(p − k)2(k · p − k2) +− 2D(k0)2p2 + 8Dk0p0k2 − 2D(p0)2k2 + Dk2p2 + 8D(k0)2(k · p) − 4Dk0p0(k · p) +− 2Dk2(k · p) − 8D(k0)k2 + Dk4 − 16D(k0)3p0 + 8D(k0)2(p0)2 + 8D(k0)4 . +(229) +Therefore, we have at least 13 integrals. But some of them vanish due to the odd +parity, we will look into details when we come across them. The integrals are much +more technically difficult than the previous ones, as now here we have to also integrate +terms regarding k0. With some observation, first we define the generic integral +I(a, b, D) = +� +dDk +1 +(k2)a[(p − k)2]b(k2 − m2 +f)((p − k)2 − m2 +f) += 2iπD/2Γ +� +a + b + 2 − D +2 +� +Γ(a + b)m2(a+b+2)−D +f +� 1 +0 +drB +� +1 − 1 +r; a + b, 2 − D +2 +� +(230) +if 0 < D < 2a+2b+4. In addition, the integral of the beta function is also convergent +for some threshold D values. Define the first integral as +I(1) +1 += +� +dDk +Dk2(p − k)2(k · p − k2) +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) += D +2 +� +dDk +(p2 − k2) − (p − k)2 +k2(p − k)2(k2 − m2 +f)((p − k)2 − m2 +f) += D +2 +� +p2I(1, 1, D) − I(0, 1, D) − I(1, 0, D) +� +(231) +The range for each integral is: for I(1, 1, D), 4 < D < 8, for I(1, 0, D) and I(0, 1, D), +2 < D < 6. Therefore the overlap D for the first integral is D = 5. +The second integral is defined to be +I(1) +2 += −2Dp2 +� +dDk +(k0)2 +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) +(232) +Then using the old trick of Feynman parameters, we have +I(1) +2 += −2Dp2Γ(6) +Γ2(2)Γ2(1) +� +dDk +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x+y+r+s−1) +(k0)2xy +([k − (y + s)p)2 − (r + s)m2 +f]6 +(233) +39 + +Note that again we have l = k−(y+s)p, ∆ = (r+s)m2 +f. And we have k0 = l0+(y+s)p0, +therefore the integral becomes, +I(1) +2 += −240Dp2 +� +dDl +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) (k0)2xy +(l2 − ∆)6 += −240Dp2 +� +dDl +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1)xy(l0 + (y + s)p)2 +(l2 − ∆)6 += −240Dp2 +� +dDl +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) +× xy((l0)2 + 2l0(y + s)p + (y + s)2p2) +(l2 − ∆)6 +(234) +Next we carry out Wick’s rotation and perform the substitution of l0 = il0 +E, +I(1) +2 += −(−1)6240iDp2 +� ∞ +0 +dlE +� +dΩD +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) +× xy[lD−1 +E +� +− (l0 +E)2 + 2il0 +E(y + s)p + (y + s)2p2)] +(l2 +E + ∆)6 +(235) +Now this integral involve the integrating on l0 +E. To tackle this integral, we concern +integration on D−sphere. +We set the spherical coordinates (l0 +E, l1 +E, · · · lD−1 +E +) in the +Euclidean l momentum space as follows: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +l0 +E += lE sin ϕ1 sin ϕ2 · · · sin ϕD−2 cos ϕD−1 +l1 +E += lE sin ϕ1 sin ϕ2 · · · sin ϕD−2 sin ϕD−1 +... +... +ln−3 +E += lE sin ϕ1 sin ϕ2 cos φ3 +ln−2 +E += lE sin ϕ1 cos ϕ2 +ln−1 +E += lE cos ϕ1 , +(236) +where +l2 +E = (l0 +E)2 + (l1 +E)2 + · · · + (lD−1 +E +)2 . +(237) +The differential volume is given by +dDlE = lD−1 +E +sinD−2 ϕ1 sinD−3 ϕ2 · · · sin ϕD−2dlEdϕ1dϕ2 · · · dϕD−1 +(238) +And the differential solid angle is given by +dΩD = sinD−2 ϕ1 sinD−3 ϕ2 · · · sin ϕD−2dϕ1dϕ2 · · · dϕD−1 , +(239) +where ϕ1, ϕ2, · · · , ϕD−2 ∈ [0, π] and ϕD−1 ∈ [0, 2π]. We first carry out the integral for +40 + +the first term involving (l0 +E)2, we have +I(1) +2a = 240iDp2 +� ∞ +0 +dlE +� +dΩD +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1)xylD−1 +E +(l0 +E)2 +(l2 +E + ∆)6 += 240iDp2 +� ∞ +0 +dlE +� +· · · +� +dϕ1dϕ2 · · · dϕD−1 sinD−2 ϕ1 sinD−3 ϕ2 · · · sin ϕD−2 +× +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1)xylD−1 +E +(lE sin ϕ1 sin ϕ2 · · · sin ϕD−2 cos ϕD−1)2 +(l2 +E + ∆)6 += 240iDp2 +� π +0 +sinD ϕ1dϕ1 +� π +0 +sinD−1 ϕ2dϕ2 +� π +0 +sinD−2 ϕ3dϕ3 · · · +� π +0 +sin3 ϕD−2 +� 2π +0 +cos2 ϕD−1dϕD−1 +× +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) +� ∞ +0 +dlE +xylD+1 +E +(l2 +E + ∆)6 +(240) +In general, +� π +0 +sinm x dx = +√πΓ +� m+1 +2 +� +Γ +� m +2 + 1 +� +if m > −1. +(241) +Also, +� ∞ +0 +dlE +lD+1 +E +(l2 +E + ∆)6 = π(D − 8)(D − 6)(D − 4)(D − 2)D csc +� πD +2 +� +7680∆5− D +2 +(242) +Then we get +I(1) +2a = iDp2π2 +32m10−D +f +� D−2 +� +k=1 +� π +0 +sinD−k+1 ϕkdϕk +� � 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) +xy +(r + s)5− D +2 +× (D − 8)(D − 6)(D − 4)(D − 2)D csc +�πD +2 +� += D2(D − 8)(D − 6)(D − 4)(D − 2)p2πD/2+1 +32 sin +� πD +2 +� +m10−D +f +� D−2 +� +k=1 +Γ +� D−k+2 +2 +� +Γ +� D−k+3 +2 +� +� +× +� 1 +0 +dr +� 1−r +0 +ds +� 1−r−s +0 +dy(1 − y − r − s)y +(r + s)5− D +2 += D2(D − 8)(D − 6)(D − 4)(D − 2)p2πD/2+1 +32 sin +� πD +2 +� +m10−D +f +Γ(2) +Γ +� D+2 +2 +� 1 +6 +� 1 +0 +dr +� 1−r +0 +ds(1 − r − s)3 +(r + s)5− D +2 +(243) +if −2 < D < 10. Our of this range the integral will diverge. Now we calculate the last +integral +� 1−r +0 +ds(1 − r − s)3 +(r + s)5− D +2 += 2 +� +r3 +D − 2 − +3r2 +D − 4 + +3r +D − 6 + +1 +8 − D +� +r +D +2 −4 ++ +96 +(D − 8)(D − 6)(D − 4)(D − 2) +(244) +And finally +� 1 +0 +dr +� 1−r +0 +ds(1 − r − s)3 +(r + s)5− D +2 += +96 +(D − 6)(D − 4)(D − 2)D +(245) +41 + +if D > 6. It is noted that if D ≤ 6, the integral does not converge. Therefore we get +I(1) +2a = 4ip2πD/2+1D(D − 8) +sin +� πD +2 +� +Γ +� D+2 +2 +� +m10−D +f +(246) +in the overall range of 6 < D < 10. +The next term to integrate will be +I(1) +2b = (−2i)(240)Dp2 +� ∞ +0 +dlE +� +dΩD +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x+y+r+s−1) l0 +E(y + s) +(l2 +E + ∆)6 +(247) +Since l0 +E = lE sin ϕ1 sin ϕ2 · · · sin ϕD−2 cos ϕD−1, when we integrate over dΩD where will +be a vanishing term of +� 2π +0 +cos ϕD−1dϕD−1 = 0 . +(248) +Therefore the whole integral vanishes. In general, since if the integral is odd +� +dDl +(2π)D lm(l0) = 0 +(249) +for any integer m. Thus when we have odd order of k0 in the integral, the integral +must vanish. Next we have the I(1) +2c integral, which is +I(1) +2c = −240iDp4 +� ∞ +0 +dlE +� +dΩD +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1)lD−1 +E +xy(y + s)2 +(l2 +E + ∆)6 += −240iDp2 +� ∞ +0 +dlE +� +· · · +� +dϕ1dϕ2 · · · dϕD−1 sinD−2 ϕ1 sinD−3 ϕ2 · · · sin ϕD−2 +× +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1)xy(y + s)2lD−1 +E +(l2 +E + ∆)6 +(250) +And as +� ∞ +0 +dlE +lD−1 +E +(l2 +E + ∆)6 = −π(D − 10)(D − 8)(D − 6)(D − 4)(D − 2) csc +� πD +2 +� +7680∆6− D +2 +(251) +if 0 < D < 12. Then +I(1) +2c = ip4(D − 10)(D − 8)(D − 6)(D − 4)(D − 2)DπD/2+1 +32 sin +� πD +2 +� +m12−D +f +� D−2 +� +k=1 +Γ +� D−k +2 +� +Γ +� D−k+1 +2 +� +� +× +� 1 +0 +dr +� 1−r +0 +ds +� 1−r−s +0 +dy(1 − y − r − s)y(y + s)2 +(r + s)6− D +2 += ip4(D − 10)(D − 8)(D − 6)(D − 4)(D − 2)D · 2πD/2+1 +32 sin +� πD +2 +� +m12−D +f +Γ(1) +Γ +� D +2 +� +× 1 +60 +� 1 +0 +dr +� 1−r +0 +ds(1 − r − s)3(3(r − 1)2 − 4(r − 1)s + 3s2) +(r + s)6− D +2 += ip4π +D +2 +1(D − 10)(D2 − 2D + 24) +6(D + 2) sin +� πD +2 +� +Γ +� D +2 +� +m12−D +f +(252) +42 + +if D > 8. When D < 8 the integral does not converge. So the I(1) +2c integral is converged +in 8 < D < 12. +Therefore, for the I(1) integral to be convergent, the spacetime +dimension constraint is 6 < D < 10 and 8 < D < 12. The solution is D = 9 which is +unique for the convergence. +The third integral is defined to be +I(1) +3 += 8Dp0 +� +dDk +(k0)k2 +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) = 0 +(253) +as the integral is odd. +The fourth integral is defined to be +I(1) +4 += −2D(p0)2 +� +dDk +k2 +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) = −2D(p0)2I(1, 2) +(254) +which only converges when 6 < D < 10. +The fifth integral is defined to be +I(1) +5 += Dp2 +� +dDk +k2 +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) = Dp2I(1, 2) +(255) +which only converges when 6 < D < 10. +The sixth integral is defined to be +I(1) +6 += 8D +� +dDk +(k0)2(k · p) +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) += 4D +� +dDk +(k0)2[(p2 − k2) − (p − k)2] +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) +(256) +Then we further define three more integrals, +I(1) +6a = 4Dp2 +� +dDk +(k0)2 +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) +(257) +which takes the same form as I(1) +2a except the front constant. We have I(1) +6a = −2I(1) +2a , +which is defined in the overall range of 6 < D < 10. Next we have, +I(1) +6b = −4D +� +dDk +(k0)2 +k2(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) +(258) +Using the similar technique as above, this amounts to give +I(1) +6b = −(−1)596iD +� ∞ +0 +dlE +� +dΩD +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) +× y[lD−1 +E +� +− (l0 +E)2 + 2il0 +E(y + s)p + (y + s)2p2)] +(l2 +E + ∆)5 += +iπD/2+1D(D − 6) +8 sin +� πD +2 +� +Γ +� D+2 +2 +� +m8−D +f ++ 2iπD/2+1p2(D − 8)(D2 + 2D + 24) +3(D + 2) sin +� πD +2 +� +Γ +� D +2 +� +m10−D +f +(259) +for 6 < D < 8. Hence only when D = 7 this integral is finite. +43 + +Finally, we have the integral of I(1) +6c , +I(1) +6c = −4D +� +dDk +(k0)2 +k4(p − k)2(k2 − m2 +f)((p − k)2 − m2 +f) . +(260) +Using the similar technique as above, this amounts to give, +I(1) +6c = −(−1)524iD +� ∞ +0 +dlE +� +dΩD +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) +× x[lD−1 +E +� +− (l0 +E)2 + 2il0 +E(y + s)p + (y + s)2p2)] +(l2 +E + ∆)5 += 2iπD/2+1(D − 8)(D − 6) +sin +� πD +2 +� +Γ +� D+2 +2 +� +m8−D +f +− +2iπD/2+1(D − 6)(D − 8)(D2 + 8) +(D + 2)(D + 4) sin +� πD +2 +� +Γ +� D +2 +� +m8−D +f +(261) +if 4 < D < 8. +The seventh integral vanishes as it is an odd integral, +I(1) +7 += −4Dp0 +� +dDk +k0(k · p) +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) = 0 . +(262) +The eighth integral is defined to be +I(1) +8 += −2D +� +dDk +k2(k · p) +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) += −D +� +dDk +(p2 − k2) − (p − k)2 +k2(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) += −D(p2I(1, 2, D) − I(0, 2, D) − I(1, 1, D)) . +(263) +The range for each integral to be finite is: for I(1, 2, D), 6 < D < 10. For I(0, 2, D), +4 < D < 8. For I(1, 1, D), 4 < D < 8. Therefore the overall range is 6 < D < 8, +which is D = 7. +The ninth integral vanishes as it is an odd integral, +I(1) +9 += −8D +� +dDk +(k0)k2 +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) = 0 . +(264) +The tenth integral is defined to be +I(1) +10 = D +� +dDk +k4 +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) = DI(0, 2, D), +(265) +which is in the range of 4 < D < 8. +The eleventh integral vanishes as it is an odd integral, +I(1) +11 = −16Dp0 +� +dDk +(k0)3 +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) = 0 . +(266) +The twelfth integral is defined to be +I(1) +12 = 8D(p0)2 +� +dDk +(k0)2 +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) = −4I(1) +2 (p0)2 +p2 +(267) +44 + +with D = 9. +The thirteenth integral is defined to be +I(1) +13 = 8D +� +dDk +(k0)4 +k4(p − k)4(k2 − m2 +f)((p − k)2 − m2 +f) += 960D +� +dDl +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1)xy((l0) + (y + s)p)4 +(l2 − ∆)6 += 960D +� +dDl +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) +× xy((l0)4 + 4(l0)3(y + s)p + 6(l0)2(y + s)2p2 + 6(l0)(y + s)3p3 + (y + s)4p4) +(l2 − ∆)6 +(268) +The integrals involving (l0)3 and (l0) vanish as they are odd integrals. We subdivide +I(1) +13 into three more integrals. +I(1) +13a = 960D +� +dDl +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) xy(l0)4 +(l2 − ∆)6 += 960iD +� ∞ +0 +dlE +� +· · · +� +dϕ1dϕ2 · · · dϕD−1 sinD−2 ϕ1 sinD−3 ϕ2 · · · sin ϕD−2 +× +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1)xylD−1 +E +(lE sin ϕ1 sin ϕ2 · · · sin ϕD−2 cos ϕD−1)4 +(l2 +E + ∆)6 += 960iD +� π +0 +sinD+2 ϕ1dϕ1 +� π +0 +sinD+1 ϕ2dϕ2 +� π +0 +sinD ϕ3dϕ3 · · · +� π +0 +sin5 ϕD−2 +� 2π +0 +cos4 ϕD−1dϕD−1 +× +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) +� ∞ +0 +dlE +xylD+3 +E +(l2 +E + ∆)6 +− 3i(D − 6)(D − 4)(D − 2)D2(D + 2)πD/2+1 +32 sin +� πD +2 +� +m8−D +f +� D−2 +� +k=1 +Γ +� D−k+4 +2 +� +Γ +� D−k+5 +2 +� +� +× +� 1 +0 +dr +� 1−r +0 +ds +� 1−r−s +0 +dy(1 − y − r − s)y +(r + s)4− D +2 += − 3iπD/2+1D(D − 6) +sin +� πD +2 +� +Γ +� D+4 +2 +� +m8−D +f +(269) +for 4 < D < 8. Next, we have +I(1) +13b = 5760iDp2 +� π +0 +sinD ϕ1 +� π +0 +sinD−1 ϕ2 · · · +� π +0 +sin3 ϕD−2dϕD−2 +� 2π +0 +cos ϕD−1dϕD−1 +× +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) +� ∞ +0 +dlE +xy(y + s)2lD+1 +E +(l2 +E + ∆)6 += 72iπD/2+1D(D − 8)p2 +sin +� πD +2 +� +Γ +� D+2 +2 +� +m10−D +f +(270) +for 6 < D < 10. +45 + +Finally, we have +I(1) +13c = 960iDp4 +� +· · · +� +dϕ1dϕ2 · · · dϕD−1 sinD−2 ϕ1 sinD−3 ϕ2 · · · sin ϕD−2 +× +� 1 +0 +� 1 +0 +� 1 +0 +� 1 +0 +dxdydrdsδ(x + y + r + s − 1) +� +dlE +xy(y + s)4lD−1 +E +(l2 +E + ∆)6 += −4iπD/2+1p4(D − 10)(D4 + 20D2 + 240D + 2304) +5(D + 2)(D + 4)(D + 6) sin +� πD +2 +� +Γ +� D +2 +� +m12−D +f +. +(271) +for 8 < D < 12. +Therefore, we have completed all the calculations of the integrals. The total am- +plitude would be +M(n) = − +m2 +f +v2 +n(2π)D +� +Integrals . +(272) +Since from the above calculations, each integral corresponds to a specific range +of D or specific value of D such that the integral converges, we see that there is no +common D for all the integrals to be convergent. The rotor mechanism can only save +the divergence for each term of the calculation with higher dimension D one by one +in the W-boson self-energy correction diagram. +8 +Conclusion +In conclusion, we have applied the rotor mechanism and quantization to the standard +model of particle physics, which naturally generates high-order derivative quantum +fields in the standard model’s Lagrangian. Upon path integral quantization, we de- +velop feynman propagators of scalar particles, gauge boson and Dirac fermion under +rotor model. When the rotor index is n = 0, this restores to the original standard +model case. Finally, we explicitly calculate quantum amplitudes of the one-loop self- +energy correction diagrams of the Higgs Boson under rotor mechanism. This include +the correction by the Higgs-self interaction, W-boson and fermion respectively. We +find that the rotor mechanism can generally remove the UV divergences, however, IR +divergence is arisen at the same time. We discover that the rotor model is able to +remove infinities (both UV and IR), or suppress divergences arise from the case of the +Higgs-self interaction. We find that the minimum spacetime dimension D for n = 1 +rotor index is 9, and for n = 2 is 17, and so on with a general formal of Dmin = 8n+1. +This suggests that the rotor mechanism can remove infinities (both UV and IR) arise +from simple integral calculation, thus give a new way to partially solve the Hierarchy +problem. However, for diagrams with more complicated integrals, such as the W-boson +loop correction and fermion loop correction, due to specific dimension range arise from +each integral term, there does not exist a general D that can cure the divergence all +at once for specific n. More future work has to be done. +References +[1] B. Grinstein, D. O’Connell, and M. B. Wise. The Lee-Wick standard model. Phys. +Rev. D 77, 025012. 2008. +46 + +[2] T.D. Lee and G.C. Wick. Negative metric and the unitarity of the S-matrix. Nuclear +Physics B, Vol 9, Issue 2. p. 209-243. 1969. +[3] T.D. Lee and G.C. Wick. Finite Theory of Quantum Electrodynamics. Phys. Rev. +D 2, 1033. 1970. +[4] B. Podolsky. A Generalized Electrodynamics Part I—Non-Quantum. Phys. Rev. +62, 68. 1942. +[5] B. Podolsky and C. Kikuchi. A Generalized Electrodynamics Part II-Quantum. +Phys. Rev. 65. 228. 1944. +[6] B. Podolsky and C. Kikuchi. Auxiliary Conditions and Electrostatic Interaction in +Generalized Quantum Electrodynamics. 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A stable higher-derivative theory with the Yang-Mills gauge sym- +metry. arXiv:2011.12928 [hep-th] +[16] P. Mukherjee, and P. Biswajit. Gauge invariances of higher derivative Maxwell- +Chern-Simons field theory: A new Hamiltonian approach. Physical Review D 85(4). +2011. +[17] A. Pais and G. E. Uhlenbeck. On Field with Non- Localized Action. Phys. Rev. +79. 1950. +[18] C. Bernard and A. Duncan. Lorentz covariance and Matthews’s theorem for +derivative coupled field theories. Phys. Rev.D 11. 1975. +[19] D. A. Eliezer and R. P. Woodard. The problem of nonlocality in string theory. +Nucl. Phys. B 325. 1989. +47 + +[20] J. Z. Simon. Higher-derivative Lagrangians, nonlocality, problems, and solutions. +Phys. Rev. D 41. 1990. +[21] T-C. Cheng, P-M. Ho, M-C. Yeh. Perturbative approach to higher derivative and +non local theories. Nucl. Phys. B 625. 2002. +[22] J. Polonyi and A. Siwek. Yang-Mills-Higgs models with higher order derivatives. +Phys Rev D.86.125006. 2012. +[23] L. Casarin. On higher-derivative gauge theories Thesis for the MSc degree in +Physics at the University of Padova. arXiv:1710.08021. 2017. +[24] E.Bergshoeff1, M.Rakowski and E.Sezgin. Higher derivative super Yang-Mills the- +ories. Physics Letters B. Vol 185, Issues 3–4, p. 371-376. 1987. +[25] T. Nakamura and S. Hamamoto. Higher Derivatives and Canonical Formalisms. +Prog. Theor. Phys. Vol 95. Issue 3. 1996. +[26] K. Andrzejewski, J. Gonera, and P. Ma´slanka, Euclidean Path Integral and +Higher- Derivative Theories, Prog. Theor. Phys. 125. 2011. +[27] C. A. Margalli and J. D. Vergara. Quantization of a Complex Higher Order Deriva- +tive Theory using Path Integrals. arXiv:1401.2487. +[28] C. Grosse-Knetter. Effective Lagrangians with higher derivatives and equations +of motion. Phys. Rev. D 49. 1994. +[29] S. W. Hawking and T. Herto. Living with ghosts. Phys. Rev. D 65. 2002. +[30] B.T.T.Wong. Generalized Yang-Mills theory under rotor mechanism. Nuc. Phys. +B. 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John Wiley & +Sons. 2008. +[40] P. W. Higgs.: Broken symmetries, massless particles and gauge fields, Phys. Lett. +12 (1964) 132. +[41] P. W. Higgs.: Broken Symmetries and the Masses of Gauge Bosons. Phys. Rev. +Lett. 13 (1964) 508. +[42] F. Englert, R. Brout.: Broken Symmetry and the Mass of Gauge Vector Mesons. +Phys. Rev. Lett. 13: 321–323. Aug. 1964. +[43] S. Weinberg.: A Model of Leptons. Phys. Rev. Lett. 19 (1967) 1264. +[44] A. Salam.: Weak and electromagnetic interactions, in: N. Svartholm (Ed.), El- +ementary Particle Physics: Relativistic Groups and Analyticity. Proceedings of the +Eighth Nobel Symposium, Almquvist and Wiskell, 1968, p. 367. +[45] T.W.B. Kibble, Symmetry breaking in non-Abelian gauge theories. Phys. Rev. +155 (1967) 1554. +[46] F. Jegerlehner. The hierarchy problem of the electroweak Standard Model revis- +ited. DESY 13-093, HU-EP-13/25. Sep. 2013. +[47] M. Holthausen, K. S. Lim, M. Lindner. Planck Scale Boundary Conditions and +the Higgs Mass. JHEP 1202, 037. 2012. +49 + diff --git a/N9FOT4oBgHgl3EQf2zTm/content/tmp_files/load_file.txt b/N9FOT4oBgHgl3EQf2zTm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2db0fee86b1e2d81deea598ea6c2b3cde6a59f20 --- /dev/null +++ b/N9FOT4oBgHgl3EQf2zTm/content/tmp_files/load_file.txt @@ -0,0 +1,1719 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf,len=1718 +page_content='Generalized Standard Model with higher-order derivatives under Rotor Mechanism and its Quantization B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Wong∗ Abstract The Standard Model is the paradigm of particle physics which gives an ac- curate theory for fundamental particle interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' However, the extension of Standard Model with higher-order derivative is not a well-studied subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This paper is a follow-up work of the previous study of the generalized Abelian gauge field theory and Yang-Mills theory under rotor mechanism of order n of higher order derivatives, and we apply to the Standard Model of particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Ro- tor mechanism on scalar field and Dirac field is also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We will study the quantization of the rotored Standard Model using path integral approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We also inherit the previous result from the path integral quantization of generalized abelian gauge field and apply it to our non-abelian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Finally, we discuss the possibility of rotor model on taming the infinities arise from the self-energy cor- rection of the Higgs boson in high spacetime dimension, thus provide a partial solution and new insight to the Hierarchy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 1 Introduction The Standard Model (SM) of particle physics is a well-established theory which de- scribes particle interaction with great precision, with quantum field theory as an under- lying mathematical foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Yet, higher-order derivative SM with a great potential to tame UV divergence is not a well studied subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Promising model includes a re- cent study in Lee-Wick Standard Model which stabilizes quadratic divergence [1], of which arises from the generalization of Lee-Wick electrodynamics [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The study of higher order derivative quantum field theory is particular interesting because it can eliminate ultraviolet (UV) divergences in scattering amplitudes [4, 5, 6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' There are studies on higher order derivative scalar field and gauge field theories, and these theories have contributions in quantum gravity and modified gravity [9, 10, 11, 12, 13, 14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Higher order derivative theory also shows its appearance in string theory [17, 18, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' In our context of study, Yang-Mills theory with higher order derivative has been studied in references [22, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Quantization of higher order derivative quantum field theory using path integral approach has been studied in [25, 26, 27, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' In our previous work, we have established the generalized, higher-order-derivative Yang-Mills theory by rotor mechanism [30], which follows upon our further previous ∗CERN, u3500478@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='hku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='hk 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='12944v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='gen-ph] 10 Jan 2023 work in the abelian counterpart [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The path integral quantization of generalized abelian gauge field theory under rotor model is conducted in our previous work [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The Yang-Mills action in D dimensional spacetime is given by [33] SYM = −1 2 � dDxTr GµνGµν = −1 4 � dDx Ga µνGµν a (1) where Gµν = ∂µTν − ∂νTµ − ig[Tµ, Tν] , (2) for which Gµν = Ga µνta and Tµ = T a µta are matrices with ta the generators of SU(N) Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Using the Lie algebra [ta, tb] = if abctc, this gives the gauge field strength as, Ga µν = ∂µT a ν − ∂νT a µ + gf abcT b µT c ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (3) We have D = 4n+4 for a renormalizable theory with unity gauge field dimension [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' By carrying our integration by parts on the kinetic term and expanding the remaining terms, equation(1) explicitly is, SYM = � dDx � T µa ˆRµνT νa − g 2f abc(∂µT νa − ∂νT µa)T b µT c ν − g2 4 f abcf adeT b µT c νT µdT νe � , (4) where ˆRµν = 1 2(□ηµν − ∂µ∂ν) is defined as the projection tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Under the rotor mechanism we introduced in our previous work in [31], which is the successive action of the projection tensors on the original gauge field, Tn µn = ˆRµnµn−1 ˆRµn−1µn−2 · · · ˆRµ3µ2 ˆRµ2µ1 ˆRµ1µ0T µ0 = 1 2n−1P µn−1 µn P µn−2 µn−1 · · · P µ2 µ3 P µ1 µ2 ˆRµ1µ0T µ0 , (5) for which P µj−1 µj = □δ µj−1 µj (6) is defined as the propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This is known as the rotor transformation which generates high-order derivative gauge fields in the action [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The n-th order Yang-Mills action after rotor transformation of gauge field under Lorentz gauge is [32] S(n) YM = −1 4 � dDx Ga n µνGµν a n = � dDx � T µa n ˆRµνT νa n − g 2f abc(∂µT νa n − ∂νT µa n )T b µT c n ν − g2 4 f abcf adeT b n µT c n νT µd n T νe n � = � dDx � 1 4n□nT µa ˆRµν□nT νa − 1 2 · 8ngf abc(∂µ□nT νa − ∂ν□nT µa)□nT b µ□nT c ν − g2 4 · 16nf abcf ade□nT b µ□nT c ν□nT µd□nT νe � (7) Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' under the rotor mechanism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' the gauge field transforms as [30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Tµ → Tn µ = 1 2n□nTµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (8) And the gauge field strength becomes [30] Ga nµν = 1 2n∂µ□nT a ν − 1 2n∂ν□nT a µ + 1 4ngf abc□nT b µ□nT c ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (9) 2 The n−th ordered covariant derivative is given by [30], Dn µ = ∂µ − i 2ng□nT c µtc , (10) The equation of motion of this generalized Yang-Mills theory is [30], Dn µGµνa n = ∂µGµνa n + 1 2ngf abc□nT b µGµνc n = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (11) The Noether’s current is [30] Jα = � − 1 4n□n ˜Gαβ k − 1 2 · 8ngf kbc(□nT αb□nT βc − □nT αc□nT βb) � δ□nT k β , (12) and the associated Noether’s charge is given by [30] Q = � dD−1xj0 = � dD−1x � − 1 4n□n ˜G0β k− 1 2 · 8ngf kbc(□nT 0b□nT βc−□nT 0c□nT βb) � δ□nT k β , (13) where ˜Gαβ k is the Maxwellian gauge field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' It can be seen that when n = 0 (no rotation), we get back the original Yang-Mills theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Under the unitary transformation by SU(N) representation, the spinor transforms as ψ′(x) = U(x)ψ(x) , (14) where U(x) = eiαa(x)ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The covariant derivative and field operator under rotor model transform, respectively as, � D′ n µ(x) = U(x)Dn µ(x)U †(x) G′ n µν(x) = U(x)Gn µν(x)U †(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (15) It follows that the rotor gauge field transforms as, □nT ′ µ = U□nTµU † + i · 2n g U∂µU † .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (16) Infinitesimally, the rotored gauge field transforms as □nT ′a µ = □nT a µ + 2n g ∂µαa + f abc□nT b µαc (17) Therefore the infinitesimal change in rotored gauge field is δ□nT a µ = □nT ′a µ − □nT a µ = 2n g (Dn µα)a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (18) The n-th ordered gauge field strength can be defined through the commutator of the n-th ordered covariant derivative, Gn µν = i g[Dn µ, Dn ν] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (19) And infinitesimally it transforms as G′ n µν = Gn µν − f abcαbGc n µν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (20) 3 It is noted that when n = 0, this gives us back all the properties of the transformation rules of the original Yang-Mills theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We will proceed the quantization of the generalized Yang-Mills theory under rotor mechanism with Feynman path integral approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The quantum amplitude can be computed as an integral of all possible field configurations over the exponential of the action [34, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Similarly to our previous work for the abelian case in [32], as in the generalized model it involves the transformation of field by T µ → □nT µ, therefore in the path integral we sum over all possible configurations of □nT µ instead of T µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' the integration measure changes as � DT µ(x) → � D□nT µ(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (21) 2 Generalized spin-0 scalar field theory under rotor mechanism In this section, we will complete the study of generalized scalar field theory under rotor mechanism, this will generate scalar fields with higher-order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' First consider the massless scalar field theory in D-dimension S = � dDx 1 2∂µφ∂µφ = � dDx φ � − □ 2 � φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (22) According to the definition of rotor mechanism in (5), we define the rotor mechanism as the successive operations of the operator that couples to the gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For the gauge field case, the operator is the projection tensor ˆRµν = 1 2(□ηµν − ∂µ∂ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For the scalar field case, we have the rotor operator as ˆ˜R = − □ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This scalar rotor operator can be recovered from tracing the the projection tensor, up to some scaling factor, ˆR = ηµν ˆRµν = ˆRµ µ = D − 1 2 □ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (23) It follows that ˆ˜R = − 1 D − 1 ˆR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (24) The rotor mechanism on scalar field is simply φ → n � j=1 � − 1 D − 1 � ˆRµj µjφ = (−1)n 2n □ · · · □□φ = (−1)n 2n □nφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (25) The generalized massless scalar field theory is therefore S = � dDx 1 2∂µ �(−1)n 2n □nφ � ∂µ �(−1)n 2n □nφ � = � dDx 1 2∂µ � 1 2n□nφ � ∂µ � 1 2n□nφ � = 1 4n � dDx □nφ � − □ 2 � □nφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (26) Notice that when n = 0, this restores back to the original case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore, we see that both the gauge field and scalar field transforms under the rotor mechanism by the same form, Tµ → Tn µ = 1 2n□nTµ , φ → φn = 1 2n□nφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (27) 4 For the massive scalar field theory, the generalized action under rotor mechanism is S = 1 2 · 4n � dDx ∂µ□nφ∂µ□nφ − m2□nφ□nφ = 1 4n � dDx □nφ � − □ + m2 2 � □nφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (28) The Euler-Lagrangian equation is ∂µ ∂L ∂µ□nφ = ∂L ∂□nφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (29) This gives the equation of motion as □n+1φ + m2□nφ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (30) Now consider the complex-scalar field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' First consider the action with two scalar fields, S = 1 2 · 4n � dDx � i=1,2 (∂µ□nφi∂µ□nφi − m2□nφi□nφi) = 1 4n � dDx � i=1,2 □nφi � − □ + m2 2 � □nφi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (31) Now define a n-rotored complex scalar field, □nΦ = 1 √ 2(□nφ1 + i□nφ2) and □nΦ† = 1 √ 2(□nφ1 − i□nφ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (32) Then the action in 31 can be written as S = 1 4n � dDx ∂µ□nΦ†∂µ□nΦ − m2□nΦ†□nΦ = 1 4n � dDx □nΦ†(−□n − m2)□nΦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (33) The two Euler-Lagrangian equations are ∂µ ∂L ∂µ□nΦ = ∂L ∂□nΦ and ∂µ ∂L ∂µ□nΦ† = ∂L ∂□nΦ† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (34) This gives the two equations of motion as follow: □n+1Φ† + m2□nΦ† = 0 and □n+1Φ + m2□nΦ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (35) Similar to the argument in our previous paper [31], the scalar field action under rotor mechanism is renormalizable in D = 4n+4 dimension with unity scalar field dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 3 Generalized spin-1/2 Dirac field theory theory under rotor mechanism Next, we will investigate how the Dirac field transforms under rotor mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' First we consider the massless Dirac action in D-dimensional spacetime, S = � dDx ¯ψiγµ∂µψ = � dDx ¯ψi/∂ψ ≡ � dDx ¯ψaiγµ ab∂µψb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (36) 5 So in analogy to the case of gauge field and scalar field, we have the rotor operator as the matrix ˆRab = iγµ ab∂µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Hence under rotor transformation, the spinor field transforms as ψa0 → (in)γµn anan−1γµn−1 an−1an−2 · · · γµ1 a1a0∂µn∂µn−1 · · · ∂µ1ψa0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (37) Now let’s see how the adjoint spinor transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Consider , γ0 a0b0ψb0 → (in)γµn anan−1γµn−1 an−1an−2 · · · γµ1 a1a0∂µn∂µn−1 · · · ∂µ1γ0 a0b0ψb0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (38) Then by taking the Hermitian conjugate (γ0 a0b0ψb0)† → (−i)n(∂µn∂µn−1 · · · ∂µ1ψ† b0γ0† b0a0)γµ1† a0a1 · · · γµn−1† an−2an−1㵆 an−1an (39) As (γ0 a0b0ψb0)† = ψ† b0γ0† b0a0 and because (γ0)† = γ0, therefore ¯ψa0 = ψ† b0γ0 b0a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Hence the adjoint spinor transforms as follow: ¯ψa0 → (−i)n(∂µn∂µn−1 · · · ∂µ1 ¯ψa0)γµ1† a0a1 · · · γµn−1† an−2an−1㵆 an−1an (40) Therefore, the generalized Dirac field theory under rotor mechanism is S = � dDx(∂νn∂νn−1 · · · ∂ν1 ¯ψ)γν1†γν2† · · · γνn† i/∂ γµnγµn−1 · · · γµ1∂µn−1∂µn−2 · · · ∂µ1ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (41) Now, let’s define short-hand tensor notation by γαβγ··· ≡ γαγβγγ · · · and ∇αβγ··· ≡ ∂α∂β∂γ · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (42) Then the action in 41 can be formally written as S = � dDx(∇νn···ν1 ¯ψ) γ†ν1···νni/∂ γµn···µ1∇µn···µ1ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (43) The Dirac action with the mass term is given by S = � dDx ¯ψiγµ∂µψ − m ¯ψψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (44) Under rotor mechanism, the whole action transforms as S = � dDx(∇νn···ν1 ¯ψ) γ†ν1···νni/∂ γµn···µ1∇µn···µ1ψ−m(∇νn···ν1 ¯ψ) γ†ν1···νn γµn···µ1∇µn···µ1ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (45) The Euler-Lagrangian equation is given by ∂µ ∂L ∂µ(∇νn···ν1 ¯ψ) γ†ν1···νn = ∂L ∂(∇νn···ν1 ¯ψ) γ†ν1···νn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (46) This gives the equation of motion as i/∂ γµn···µ1∇µn···µ1ψ − mγµn···µ1∇µn···µ1ψ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (47) 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='1 Generalized Quantum electrodynamics under rotor mech- anism Now we proceed to develop the theory of higher-order derivative quantum electrody- namics (QED) by rotor mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' From [31], the general Maxwell action under rotor mechanism is S = − 1 4n+1 � dDx □nGµν□nGµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (48) From the Dirac action under rotor mechanism, we expect the interactive term can be achieved by replacing the ordinary partial derivative into covariant derivative Dn S = � dDx(∇νn···ν1 ¯ψ) γ†ν1···νni /Dn γµn···µ1∇µn···µ1ψ , (49) where Dn α = ∂α + ie 2n□nTα (50) under Lorentz gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore, the full QED action with interaction under rotor mechanism is given by SQED = � dDx � − 1 4n+1□nGµν□nGµν + (∇νn···ν1 ¯ψ) γ†ν1···νn(iγα∂α − m) γµn···µ1∇µn···µ1ψ − e 2n(∇νn···ν1 ¯ψ) γ†ν1···νnγα□nTαγµn···µ1∇µn···µ1ψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (51) 4 Path integral quantization under rotor mecha- nism 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='1 Generalized Yang-Mills Theory In this section, we will study the quantization of general Yang-Mills theory by path integral approach in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' From now on, we take the transformed □nT µ field as the field variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Simply speaking, the physics is changed by T µ → □nT µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The quantum amplitude of the □nT µ field in the renormalizable 4n + 4 dimension is ⟨□nT µ f (tf,xxx)|e−i ˆH(tf−ti)|□nT µ i (ti,xxx)⟩ = � D□nT µ(x) exp � iS(n) YM[□nT µ] � , (52) where |□nT µ i (ti,xxx)⟩ is the field state at initial time ti and |□nT µ i (tf,xxx)⟩ is the field state at final time tf, and ˆH is the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The sourced generating functional is a functional of 4-(covariant) transformed vector current □nJµ(x), Z[□nJµ(x)] = � D□nT µ(x) exp � iS(n) YM[□nT µ] + 2i � d4n+4x Tr � □nJµ(x)□nT µ(x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (53) The normalized generating functional for gauge field is Z[□nJµ] = Z[□nJµ] Z[0] = � D□nT µ(x) exp � iS(n) YM[□nT µ] + 2i � d4n+4xTr � □nJµ(x)□nT µ(x) �� � D□nT µ(x) exp � iS(n) YM[□nT µ] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (54) 7 The free n-point correlation function is given by ⟨0|T□n ˆT ν1(x1)□n ˆT ν2(x2) · · · □n ˆT νn(xn)|0⟩ = 1 in δn δ□nJν1(x1)δ□nJν2(x2) · · · δ□nJνn(xn)Z[□nJµ] ���� □nJµ=0 , (55) where noting that each gauge field has different Lorentz indices and T means the time ordering operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Then the path integral representation is ⟨0|T□n ˆT ν1(x1) · · · □n ˆT νn(xn)|0⟩ = � D□nT µ(x) □nT ν1(x1) · · · □nT νn(xn) exp � iS(n) YM[□nT µ] � � D□nT µ(x) exp � iS(n) YM[□nT µ] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (56) Since the projection tensor in the action S(n) YM is not invertible, we need to perform gauge fixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We will use the Fadeev-Popov method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Recall the gauge field transforms as (16) □nT ′ µ = □nT U µ = U□nTµU † + i · 2n g U∂µU † .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (57) Now we need to choose a gauge fixing functional, G(□nT µ) = ∂µ□nT µ(x) − w(x) , (58) where w(x) is some arbitrary matrix (w(x) = wa(x)ta ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Now we use the identity, 1 = � DUδ[G(□nT µU)] det �δG[□nT µU(x)] δU(y) � (59) Next we insert this into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='1), Z[□nJµ(x)] = � DUD□nT µ(x) δ[G(□nT Uµ)] det �δG[□nT Uµ(x)] δU(y) � × exp � iS(n) YM[□nT µ] + 2i � d4n+4x Tr � □nJµ(x)□nT µ(x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (60) But since the action is gauge invariant, S(n) YM[□nT Uµ] = S(n) YM[□nT µ], also the integration measure remains unchanged, D□nT µU = D□nT µ, therefore we can write Z[□nJµ(x)] = � DUD□nT Uµ(x) δ[G(□nT Uµ)] det �δG[□nT Uµ(x)] δU(y) � × exp � iS(n) YM[□nT Uµ] + 2i � d4n+4x Tr � □nJµ(x)□nT µ(x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (61) It is remarked that the source term is not gauge invariant unless Dµ□nTµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Now we can relabel the transformed, rotored gauge field variable to □nT Uµ = □nT ′µ, Z[□nJµ(x)] = � DUD□nT ′µ(x) δ[G(□nT ′µ)] det �δG[□nT ′µ(x)] δU(y) � × exp � iS(n) YM[□nT ′µ] + 2i � d4n+4x Tr � □nJµ(x)□nT µ(x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (62) 8 Now we choose some normalization N(ξ) which is dependent on the gauge fixing parameter ξ such that 1 = N(ξ) � Dw(x) exp � − i � d4n+4x Tr �w2(x) 2ξ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (63) Then we relate the rotored field variable □nT ′µ(x) to □nT µ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' And inserting equa- tion(63) to (62), now we obtain, Z[□nJµ(x)] = � N(ξ) � DU � � Dw(x)D□nT µ(x) δ[G(□nT µ)] det �δG[□nT µ(x)] δU(y) � × exp � iS(n) YM[□nT µ] + 2i � d4n+4x Tr � □nJµ(x)□nT µ(x) � − i � d4n+4x Tr �w2(x) ξ �� , (64) where N(ξ) � DU is some constant and can be integrated out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Now employing the gauge condition in 58, in which the Dirac delta function picks out the term of ∂µ□nT µ(x), we obtain, Z[□nJµ(x)] ∝ � D□nT µ(x) det �δG[□nT µ(x)] δU(y) � × exp � iS(n) YM[□nT µ] + 2i � d4n+4x Tr � □nJµ(x)□nT µ(x) � − i ξ � d4n+4x Tr � ∂µ□nT µ(x)∂µ□nT µ(x) �� , (65) where the last term in equation (65) is the gauge fixing term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We define the gauge-fixed action as S(n) ξ,YM = � d4n+4x � − 1 2Tr Gn µνGµν n − 1 ξ Tr (∂µ□nT µ(x)∂µ□nT µ(x)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (66) Now we proceed to calculate the terms in the determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' First notice that δ□nT U µ = −1 g � ∂µα + ig � 1 2n□nT U µ , α �� = −1 gDn µα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (67) For G[□nTµ(x)] = g∂µ□nTµ(x), then δG[□nTµ(x)] = −∂µDn µα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (68) Since iα = δU · U −1, we have δG[□nT µ(x)] δU(y) = iδ4n+4(x − y)∂µDn µ · U(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (69) Since the determinant of U(y) is unity, it follows that det �δG[□nT µ(x)] δU(y) � = det � iδ4n+4(x − y)∂µDn µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (70) Now we can implement the Grassmann action as follow det(i∂µDn µ) = � D□nc∗D□nc exp � − 2i � d4n+4x Tr(□nc∗∂µDn µ□nc) � , (71) 9 where □nc∗, □nc are rotored transformed Grassmann fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore, the full sourced partition functional now gives Z[□nJµ(x)] ∝ � D□nT µ(x)D□nc∗(x)D□nc(x) × exp � iS(n) YM[□nT µ] + 2i � d4n+4x Tr � □nJµ(x)□nT µ(x) � − i ξ � d4n+4x Tr � ∂µ□nT µ(x)∂µ□nT µ(x) �� − 2i � d4n+4x Tr � □nc∗(x)∂µDn µ□nc(x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (72) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 Generalized Scalar field theory The quantization of generalized scalar field theory follows similarly to that of the abelian gauge field case in our previous paper [30], where we will not repeat the steps here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Generally speaking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' the quantum amplitude follows the form of ⟨□nφf(tf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='xxx)|e−i ˆH(tf−ti)|□nφi(ti,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='xxx)⟩ = � D□nφ(x) exp � i 4n � d4n+4x □nφ � −□ + m2 2 � □nφ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (73) The sourced partition functional is then given by Z[□nJ(x)] = � D□nφ(x) exp � i 4n � d4n+4x □nφ � −□ + m2 2 � □nφ+i � d4n+4x □nJ□nφ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (74) The normalized generating functional for the scalar field is Z[□nJ] = Z[□nJ] Z[0] = � D□nφ(x) exp � i 4n � d4n+4x □nφ � − □+m2 2 � □nφ + i � d4n+4x □nJ□nφ � � D□nφ(x) exp � i 4n � d4n+4x □nφ � − □+m2 2 � □nφ � (75) The free n-point correlation function is given by ⟨0|T□nφ(x1)□nφ(x2) · · · □nφ(xn)|0⟩ = 1 in δn δ□nJ(x1)δ□nJ(x2) · · · δ□nJ(xn)Z[□nJ] ���� □nJ=0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (76) which is evaluated to be ⟨0|T□nφ(x1)□nφ(x2) · · · □nφ(xn)|0⟩ = � D□n(x)□nφ(x1)□nφ(x2) · · · □nφ(xn) exp � i 4n � d4n+4x □nφ � − □+m2 2 � □nφ � � D□nφ(x) exp � i 4n � d4n+4x □nφ � − □+m2 2 � □nφ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (77) Using the same technique in [32] similar to the case of gauge field, one finds the Feynman propagator as ⟨0|T□nφ(x)□nφ(y)|0⟩ = � d4n+4p (2π)4n+4 i · 4n p4n(p2 − m2)e−ip·(x−y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (78) 10 Or in the most generalized case, for the case of inhomogeneous order, we have ⟨0|T□nφ(x)□kφ(y)|0⟩ = � d2n+2k+4p (2π)2n+2k+4 i · 2n+k p2n+2k(p2 − m2)e−ip·(x−y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (79) From 78, when n = 0 which is the unrotored case this restores us back to the original two point correlation and the Feynman propagator of the scalar fields, ⟨0|Tφ(x)φ(y)|0⟩ = � d4p (2π)4 i (p2 − m2)e−ip·(x−y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (80) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='3 Generalized Dirac field and Electrodynamics For the quantization of Dirac field under rotor mechanism, first we consider the quan- tum amplitude as follow: ⟨γµn···µ1∇µn···µ1ψ(tf,xxx)|e−i ˆH(tf−ti)|∇νn···ν1 ¯ψ(ti,xxx) γ†ν1···νn⟩ = � D[(∇νn···ν1 ¯ψ) γ†ν1···νn]D[γµn···µ1∇µn···µ1ψ] × exp � i � dDx (∇νn···ν1 ¯ψ) γ†ν1···νn(i/∂ − m) γµn···µ1∇µn···µ1ψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (81) The sourced partition functional is then given by Z[∇βn···β1 ¯J(x) γ†β1···βn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' γαn···α1∇αn···α1J(x)] = � D[(∇νn···ν1 ¯ψ) γ†ν1···νn]D[γµn···µ1∇µn···µ1ψ] × exp � i � dDx (∇νn···ν1 ¯ψ) γ†ν1···νn(i/∂ − m) γµn···µ1∇µn···µ1ψ + i � dDx(∇νn···ν1 ¯ψ) γ†ν1···νnγαn···α1∇αn···α1J(x) + i � dDx∇βn···β1 ¯J(x) γ†β1···βnγµn···µ1∇µn···µ1ψ � (82) And it is remarked that iS[(∇νn···ν1 ¯ψ) γ†ν1···νn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' γµn···µ1∇µn···µ1ψ] = �� dDxdDy(∇νn···ν1 ¯ψ(x)) γ†ν1···νniδD(x − y)(i/∂ − m)γµn···µ1∇µn···µ1ψ(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (83) Now perform fourier transform, ψ(x) = � dDp (2π)D ˜ψ(p)e−ip ˙x and ¯ψ(x) = � dDp (2π)D ¯˜ψ(p)e+ip ˙x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (84) 11 Equation 83 yields iS[(∇νn···ν1 ¯ψ) γ†ν1···νn, γµn···µ1∇µn···µ1ψ] = �� dDxdDy �� dDp (2π)D dDq (2π)D (in)(−i)npνn · · · pν1 ¯˜ψ(p)γ†ν1···νniδD(x − y)(/q − m)γµn···µ1 × qµn · · · qµ1 ˜ψ(q)e−ip·x+iq·y = � dDx �� dDp (2π)D dDq (2π)D pνn · · · pν1 ¯˜ψ(p)γν1† · · · γνn†i(/q − m)γµn · · · γµ1qµn · · · qµ1 ˜ψ(q)e−i(p−q)·x = �� dDp (2π)D dDq (2π)D (/p†)n ¯˜ψ(p)i(2π)DδD(p − q)(/q − m)/qn ˜ψ(q) = � dDp (2π)D ¯˜ψ(p)(/p†)ni(/p − m)(/p)n ˜ψ(p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (85) Therefore, we have the matrix element as ˜ M(p, q) = −i(2π)Dδ(p − q)(/p†)n(/q − m)(/q)n (86) Therefore, the momentum green’s function for Dirac field under rotor mechanism is ˜ M −1(p, q) = i (/p†)n(/q − m)(/q)n(2π)Dδ(p − q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (87) Hence, the Feynman propagator in momentum space is S(n) F (p) = i (/p†)n(/p − m)(/p)n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (88) Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' let’s evaluate the sourceless partition functional,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' which is Z[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 0] = � D[(∇νn···ν1 ¯ψ) γ†ν1···νn]D[γµn···µ1∇µn···µ1ψ] × exp � i � dDx (∇νn···ν1 ¯ψ) γ†ν1···νn(i/∂ − m) γµn···µ1∇µn···µ1ψ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (89) which is Z[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 0] = � D[(∇νn···ν1 ¯ψ) γ†ν1···νn]D[γµn···µ1∇µn···µ1ψ] × exp � i �� dDxdDy (∇νn···ν1 ¯ψ(x)) γ†ν1···νnδD(x − y)(i/∂ − m) γµn···µ1∇µn···µ1ψ(y) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (90) They by using the identity from Gaussian integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' � � n � j=1 dθ∗ jdθj � exp(−θ∗TMθ) = det(M) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (91) we obtain Z[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 0] = det[δD(x − y)(i/∂ − m)] (92) 12 Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' using the usual fact from Gaussian integral with sourced term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' � � n � j=1 dθ∗ jdθj � exp(−θ∗TMθ + η∗Tθ + θ∗Tη) = det(M) exp(η∗TM −1η) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (93) and with equations (82) and (83),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' the sourced partition functional for the Dirac field is evaluated to be Z[∇βn···β1 ¯J(x) γ†β1···βn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' γαn···α1∇αn···α1J(x)] = det[δD(x − y)(i/∂ − m)] exp � �� dDxdDy∇βn···β1 ¯J(x) γ†β1···βnM −1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' y)γαn···α1∇αn···α1J(y) � = Z[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 0] exp � �� dDxdDy∇βn···β1 ¯J(x) γ†β1···βnM −1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' y)γαn···α1∇αn···α1J(y) � (94) The normalized generating functional for the Dirac field is Z[∇βn···β1 ¯J(x) γ†β1···βn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' γαn···α1∇αn···α1J(x)] = Z[∇βn···β1 ¯J(x) γ†β1···βn ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' γαn···α1∇αn···α1J(x)] Z[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 0] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (95) The two point correlation function is given by ⟨0|T[γµn···µ1∇µn···µ1ψ(x1)][(∇νn···ν1 ¯ψ(x2)) γ†ν1···νn]|0⟩ = 1 i2 δ2Z[∇βn···β1 ¯J(x) γ†β1···βn , γαn···α1∇αn···α1J(x)] δ[γµn···µ1∇µn···µ1J(x1)]δ[(∇νn···ν1 ¯J(x2)) γ†ν1···νn] ���� γµn···µ1∇µn···µ1J(x1)=(∇νn···ν1 ¯J(x2)) γ†ν1···νn=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (96) The variational principle follows that δγαn···α1∇αn···α1J(x) δγµn···µ1∇µn···µ1J(y) = δD(x − y) (97) and δ∇βn···β1 ¯ψ(x) γ†β1···βn δ∇νn···ν1 ¯ψ(y) γ†ν1···νn = δD(x − y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (98) This would give ⟨0|T[γµn···µ1∇µn···µ1ψ(x1)][(∇νn···ν1 ¯ψ(x2)) γ†ν1···νn]|0⟩ = M −1(x, y) = � dDp (2π)D i (/p†)n(/p − m)(/p)ne−ip·(x−y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (99) For the inhomogeneous case, we will have, ⟨0|T[γµn···µ1∇µn···µ1ψ(x1)][(∇νk···ν1 ¯ψ(x2)) γ†ν1···νk]|0⟩ = � dDp (2π)D i (/p†)n(/p − m)(/p)k e−ip·(x−y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (100) From equation (99), when n = 0, this restores us back the fermion two-point correlation function and Feynman propagator ⟨0|Tψ(x) ¯ψ(y)|0⟩ = � dDp (2π)D i /p − me−ip·(x−y) (101) 13 To quantize the generalized electrodynamics under rotor mechanism, we first con- sider the sourceless functional as follow: Z[0, 0, 0] = � D[(∇νn···ν1 ¯ψ) γ†ν1···νn]D[γµn···µ1∇µn···µ1ψ]D□nT µ exp � iSQED � , (102) where SQED is the quantum electrodynamic action in 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The normalized generating functional for QED is ZQED[∇βn···β1 ¯J(x) γ†β1···βn , γαn···α1∇αn···α1J(x), □nJµ] = exp (i � dDzLint[source])Z[∇βn···β1 ¯J(x) γ†β1···βn , γαn···α1∇αn···α1J(x), □nJµ] {exp (i � dDzLint[source])Z[∇βn···β1 ¯J(x) γ†β1···βn , γαn···α1∇αn···α1J(x), □nJµ]}|sourced terms=0 , (103) where exp � i � d4z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Lint[source] � = exp � i � dDz 1 iδ∇βn···β1 ¯J(z) γ†β1···βn 1 iδγαn···α1∇αn···α1J(y) 1 iδ□nJµ(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (104) Then the full physical two-point correlation function is ⟨Ω|Tψ(x) ¯ψ(y)|Ω⟩ = 1 i2 δ2ZQED δ∇βn···β1 ¯J(x) γ†β1···βnδγαn···α1∇αn···α1J(y) ���� source terms=0 (105) with |Ω⟩ the physical vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4 A summary of Feynman propagators under rotor mecha- nism Using the path integral quantization technique approach, we can obtain the Feynman propagators under rotor mechanism ( n-rotors of □n operators) as follow: Scalar spin-0 boson: ∆(n) F (x − y) = � dDp (2π)D i · 4n p4n(p2 − m2)e−ip·(x−y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (106) Massless spin-1 gauge boson: D(n) µν (x − y) = � dDp (2π)D −i · 4ngµν p2+4n e−ip·(x−y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (107) Massless spin-1 gauge boson in Lorentz gauge: D(n) µν (x − y) = � dDp (2π)D −i · 4n p2+4n � gµν − (1 − ξ)pµpν p2 � e−ip·(x−y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (108) Massive spin-1 gauge boson: D(n) µν (x − y) = � dDp (2π)D −i · 4n p4n(p2 − M 2) � gµν − pµpν M 2 � e−ip·(x−y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (109) 14 Massive spin-1 gauge boson in Lorentz gauge D(n) µν (x−y) = � dDp (2π)D −i · 4n p4n � gµν p2 − M 2− 1 p2 − M 2 � 1−1 ξ �� 1 M 2 − p2 ξ � pµpν � e−ip·(x−y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (110) Dirac spin-1/2 fermion: SF(x − y) = � dDp (2π)D i (/p†)n(/p − m)(/p)ne−ip·(x−y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (111) Massless spin-1 gluon: Dab (n) µν (x − y) = δabD(n) µν (x − y) = � dDp (2π)D −iδab · 4n p2+4n � gµν − (1 − ξ)pµpν p2 � e−ip·(x−y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (112) Ghost field: Gab (n)(x − y) = � dDp (2π)D iδab p2+4ne−ip·(x−y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (113) 5 Generalized Higgs mechanism under rotor mech- anism The Higgs mechanism is responsible for the explanation of mass acquisition of gauge boson through the process of spontaneous symmetry breaking, it also explains how the fermions couple with the Higgs field to main mass [40, 41, 42, 43, 44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' In addition, it also predicts all possible Higgs interactions and decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Under rotor mechanism, the potential term is V = µ2 4n □nφ†□nφ + λ 16n(□nφ†□nφ)2 , (114) where µ2 < 0 and λ > 0 for spontaneous symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The Higgs action is S = � dDx � − 1 4n+1□nGµν□nGµν+ 1 4n(Dn µ□nφ)†(Dn µ□nφ)−µ2 4n □nφ†□nφ− λ 16n(□nφ†□nφ)2 � , (115) This Lagrangian is invariant under U(1) transformation, (□nφ) → (□nφ)′ = eiθ(□nφ) (116) When expressed in terms of two real rotored scalar fields □nφ1 and □nφ2, action reads V (□nφ1, □nφ2) = µ2 2 · 4n � (□nφ1)2 + (□nφ2)2� + λ 4 · 16n � (□nφ1)2 + (□nφ2)2�2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (117) The potential has an infinite minimum when (□nφ1)2 + (□nφ2)2 = −4n · µ2 λ = v2 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (118) 15 Upon the elimination of Goldstone boson by Unitary gauge and by perturbation of the non-zero vacuum expectation value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' we have □nφ(x) = 1 √ 2 � vn + □nh(x) � (119) Then after spontaneously symmetry breaking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' we have S = � dDx � − 1 4n+1□nGµν□nGµν + 1 2 · 4n � ∂µ − i 2ng□nTµ(vn + □nh) �� ∂µ + i 2ng□nTµ(vn + □nh) � − µ2 2 · 4n(vn + □nh)2 − λ 4 · 16n(vn + □nh)4 � = � dDx � − 1 4n+1□nGµν□nGµν + 1 2 · 4ng2v2 n□nTµ□nT µ + 1 2 · 4n∂µ□nh∂µ□nh − 1 4nλv2 n□nh□nh + 1 8ng2vn□nTµ□nT µ□nh + 1 2 · 8ng2□nTµ□nT µ□nh□nh − 1 8nλvn□nh□nh□nh − 1 4 · 16nλ□nh□nh□nh□nh (120) The mass term of the rotored massive gauge boson is identified as Mn = gvn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' and the mass term of the rotored Higgs boson is mn = √ 2λvn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' It is noted that when n = 0, the action (120) will restore back to the normal Higgs action with spontaneous symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' From the third line in (120), this gives the interactions between the rotored Higgs boson and rotored gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We can rewrite them in terms of the metric, 1 8ngMngµν□nT µ□nT ν□nh , 1 2 · 8ng2gµν□nT µ□nT ν□nh□nh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (121) In terms of vertex diagram, we have the following feynman rules, Figure 1: Feyman rules for rotored Higgs boson decaying into two rotored bosons (left) and rotored scattering process (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' From the fourth line in 120, this gives the rotored Higgs boson self interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The Feynman rules are as follow: 16 Figure 2: Feyman rules for third order and fourth order rotored Higgs boson interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 6 A discussion of Hierarchy Problem and the par- tial solution provided by the rotor mechanism It has been known that high-order derivative quantum field theory makes a good job in eliminating UV divergence [4, 5, 6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' In this section, we will investigate how the high-order-derivative quantum field theory by the rotor mechanism can suppress UV divergence in the quantum loop processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Although the standard model is renormal- izable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' the divergences in the Lagrangian can be treated by adding finite number of counter terms, the Hierarchy Problem which involves fine tunning of the Higgs mass remains a long-lasting problem in high-energy particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Recently, reference [1] generalizes Lee-Wick electrodynamics with high-order field derivatives to the Standard Model and offers a solution to the Hierarchy Problem to tame UV divergences in one- loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Here, we will see how high-order derivative field theory by rotor mechanism can eliminate the UV divergence in amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Let us have a revisit to the Hierarchy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For example, taking the 4-fields self-interaction of the Higgs boson the one-particle irreducible (1 PI) function can be easily calculated by integrating through the loop momentum k, ˜Γ(p) = −iλ2 8 � d4k (2π)4 i k2 − m2 H = −iλ2 128π2 � Λ2 − 2m2 H ln � Λ mH � + O �m2 H Λ4 � � + O(λ4), (122) which has a dominant quadratic divergence of high energy scale Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Hence, the renor- malization of the bare Higgs boson’s mass for the 4-field self interaction is, ˜m2 H = m2 H + i˜Γ(p) = m2 H + λ2 128π2 � Λ2 − 2m2 H ln � Λ mH � + O �m2 H Λ2 � � + O(λ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (123) It is noted that mH = µ above is the bare mass, while ˜mH is the physical, observed mass-the renormalized mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The full calculation of all quantum loop corrections from electroweak and Yukawa interaction to the nth-order loop takes the general form [46], ˜m2 H = m2 H + Λ2 16π2Cn(µ2 = m2 H) , (124) where Cn is the a polynomial expansion of the bare Higgs mass scale and it is a function of the Higgs self-coupling λ, electroweak couplings gW, g′ and the Yukawa coupling Y f ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' To the first-order loop calculation [46], ˜m2 H = m2 H + Λ2 32π2(4λ2 + 3g′2 + 9g2 W − 24Y f 2 ij ) + O(λ2, g′4, g4 W, Y f 4 ij ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (125) 17 The physical Higgs boson mass is measured to be ˜mH ≃ 126 GeV and hence ˜m2 H ∼ 104, GeV2, while the quantum corrections lead to the quadratic divergence of the Planck scale Λ ∼ 1019 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' From equation (125) we can see that the physical Higgs mass depends strictly on the strength of each of the coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The nature requires an extremely precise fine-tuning cancellation of the couplings and the bare Higgs mass to each nth order of quantum loop divergence by 1017 GeV, so as to obtain the low-energy, EW-scale Standard Model we observe today [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This is known as the Hierarchy Prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The Hierarchy Problem gives a strong motivation to search for New Physics (NP) particles with mass scales ΛNP > ΛEW such that they can cancel these loop divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' One important scheme is supersymmetry, which introduces new particles between bosonic and fermonic symmetry, such that each divergent bosonic loop has a fermionic counterpart, vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Since fermonic amplitude has an extra factor of minus sign, this can cancels the bosonic loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Using the rotor mechanism, we will see that the divergent loop processes are sup- pressed and this will lead to convergent result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The generic problematic 1-loop di- vergent process includes, for example, the self-correction of the Higgs boson by the W-boson loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The Feynman diagram is illustrated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Figure 3: Feyman diagram of self-energy correction of the Higgs boson by a W-boson loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The n-rotored amplitude is given by M(n) = g2 W(16)nM 2 W 64n(2π)D � dDk gαβgδγ gβδ − kβkδ/M 2 W k4n(k2 − M 2 W) gγα − (p − k)γ(p − k)α)/M 2 W (p − k)4n((p − k)2 − M 2 W) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (126) By expanding, we obtain M(n) = g2 WM 2 W 4n(2π)D � � dDk gαβgβδgδγgγα k4n(p − k)4n(k2 − M 2 W)((p − k)2 − M 2 W) − 1 M 2 W � dDk (p − k)α(p − k)α k4n(p − k)4n(k2 − M 2 W)((p − k)2 − M 2 W) − 1 M 2 W � dDk kαkα k4n(p − k)4n(k2 − M 2 W)((p − k)2 − M 2 W) + 1 M 4 W � dDk kδ(p − k)δkα(p − k)α k4n(p − k)4n(k2 − M 2 W)((p − k)2 − M 2 W) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (127) First, we can give a rough estimate of the amplitude by the leading term, M(n) 1 ∼ g2 WM 2 W 4n(2π)D � dDk gαβgβδgδγgγα k4n(p − k)4n(k2 − M 2 W)((p − k)2 − M 2 W) ∼ 21−DDg2 WM 2 W 4n√ πDΓ � D 2 � � Λ kD−1dk k8n+4 , (128) 18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='.where have have used the fact that of the integration of D-dimensional solid angle is given by � dΩD = 2πD/2 Γ(D/2) with Γ(x) the Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' When n = 0 and in four spacetime dimension D = 4 which is the unrotored case in 4D spacetime (which is just the normal case), we have logarithmic divergence M(0) 1 ∼ g2 WM 2 W 2π2 ln Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (129) Hence in the normal case, the self-correction of the Higgs boson by the W boson is logarithmic divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' But under rotor mechanism, by the second line of 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' we have M(n) 1 ∼ 21−DDg2 WM 2 W 4n√ πDΓ � D 2 � (D − 8n − 4) ΛD−8n−4 , (130) In our 4D universe, we simply have M(n) 1 ∼ − g2 WM 2 W 22n+4π2n 1 Λ8n (131) As Λ is a very large value, which can be up to Planck scale 1019 GeV, even for n = 1 M(1) 1 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For the second term, M(n) 2 = g2 W 4n(2π)D � dDk 1 k4n(p − k)4n−2(k2 − M 2 W)((p − k)2 − M 2 W) ∼ 21−Dg2 W 4n√ πDΓ � D 2 � � Λ kD−1dk k8n+2 = 21−Dg2 W 4n√ πDΓ � D 2 � (D − 8n − 2) ΛD−8n−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (132) For the unrotored case n = 0 and in D = 4, M(0) 2 ∼ g2 W 16π2Λ2 , (133) which contributes to a quadratic divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' But under rotor mechanism, M(n) 2 ∼ g2 W 2(2n+4)π2(1 − 4n)Λ2−8n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (134) For example for n = 1, M(1) 2 ∝ 1 Λ6 which is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For the third term, M(n) 3 = g2 W 4n(2π)D � dDk 1 k4n−2(p − k)4n(k2 − M 2 W)((p − k)2 − M 2 W) ∼ 21−Dg2 W 4n√ πDΓ � D 2 � � Λ kD−1dk k8n+2 = 21−Dg2 W 4n√ πDΓ � D 2 � (D − 8n − 2) ΛD−8n−2 , (135) 19 which has the same result as the M(n) 2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For the last term, M(n) 4 ∼ g2 W 4n(2π)DM 2 W � dDk (k · p − k2)2 k4n(p − k)4n(k2 − M 2 W)((p − k)2 − M 2 W) = g2 W 4n(2π)DM 2 W � dDk (k · p)2 − 2(k · p)k2 + k4 k4n(p − k)4n(k2 − M 2 W)((p − k)2 − M 2 W) ∼ g2 W 4n(2π)DM 2 W � dDk k2p2 − 2(k · p)k2 + k4 [k2(p − k)2]2n(k2 − M 2 W)((p − k)2 − M 2 W) = g2 W 4n(2π)DM 2 W � dDk 1 [k2(p − k)2]2n−1(k2 − M 2 W)((p − k)2 − M 2 W) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (136) From the second line to the third line, we have made the order approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' No- tice that (k · p)2 = kµpµkαpα = (kµpα)pµkα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Then notice that k2p2 = kµkµpαpα = (kµpα)pαkµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' They are obviously different but they have the same order in k and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' As here we are interested in order calculation, to simplify the analysis we make the above approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Then we have M(n) 4 ∼ 21−Dg2 W 4n√ πDΓ � D 2 � M 2 W � Λ kD−1dk k8n = 21−Dg2 W 4n√ πDM 2 WΓ � D 2 � (D − 8n) ΛD−8n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (137) For the unrotored case n = 0 and in D = 4, M(0) 4 ∼ g2 W 32π2Λ4 , (138) which is a seriously 4-th order divergent term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' But under rotor mechanism, M(n) 4 ∼ g2 W 2(2n+5)π2M 2 W(1 − 2n)Λ4−8n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (139) For example for n = 1, M(1) 2 ∝ 1 Λ4 which is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore, the amplitude is M(n) = M(n) 1 + M(n) 2 + M(n) 3 + M(n) 4 ∼ g2 W 4n+2π2Λ8n �M 2 W n + Λ2 1 − 4n + Λ4 4M 2 W(1 − 2n) � (140) in D = 4 spacetime dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We can see that when n = 1 under rotor mechanism, the amplitude is already convergent in the high-energy regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore, we see how the rotor mechanism can remove infinities in loop diagrams at high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Next, consider the self-correction of Higgs Boson by a fermion loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The Feynman diagram is as follow: Figure 4: Feyman diagram of self-energy correction of the Higgs boson by a fermionic loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 20 EASince the vertex factor is just −i mf vn , the n-rotored amplitude is given by M(n) = − m2 f v2 n(2π)D � dDk Tr � 1 (/k†)n(/k − mf)(/k)n 1 (/p† − /k†)n(/p − /k − mf)(/p − /k)n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (141) The minus sign is due to the Feynman rule for the fermion loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Now using the following fact that γµγµ = 4ID (γµγµ)† = 4I† D 㵆ㆠµ = 4ID .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (142) Equivalently, the amplitude in (141) reads M(n) = − m2 f v2 n(2π)D � dDk Tr �(/k)n(/k + mf)(/k†)n k4n(k2 − m2 f) (/p − /k)n(/p − /k + mf)(/p† − /k†)n (p − k)4n((p − k)2 − m2 f) � (143) The rough estimation of (141) gives M(n) ∼ − m2 f v2 n(2π)D � dDk Tr � 1 /k2+4n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (144) For the unrotored case, n = 0 and in 4D spacetime, we have M(n) ∼ − m2 f v2(2π)D Λ2 (145) which has a quadratic divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' But under rotor mechanism we have M(n) ∼ − m2 f v2 n(2π)D Λ2−4n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (146) When n = 1, we can see that M(n) ∝ 1 Λ2, which is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Thus the divergence is eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Next, we consider the self-energy correction of the Higgs boson by the 3rd order self-interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The vertex factor is i 2·8n m2 H vn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The Feynman diagram is Figure 5: Feyman diagram of self-energy correction of the Higgs boson by a 3rd-order self interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The amplitude for such process is M(n) = (16)nm4 H 4 · 64n(2π)Dv2 n � dDk 1 k4n(k2 − m2 H) 1 (p − k)4n((p − k)2 − m2 H) (147) 21 By rough estimation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' we have M(n) ∼ m4 H 2D+2n+1√ πDv2 n � Λ kD−1dk k4+8n ∼ m4 H 2D+2n+1√ πD(D − 8n − 4)v2 n ΛD−8n−4 (148) For the unrotored case in 4D spacetime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' n = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' we have logarithmic divergence M(0) ∼ m4 H 32π2v2 n ln Λ (149) For the general case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' if we consider D = 4 for our universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' we have M(n) ∼ − m4 H 2D+4n+4√ πDnv2 n 1 Λ8n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (150) For n = 1, the matrix element goes for M(n) ∝ 1 Λ8, which drops very quickly to zero for large Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore, the problem of divergence vanishes in the rotor mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' In general, the above method of power counting used by the above example applies to other diagrams with higher number of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The generic form takes the following: M(n) ∼ � · · � dDk1dDk2 · · · dDkL k4n i · · · k4n l k2 i · · · k2 l (/k† j)n(/kj)n+1 · · · (/k† p)n(/kp)n+1 (151) Let S be the superficial degree of divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' And let L be the number of loops, NB be the number of internal spin-1 gauge boson propagator(s), NH be the number of internal Higgs boson propagator(s) and Nf be the number of internal fermion propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Then we have S = (power of momentum in numerator)- (power of momentum in demoninator ) = DL − (2NB + 4NBn) − (2NH + 4NHn) − (Nf + 2Nfn) = DL − (2NB + 2NH + Nf) − 2n(2NB + 2NH + Nf) = DL − (2n + 1)(2NB + 2NH + Nf) (152) Naively, we expect a diagram to have a divergence proportional to LambdaS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We expect logarithmic divergence log Λ when S = 0, and no divergence when S < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This takes place when the following inequality holds, DL − (2n + 1)(2NB + 2NH + Nf) < 0 n > DL (2NB + 2NH + Nf) − 1 2 , (153) and also we demand n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' In addition, as the number of loops satisfy the following equation, L = I − V + 1 , (154) where I is the number of internal lines, V is the number of vertices, it follows that L = NB + NH + Nf − V + 1 (155) 22 Let us illustrate by an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Consider the following 3rd-order self-energy correction of the Higgs boson, Figure 6: Feyman diagram of self-energy correction of the Higgs boson by a 3-loop interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The Feynman amplitude can be easily written down as M(n) = −M 3 Wm2 fm2 H 2(2π)3Dv3 n ��� dDk dDq dDl gβαgγδgµν × gαγ − (p − k)α(p − k)γ/M 2 W (p − k)4n((p − k)2 − M 2 W) gδµ − (p − k − q)δ(p − k − q)µ/M 2 W (p − k − q)4n((p − k − q)2 − m2 W) gνβ − kνkβ/M 2 W k4n(k2 − M 2 W) × � 1 q4n(q2 − m2 H) �2 1 (p − q)4n(p − q)2 − m2 H × Tr � 1 (/l ∗)n(/l − m)(/l n) 1 (/q∗ − /l ∗)n(/q − /l − m)(/q − /l)n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (156) To compute the superficial degree of UV divergence, we utilize the power counting formula in (152).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' First notice that NB = 3, NH = 3, Nf = 2, V = 6, then L = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Hence it follows that S = 3D − 14(2n + 1) = 3D − 28n − 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We can see that when n = 0 for the unrotored case and D = 4 spacetime S = −2, thus this diagram is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We can also see that in D = 4 spacetime S = −2 − 28n < 0, so there is no UV divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore, in this section, we have seen how infinities arise from high-energy ends of loop diagrams are tamed by higher-order-derivatives fields under rotor mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' In such way, the UV divergences of from the 1-loop self-correction of Higgs boson propagator are eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 7 Explicit calculation of 1-loop self-correction of Higgs boson under rotor mechanism and a dis- cussion on the Hierarchy Problem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='1 Correction by the 3rd order Higgs vertex In this section, we will demonstrate the explicit calculation of 1-loop self correction of Higgs boson by third order Higgs vertex under rotor mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We will show that although the rotor mechanism can tame ultraviolet UV divergence at high ener- gies, it introduces Infra-red IR divergence at low energies in 4-dimensional spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' However, we will demonstrate that the rotor mechanism can remove all infinities in 23 higher dimension using the example of 1-loop self-correction of Higgs Boson by third order Higgs vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' According to equation (147), the n-th order rotored amplitude is proportional to the following integral, M(n) ∝ � dDk 1 k4n(k2 − m2 H) 1 (p − k)4n((p − k)2 − m2 H) (157) Define the integral as I(n) = � dDk 1 k4n(k2 − m2 H) 1 (p − k)4n((p − k)2 − m2 H) (158) We will first demonstrate the calculation of n = 1 case, then the general n rotored case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Using the Feynman parameter method given in [36] 1 A1A2 · · · An = � 1 0 · · � 1 0 dx1dx2 · · · dxn δ � n � i=1 xi − 1 � (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (x1A1 + x2A2 + · · · xnAn)n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (159) For the n = 1 rotored case, we can rewrite the Feynman integral using equation (159) 1 k4(p − k)4(k2 − m2 H)((p − k)2 − m2 H) = � 1 0 � 1 0 � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsdudvδ(x + y + r + s + u + v − 1) × 5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' [xk2 + yk2 + r(p − k)2 + s(p − k)2 + u(k2 − m2 H) + v((p − k)2 − m2 H)]6 (160) Now let us evaluate the terms in the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' xk2 + yk2 + r(p − k)2 + s(p − k)2 + u(k2 − m2 H) + v((p − k)2 − m2 H) = xk2 + yk2 + r(p2 − 2p · k + k2) + s(p2 − 2p · k + k2) + u(k2 − m2 H) + v(p2 − 2p · k + k2 − m2 H) = (x + y + r + s + u + v)k2 + (r + s + v)p2 − 2(r + s + v)(p · k) − (u + v)m2 H = k2 − 2(r + s + v) + (r + s + v)p2 − (u + v)m2 H = (k − (r + s + v)p)2 − (u + v)m2 H , (161) where in the fourth line we used the fact that x + y + r + s + u + v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We define new variable l = k − (r + s + v)p and the effective mass ∆ = (u + v)m2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Also it is clearly that dDk = dDl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore, now the integral reads I(1) = � dDk � 1 0 � 1 0 � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsdudv × δ(x + y + r + s + u + v − 1) 120 [(k − (r + s + v)p)2) − (u + v)m2 H]6 (162) Then we have I(1) = � dDl � 1 0 � 1 0 � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsdudvδ(x + y + r + s + u + v − 1) 120 (l2 − ∆)6 (163) 24 Now we carry out Wick’s rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The Wick’s rotation simply amount to substitute l0 = il0 E and lll = lllE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Then the integral now reads I(1) = 120i(−1)6 � dΩ4 � ∞ 0 dlE � 1 0 � 1 0 � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsdudv × δ(x + y + r + s + u + v − 1) l3 E (l2 E + ∆)6 = 120i(2π2) Γ(2) � ∞ 0 dlE � 1 0 � 1 0 � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsdudv × δ(x + y + r + s + u + v − 1) l3 E (l2 E + ∆)6 (164) Next using the integral fact that � ∞ 0 x3 (x2 + a)6dx = − a + 5x2 40(a + x2)5 ���� ∞ 0 = 1 40a4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (165) Therefore, now the integral reads I(1) = 240iπ2 40m8 H � 1 0 � 1 0 � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsdudvδ(x + y + r + s + u + v − 1) 1 (u + v)4 = 6iπ2 m8 H � 1 0 du � 1−u 0 dv � 1−u−v 0 ds � 1−s−u−v 0 dr � 1−r−s−u−v 0 dy 1 (u + v)4 = 6iπ2 m8 H lim u→0 � − 1 4 + 1 36u2 − 1 4u − 1 2 ln u � = −3iπ2 2m8 H + infinity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (166) We can see that the integral diverges when u → 0, this is referred as the infra-red IR divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Finally, we obtain the amplitude as M(1) = 3i 512π2m4 Hv2 n + infinity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (167) Therefore, we can see that although the rotor mechanism resolves the high-energy infinity problem, it introduces infra-red IR divergences at low energy in D = 4 space- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The General n case To calculate the amplitude of general n-rotored case, we need to use the following Feynman integral formula given in [36] 1 Am1 1 Am2 2 · · Amn n = � 1 0 · · � 1 0 � n � i=1 dxi � δ � n � i=1 xi − 1 � �n i=1 xmi−1 i � �n i=1 xiAi ��n i=1 mi × Γ(m1 + m2 + · · · + mn) Γ(m1)Γ(m2) · · · Γ(mn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(168) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='The n-rotored integral is calculated to be ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='I(n) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dDk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(k2)2n(k2 − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='H) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='[(p − k)2]2n((p − k)2 − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='H) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dDk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dxdydrdsδ(x + y + r + s − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='x2n−1y2n−1r0s0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='[xk2 + y(p − k)2 + r(k2 − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='H) + s((p − k)2 − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='H)]4n+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ(4n + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ(2n)Γ(2n)Γ(1)Γ(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dDk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='δ(x + y + r + s − 1)dxdydrds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='x2n−1y2n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='[(k − (y + s)p)2 − (r + s)m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='H]4n+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ(4n + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ2(2n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(169) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='This time take l = k − (y + s)p and the effective mass ∆(r + s) = m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Then we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='I(n) = Γ(4n + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ2(2n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dDl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dxdydrdsδ(x + y + r + s − 1) x2n−1y2n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(l2 − ∆)4n+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= i(−1)4n+2Γ(4n + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ2(2n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dlE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dΩD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dxdydrdsδ(x + y + r + s − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× x2n−1y2n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='lD−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(l2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E + ∆)4n+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= 2i(−1)4n+2πD/2Γ(4n + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ2(2n)Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dlE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dxdydrdsδ(x + y + r + s − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× x2n−1y2n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='lD−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(l2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E + ∆)4n+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(170) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Next we calculate the lE integral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dlE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='lD−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(l2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E + ∆)4n+2 = Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4n + 2 − D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2∆4n+2− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 Γ(4n + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='if 0 < D < 8n + 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (171) Not D is out of the range, the integral does not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' now the I(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='integral is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='I(n) = i(−1)4n+2πD/2Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4n + 2 − D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ2(2n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dxdydrdsδ(x + y + r + s − 1)x2n−1y2n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='∆4n+2− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= i(−1)4n+2πD/2Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4n + 2 − D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ2(2n)m8n+4−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dxdydrdsδ(x + y + r + s − 1) x2n−1y2n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(r + s)4n+2− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= i(−1)4n+2πD/2Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4n + 2 − D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ2(2n)m8n+4−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r−s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dy(1 − y − r − s)2n−1y2n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(r + s)4n+2− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= i(−1)4n+2πD/2Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4n + 2 − D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ2(2n)m8n+4−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ2(2n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ(4n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='ds(1 − r − s)4n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(r + s)4n+2− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= i(−1)4n+2πD/2Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4n + 2 − D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(−1)4nΓ(4n)m8n+4−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='drB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='1 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 4n, 2 − D 2 � , (172) 26 where B � 1 − 1 r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 4n, 2 − D 2 � = � 1− 1 r 0 t4n−1(1 − t)1− D 2 dt (173) is the incomplete Beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Hence, we need to compute the following integral I(n)(n, D) = iπD/2Γ � 4n + 2 − D 2 � Γ(4n)m8n+4−D H � 1 0 dr � 1− 1 r 0 t4n−1(1 − t)1− D 2 dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (174) Let us define the integral J(n, D) = � 1 0 dr � 1− 1 r 0 t4n−1(1 − t)1− D 2 dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (175) First we analyse the case when n = 1/4, which is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Although this corresponds to an unphysical rotor number, it is worth to study this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' It is found that, when n = 1/4, the integral is computed to be J �1 4, D � = 2 D − 4 � 2 D − 2 − 1 � if D > 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (176) The plot is as follow: Figure 7: The plot of J � 1 4, D � against spacetime dimension D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' When n = 1/4, so the converged amplitude takes place only when −iM(1/4) �1 4, D � = Γ � 3 − D 2 � 2D+2√ 2πDm2−D H v2 1 4 � 2 D − 4 � 2 D − 2 − 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (177) The amplitude against spacetime dimension is plotted as follow: 27 I,d 4 5 4 6 7 8 9 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0Figure 8: The plot of −iM � 1 4, D � against spacetime dimension D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' In particular, in our living spacetime dimension D = 4, using L-Hospital rule, J �1 4, 4 � = lim D→4 2 D − 4 � 2 D − 2 − 1 � = −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (178) We thus obtain a convergent amplitude for n = 1/4, D = 4 iM(1/4) �1 4, 4 � = m2 H 64 √ 2π2v2 1 4 = m2 H 128π2v2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='044 × 10−4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (179) where we have used the fact that v2 n = 4nv2 in equation (118) and v = 246 GeV is the vacuum expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The amplitude is inversely proportional to the square of the Higgs boson mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' It is noted that the amplitude is ill-defined when D = 2, for which it diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Now, we investigate the physical n, D ∈ Z+ cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We find that for each integer n, the minimum dimension D that gives convergent J integral and amplitude iM(n) is given by the formula Dmin(n) = 8n + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (180) For example, for n = 1 case, Dmin(1) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' That means the lowest spacetime dimension for which the rotor mechanism to give finite result is nine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For n = 1, if D < 9, the result will be all infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For example, for the n = 2 case, Dmin(2) = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' That means the lowest spacetime dimension for which the rotor mechanism to give finite result is seventeen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For n = 2, if D < 17, this will be all infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This applies to higher positive integer n values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' And we conclude that for the Higgs boson self-correction by 3rd order Higgs-self interaction, the minimum physical dimension for taming the divergences (for both IR and UV) of the lowest-order rotor model (n = 1) is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Now we numerically work out the pair of (n, d) which contributes the finite con- vergent J(n, D) integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The result is plotted in figure (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 28 iM 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 100 200 300Figure 9: The plot of J(n, D) against rotor number n and spacetime dimension D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The full amplitude is given by −iM(n)(n, D) = Γ � 4n + 2 − D 2 � 2D+4n+2Γ(4n) √ πDm8n−D H v2J(n, D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (181) In general, in the nominator, the Gamma function Γ � 4n + 2 − D 2 � imposes constraints such that the amplitude is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' First notice that Γ(−m) is infinity for m equal to positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore, such condition occurs when D = 2k and 4n + 2 − k ≤ 0, so it takes place when k ≥ 4n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Together with D ≥ 8n + 1 and D < 8n + 4, the conditions for finite amplitude are � � � � � � � � � D < 8n + 4 D ≥ 8n + 1 k ≱ 4n + 2 D ̸= 2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (182) For example n = 1, we have k ≥ 6 , D ≥ 12, this will give Γ(−m), which is divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' So for n = 1, the required (n, D) pair for finite amplitude is (1, 9), (1, 10), (1, 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For example n = 2, the required (n, D) pair for finite amplitude is (2, 17), (2, 18), (2, 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For example, −iM(1)(1, 9) = Γ � 3 2 � 215Γ(4)π9/2m−1 H v2J(1, 9) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='931 × 10−11 GeV−1 , (183) which is a very small value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore, under the rotor mechanism with rotor index n = 1, at D = 9, the divergence is removed in high dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Thus this demonstrates how high-order-derivative field theory under rotor mechanism can remove both UV and IR divergences in high spacetime dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This shows that the Hierarchy problem can be solved by rotor mechanism in high spacetime dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 29 2 n 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='003 J(n,d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='000 60 40 20 d7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 Correction by the W boson Next, we will study the self-energy correction of Higgs boson by massive W boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Using the result in (136), we have the amplitude as M(n) = g2 WM 2 W 4n(2π)D � � dDk D k4n(p − k)4n(k2 − M 2 W)((p − k)2 − M 2 W) − 1 M 2 W � dDk (p − k)2 k4n(p − k)4n(k2 − M 2 W)((p − k)2 − M 2 W) − 1 M 2 W � dDk k2 k4n(p − k)4n(k2 − M 2 W)((p − k)2 − M 2 W) + 1 M 4 W � dDk [k · (p − k)]2 k4n(p − k)4n(k2 − M 2 W)((p − k)2 − M 2 W) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (184) From this we define the following integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' First from the first line, we have I(n) 1 = � dDk D (k2)2n[(p − k)2]2n(k2 − M 2 W)((p − k)2 − M 2 W) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (185) From the second line we have I(n) 2 = 1 M 2 W � dDk 1 (k2)2n[(p − k)2]2n−1(k2 − M 2 W)((p − k)2 − M 2 W) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (186) From the third line we have I(n) 3 = 1 M 2 W � dDk 1 (k2)2n−1[(p − k)2]2n(k2 − M 2 W)((p − k)2 − M 2 W) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (187) For the last time, we use the following standard trick to convert the inner product of k · p to squared values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The following identity is used: a · b = 1 2[a2 + b2 − (a − b)2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (188) Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' I(n) 4 = 1 M 4 W � dDk (k · p − k2)2 (k2)2n[(p − k)2]2n(k2 − M 2 W)((p − k)2 − M 2 W) = 1 M 4 W � dDk [ 1 2 � (k2 + p2) − (k − p)2� − k2]2 (k2)2n[(p − k)2]2n(k2 − M 2 W)((p − k)2 − M 2 W) = 1 4M 4 W � dDk (p2 − k2)2 − 2(p2 − k2)(p − k)2 + (p − k)4 (k2)2n[(p − k)2]2n(k2 − M 2 W)((p − k)2 − M 2 W) = 1 4M 4 W � dDk(p4 − 2p2k2 + k4) − 2(p2 − k2)(p − k)2 + (p − k)4 (k2)2n[(p − k)2]2n(k2 − M 2 W)((p − k)2 − M 2 W) (189) For each term in the last line,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' we define six more integrals: I(n) 4a = p4 4M 4 W � dDk 1 (k2)2n[(p − k)2]2n(k2 − M 2 W)((p − k)2 − M 2 W) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (190) I(n) 4b = −2p2 4M 4 W � dDk 1 (k2)2n−1[(p − k)2]2n(k2 − M 2 W)((p − k)2 − M 2 W) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (191) 30 I(n) 4c = 1 4M 4 W � dDk 1 (k2)2n−2[(p − k)2]2n(k2 − M 2 W)((p − k)2 − M 2 W) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (192) I(n) 4d = −2p2 4M 4 W � dDk 1 (k2)2n[(p − k)2]2n−1(k2 − M 2 W)((p − k)2 − M 2 W) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (193) I(n) 4e = 2 4M 4 W � dDk 1 (k2)2n−1[(p − k)2]2n−1(k2 − M 2 W)((p − k)2 − M 2 W) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (194) I(n) 4f = 1 4M 4 W � dDk 1 (k2)2n[(p − k)2]2n−2(k2 − M 2 W)((p − k)2 − M 2 W) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (195) Then now the computation becomes straight forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The computation is similar to that of the 3rd order Higgs vertex case, therefore we will just show the final result in case some special issues are noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' First of all, I(n) 1 = iDπD/2Γ � 4n + 2 − D 2 � Γ(4n)m8n+4−D W � 1 0 B � 1 − 1 r, 4n, 2 − D 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (196) if 8n + 1 ≤ D < 8n + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Secondly, I(n) 2 = iπD/2Γ � 4n + 1 − D 2 � Γ(4n − 1)m8n+4−D W � 1 0 B � 1 − 1 r, 4n − 1, 2 − D 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (197) if 8n − 1 ≤ D < 8n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Thirdly, I(n) 3 = iπD/2√πΓ � 4n + 1 − D 2 � 42n−1Γ(2n)Γ � 2n − 1 2 � m8n+4−D W � 1 0 B � 1 − 1 r, 4n − 1, 2 − D 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (198) if 8n − 1 ≤ D < 8n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Then I(n) 4a = p4 4M 4 W × I(n) 1 D = iπD/2Γ � 4n + 2 − D 2 � p4 4Γ(4n)m8n+8−D W � 1 0 B � 1 − 1 r, 4n, 2 − D 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (199) if 8n + 1 ≤ D < 8n + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Then I(n) 4b = − 2p2 4M 2 W I(n) 3 = − 2iπD/2√πΓ � 4n + 1 − D 2 � p2 42nΓ(2n)Γ � 2n − 1 2 � m8n+6−D W � 1 0 B � 1−1 r, 4n−1, 2−D 2 � (200) 31 For the I(n) 4c integral, the situation is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' It it is worth to go through the whole computation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='I(n) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4c = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4M 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='W ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(l2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E + ∆)4n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='i(−1)4nΓ(4n)(2πD/2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4M 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='WΓ(2n − 2)Γ(2n)Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dxdydrdsδ(x + y + r + s − 1)x2n−3y2n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× 2nΓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4n − D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='∆4n− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 Γ(4n + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='i(−1)4nnπD/2Γ(4n)Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4n − D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ(2n − 2)Γ(2n)Γ(4n + 1)M 8n+4−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dxdydrdsδ(x + y + r + s − 1) x2n−3y2n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(r + s)4n− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='i(−1)4nnπD/2Γ(4n)Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4n − D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ(2n − 2)Γ(2n)Γ(4n + 1)M 8n+4−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r−s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dy(1 − y − r − s)2n−3y2n−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(r + s)4n− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='i(−1)4nnπD/2Γ(4n)Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4n − D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ(2n − 2)Γ(2n)Γ(4n + 1)M 8n+4−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='ds(1 − r − s)4n−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(r + s)4n− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ(2n)Γ(2n − 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ(4n − 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= i(−1)4nnπD/2Γ(4n)Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='4n − D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ(4n + 1)Γ(4n − 2)M 8n+4−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='ds(1 − r − s)4n−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(r + s)4n− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(201) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='if 0 < D < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Now we evaluate the last integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This gives � 1−r 0 ds(1 − r − s)4n−3 (r + s)4n− D 2 = Γ(4n − 2) (1 − r)4n−2r4n− D 2 2 ˜F1 � 1, 4n − D 2 , 4n − 1, 1 − 1 r � , (202) where 2 ˜F1 is the regularized generalized hypergeometric (2, 1) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We will give the definition of generalized hypergeometric function and regularized generalized hy- pergeometric (p, q) function here, which is formally denoted as pFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' It is given that pFq(a1, · · · ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' b1, · · · bq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' z) = ∞ � n=0 �p k=1(ak)(n) �q k=1(bk)(n) zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' , (203) where the Pochhammer symbol for rising factorial is used, (a)(0) = 1, · · · , (a)(n) = a(a + 1)(a + 2) · · · (a + n − 1) = n−1 � l=0 (a + l) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (204) The corresponding regularized generalized hypergeometric (p, q) function is defined by p ˜Fq(a1, · · · ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' b1, · · · bq : z) = pFq(a1, · · · ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' b1, · · · bq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' z) Γ(b1) · · · Γ(bq) (205) For our case, we are concerned with regularized generalized hypergeometric (2, 1) func- tion, which is 2 ˜F1(a1, a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' b1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' z) = 2F1(a1, a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' b1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' z) Γ(b1) = 1 Γ(b1) ∞ � n=0 (a1)(n)(a2)(n) (b1)(n) zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (206) 32 Therefore, for our case in (202), we have 2 ˜F1 � 1, 4n − D 2 , 4n − 1, 1 − 1 r � = 1 Γ(4n − 1) ∞ � m=0 1(m)� 4n − D 2 �(m) m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (4n − 1)(m) � 1 − 1 r �m = 1 Γ(4n − 1) ∞ � m=0 � 4n − D 2 �(m) (4n − 1)(m) � 1 − 1 r �m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (207) Therefore, the last line of (201) yields I(n) 4c = i(−1)4nnπD/2Γ(4n)Γ � 4n − D 2 � Γ(4n + 1)Γ(4n − 1)M 8n+4−D W � 1 0 dr 1 (1 − r)4n−2r4n− D 2 2 ˜F1 � 1, 4n − D 2 , 4n − 1, 1 − 1 r � = i(−1)4nnπD/2Γ(4n)Γ � 4n − D 2 � Γ(4n + 1)Γ(4n − 1)M 8n+4−D W � 1 0 dr 1 (1 − r)4n−2r4n− D 2 ∞ � m=0 � 4n − D 2 �(m) (4n − 1)(m) � 1 − 1 r �m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (208) Thus this completes the mathematical calculation of I(n) 4c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Then we continue to calculate the integral I(n) 4d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This is easy, I(n) 4d = −2p2 4M 2 W I(n) 2 = −iπD/2Γ � 4n + 1 − D 2 � p2 2Γ(4n − 1)m8n+6−D W � 1 0 B � 1 − 1 r, 4n − 1, 2 − D 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (209) Next, I(n) 4e = i(−1)4nπD/2√πΓ(4n)Γ(4n − 2)Γ � 4n − D 2 � 24n−4Γ(2n − 1)Γ(4n + 1)Γ � 2n − 1 2 � M 8n+4−D W � 1 0 dr 2 ˜F1 � 1, 4n − D 2 , 4n − 1, 1 − 1 r � (1 − r)4n−2r4n− D 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (210) Finally, I(n) 4f = i(−1)4nΓ(4n)Γ � 4n − D 2 � 4Γ(4n + 1)M 8n+4−D W � 1 0 dr 2 ˜F1 � 1, 4n − D 2 , 4n − 1, 1 − 1 r � (1 − r)4n−2r4n− D 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (211) The amplitude of the Higgs-self energy correction by W boson is hence M(n) = g2M 2 W 4n(2π)D � I(n) 1 − I(n) 2 − I(n) 3 + I(n) 4a + I(n) 4b + I(n) 4c + I(n) 4d + I(n) 4e + I(n) 4f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (212) It is worth to analyse the properties of these integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We can see that unlike the case of the self-energy correction for the Higgs-boson from the 3rd order Higgs interaction, here the amplitude is ill-defined for the n = 1/4 case, as we have the denominator of Γ(4n − 1) in I(n) 2 , I(n) 4c , I(n) 4d , I(n) 4e , I(n) 4f , which diverges when n = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Now, in order to get an idea of what pairs of (n, D) give us convergent result, first we would like to analyse the integral in (208).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Define p(n, D, r) = 1 (1 − r)4n−2r4n− D 2 2 ˜F1 � 1, 4n − D 2 , 4n − 1, 1 − 1 r � (213) Then we plot p(n, D, r) with different n and D values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 33 Figure 10: The plot of p(n, D, r) with of n = 1 and different values of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Figure 11: The plot of p(n, D, r) with of n = 2 and different values of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 34 — p(1, 1, r) p(2, 1, r) 5 p(3, 1, r) — p(4, 1, r) 4 — p(5, 1, r) 3 — p(6, 1, r) — p(7, 1, r) p(8, 1, r) p(9, 1, r) — p(10, 1, r) 2 3 4 5 p(10, 2, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='07 p(11, 2, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='06 p(12, 2, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='05 p(13, 2, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='04 p(14, 2, r) p(15, 2, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='03 p(16, 2, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='02 p(17, 2, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='01 p(18, 2, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 — p(19, 2, r)Figure 12: The plot of p(n, D, r) with of n = 3 and different values of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Figure 13: The plot of p(n, D, r) with of n = 4 and different values of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 35 p(20, 3, r) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='×10-8 p(21, 3, r) p(22, 3, r) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× 10-8 p(23, 3, r) p(24, 3, r) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' × 10-8 p(25, 3, r) p(26, 3, r) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' × 10-8 p(27, 3, r) p(28, 3, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 — p(29, 3, r)— p(28, 4, r) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5× 10-12 p(29, 4, r) p(30, 4, r) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× 10-12 p(31, 4, r) p(32,4,r) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× 10-13 p(33, 4, r) p(34, 4, r) p(35, 4, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0Figure 14: The plot of p(n, D, r) with of n = 5 and different values of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We study the general properties of the p(n, D, r) function and the integration of it from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Let’s first study the (1, D) case in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For each set of (1, D), we compute p(1, D, r) and � 1 0 p(1, D, r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (n, D) p(n, D, r) � 1 0 p(1, D, r)dr (1, 1) − 2 3r−3/2 + 2 5r−5/2 + 4 15 ∞ (1, 2) (r−1)2 2r2 ∞ (1, 3) − 2 √r + 2 3r−3/2 + 4 ∞ (1, 4) 1 r + ln r − 1 ∞ (1, 5) 2√r �� 1 r − 1 �2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='333 (1, 6) r − 1 − ln r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 (1, 7) 2 3 √r � r + 2 � 1 r − 3 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='267 (1, 8) 1 2(r − 1)2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='167 (1, 9) 2 15 � 2 � 1 r − 5r + 3r2 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='1143 (1, 10) 1 6(r − 1)2(2r + 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0833 We can see that for the n = 1 case, the integral only converges when D > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' And in general, for higher dimension D the integral decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This pattern also applies for other n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For each n, there exists a threshold dimension Dmin(n) such that below this dimension the integral diverges, upon computation we find that Dmin(n) = 8n − 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (214) When n = 1, the threshold is Dmin = 5, which is confirmed by the above table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' And when n = 2, Dmin = 13, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' From all the plots above, we can see that larger n with larger D results in smaller values of the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This shows that we will obtain smaller amplitudes with increasing (n, D) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 36 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='x10 p(36, 5, r) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' × 10-18 p(37, 5, r) p(38, 5, r) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' × 10-18 p(39, 5, r) p(40,5,r) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' × 10-18 p(41, 5, r) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' × 10-18 p(42, 5, r) p(43, 5, r) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0The full amplitude is now given by M(n) = ig2 W 22n+D√ πDM 8n+2−D W �Γ � 4n + 2 − D 2 � Γ(4n) � D + p4 4M 4 W � � 1 0 B � 1 − 1 r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 4n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 2 − D 2 � − Γ � 4n + 1 − D 2 �� 1 Γ(4n − 1) � 1 + p2 2M 2 W � + √π 42nΓ(2n)Γ � 2n − 1 2 � � 4 + 2p2 M 2 W �� × � 1 0 B � 1 − 1 r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 4n − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 2 − D 2 � + (−1)4nΓ(4n)Γ � 4n − D 2 � Γ(4n + 1) � n Γ(4n − 1) + √πΓ(4n − 2) Γ(2n − 1)Γ � 2n − 1 2 � + 1 4 � × � 1 0 dr 2 ˜F1 � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 4n − D 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 4n − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 1 − 1 r � (1 − r)4n−2r4n− D 2 � (215) The full amplitude is a function of n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='D and incoming momentum p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' M(n) ≡ M(n)(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' It is found that for the unrotored case (n = 0) in our D = 4 spacetime, this amplitude diverges to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This is obvious due to the term Γ(4n) appearing in the denom- inator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This result is expected as in the Standard Model, one-loop correction of the Higgs boson by W boson diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' However, as by the above results, we see that the integrals converge only in different set of (n, D) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We want to find the common (n, D) values for all the integrals, which amount to solve the three inequalities: � � � � � � � � � 8n + 1 ≤ D < 4n + 4 8n − 1 ≤ D < 8n + 2 0 ≤ D < 8 D ≥ 8n − 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (216) However, there is no solution for this set of inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore, the amplitude in (215) means to be diverged in any particular set of (n, D) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='3 Correction by the fermion loop Finally, we remain to calculate the self-energy correction of Higgs Boson by fermion loop under rotor mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This is essentially to compute the amplitude calculation in (143).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We need to develop some Dirac algebra and trace identities involving 㵆, as well as γµ and 㵆 together in the most general D spacetime dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' First notice that the complex conjugate of the γµ matrix is defined by 㵆 = γ0γµγ0† = γ0γµγ0 , (217) where we have used the fact that γ0† = γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Then we immediately obtain our first identity : Tr(㵆) = Tr(γ0γµγ0) = Tr(γ0γ0γµ) = Tr(γµ) = 0 , (218) where we have used the trace identity Tr(ABC) = Tr(CBA) and the fact that (γ0)2 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Next, we prove the Dirac algebra for hermitian gamma matrices {㵆, γν†} = 2ηµνID .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (219) 37 The proof is straight forward, {㵆, γν†} = 㵆γν† + γν†γµ† = γ0γµγ0γ0γνγ0 + γ0γνγ0γ0γµγ0 = γ0(γµγν + γνγµ)γ0 = γ0(2ηµνID×D)γ0 = 2ηµνγ0ID×Dγ0 = 2ηµνID×D , (220) where we have used the fact that (γ0)2 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This result implies /k/k = /k†/k† = k2ID×D and /k/p = /k†/p† = (k · p)ID×D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (221) Next we will prove the following identity: Tr(γµγρ†) = D(2η0µη0ρ − ηµρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (222) By the definition (217), it is noticed that Tr(γµγρ†) = Tr(γµγ0γργ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Then using the identity of Tr(γµγνγργσ) = D(ηρσηµν − ηνσηµρ + ηµσηνρ), by putting ν and σ equal to 0, then we get Tr(γµγρ†) = Tr(γµγ0γργ0) = D(ηρ0ηµ0 − η00ηµρ + ηµ0η0ρ) = D(2η0µη0ρ − ηµρ) , (223) where η00 = 1 in our diag(+ − −−) metric convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Next we verify the following computationally, Tr(㵆γνγρ†) = Tr(γ0γµγ0γνγ0γργ0) = Tr(γµγν†γρ) = 0 (224) that the trace of a mixture of odd number of γµ and 㵆 matrices is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Next, another useful identity we need to use later is Tr(γµγν†γργσ†) = Tr(γµγ0γνγ0γργ0γσγ0) = D(8η0µη0νη0ρη0σ − 2η0ρη0σηµν − 2η0µη0σηνρ − 2η0νη0ρηµσ − 2η0µη0νηρσ + ηµνηρσ − ηµρηνσ + ηµσηνρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (225) For simplicity, we will consider the n = 1 case first in (143).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Now we can evaluate the nominator of (143) (/k)(/k + mf)(/k†)(/p − /k)(/p − /k + mf)(/p† − /k†) = (k2/k† + mf/k/k†) [(p − k)2(/p† − /k†) + mf(/p − /k)(/p† − /k†)] = k2(p − k)2(k · p − k2)ID×D + mf(p − k)2/k/k†(/p† − /k†) + mfk2/k†(/p − /k)(/p† − /k†) + m2 f/k/k†(/p − /k)(/p† − /k†) = k2(p − k)2(k · p − k2)ID×D + mf(p − k)2/k(k · p − k2) + mfk2/k†(/p/p† − /p†/k† − /k†/p† + /k/k†) + m2 f/k/k†(/p/p† − /p†/k† − /k†/p† + /k/k†) (226) Now we need to take the trace of the above expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' First notice that, Tr(/k) = kµTr(γµ) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (227) 38 Then using the identity of 224, Tr(/k†/p/q†) = kµpνqρTr(㵆γνγρ†) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(228) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Therefore using these two results and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='k2(p − k)2(k · p − k2)ID×D + mf(p − k)2/k(k · p − k2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='+ mfk2/k†(/p/p† − /p†/k† − /k†/p† + /k/k†) + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='f/k/k†(/p/p† − /p†/k† − /k†/p† + /k/k†) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= Dk2(p − k)2(k · p − k2) + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='fTr(/k/k†/p/p† − /k/k†/p/k† − /k/k†/k/p†) + /k/k†/k/k†) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= Dk2(p − k)2(k · p − k2) + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='f(kµkνpρpσ − kµkνpρkσ − kµkνkρpσ + kµkνkρkσ)Tr(γµγν†γργσ†) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= Dk2(p − k)2(k · p − k2) + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='f(kµkνpρpσ − kµkνpρkσ − kµkνkρpσ + kµkνkρkσ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× D(8η0µη0νη0ρη0σ − 2η0ρη0σηµν − 2η0µη0σηνρ − 2η0νη0ρηµσ − 2η0µη0νηρσ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='+ ηµνηρσ − ηµρηνσ + ηµσηνρ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= Dk2(p − k)2(k · p − k2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='− 2D(k0)2p2 + 8Dk0p0k2 − 2D(p0)2k2 + Dk2p2 + 8D(k0)2(k · p) − 4Dk0p0(k · p) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='− 2Dk2(k · p) − 8D(k0)k2 + Dk4 − 16D(k0)3p0 + 8D(k0)2(p0)2 + 8D(k0)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (229) Therefore, we have at least 13 integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' But some of them vanish due to the odd parity, we will look into details when we come across them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The integrals are much more technically difficult than the previous ones, as now here we have to also integrate terms regarding k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' With some observation, first we define the generic integral I(a, b, D) = � dDk 1 (k2)a[(p − k)2]b(k2 − m2 f)((p − k)2 − m2 f) = 2iπD/2Γ � a + b + 2 − D 2 � Γ(a + b)m2(a+b+2)−D f � 1 0 drB � 1 − 1 r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' a + b, 2 − D 2 � (230) if 0 < D < 2a+2b+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' In addition, the integral of the beta function is also convergent for some threshold D values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Define the first integral as I(1) 1 = � dDk Dk2(p − k)2(k · p − k2) k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) = D 2 � dDk (p2 − k2) − (p − k)2 k2(p − k)2(k2 − m2 f)((p − k)2 − m2 f) = D 2 � p2I(1, 1, D) − I(0, 1, D) − I(1, 0, D) � (231) The range for each integral is: for I(1, 1, D), 4 < D < 8, for I(1, 0, D) and I(0, 1, D), 2 < D < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore the overlap D for the first integral is D = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The second integral is defined to be I(1) 2 = −2Dp2 � dDk (k0)2 k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) (232) Then using the old trick of Feynman parameters, we have I(1) 2 = −2Dp2Γ(6) Γ2(2)Γ2(1) � dDk � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsδ(x+y+r+s−1) (k0)2xy ([k − (y + s)p)2 − (r + s)m2 f]6 (233) 39 Note that again we have l = k−(y+s)p, ∆ = (r+s)m2 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' And we have k0 = l0+(y+s)p0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' therefore the integral becomes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' I(1) 2 = −240Dp2 � dDl � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsδ(x + y + r + s − 1) (k0)2xy (l2 − ∆)6 = −240Dp2 � dDl � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsδ(x + y + r + s − 1)xy(l0 + (y + s)p)2 (l2 − ∆)6 = −240Dp2 � dDl � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsδ(x + y + r + s − 1) × xy((l0)2 + 2l0(y + s)p + (y + s)2p2) (l2 − ∆)6 (234) Next we carry out Wick’s rotation and perform the substitution of l0 = il0 E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' I(1) 2 = −(−1)6240iDp2 � ∞ 0 dlE � dΩD � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsδ(x + y + r + s − 1) × xy[lD−1 E � − (l0 E)2 + 2il0 E(y + s)p + (y + s)2p2)] (l2 E + ∆)6 (235) Now this integral involve the integrating on l0 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' To tackle this integral, we concern integration on D−sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We set the spherical coordinates (l0 E, l1 E, · · · lD−1 E ) in the Euclidean l momentum space as follows: � � � � � � � � � � � � � � � � � � � � � l0 E = lE sin ϕ1 sin ϕ2 · · · sin ϕD−2 cos ϕD−1 l1 E = lE sin ϕ1 sin ϕ2 · · · sin ϕD−2 sin ϕD−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' ln−3 E = lE sin ϕ1 sin ϕ2 cos φ3 ln−2 E = lE sin ϕ1 cos ϕ2 ln−1 E = lE cos ϕ1 , (236) where l2 E = (l0 E)2 + (l1 E)2 + · · · + (lD−1 E )2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (237) The differential volume is given by dDlE = lD−1 E sinD−2 ϕ1 sinD−3 ϕ2 · · · sin ϕD−2dlEdϕ1dϕ2 · · · dϕD−1 (238) And the differential solid angle is given by dΩD = sinD−2 ϕ1 sinD−3 ϕ2 · · · sin ϕD−2dϕ1dϕ2 · · · dϕD−1 , (239) where ϕ1, ϕ2, · · · , ϕD−2 ∈ [0, π] and ϕD−1 ∈ [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We first carry out the integral for 40 the first term involving (l0 E)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='I(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2a = 240iDp2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dlE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dΩD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dxdydrdsδ(x + y + r + s − 1)xylD−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(l0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(l2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E + ∆)6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= 240iDp2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dlE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='· · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dϕ1dϕ2 · · · dϕD−1 sinD−2 ϕ1 sinD−3 ϕ2 · · · sin ϕD−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dxdydrdsδ(x + y + r + s − 1)xylD−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(lE sin ϕ1 sin ϕ2 · · · sin ϕD−2 cos ϕD−1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(l2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E + ∆)6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= 240iDp2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='sinD ϕ1dϕ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='sinD−1 ϕ2dϕ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='sinD−2 ϕ3dϕ3 · · · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='sin3 ϕD−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='cos2 ϕD−1dϕD−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dxdydrdsδ(x + y + r + s − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dlE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='xylD+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(l2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E + ∆)6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(240) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='In general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' � π 0 sinm x dx = √πΓ � m+1 2 � Γ � m 2 + 1 � if m > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (241) Also,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dlE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='lD+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(l2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='E + ∆)6 = π(D − 8)(D − 6)(D − 4)(D − 2)D csc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� πD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='7680∆5− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(242) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Then we get ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='I(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2a = iDp2π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='32m10−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='sinD−k+1 ϕkdϕk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� � 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dxdydrdsδ(x + y + r + s − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='xy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(r + s)5− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× (D − 8)(D − 6)(D − 4)(D − 2)D csc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='�πD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= D2(D − 8)(D − 6)(D − 4)(D − 2)p2πD/2+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='32 sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� πD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='m10−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D−k+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D−k+3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r−s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dy(1 − y − r − s)y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(r + s)5− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= D2(D − 8)(D − 6)(D − 4)(D − 2)p2πD/2+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='32 sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� πD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='m10−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='ds(1 − r − s)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(r + s)5− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(243) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='if −2 < D < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Our of this range the integral will diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Now we calculate the last integral � 1−r 0 ds(1 − r − s)3 (r + s)5− D 2 = 2 � r3 D − 2 − 3r2 D − 4 + 3r D − 6 + 1 8 − D � r D 2 −4 + 96 (D − 8)(D − 6)(D − 4)(D − 2) (244) And finally � 1 0 dr � 1−r 0 ds(1 − r − s)3 (r + s)5− D 2 = 96 (D − 6)(D − 4)(D − 2)D (245) 41 if D > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' It is noted that if D ≤ 6, the integral does not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore we get I(1) 2a = 4ip2πD/2+1D(D − 8) sin � πD 2 � Γ � D+2 2 � m10−D f (246) in the overall range of 6 < D < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The next term to integrate will be I(1) 2b = (−2i)(240)Dp2 � ∞ 0 dlE � dΩD � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsδ(x+y+r+s−1) l0 E(y + s) (l2 E + ∆)6 (247) Since l0 E = lE sin ϕ1 sin ϕ2 · · · sin ϕD−2 cos ϕD−1, when we integrate over dΩD where will be a vanishing term of � 2π 0 cos ϕD−1dϕD−1 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (248) Therefore the whole integral vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' In general, since if the integral is odd � dDl (2π)D lm(l0) = 0 (249) for any integer m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Thus when we have odd order of k0 in the integral, the integral must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Next we have the I(1) 2c integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' which is I(1) 2c = −240iDp4 � ∞ 0 dlE � dΩD � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsδ(x + y + r + s − 1)lD−1 E xy(y + s)2 (l2 E + ∆)6 = −240iDp2 � ∞ 0 dlE � · · � dϕ1dϕ2 · · · dϕD−1 sinD−2 ϕ1 sinD−3 ϕ2 · · · sin ϕD−2 × � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsδ(x + y + r + s − 1)xy(y + s)2lD−1 E (l2 E + ∆)6 (250) And as � ∞ 0 dlE lD−1 E (l2 E + ∆)6 = −π(D − 10)(D − 8)(D − 6)(D − 4)(D − 2) csc � πD 2 � 7680∆6− D 2 (251) if 0 < D < 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='I(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2c = ip4(D − 10)(D − 8)(D − 6)(D − 4)(D − 2)DπD/2+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='32 sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� πD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='m12−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D−k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D−k+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r−s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dy(1 − y − r − s)y(y + s)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(r + s)6− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= ip4(D − 10)(D − 8)(D − 6)(D − 4)(D − 2)D · 2πD/2+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='32 sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� πD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='m12−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='ds(1 − r − s)3(3(r − 1)2 − 4(r − 1)s + 3s2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(r + s)6− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= ip4π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 +1(D − 10)(D2 − 2D + 24) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='6(D + 2) sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� πD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='m12−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(252) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='if D > 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' When D < 8 the integral does not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' So the I(1) 2c integral is converged in 8 < D < 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore, for the I(1) integral to be convergent, the spacetime dimension constraint is 6 < D < 10 and 8 < D < 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The solution is D = 9 which is unique for the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The third integral is defined to be I(1) 3 = 8Dp0 � dDk (k0)k2 k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) = 0 (253) as the integral is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The fourth integral is defined to be I(1) 4 = −2D(p0)2 � dDk k2 k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) = −2D(p0)2I(1, 2) (254) which only converges when 6 < D < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The fifth integral is defined to be I(1) 5 = Dp2 � dDk k2 k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) = Dp2I(1, 2) (255) which only converges when 6 < D < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The sixth integral is defined to be I(1) 6 = 8D � dDk (k0)2(k · p) k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) = 4D � dDk (k0)2[(p2 − k2) − (p − k)2] k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) (256) Then we further define three more integrals, I(1) 6a = 4Dp2 � dDk (k0)2 k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) (257) which takes the same form as I(1) 2a except the front constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We have I(1) 6a = −2I(1) 2a , which is defined in the overall range of 6 < D < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Next we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' I(1) 6b = −4D � dDk (k0)2 k2(p − k)4(k2 − m2 f)((p − k)2 − m2 f) (258) Using the similar technique as above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' this amounts to give I(1) 6b = −(−1)596iD � ∞ 0 dlE � dΩD � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsδ(x + y + r + s − 1) × y[lD−1 E � − (l0 E)2 + 2il0 E(y + s)p + (y + s)2p2)] (l2 E + ∆)5 = iπD/2+1D(D − 6) 8 sin � πD 2 � Γ � D+2 2 � m8−D f + 2iπD/2+1p2(D − 8)(D2 + 2D + 24) 3(D + 2) sin � πD 2 � Γ � D 2 � m10−D f (259) for 6 < D < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Hence only when D = 7 this integral is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 43 Finally, we have the integral of I(1) 6c , I(1) 6c = −4D � dDk (k0)2 k4(p − k)2(k2 − m2 f)((p − k)2 − m2 f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (260) Using the similar technique as above, this amounts to give, I(1) 6c = −(−1)524iD � ∞ 0 dlE � dΩD � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsδ(x + y + r + s − 1) × x[lD−1 E � − (l0 E)2 + 2il0 E(y + s)p + (y + s)2p2)] (l2 E + ∆)5 = 2iπD/2+1(D − 8)(D − 6) sin � πD 2 � Γ � D+2 2 � m8−D f − 2iπD/2+1(D − 6)(D − 8)(D2 + 8) (D + 2)(D + 4) sin � πD 2 � Γ � D 2 � m8−D f (261) if 4 < D < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The seventh integral vanishes as it is an odd integral, I(1) 7 = −4Dp0 � dDk k0(k · p) k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (262) The eighth integral is defined to be I(1) 8 = −2D � dDk k2(k · p) k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) = −D � dDk (p2 − k2) − (p − k)2 k2(p − k)4(k2 − m2 f)((p − k)2 − m2 f) = −D(p2I(1, 2, D) − I(0, 2, D) − I(1, 1, D)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (263) The range for each integral to be finite is: for I(1, 2, D), 6 < D < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For I(0, 2, D), 4 < D < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' For I(1, 1, D), 4 < D < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore the overall range is 6 < D < 8, which is D = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The ninth integral vanishes as it is an odd integral, I(1) 9 = −8D � dDk (k0)k2 k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (264) The tenth integral is defined to be I(1) 10 = D � dDk k4 k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) = DI(0, 2, D), (265) which is in the range of 4 < D < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The eleventh integral vanishes as it is an odd integral, I(1) 11 = −16Dp0 � dDk (k0)3 k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (266) The twelfth integral is defined to be I(1) 12 = 8D(p0)2 � dDk (k0)2 k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) = −4I(1) 2 (p0)2 p2 (267) 44 with D = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The thirteenth integral is defined to be I(1) 13 = 8D � dDk (k0)4 k4(p − k)4(k2 − m2 f)((p − k)2 − m2 f) = 960D � dDl � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsδ(x + y + r + s − 1)xy((l0) + (y + s)p)4 (l2 − ∆)6 = 960D � dDl � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsδ(x + y + r + s − 1) × xy((l0)4 + 4(l0)3(y + s)p + 6(l0)2(y + s)2p2 + 6(l0)(y + s)3p3 + (y + s)4p4) (l2 − ∆)6 (268) The integrals involving (l0)3 and (l0) vanish as they are odd integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We subdivide I(1) 13 into three more integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='I(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='13a = 960D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� 1−r−s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='dy(1 − y − r − s)y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(r + s)4− D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='= − 3iπD/2+1D(D − 6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� πD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� D+4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='m8−D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='(269) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='for 4 < D < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Next, we have I(1) 13b = 5760iDp2 � π 0 sinD ϕ1 � π 0 sinD−1 ϕ2 · · · � π 0 sin3 ϕD−2dϕD−2 � 2π 0 cos ϕD−1dϕD−1 × � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsδ(x + y + r + s − 1) � ∞ 0 dlE xy(y + s)2lD+1 E (l2 E + ∆)6 = 72iπD/2+1D(D − 8)p2 sin � πD 2 � Γ � D+2 2 � m10−D f (270) for 6 < D < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 45 Finally, we have I(1) 13c = 960iDp4 � · · � dϕ1dϕ2 · · · dϕD−1 sinD−2 ϕ1 sinD−3 ϕ2 · · · sin ϕD−2 × � 1 0 � 1 0 � 1 0 � 1 0 dxdydrdsδ(x + y + r + s − 1) � dlE xy(y + s)4lD−1 E (l2 E + ∆)6 = −4iπD/2+1p4(D − 10)(D4 + 20D2 + 240D + 2304) 5(D + 2)(D + 4)(D + 6) sin � πD 2 � Γ � D 2 � m12−D f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (271) for 8 < D < 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Therefore, we have completed all the calculations of the integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The total am- plitude would be M(n) = − m2 f v2 n(2π)D � Integrals .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' (272) Since from the above calculations, each integral corresponds to a specific range of D or specific value of D such that the integral converges, we see that there is no common D for all the integrals to be convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The rotor mechanism can only save the divergence for each term of the calculation with higher dimension D one by one in the W-boson self-energy correction diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 8 Conclusion In conclusion, we have applied the rotor mechanism and quantization to the standard model of particle physics, which naturally generates high-order derivative quantum fields in the standard model’s Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Upon path integral quantization, we de- velop feynman propagators of scalar particles, gauge boson and Dirac fermion under rotor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' When the rotor index is n = 0, this restores to the original standard model case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Finally, we explicitly calculate quantum amplitudes of the one-loop self- energy correction diagrams of the Higgs Boson under rotor mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This include the correction by the Higgs-self interaction, W-boson and fermion respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We find that the rotor mechanism can generally remove the UV divergences, however, IR divergence is arisen at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We discover that the rotor model is able to remove infinities (both UV and IR), or suppress divergences arise from the case of the Higgs-self interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' We find that the minimum spacetime dimension D for n = 1 rotor index is 9, and for n = 2 is 17, and so on with a general formal of Dmin = 8n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' This suggests that the rotor mechanism can remove infinities (both UV and IR) arise from simple integral calculation, thus give a new way to partially solve the Hierarchy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' However, for diagrams with more complicated integrals, such as the W-boson loop correction and fermion loop correction, due to specific dimension range arise from each integral term, there does not exist a general D that can cure the divergence all at once for specific n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' More future work has to be done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Grinstein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' O’Connell, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' The Lee-Wick standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' D 77, 025012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 46 [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Lee and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Wick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Negative metric and the unitarity of the S-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Nuclear Physics B, Vol 9, Issue 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 209-243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Lee and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Wick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Finite Theory of Quantum Electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' D 2, 1033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Podolsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' A Generalized Electrodynamics Part I—Non-Quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 62, 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 1942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Podolsky and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Kikuchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' A Generalized Electrodynamics Part II-Quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 1944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' [6] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Podolsky and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Kikuchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Auxiliary Conditions and Electrostatic Interaction in Generalized Quantum Electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 67, 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 1945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Podolsky and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Schwed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Review of a Generalized Electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 20, 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 1948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Montgomery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' Relativistic 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} +page_content=' 49' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FOT4oBgHgl3EQf2zTm/content/2301.12944v1.pdf'} diff --git a/N9FRT4oBgHgl3EQfHDeZ/content/tmp_files/2301.13487v1.pdf.txt b/N9FRT4oBgHgl3EQfHDeZ/content/tmp_files/2301.13487v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d5fbdd8b63560463033c7cd3ae170f1bb853221 --- /dev/null +++ b/N9FRT4oBgHgl3EQfHDeZ/content/tmp_files/2301.13487v1.pdf.txt @@ -0,0 +1,2210 @@ +Published as a conference paper at ICLR 2023 +ADVERSARIAL +TRAINING +OF +SELF-SUPERVISED +MONOCULAR +DEPTH +ESTIMATION +AGAINST +PHYSICAL-WORLD ATTACKS +Zhiyuan Cheng +Purdue University +cheng443@purdue.edu +James Liang +Rochester Institute of Technology +jcl3689@rit.edu +Guanhong Tao +Purdue University +taog@purdue.edu +Dongfang Liu∗ +Rochester Institute of Technology +dongfang.liu@rit.edu +Xiangyu Zhang∗ +Purdue University +xyzhang@cs.purdue.edu +ABSTRACT +Monocular Depth Estimation (MDE) is a critical component in applications such +as autonomous driving. There are various attacks against MDE networks. These +attacks, especially the physical ones, pose a great threat to the security of such sys- +tems. Traditional adversarial training method requires ground-truth labels hence +cannot be directly applied to self-supervised MDE that does not have ground- +truth depth. Some self-supervised model hardening techniques (e.g., contrastive +learning) ignore the domain knowledge of MDE and can hardly achieve optimal +performance. In this work, we propose a novel adversarial training method for +self-supervised MDE models based on view synthesis without using ground-truth +depth. We improve adversarial robustnessagainstphysical-worldattacksusingL0- +norm-bounded perturbation in training. We compare our method with supervised +learning based and contrastive learning based methods that are tailored for MDE. +ResultsontworepresentativeMDEnetworksshowthatweachievebetterrobustness +against various adversarial attacks with nearly no benign performance degradation. +1 +INTRODUCTION +Monocular Depth Estimation (MDE) is a technique that estimates depth from a single image. It +enables 2D-to-3D projection by predicting the depth value for each pixel in a 2D image and serves +as a very affordable replacement for the expensive Lidar sensors. It hence has a wide range of +applications such as autonomous driving (Karpathy, 2020), visual SLAM (Wimbauer et al., 2021), +and visual relocalization (von Stumberg et al., 2020), etc. In particular, self-supervised MDE gains +fast-growing popularity in the industry (e.g., Tesla Autopilot (Karpathy, 2020)) because it does not +require the ground-truth depth collected by Lidar during training while achieving comparable accu- +racy with supervised training. Exploiting vulnerabilities of deep neural networks, multiple digital- +world (Zhang et al., 2020; Wong et al., 2020) and physical-world attacks (Cheng et al., 2022) against +MDE have been proposed. They mainly use optimization-based methods to generate adversarial ex- +amples to fool the MDE network. Due to the importance and broad usage of self-supervised MDE, +these adversarial attacks have posed a great threat to the security of applications such as autonomous +driving, which makes the defense and MDE model hardening an urgent need. +Adversarial training (Goodfellow et al., 2014) is the most popular and effective way to defend ad- +versarial attacks. However, it usually requires ground truth labels in training, making it not directly +applicable to self-supervised MDE models with no depth ground truth. Although contrastive learn- +ing gains a lot of attention recently and has been used for self-supervised adversarial training (Ho & +Nvasconcelos, 2020; Kim et al., 2020), it does not considerthedomainknowledgeofdepth estimation +and can hardly achieve optimal results (shown in Section 4.2). In addition, many existing adversarial +training methods do not consider certain properties of physical-world attacks such as strong pertur- +∗Corresponding authors. +1 +arXiv:2301.13487v1 [cs.CV] 31 Jan 2023 + +Published as a conference paper at ICLR 2023 +(a) Conceptual illustration. +(b) Different Views. +Figure 1: Self-supervised adversarial training of MDE with view synthesis. +bations. Hence in this paper, we focus on addressing the problem of hardening self-supervised MDE +models against physical-world attacks without requiring the ground-truth depth. +A straightforward proposal to harden MDE models is to perturb 3D objects in various scenes and +ensure the estimated depths remain correct. However, it is difficult to realize such adversarial train- +ing. First, 3D perturbations are difficult to achieve in the physical world. While one can train the +model in simulation, such training needs to be supported by a high-fidelity simulator and a powerful +scene rendering engine that can precisely project 3D perturbations to 2D variations. Second, since +self-supervised MDE training does not have the ground-truth depth, even if realistic 3D perturba- +tions could be achieved and used in training, the model may converge on incorrect (but robust) depth +estimations. In this paper, we propose a new self-supervised adversarial training method for MDE +models. Figure 1a provides a conceptual illustration of our technique. A board A printed with the +2D image of a 3D object (e.g., a car) is placed at a fixed location (next to the car at the top-right +corner). We use two cameras (close to each other at the bottom) Ct and Cs to provide a stereo +view of the board (images It and Is in Figure 1b). Observe that there are fixed geometric relations +between pixels in the two 2D views produced by the two respective cameras such that the image in +one view can be transformed to yield the image from the other view. Intuitively, It can be acquired +by shifting Is to the right. Note that when two cameras are not available, one can use two close-by +frames in a video stream to form the two views as well. During adversarial training, camera Ct takes +a picture It of the original 2D image board A. Similarly, camera Cs takes a picture Is of the board +too (step ‚). The bounding box of the board A is recognized in It and the pixels in the bounding box +corresponding to A are perturbed (step ƒ). Note that these are 2D perturbations similar to those in +traditional adversarial training. At step „, the perturbed image It+perturbations is fed to the MDE +model to make depth estimation, achieving a 3D projection of the object. Due to the perturbations, a +vulnerable model generates distance errors as denoted by the red arrow between A and the projected +3D object in Figure 1a. At step …, we try to reconstruct It from Is. The reconstruction is parameter- +ized on the cameras’ relative pose transformations and the estimated distance of the object from the +camera. Due to the distance error, the reconstructed image Is→t (shown in Figure 1b) is different +from It. Observe that part of the car (the upper part inside the red circle) is distorted. In comparison, +Figure 1b also shows the reconstructed image Iben +s→t without the perturbation, which is much more +similar to It. The goal of our training (of the subject MDE model) is hence to reduce the differences +betweenoriginalandreconstructedimages.Theaboveprocessisconceptual,whosefaithfulrealization +entails a substantial physical-world overhead. In Section 3, we describe how to avoid the majority +of the physical-world cost through image synthesis and training on synthesized data. +While traditional adversarial training assumes bounded perturbations in L2 or L∞ norm (i.e., mea- +suring the overall perturbation magnitude on all pixels), physical-world attacks are usually un- +bounded in those norms. They tend to be stronger attacks in order to be persistent with environ- +mental condition variations. To harden MDE models against such attacks, we utilize a loss function +that can effectively approximate the L0 norm (measuring the number of perturbed pixels regardless +of their perturbation magnitude) while remaining differentiable. Adversarial samples generated by +minimizing this loss can effectively mimic physical attacks. We make the following contributions: +• We develop a new method to synthesize 2D images that follow physical-world constraints (e.g., +relative camera positions) and directly perturb such images in adversarial training. The physical +world cost is hence minimized. +2 + +Distance Error +Reconstructed View Is-t +3D Projection +① +Loss +① +2D image A +of the object ++Adv. Noise +View Is +Depth Estimation +View It + perturbation +Camera Pose Trans +CtIt +Is +Distorted +Is-t +jben +s-tPublished as a conference paper at ICLR 2023 +• Our method utilizes the reconstruction consistency from one view to the other view to enable +self-supervised adversarial training without the ground-truth depth labels. +• We generate L0-bounded perturbations with a differentiable loss and randomize the camera and +objectsettingsduringsynthesistoeffectivelymimicphysical-worldattacksandimproverobustness. +• We evaluate the method and compare it with a supervised learning baseline and a contrastive +learning baseline, adapted from state-of-the-art adversarial contrastive learning (Kim et al., 2020). +Results show that our method achieves better robustness against various adversarial attacks with +nearly no model performance degradation. The average depth estimation error of an adversarial +object with 1/10 area of perturbation is reduced from 6.08 m to 0.25 m by our method, better +than 1.18 m by the supervised learning baseline. Moreover, the contrastive learning baseline +degrades model performance a lot. Our physical-world experiments video is available at https: +//youtu.be/_b7E4yUFB-g. +2 +RELATED WORKS +Self-supervised MDE. Due to the advantage of training without the depth ground truth, self- +supervised MDE has gained much attention recently. In such training, stereo image pairs and/or +monocular videos are used as the training input. Basically, two images taken by camera(s) from +adjacent poses are used in each optimization iteration. A depth network and a pose network are +used to estimate the depth map of one image and the transformation between the two camera poses, +respectively. With the depth map and pose transformation, it further calculates the pixel correspon- +dence across the images and then tries to rearrange the pixels in one image to reconstruct the other. +The pose network and the depth network are updated simultaneously to minimize the reconstruction +error. Garg et al. (2016) first propose to use color consistency loss between stereo images in train- +ing. Zou et al. (2018) enable video-based training with two networks (one depth network and one +pose network). Many following works improve the self-supervision with new loss terms (Godard +et al., 2017; Bian et al., 2019; Wang et al., 2018; Yin & Shi, 2018; Ramamonjisoa et al., 2021; Yang +et al., 2020) or include temporal information (Wang et al., 2019; Zou et al., 2020; Tiwari et al., 2020; +Watson et al., 2021). Among them, Monodepth2 (Godard et al., 2019) significantly improves the +performance with several novel designs such as minimum photometric loss selection, masking out +static pixels, and multi-scale depth estimation. Depthhints (Watson et al., 2019) further improves it +via additional depth suggestions obtained from stereo algorithms. While such unsupervised training +is effective, how to improve its robustness against physical attack remains an open problem. +MDE Attack and Defense. Mathew et al. (2020) use a deep feature annihilation loss to launch +perturbation attack and patch attack. Zhang et al. (2020) design a universal attack with a multi- +task strategy and Wong et al. (2020) generate targeted adversarial perturbation on images which +can alter the depth map arbitrarily. Hu & Okatani (2019) propose a defense method against pertur- +bation attacks by masking out non-salient pixels. It requires another saliency prediction network. +For physical-world attacks, Cheng et al. (2022) generates a printable adversarial patch to make the +vehicle disappear. To the best of our knowledge, we are the first work focusing on improving the +robustness of self-supervised MDE models against physical-world attacks. +Adversarial Robustness. It is known that deep neural networks (DNN) are vulnerable to adversar- +ial attacks. Imperceptible input perturbations could lead to model misbehavior (Szegedy et al., 2013; +Madry et al., 2018; Moosavi-Dezfooli et al., 2016). Typically, adversarial training is used to improve +the robustness of DNN models. It uses both benign and adversarial examples for training (Madry +et al., 2018; Carlini & Wagner, 2017; Tramèr et al., 2018). Adversarial training has been applied to +many domains like image classification (Carlini & Wagner, 2017; Madry et al., 2018), object detec- +tion (Zhang & Wang, 2019; Chen et al., 2021b;a), and segmentation (Xu et al., 2021; Hung et al., +2018; Arnab et al., 2018) etc. A common requirement for adversarial training is supervision because +generating adversarial examples needs ground truth, and most tasks require labels for training. Some +semi-supervised adversarial learning methods (Carmon et al., 2019; Alayrac et al., 2019) use a small +portion of labeled data to enhance robustness. Contrastive learning (Ho & Nvasconcelos, 2020; Kim +et al., 2020) is also used with adversarial examples either for better self-supervised learning or to +improve robustness. In this work, we explore the adversarial training of MDE without using ground- +truth depth and compare our method with contrastive learning-based and supervised learning-based +methods that are specifically tailored for MDE. There are other defense techniques such as input +transformations. However, Athalye et al. (2018a) point out that these techniques largely rely on +obfuscated gradients which may not lead to true robustness. In the scenarios of autonomous driving, +3 + +Published as a conference paper at ICLR 2023 +Figure 2: (a) Bird view of the relative positions of the camera and the target object. (b) 3D coordinates of the +four corners of the object in the camera frame. (c) Projection from the physical-world object to the two views. +there are other works focusing on the security of Lidar or multi-sensor fusion-based systems (Tu +et al., 2020; 2021; Cao et al., 2019; 2021). They use sensor spoofing or adversarial shapes to fool +the Lidar hardware or AI model, while in this work, we consider fully-vision-based autonomous +driving systems in which MDE is the key component. +3 +OUR DESIGN +Our technique consists of a few components. The first one (Section 3.1) is view synthesis that +generates the two views It and Is of the object and is equivalent to step ‚ in Figure 1a. The second +one (Section 3.2) is robust adversarial perturbation that perturbs It to induce maximum distance +errors (step ƒ in Figure 1a). The third one (Section 3.3) is the self-supervised adversarial training +(steps „ and …). We will discuss the details of these components and then present the training +pipeline at the end. +3.1 +VIEW SYNTHESIS TO AVOID PHYSICAL WORLD SCENE MUTATION +As mentioned in Section 1, conceptually we need to place an image board of the 3D object at some +physical locations on streets and use two cameras to take pictures of the board. In order to achieve +robustness in training, we ought to vary the image of the object (e.g., different cars), the position of +the image board, the positions and angles of the cameras, and the street view. This entails enormous +overhead in the physical world. Therefore, we propose a novel view synthesis technique that requires +minimal physical world overhead. Instead, it directly synthesizes Is and It with an object in some +street view, reflecting different settings of the aforementioned configurations. +Specifically, we use one camera to take a picture It of a 2D image board A of the object, with the +physical width and height W and H, respectively. As shown in Figure 2 (b), the four corners of the +board have the 3D coordinates (xt +0, yt +0, zt +0), ..., and (xt +3, yt +3, zt +3), respectively, in the camera frame, +namely, the coordinate system with the camera as the origin. The camera is placed at zc distance +away from the board with an angle of α, as shown in Figure 2 (a). The size of the board is true to +the rear of the object. This is important for the later realistic synthesis step. +After that, we derive a projection function that can map a pixel in A to a pixel in It. The function is +parameterized on W, H, zc, α, etc. such that we can directly use it to synthesize a large number of +It’s with different A’s by changing those parameter settings. To acquire Is that is supposed to form +a stereo view with It, we do not necessarily need another camera in the physical world. Instead, +we can use two neighboring video frames of some street view (e.g., from the KITTI dataset (Geiger +et al., 2013)), denoted as Rt and Rs, to approximate a stereo pair taken by two close-by cameras. +Note that the differences between the two images implicitly encode the relative positions of the two +cameras. A prominent benefit of such approximation is that a large number of camera placements +and street views can be easily approximated by selecting various neighboring video frames. This +is consistent with existing works (Godard et al., 2019; Watson et al., 2019). We replace the area in +It that does not correspond to A, i.e., the background of the object, with the corresponding area in +Rt. Intuitively, we acquire a realistic It by stamping the object’s image to a background image. The +synthesis respects physical world constraints. A projection function parameterized by Rt and Rs +can be derived to map a pixel in one camera’s view to a pixel in the other. Then, we project the part +of It that corresponds to A using the projection function and stamp it on Rs, acquiring Is. As such, +the resulted view of A (in Is) is consistent with the camera pose denoted by Rs. It and Is are then +used in model hardening (discussed later in Section 3.3). +4 + +W +(ub, va) +(us, v) +(> +(xt, yt, zt) +α +(xt,yt,z +(ul,vt) +(uz, v2) +Zc +0,0;2) +H< +X +(us;v) +(us, vs) +(xi, yi,zi) +(x2, y2, z2) +(ui,vi) +(uz, v2) +y +A +Is +(a) +(b) +(c)Published as a conference paper at ICLR 2023 +Formally, if the center of the physical camera’s view aligns with the center of the image board A, +the correlation between a pixel (uA, vA) in A and its 3D coordinate (xt, yt, zt) is: +� +�� +xt +yt +zt +1 +� +�� = +� +�� +cos α +0 +− sin α +0 +0 +1 +0 +0 +sin α +0 +cos α +zc +0 +0 +0 +1 +� +�� · +� +�� +W/w +0 +−W/2 +0 +H/h +−H/2 +0 +0 +0 +0 +0 +1 +� +�� · +� +� +uA +vA +1 +� +� , +(1) +where w and h are the width and height of A in pixels. The other variables (e.g., α, zc, W, H, etc.) +are defined in Figure 2. The 3D coordinates can be further projected to pixels in It and Is as: +�ut vt 1�⊤ = 1/zt · K · +�xt yt zt 1�⊤ , +[us vs 1]⊤ = 1/zs · K · [xs ys zs 1]⊤ , +[xs ys zs 1]⊤ = Tt→s · +�xt yt zt 1�⊤ , +(2) +where Tt→s is the camera pose transformation (CPT) that projects 3D coordinates in the phys- +ical camera Ct’s coordinate system to coordinates in the other (virtual) camera Cs’s coordinate +system. It is determined by Rs and Rt as mentioned before. K is the camera intrinsic. Com- +bining Equation 1 and Equation 2, we know the projections from pixel (uA, vA) of the object +image to pixel (ut, vt) in It and to pixel (us, vs) in Is. Let +�ut vt 1�⊤ = P A→t +zc,α (uA, vA) and +[us vs 1]⊤ = P A→s +zc,α,Tt→s(uA, vA). We synthesize It and Is as: +It[u, v] = +� +A[uA, vA], +[u v 1]⊤ = P A→t +zc,α (uA, vA) +Rt[u, v], +otherwise +, +(3) +Is[u, v] = +� +A[uA, vA], +[u v 1]⊤ = P A→s +zc,α,Tt→s(uA, vA) +Rs[u, v], +otherwise +, +(4) +where Rt and Rs are the background images implicitly encoding the camera relative poses. A large +number of It and Is are synthesized by varying Rt, Rs, zc, α, A, and used in hardening. The creation +induces almost zero cost compared to creating a physical world dataset with similar diversity. +3.2 +ROBUST ADVERSARIAL PERTURBATIONS +We use an optimization based method to generate robust adversarial perturbations δ on the object +image A composing the corresponding adversarial object A + δ and synthesize I′ +t by replacing +A with A + δ in Equation 3. The synthesized I′ +t is then used in adversarial training. We bound +the perturbations with L0-norm, which is to constrain the number of perturbed pixels. Compared +with digital-world attacks that use traditional L∞-norm or L2-norm-bounded perturbations (e.g., +FGSM (Goodfellow et al., 2014), Deepfool (Moosavi-Dezfooli et al., 2016), and PGD (Madry et al., +2018)), physical-world attacks usually use adversarial patches (Brown et al., 2017) without restric- +tions on the magnitude of perturbations in the patch area because stronger perturbations are needed +to induce persistent model misbehavior in the presence of varying environmental conditions (e.g., +lighting conditions, viewing angles, distance and camera noise). Hence, L0-norm is more suitable in +physical-world attacks because it restricts the number of pixels to perturb without bounding the per- +turbation magnitude of individual pixels. However, the calculation of L0-norm is not differentiable +by definition and hence not amenable for optimization. We hence use a soft version of it as proposed +in Tao et al. (2022). The main idea is to decompose the perturbations into positive and negative +components and use the long-tail effects of tanh function, in the normalization term, to model the +two ends of a pixel’s value change (i.e., zero perturbation or arbitrarily large perturbation). As such, +a pixel tends to have very large perturbation or no perturbation at all. +δ = maxp · (clip(bp, 0, 1) − clip(bn, 0, 1)). +(5) +Lpixel = +� +h,w +� +max +c +�1 +2(tanh(bp +γ ) + 1) +�� ++ +� +h,w +� +max +c +�1 +2(tanh(bn +γ ) + 1) +�� +. +(6) +Specifically, the perturbation is defined in Equation 5 and the normalization term is Lpixel in Equa- +tion 6, where bp and bn are the positive and negative components; clip() bounds the variable to a +5 + +Published as a conference paper at ICLR 2023 +Figure 3: Pipeline of adversarial training of self-supervised monocular depth estimation. Solid lines denote +data flow and dashed lines denote back propagation paths. +range of [0,1]; h, w, c are the height, width and channels of image and γ is a scaling factor. We refer +readers to Tao et al. (2022) for detailed explanations. +Equation 7 gives the formal definition of our perturbation generation method, where Sp is a distri- +bution of physical-world distance and view angles (e.g., reflecting the relations between cameras +and cars during real-world driving); SR is the set of background scenes (e.g., various street views); +D() is the MDE model which outputs the estimated depth map of given scenario and MSE() is the +mean square error. +min +bn,bp Ezc,α∼Sp,Rt∼SR +� +MSE +� +D (I′ +t)−1 , 0 +�� ++ Lpixel, +s.t. L0(δ) ≤ ϵ. +(7) +Our adversarial goal is to make the target object further away, so we want to maximize the depth +estimation (i.e., minimize the reciprocal). Intuitively, we synthesize I′ +t with random Rt and differ- +ent α and zc of object A and use expectation of transformations (EoT) (Athalye et al., 2018b) to +improve physical-world robustness. We minimize the mean square error between zero and the re- +ciprocal of synthesized scenario’s depth estimation in the adversarial loss term and use Lpixel as the +normalization term of perturbations. Parameter ϵ is a predefined L0-norm threshold of perturbations +and it denotes the maximum ratio of pixels allowed to perturb (e.g., ϵ = 1/10 means 1/10 pixels can +be perturbed at most). +3.3 +SELF-SUPERVISED MDE TRAINING. +In each training iteration, we first synthesize It and Is as mentioned earlier. Perturbations are then +generated to It to acquire I′ +t following Equation 7. As illustrated in Figure 1a, we reconstruct a +version of It from Is using the depth information derived from I′ +t. We call the resulted image Is→t. +Intuitively, I′ +t causes depth errors which distort the projection from Is to Is→t such that the latter +appears different from It. Similar to how we project (uA, vA) to (ut, vt) or (us, vs) earlier in this +section (i.e., Equation 2), we can project (us, vs) in Is to a pixel (us→t, vs→t) in Is→t. This time, +we use the depth information derived from I′ +t. Formally, the projection is defined as: +�xs→t ys→t zs→t 1�⊤ = K−1 · DI′ +t(us→t, vs→t) · +�us→t vs→t 1�⊤ , +[xs ys zs 1]⊤ = Tt→s · +�xs→t ys→t zs→t 1�⊤ , +[us vs 1]⊤ = 1/zs · K · [xs ys zs 1]⊤ . +(8) +Intuitively, there are relations between 2D image pixels and 3D coordinates, i.e., the first and the +third formulas in Equation 8. The 3D coordinates also have correlations decided by camera poses, +i.e., the second formula. Observe that, the first 2D-to-3D relation is parameterized on DI′ +t, the depth +estimation of I′ +t. Let [us vs]⊤ = PDI′ +t,Tt→s(us→t, vs→t) be the transformation function that projects +a pixel in Is→t to a pixel in Is derived from Equation 8. Is→t is synthesized as: +Is→t[u, v] = Is[PDI′ +t,Tt→s(u, v)]. +(9) +Intuitively, it rearranges the pixels in Is to form Is→t. We then compare Is→t with It and minimize +their difference to guide the training. +Our training pipeline is shown in Figure 3. There are two trainable networks, the MDE model D and +a camera transposing model TP. Recall that we need the camera pose transformation matrix Tt→s +6 + +→ Data Flow +→ Back Propagation +α,Zc,Rs +IS +A +α,Zc, Rt +D(It) +pe(It,l ++8Published as a conference paper at ICLR 2023 +Table 1: Benign performance of original and hardened models on depth estimation. +Monodepth2 +DepthHints +Models +ABSE↓ +RMSE↓ +ABSR↓ +SQR↓ +δ ↑ +ABSE↓ +RMSE↓ +ABSR↓ +SQR↓ +δ ↑ +Original +2.125 +4.631 +0.106 +0.807 +0.877 +2.021 +4.471 +0.100 +0.728 +0.886 +L0+SelfSup (Ours) +2.16 +4.819 +0.105 +0.831 +0.874 +2.123 +4.689 +0.103 +0.777 +0.877 +L0+Sup +2.162 +4.648 +0.110 +0.846 +0.876 +2.015 +4.453 +0.100 +0.734 +0.887 +L0+Contras +3.218 +6.372 +0.155 +1.467 +0.782 +3.626 +6.742 +0.209 +1.561 +0.694 +PGD+SelfSup +2.169 +4.818 +0.105 +0.826 +0.874 +2.120 +4.680 +0.103 +0.774 +0.877 +PGD+Sup +2.153 +4.637 +0.109 +0.838 +0.876 +2.019 +4.460 +0.101 +0.736 +0.886 +PGD+Contras +3.217 +6.083 +0.194 +1.825 +0.756 +3.928 +7.526 +0.213 +2.256 +0.701 +* For hardened models, A+B denotes generating adversarial perturbation with method A and training with method B. +between the two cameras’ coordinate systems. We hence train the TP network that predicts Tt→s +from a given pair of background images Rs and Rt. We denote it as: Tt→s = TP(Rt, Rs). Observe +that in Figure 3, from left to right, the pipeline takes the object image A and synthesizes images It +and Is. It and A are further used to derive adversarial sample I′ +t, which is fed to the depth network +D to acquire depth estimation. The depth information, the TP network’s output Tt→s, and Is are +used to derive Is→t. Two outputs Is→t and It are compared. The training objective is hence as: +min +θD,θTP Lp = pe(It, Is→t), +(10) +which is to update the weight values of D and TP to minimize the photometric reconstruction error +of the two outputs, denoted by pe(). Specific designs of pe() differ in literature but our model +hardening technique is general to all self-supervised MDE methods. +4 +EVALUATION +In this section, we evaluate the performance of our method in white-box, black-box, and physical- +world attack scenarios, and discuss the ablations. Our code shall be available after acceptance. +4.1 +EXPERIMENTAL SETUP. +Networks and Dataset. We use Monodepth2 (Godard et al., 2019) and DepthHints (Watson et al., +2019) as our subject networks to harden. They are representative and popular self-supervised MDE +models that are widely used as benchmarks in the literature. Both models are trained on the KITTI +dataset (Geiger et al., 2013) and our methods fine-tune the original models publicly available. +Baselines. There are no direct baselines available since no prior works have been focusing on harden- +ing MDE models as far as we know. Hence we extend state-of-the-art contrastive learning-based and +supervised learning-based adversarial training methods to MDE and use them as our baselines. They +do not require ground-truth depth, same as our self-supervised method. Details are in Appendix A +Training Setup. In adversarial training, the ranges of distance zc and viewing angle α are sam- +pled randomly from 5 to 10 meters and -30 to 30 degrees, respectively. The view synthesis uses +EoT (Athalye et al., 2018b). We generate the adversarial perturbations with two methods: L0-norm- +bounded with ϵ = 1/10 and L∞-norm-bounded (i.e., PGD (Madry et al., 2018)) with ϵ = 0.1. The +latter is for comparison purposes. We train with our self-supervised method and two baseline meth- +ods based on contrastive learning and supervised learning. Hence, there are 6 approaches combining +the 2 perturbation generation methods with the 3 training methods. With these approaches, we fine- +tune the original model for 3 epochs on the KITTI dataset and produce 6 hardened models for each +network. Other detailed configurations and the selection of 2D object images are in Appendix B. +Attacks. We conduct various kinds of attacks to evaluate the robustness of different models. They +are L0-norm-bounded attacks with ϵ = 1/20, 1/10, 1/5 and 1/3, L∞-norm-bounded (PGD) attacks +with ϵ = 0.05, 0.1 and 0.2 (image data are normalized to [0,1]), and an adversarial patch attack +in Mathew et al. (2020). Adversarial perturbation or patch is applied to an object image. The +patch covers 1/10 of the object at the center. Each attack is evaluated with 100 randomly selected +background scenes. The object is placed at a distance range of 5 to 30 meters and a viewing angle +range of -30 to 30 degrees. We report the average attack performance over different background +scenes, distances, and viewing angles for each attack. In addition, we conduct the state-of-the-art +physical-world attack (Cheng et al., 2022) with the printed optimal patch and a real vehicle in driving +scenarios. Adversarial examples are in Appendix D. Evaluation with more attacks are in Appendix J. +Metrics. We use the mean absolute error (ABSE), root mean square error (RMSE), relative absolute +error (ABSR), relative square error (SQR), and the ratio of relative absolute error under 1.25 (δ) as +7 + +Published as a conference paper at ICLR 2023 +Table 2: Defence performance of original and hardened models under attacks. +Attacks +Original +L0+SelfSup +(Ours) +L0+Sup +L0+Contras +PGD+SelfSup +PGD+Sup +PGD+Contras +ABSE↓ +δ ↑ +ABSE↓ +δ ↑ +ABSE↓ +δ ↑ +ABSE↓ +δ ↑ +ABSE↓ +δ ↑ +ABSE↓ +δ ↑ +ABSE↓ +δ ↑ +Monodepth2 +L0 1/20 +4.71 +0.65 +0.18 +0.99 +0.44 +0.98 +1.30 +0.67 +0.49 +0.95 +0.58 +0.94 +0.69 +0.94 +L0 1/10 +6.08 +0.51 +0.25 +0.98 +0.94 +0.93 +1.75 +0.54 +0.82 +0.91 +1.18 +0.79 +0.96 +0.89 +L0 1/5 +8.83 +0.39 +0.34 +0.9 +1.59 +0.85 +2.32 +0.46 +2.33 +0.70 +2.72 +0.51 +1.11 +0.85 +L0 1/3 +9.99 +0.34 +0.52 +0.96 +2.08 +0.78 +2.65 +0.41 +4.32 +0.51 +4.09 +0.41 +1.75 +0.70 +PGD 0.05 +4.74 +0.56 +0.82 +0.97 +1.29 +0.80 +6.61 +0.38 +0.67 +0.98 +0.82 +0.95 +1.82 +0.67 +PGD 0.1 +11.68 +0.34 +1.53 +0.85 +2.53 +0.71 +12.74 +0.24 +1.38 +0.95 +1.64 +0.76 +2.66 +0.53 +PGD 0.2 +17.10 +0.23 +3.46 +0.69 +6.14 +0.50 +20.14 +0.15 +3.81 +0.58 +5.04 +0.32 +3.97 +0.42 +Patch +2.71 +0.77 +0.39 +0.98 +1.35 +0.89 +6.40 +0.52 +0.40 +0.98 +0.84 +0.92 +0.50 +0.95 +DepthHints +L0 1/20 +2.33 +0.66 +0.19 +0.99 +0.34 +0.96 +1.06 +0.83 +0.22 +0.99 +0.58 +0.89 +0.40 +0.99 +L0 1/10 +3.19 +0.59 +0.27 +0.99 +0.48 +0.95 +1.56 +0.77 +0.42 +0.98 +1.03 +0.79 +0.60 +0.97 +L0 1/5 +4.77 +0.42 +0.40 +0.98 +0.96 +0.82 +1.85 +0.75 +0.83 +0.92 +1.93 +0.68 +0.66 +0.95 +L0 1/3 +6.03 +0.36 +0.48 +0.98 +1.64 +0.68 +2.60 +0.69 +1.45 +0.79 +3.06 +0.57 +1.16 +0.82 +PGD 0.05 +3.11 +0.48 +0.62 +0.98 +1.23 +0.75 +4.05 +0.55 +0.64 +0.98 +0.93 +0.79 +1.16 +0.79 +PGD 0.1 +6.44 +0.36 +1.27 +0.86 +2.37 +0.62 +7.59 +0.36 +1.21 +0.92 +1.76 +0.67 +1.74 +0.62 +PGD 0.2 +18.37 +0.23 +3.09 +0.60 +7.13 +0.41 +13.59 +0.24 +6.14 +0.68 +4.22 +0.37 +2.60 +0.49 +Patch +0.70 +0.91 +0.46 +0.95 +0.53 +0.93 +6.90 +0.49 +0.46 +0.95 +0.36 +0.99 +0.34 +0.98 +*Bold and underlining indicate the best and second best performance in each row. Hardened models are named the same as Table 1. +the evaluation metrics. These metrics are widely used in evaluating depth estimation performance. +Metric δ denotes the percentage of pixels of which the ratio between the estimated depth and ground- +truth depth is smaller than 1.25. It is the higher, the better and the others are the lower, the better. +The definition of each metric can be found in Appendix C. +4.2 +MAIN RESULTS +Benign Performance. +Together with the original model, we have 7 models under test for each +network. We evaluate the depth estimation performance on the KITTI dataset using the Eigen split +and report the results in Table 1. As shown, self-supervised and supervised methods have little +influence on the models’ depth estimation performance, which means these approaches can harden +the model with nearly no benign performance degradation. In contrast, the contrastive learning- +based approach performs the worst. The ABSE of estimated depth is over 1 m worse than the +original model. The reason could be that contrastive learning itself does not consider the specific task +(i.e., MDE) but fine-tunes the encoder to filter out the adversarial perturbations. Thus the benign +performance is sacrificed during training. The other two methods consider the depth estimation +performance either by preserving the geometric relationship of 3D space in synthesized frames or +by supervising the training with estimated depth. +White-box Attacks. We conduct various white-box attacks on each model to evaluate the robustness +of hardened models. Specifically, for each model, we compare the estimated depth of the adversar- +ial scene (i.e., I′ +t) with that of the corresponding benign scene (i.e., It in Equation 3) and larger +difference means worse defense performance. Table 2 shows the result. As shown, all the hardened +models have better robustness than the original models against all kinds of attacks, and it is generic +on the two representative MED networks, which validates the effectiveness of our adversarial train- +ing approaches. Comparing different approaches, L0+SelfSup has the best performance in gen- +eral. It reduces the ABSE caused by all-level L0-norm-bounded attacks from over 4.7 m to less than +0.6 m. Specifically,theself-supervision-basedmethodoutperformsthecontrastivelearning-basedand +the supervision-based methods regardless of the perturbation generation method used. It is because +the self-supervision-based method follows the original training procedure that is carefully designed +for the network and has been evaluated thoroughly. It is not surprising that models adversarially +trainedwithL0-norm-boundedperturbation(ourmethod)achieve better robustness against L0-norm- +bounded attacks and so do PGD-based methods, but more importantly, L0-norm-based training +also has good defense performance against PGD attacks. The robustness of L0+SelfSup is only +slightly worse than PGD+SelfSup on some PGD attacks and even better than it on stronger PGD +attacks. An explanation is that L0-norm does not restrict the magnitude of perturbation on each +pixel, and stronger PGD attacks are closer to this setting (i.e., high-magnitude perturbations) and +can be well-defended using the L0-norm-based adversarial training. Monodepth2 is vulnerable to +the patch attack, and this kind of attack can be well-defended by our methods. L0+SelfSup also +performs the best. Depthhints itself is relatively robust to the patch attack, and our methods can fur- +ther reduce the attack effect. Our defense generalizes well to complex scenes including various road +types, driving speed, and the density of surrounding objects. Qualitative results are in Appendix D. +8 + +Published as a conference paper at ICLR 2023 +Figure 4: +Physical-world +attack and defence. +Video: +https://youtu.be/ +_b7E4yUFB-g +Training Distance Range (m) +Mean Absolute Error (m) +0 +2 +4 +6 +5 +5~10 5~15 5~20 5~25 5~30 +L0-Norm Attack 1/20 +L0-Norm Attack 1/10 +L0-Norm Attack 1/5 +L0-Norm Attack 1/3 +(a) L0-norm attack. +Training Distance Range (m) +Mean Absolute Error (m) +0 +2 +4 +6 +5 +5~10 5~15 5~20 5~25 5~30 +PGD Performance 0.05 +PGD Performance 0.1 +PGD Performance 0.2 +Physical World ATK +(b) PGD and patch attack. +Figure 5: Robustness with different training distance ranges. +Table 3: Defence performance of original and hardened models +under black-box attacks. +Target +Original +L0+SelfSup +(Ours) +L0+Sup +L0+Contras +Source +ABSE +δ ↑ +ABSE +δ ↑ +ABSE +δ ↑ +ABSE +δ ↑ +Original +- +- +0.25 +0.99 +0.55 +0.95 +1.35 +0.71 +L0+SelfSup +0.52 +0.93 +- +- +0.24 +0.98 +0.45 +0.95 +L0+Sup +1.09 +0.80 +0.29 +0.99 +- +- +0.65 +0.89 +L0+Contras +2.65 +0.50 +0.22 +0.99 +0.27 +0.98 +- +- +*Bold indicates the best performance in each row or column. +Black-box Attacks. +We also vali- +date our methods against black-box +attacks. We use the original Mon- +odepth2 model and the models fine- +tuned with L0-norm-bounded pertur- +bations and the three training methods. +We perform L0-norm-bounded attacks +on each model with ϵ = 1/10 and ap- +ply the generated adversarial object to +other models evaluating the black-box +attack performance. The first column in Table 3 denotes the source models and other columns are the +target models’ defense performance. Looking at each column, adversarial examples generated from +L0+SelfSup have the worst attack performance, which indicates the low transferability of ad- +versarial examples generated from the model trained with our self-supervised methods. From each +row, we can observe that L0+SelfSup has the best defense performance against adversarial ex- +amples generated from each source model, which further validates the robustness of L0+SelfSup. +In summary, the self-supervised training method can produce safer and more robust models. Their +adversarial examples have low transferability and they defend against black-box attacks well. +Physical-world Attacks. +Our evaluation with the state-of-the-art physical-world MDE at- +tack (Cheng et al., 2022) validates the effectiveness of our method in various real-world lighting +conditions, driving operations, road types, etc. The experimental settings are the same as Cheng +et al. (2022). Figure 4 shows the result. The first row is the real-world adversarial scene, in which +a car is moving with an adversarial patch attached to the rear. The second row is the depth pre- +dicted by the original Monodepth2 model and the third row is predicted by our hardened model +(L0+SelfSup). The “hole” in the output of the original model caused by the adversarial patch is +fixed in our hardened model. It is known that the adversarial attacks designed for the physical world +are still generated in the digital world and they have better digital-world attack performance than +physical-world performance because additional environmental variations in the physical world de- +grade the effectiveness of adversarial patterns (Braunegg et al., 2020; Wu et al., 2020; Cheng et al., +2022). Hence, defending attacks in the digital world is more difficult and our success in digital-world +defense in previous experiments has implied effectiveness in the physical world. +4.3 +ABLATIONS +Distance Range in Training. While synthesizing views in training, the range of distance zc of the +target object is randomly sampled from d1 to d2 meters. In this ablation study, we evaluate the effect +of using different ranges of distance in training. We use L0+SelfSup to fine-tune the original +Monodepth2 model. The ranges of distance we use in training are [5, 5], [5, 10], [5, 15], [5, 20], [5, +25] and [5, 30] (Unit: meter). Note that, for a fair comparison, the range of distance we use in model +evaluation is always from 5 to 30 meters. The results are shown in Figure 5. As shown, the model +trained with a distance range of 5-10 meters has the best robustness and a larger or smaller distance +range could lead to worse performance. It is because further distances lead to smaller objects on the +image and fewer pixels are affected by the attack. Thus the attack performance is worse at further +distances and training with these adversarial examples is not the most effective. If the distance in +training is too small (e.g., 5 meters), the model cannot defend various scales of attack patterns and +9 + +Real-world Scene +Original Model +HardenedModePublished as a conference paper at ICLR 2023 +cannot generalize well to further distances. In our experiments, the range of 5-10 meters makes a +good balance between training effectiveness and generality. +Other ablation studies about viewing angles range in training are in Appendix E, transferability +to unseen target objects is in Appendix F, comparing training from scratch and fine-tuning is in +Appendix G and the performance of method combinations can be found in Appendix H. +5 +CONCLUSION +We tackle the problem of hardening self-supervised Monocular Depth Estimation (MDE) mod- +els against physical-world attacks without using the depth ground truth. +We propose a self- +supervised adversarial training method using view synthesis considering camera poses and use L0- +norm-bounded perturbation generation in training to improve physical-world attacks robustness. +Compared with traditional supervised learning-based and contrastive learning-based methods, our +method achieves better robustness against various adversarial attacks in both the digital world and +the physical world with nearly no benign performance degradation. +10 + +Published as a conference paper at ICLR 2023 +6 +REPRODUCIBILITY STATEMENT +To help readers reproduce our results, we have described the implementation details in our experi- +mental setup (Section 4.1 and Appendix B). 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In ECCV, 2020. +14 + +Published as a conference paper at ICLR 2023 +Appendix +Adversarial Training of Self-supervised Monocular Depth +Estimation against Physical-World Attacks +This document provides more details about our work and additional experimental settings and result. +We organize the content of our appendix as follows: +• Section A: the baseline methods we tailored for MDE specifically. +• Section B: more details about the training configurations. +• Section C: the formal definition of the metrics we used in our evaluation. +• Section D: the adversarial attack examples and qualitative results of defense. +• Section E: the effect of different ranges of viewing angles in training. +• Section F: the transferability evaluation of our hardened models to other target objects. +• Section G: the difference between fine-tuning and training from scratch. +• Section H: the model performance of combining different methods. +• Section I: comparing the supervised baseline trained with pseudo-depth labels and that +trained with ground-truth depth labels. +• Section J: the robustness against more attacks. +• Section K: human evaluation of the quality of our synthesized images. +• Section L: the influence of inaccurate projections. +• Section M: extension of our method to indoor scenes and more advanced MDE networks. +• Section N: the broader impact and limitations. +A +BASELINES +Adversarial Contrastive Learning. Contrastive learning is a widely used technique specifically +tailored to self-supervised learning scenarios (Chen et al., 2020; He et al., 2020; Wu et al., 2018; +Tian et al., 2020; Ye et al., 2019; Misra & Maaten, 2020). It has been used with adversarial exam- +ples either to improve the robustness of models against adversarial attacks (Kim et al., 2020) or to +enhance the performance of contrastive learning itself (Ho & Nvasconcelos, 2020). In this work, we +extend a state-of-the-art contrastive learning-based adversarial training method (Kim et al., 2020) to +harden MDE models against physical attacks. We use it as a baseline to compare with our method. +Figure 6: (a) Adversarial contrastive learning of model encoder. The color-augmented benign (It) and adver- +sarial (I′ +t) examples are fed to the depth model encoder (the grey block) and one embedding (e′) is then fed to +a prediction multi-layer perceptron (MLP) that transcribes the embedding to another embedding p′. We maxi- +mize the similarity of the output (p′) with the other embedding (e). Backpropagation is only calculated along +one side to update the encoder. (b) Supervised adversarial training with the estimated depth as the pseudo- +ground truth. We use the output of the original model on benign examples as the ground truth to supervise the +training of the subject model with adversarial examples. The solid lines denote data flow and the dashed lines +denote back propagation paths. +15 + +Adversarial Image(It) +Benign Image(It) +(a) Adversarial +Contrastive Learning +(a) +Stop Grad +Depth CNN +Depth CNN +e +MLP +Sim +(b) Supervised +Adversarial Learning +D(lt) +D(It) +MSE +(b)Published as a conference paper at ICLR 2023 +Figure 7: The various target objects in the transferability evaluation. +Similar to Kim et al. (2020), the positive pairs in our contrastive learning are the benign examples +(It) and the corresponding adversarial examples (I′ +t), and we further augment those examples by +changing the color. Different from Kim et al. (2020), we do not need negative pairs. Instead, we +use a learning method proposed in SimSiam (Chen & He, 2021) that only requires positive pairs and +can achieve competitive performance with smaller batch sizes. Figure 6 (a) shows the procedure. +The key point is to maximize the similarity between the embeddings of the benign and adversarial +examples so that their depth map outputs from the decoder network are similar. The parameters +of the subject MDE model’s encoder and the prediction MLP network are updated iteratively in +training. We use color augmentation instead of other transformations (e.g., resizing and rotation) +because the embeddings should be similar among positive samples and the change of color would +not affect the depth map output (but other transformations would). Other settings such as the MLP +network structure are the same as SimSiam (Chen & He, 2021) and we refer readers to it for detailed +explanations. +Supervised Adversarial Learning with Estimated Depth. Since we do not have depth ground +truth in the self-supervised scenario, one alternative way to do adversarial training is to use the +estimated depth by the original model with inputs of benign samples as the pseudo ground truth +(i.e., pseudo labels) and perform supervised adversarial training. As shown in Figure 6 (b), we use +mean square error (MSE) as the loss function to update MDE model parameters and minimize the +difference between the model output of adversarial samples and the pseudo ground truth. +Using the pseudo ground truth predicted by an existing model is proved to be a simple and effective +method in the field of semi-supervised learning (SSL) (Lee et al., 2013) and it has been used in +adversarial training (Deng et al., 2021) and self-supervised MDE (Petrovai & Nedevschi, 2022) to +boost model performance. Particularly, in the field of MDE, using pseudo-ground truth is good +enough compared with using the real ground truth (Petrovai & Nedevschi, 2022). Same as our +supervised baseline, Petrovai & Nedevschi (2022) uses the depth estimated by an existing MDE +model (i.e., pseudo depth labels) to supervise the following MDE model training. Results show +that the pseudo-supervised model has similar or better performance than the reference model trained +with ground-truth depth. We also conduct experiments comparing the performance of supervised +baseline trained with pseudo depth labels and ground-truth depth labels, which proves that a pseudo- +supervised baseline is not a weak choice. The results can be found in Appendix I. +B +TRAINING CONFIGURATIONS +We train our model with one GPU (Nvidia RTX A6000) that has a memory of 48G and the CPU is +Intel Xeon Silver 4214R. For each model, doing adversarial training from scratch takes around 70 +hours. It includes 20 epochs of training on the KITTI dataset. The fine-tuning of 3 epochs takes about +10 hours. The input resolution of our MDE model is 1024*320 and the original monodepth2 and +depthhints models we used for fine-tuning are the official versions trained with both stereo images +and videos. In our hardening, we use stereo images with fixed camera pose transformation Tt→s. In +perturbation generation, we use 10 steps and a step size of 2.5 · ϵ/10 in L2 and L∞-bounded attacks +to ensure that we can reach the boundary of the ϵ-ball from any starting point within it (Madry et al., +2018) and a batch size of 12. In MDE training, the batch size is 32, and the learning rate is 1e-5. We +use Adam as the optimizer and other training setups are the same as the original model. +As for the selection of 2D images of objects, as shown in Figure 2 (a) and Figure 2 (b), we have +assumptions about the initial relative positions between the target object and the camera (i.e., the +3D coordinates of the center of the object is (0, 0, zc) in the camera’s coordinate system and the +16 + +(a) BMW SUV Black +(b) Toyota Sedan Blue +(c) Subaru Sedan White +(d) Volvo SUV Grey +(e) Traffic BarrierPublished as a conference paper at ICLR 2023 +Figure 8: Examples of adversarial attacks in our robustness evaluation. +viewing angle α of the camera is 0 degree). Hence, for a more realistic and high-quality synthesis, +the camera should look at the center of the target object at the same height while taking the 2D +image of the object. The width w and height h of the 2D image of the object should be proportional +to the physical size W and H of it: w/W = h/H. Moreover, when we prepared the 2D image +of the object, we also prepared a corresponding mask to “cut out” the main body of the object for +projection and we take the object together with its shadow to preserve reality. Examples of object +masks can be found in Figure 11. +We train models with L0 and L∞-bounded (i.e., PGD) perturbations in our evaluation but not L2 +norm because Madry et al. (2018) has demonstrated that models hardened with L∞-bounded pertur- +bations are also robust against L2-bounded attacks and our experiments in Appendix J also validate +the robustness of our models. In addition, physical-world attacks with adversarial patches have +more resemblance to L0-bounded attacks that only restrict the ratio of perturbed pixels rather than +the magnitude of the perturbation. +C +EVALUATION METRICS +The evaluation metrics we used in our evaluation are defined as follows, where we use X = +{x1, x2, ..., xn} to denote the estimated depth map and Y = {y1, y2, ..., yn} to denote the refer- +ence depth map and I() is the indicator function that evaluates to 1 only when the condition is +satisfied and 0 otherwise. +ABSE = 1 +n +n +� +i=1 +|xi − yi| +(11) +RMSE = +� +� +� +� 1 +n +n +� +i=1 +(xi − yi)2 +(12) +ABSR = 1 +n +n +� +i=1 +(|yi − xi| +yi +) +(13) +SQR = 1 +n +n +� +i=1 +(yi − xi)2 +yi +(14) +17 + +(a) Lo-norm bounded perturbation. (ε = 1/10) +(b) Lo-norm bounded perturbation. (ε = 0.1) +(c) Unbounded adversarial patch attack. +(d) Optimal physical-world patch attackPublished as a conference paper at ICLR 2023 +Figure 9: Qualitative results of the defensive performance of our hardened model. +δ = 1 +n +n +� +i=1 +I(max{xi +yi +, yi +xi +} < 1.25) +(15) +The mean absolute error (ABSE) and root mean square error (RMSE) are common metrics and are +easy to understand. Intuitively, the relative absolute error (ABSR) is the mean ratio between the +error and the ground truth value, and the relative square error (SQR) is the mean ratio between the +square of error and the ground truth value. δ denotes the percentage of pixels of which the ratio +between the estimated depth and ground-truth depth is smaller than 1.25. +D +ADVERSARIAL ATTACK EXAMPLES +Figure 8 gives examples of the three kinds of adversarial attacks we conducted in our robustness +evaluation. The first column is the original object; the second column is the adversarial one and +the third column is the adversarial perturbations. We scale the adversarial perturbations of the L∞- +norm-bounded attack for better visualization. L0-norm restricts the number of perturbed pixels. +L∞-norm restricts the magnitude of perturbation of each pixel. Adversarial patch attack perturbs +pixels within the patch area. +Figure 9 shows the qualitative results of our evaluation. The vertical axis denotes different street +views and the horizontal axis denotes different models and attacks. Here we compare the perfor- +18 + +Original Model +Hardened Model +Lo Attack 1/10 +Adv. Patch Attack +Lo Attack 1/10 +Ady. Patch Attack + Scene +Depth +Error + Scene +Depth +ErrorScene +Depth +Error + Scene +Depth +ErrorPublished as a conference paper at ICLR 2023 +ABSE +0 +1 +2 +3 +4 +L0-Norm +L0-Norm +L0-Norm 1/5 +L0-Norm 1/3 +PGD 0.05 +PGD 0.1 +PGD 0.2 +Patch Attack +Fine Tune +From Scratch +Figure 10: +Robustness performance of +fine-tuning and training from scratch. +Table 4: Benign performance of models trained with methods +combinations. +ABSE +RMSE +ABSR +SQR +δ ↑ +Original +2.125 +4.631 +0.106 +0.807 +0.877 +SelfSup+Con +2.172 +4.84 +0.105 +0.847 +0.875 +SelfSup+Sup +2.161 +4.819 +0.105 +0.836 +0.875 +Con+Sup +2.408 +4.986 +0.122 +0.93 +0.849 +All +2.171 +4.83 +0.105 +0.841 +0.875 +Table 5: Ablations on the range of angles in adversarial training with L0+SelfSup. +0◦ +−10◦ − 10◦ +−20◦ − 20◦ +−30◦ − 30◦ +−40◦ − 40◦ +Attacks +ABSE +ABSE +ABSE +ABSE +ABSE +L0 1/10 +0.271 +0.260 +0.253 +0.251 +0.249 +L0 1/5 +0.363 +0.352 +0.351 +0.347 +0.350 +PGD 0.1 +1.662 +1.543 +1.544 +1.539 +1.536 +PGD 0.2 +3.587 +3.472 +3.465 +3.467 +3.470 +mance of the original Monodepth2 model with our model hardened by L0+SelfSup. For each street +view, model, and attack, the first row is the adversarial scene (i.e., the scene with the adversarial +object), the second row is the corresponding depth estimation and the third row is the depth estima- +tion error caused by the adversarial perturbations. As shown, our method mitigates the effectiveness +of attacks significantly and the attacks can hardly cause any adversarial depth estimation error on +our hardened model. In addition, our method works on different scenes including complex ones +that have multiple pedestrians or vehicles at different speeds. It is because we do not have assump- +tions about the scene geometry or surrounding objects in our method. Both the scene synthesis and +depth estimation are single-image-based and the scene geometry or moving speed will not affect the +quality of synthesis. +E +VIEWING ANGLES RANGE IN TRAINING +Randomizing the degree of synthesis will make the model more robust in the physical-world settings +because the camera’s viewing angle is not fixed toward the target object in practice, and the range we +use (-30 degrees to 30 degrees) covers the most common situations. Ablations of the range of angles +during adversarial training do not have as much effect as changing the range of distance because +distance has a dominant influence on the size (i.e., number of pixels) of the adversarial object on the +synthesized scene image (i.e., the further object looks smaller) and the number of adversarial pixels +affects the attack performance a lot (Brown et al., 2017). We conduct experiments with different +viewing angle ranges, and other settings are the same as the ablation study of distance ranges in +Training. Table 5 shows the result. As shown, ranges of viewing angles have less influence on +the defensive performance, and using a fixed setting (i.e., 0◦) still performs the worst, which is +consistent with our claims above. +F +TRANSFER TO OTHER TARGET OBJECTS +The target object under attack may not be used in training. In this experiment, we evaluate how the +adversarially trained models perform on unseen target objects. We evaluate with four vehicles and +one traffic barrier using the original and the three hardened Monodepth2 models and we perform +L0-norm-bounded attack on each object with ϵ = 1/10. Figure 7 shows the examples of target +19 + +Published as a conference paper at ICLR 2023 +Table 6: Defence performance of attacks on various target objects. +Models +Original +L0+SelfSup +(Ours) +L0+Sup +L0+Contras +Objects +ABSE ↓ +δ ↑ +ABSE ↓ +δ ↑ +ABSE ↓ +δ ↑ +ABSE ↓ +δ ↑ +BMW SUV Black +6.079 +0.512 +0.248 +0.988 +0.949 +0.934 +1.642 +0.552 +Toyota Sedan Blue +4.259 +0.456 +0.651 +0.905 +1.097 +0.833 +1.701 +0.606 +Subaru Sedan White +4.262 +0.437 +1.408 +0.776 +2.602 +0.632 +2.324 +0.505 +Volvo SUV Grey +6.379 +0.535 +1.565 +0.889 +1.859 +0.836 +2.476 +0.552 +Traffic Barrier +4.479 +0.375 +1.619 +0.54 +2.3 +0.483 +0.813 +0.767 +*Bold indicates the best performance in each row and underlining means the second best. +objects that we used in our transferability evaluation. Figure 7(a) is the object we used in adversarial +training and others are target objects under attack in the robustness evaluation. Table 6 shows the +result. The first column is the target objects under attack and the following columns are the perfor- +mance of different models. The first object (BMW SUV Black) is an object used in training and +our hardened models are most robust on it. The others are unseen objects during training and the +hardened models can still mitigate the attack effect a lot compared with the original model, which +validates the transferability of our hardened models. L0+SelfSup still has the best performance +in general. +G +TRAINING FROM SCRATCH +We also compare the robustness performance of training from scratch and fine-tuning an existing +model with our self-supervised adversarial training method using L0-norm-bounded perturbation. +We evaluate on Monodepth2 and Figure 10 shows the ABSE result. As shown, training from scratch +can provide more robustness, especially for higher-level attacks. As for the benign performance +(i.e., depth estimation performance), the ABSE of the fine-tuned model is 2.16 while the ABSE of +the model trained from scratch is 2.468, meaning that the model trained from scratch has slightly +worse than benign performance. +H +ADVERSARIAL TRAINING METHODS COMBINATION +We introduced three adversarial training methods in Section 3 and evaluated them separately in +our main experiments. In this experiment, we explore the method combinations. +Specifically, +we combine different loss terms together as a total loss in adversarial training and there are four +combinations in total. The perturbation in training is L0-norm-bounded with ϵ = 1/10 and other +evaluation setups are the same as our main experiments. The depth estimation performance (i.e., +clean performance) of each model is shown in Table 4. Results show that models trained with the +self-supervised method have equivalent performance as the original model and Con+Sup performs +slightly worse than the original one. Table 7 shows the robustness performance under attacks. As +shown, combining self-supervised learning with contrastive learning could achieve better robustness +than self-supervised learning itself, and combining all three methods further improves the robustness +under some attacks. +I +SUPERVISED BASELINE WITH GROUND TRUTH DEPTH +Although in the scope of this paper, we discuss self-supervised scenarios and assume the ground- +truth depth is not available in training, we still conduct experiments comparing the defensive perfor- +mance of our supervised baseline trained with pseudo-ground truth and that trained with ground- +truth depth. +We use Monodepth2 as the subject model. +The results in Table 8 show that us- +ing ground-truth depth (L0+Sup(GT)) has a similar performance to using pseudo ground truth +(L0+Sup(Pseudo)), and they are still worse than our self-supervised approach. Hence, whether +to use ground-truth depth is not the bottleneck, and our pseudo-supervised baseline is not a weak +choice, which also has significant defensive performance against adversarial attacks. +20 + +Published as a conference paper at ICLR 2023 +Table 7: Defence performance of models trained with methods combinations. +Attacks +Original +SelfSup+Con +SelfSup+Sup +Con+Sup +All +ABSE +δ ↑ +ABSE +δ ↑ +ABSE +δ ↑ +ABSE +δ ↑ +ABSE +δ ↑ +Monodepth2 +L0 1/20 +4.711 +0.652 +0.182 +0.998 +0.193 +0.989 +0.529 +0.917 +0.187 +0.998 +L0 1/10 +6.088 +0.516 +0.244 +0.991 +0.256 +0.988 +0.849 +0.761 +0.24 +0.991 +L0 1/5 +8.83 +0.393 +0.357 +0.967 +0.394 +0.969 +1.293 +0.714 +0.288 +0.986 +L0 1/3 +9.996 +0.344 +0.491 +0.958 +0.568 +0.954 +1.803 +0.667 +0.419 +0.97 +PGD +4.747 +0.56 +0.681 +0.988 +0.816 +0.982 +1.666 +0.72 +0.709 +0.986 +PGD +11.684 +0.343 +1.332 +0.869 +1.586 +0.839 +2.832 +0.646 +1.637 +0.884 +PGD +17.109 +0.233 +3.382 +0.591 +3.546 +0.65 +4.612 +0.472 +3.056 +0.621 +Patch +2.714 +0.778 +0.758 +0.945 +0.331 +0.977 +1.106 +0.798 +0.277 +0.995 +*Bold indicates the best performance in each row and underlining means the second best. +Table 8: Performance of the supervised baseline trained with pseudo depth label (L0+Sup(Pseudo)) and +ground-truth depth label (L0+Sup(GT)). +Attacks +Original +L0+SelfSup (Ours) +L0+Sup(Pseudo) +L0+Sup(GT) +ABSE ↓ +δ ↑ +ABSE ↓ +δ ↑ +ABSE ↓ +δ ↑ +ABSE ↓ +δ ↑ +L0 1/10 +6.08 +0.51 +0.25 +0.98 +0.94 +0.93 +0.75 +0.92 +L0 1/5 +8.83 +0.39 +0.34 +0.9 +1.59 +0.85 +0.98 +0.83 +PGD 0.1 +11.68 +0.34 +1.53 +0.85 +2.53 +0.71 +2.44 +0.68 +PGD 0.2 +17.1 +0.23 +3.46 +0.69 +6.14 +0.50 +5.11 +0.32 +J +ROBUSTNESS AGAINST MORE ATTACKS +In addition to the attacks evaluated in our main paper (PGD attacks, l0 attacks, adversarial patch at- +tack and physical-world optimal patch attack), we have also evaluated the robustness of our models +against more attacks: the L2-bounded PGD attack (Madry et al., 2018), APGD attacks in AutoAt- +tack (Croce & Hein, 2020), Square attack (Andriushchenko et al., 2020)(a query-based black-box +attack), Gaussian Blur (Rauber et al., 2020) and AdvLight (Duan et al., 2021). The subject network +is Mondepth2. Results can be found in Table 9. As shown, Our model is still more robust than the +original model in all cases, and our method outperforms all others. Attacks with an L∞-bound of +0.1 can only cause less than 1 m depth estimation error on our hardened models while more than 10 +m on the original model. The black-box attack does not have a good performance in the MDE task +and only causes a mean depth estimation error of 0.924 m with 5000 queries to the original model. +Although the Gaussian Blur and AdvLight are more stealthy and natural attacks, they are not as as +effective as other methods in the task of MDE (i.e., per-pixel-based regression task). +K +QUALITY OF THE VIEW SYNTHESIS +To demonstrate the quality of our view synthesis, we show more examples of the synthesized images +in Figure 11. Is and It are two adjacent views of the same scene. Different target objects (e.g., the +white sedan, black SUV and traffic barrier) are synthesized into the views with various distances +zc and background scenes. To evaluate the veracity of these images, we use Amazon Mechanical +Turk to conduct human evaluations of our synthesized images. Participants are requested to evaluate +the quality of the synthesized objects from 4 distinct perspectives: size, consistency of location, +lighting, and overall quality. The score ranges from 1 to 10 for each metric. We use the score of 1 +to indicate scenes synthesized with random projections and the score of 10 for real scenes taken by +stereo cameras. Examples of perfect (score of 10) and unrealistic (score of 1) scenes for each metric +are provided as references (See Figure 12). The results of the experiment with 100 participants can +be found in Table 10. In each row, we show the number of participants who gave a score within +the corresponding range and summarize the average in the last row. As demonstrated, We got an +average score of 7.7 regarding the size, 7.21 regarding the location, 7.43 regarding the lighting, and +7.74 regarding the overall quality. Compared with the real scene, our synthesized scene is slightly +inferior but still much better than random projection. More importantly, it works well to harden the +model against physical-world attacks at a low cost. +21 + +Published as a conference paper at ICLR 2023 +Table 9: Defensive performance of original and hardened models under more attacks. +Attacks +Original +L0+SelfSup (Ours) +L0+Sup +L0+Contras +ABSE +δ ↑ +ABSE +δ ↑ +ABSE +δ ↑ +ABSE +δ ↑ +L2-PGD ϵ = 8 +1.403 +0.76 +0.294 +0.996 +1.161 +0.741 +0.66 +0.919 +L2-PGD ϵ = 16 +6.491 +0.522 +0.597 +0.984 +2.516 +0.479 +1.437 +0.734 +L2-PGD ϵ = 24 +13.018 +0.354 +0.932 +0.913 +3.613 +0.387 +2.92 +0.7 +APGD ϵ = 0.05 +5.557 +0.739 +0.423 +0.976 +1.614 +0.793 +2.71 +0.859 +APGD ϵ = 0.1 +10.216 +0.46 +0.928 +0.945 +3.279 +0.603 +4.578 +0.793 +Square Attack +ϵ = 0.1 N=5000 +0.924 +0.924 +0.422 +0.991 +0.712 +0.934 +0.568 +0.973 +Gaussian Blur +0.323 +0.996 +0.191 +0.997 +0.288 +0.997 +0.264 +0.997 +AdvLight +0.512 +0.988 +0.493 +0.991 +0.513 +0.987 +0.504 +0.988 +Figure 11: More examples of view synthesis with different background scenes, target objects and +distance zc of the object. Object mask is used to remove background of the 2D object image. +L +INFLUENCE OF INACCURATE PROJECTIONS +As stated in Section 3.1, we project a 2D image of the target object to two adjacent views considering +the camera pose transformation Tt→s between Cs and Ct (Equation 3 and 4), then we enable the +self-supervised training of MDE network by reconstructing It from Is using the estimated depth +DI′ +t (Equation 9). In this section, we explore what influence the inaccurate projections of the two +camera views would have on training results. In this experiment, we only consider half of the camera +pose transformation (i.e., Tt→s/2) while projecting the object to Is instead of the true value Tt→s, +which leads to a wrong Is intentionally. For example, the distance between the stereo cameras in +the KITTI dataset is 0.54 m, but we use 0.27 m to synthesize Is. It is still correctly synthesized. We +use L0-bounded perturbation with ϵ = 1/10 and self-supervised training to harden the Monodepth2 +model. Figure 13 shows the result. As shown, the outline of the estimated depth of the target object +22 + +It +Is +Zc = 5 +White +Zc = 10 +Zc = 15 +It +Is +Toyota Sedan +Object Mask +Zc = 5 +Zc = 10 +Zc = 15 +Blue +It +Is +Volvo SUV +Object Mask +Zc = 5 +Zc = 10 +Zc = 15 +Black +It +Is +Traffic +Object Mask +Zc = 5 +Zc = 10 +Zc = 15 +Barrier +Figure 1l: More examples of view synthesisPublished as a conference paper at ICLR 2023 +Figure 12: The reference images used in our human study. +Table 10: Human evaluations of the quality of our synthesized images. We show the number of +participants who gave a score in the corresponding range in each row. +Score Ranges +Size +Location +Lightning +Overall +1-2 +0 +0 +1 +0 +5-6 +4 +6 +6 +3 +7-8 +16 +24 +23 +18 +9-10 +42 +48 +40 +45 +Total +100 +100 +100 +100 +Average Score +7.7 +7.21 +7.43 +7.74 +is blurred when the MDE model is trained with inaccurate projection. It remains clear for the model +trained with accurate projection. +M +EXTENSION TO INDOOR SCENES AND ADVANCED NETWORK +Although we focus on outdoor scenarios like autonomous driving, a domain where self-supervised +MDE is widely adopted (e.g., Tesla Autopilot), our technique hardens the MDE networks and does +not have any assumptions about indoor or outdoor scenes. Actually, both Monodepth2 and Depth- +Hints models have been proven to be directly applicable on indoor scenes (e.g., NYU-Depth-v2 +Dataset) without any retraining (Peng et al., 2021; Zhou et al., 2022), which implies the indoor- +scene capability of our hardened models. We further conduct additional experiments with indoor +scenes in the NYU-Depth-v2 Dataset. We launch adversarial attacks in a square area at the center +of each scene to mimic a physical patch in the scene and evaluate the defensive performance of the +original and hardened Monodepth2 models. Results are shown in Table 11. As shown, our hardened +models reduce the depth estimation error caused by the attack significantly and the model trained +with our self-supervised method has the best defensive performance. Hence, the robustness of our +hardened model still holds for indoor scenes. +Monodepth2 and DepthHints are the two popular and representative self-supervised MDE networks, +and they are widely used as baselines in the literature, so we use them as our subject networks. How- +ever, we do not have any assumptions about the MDE network structure, and our method should +also work on state-of-the-art MDE models. Hence we conduct additional experiments with Many- +depth (Watson et al., 2021). We use our L0-bounded perturbation with self-supervised adversarial +training to harden the original Manydepth model, and Table 12 shows the defensive performance of +the original and hardened models. As shown, the original models are still vulnerable to both L0- +Table 11: Defensive performance on the indoor scenes with the NYU-Depth-v2 Dataset. +Attacks +Original +L0+SelfSup (Ours) +L0+Sup +L0+Contras +ABSE +δ ↑ +ABSE +δ ↑ +ABSE +δ ↑ +ABSE +δ ↑ +L0 1/20 +1.243 +0.772 +0.099 +0.993 +0.363 +0.976 +0.695 +0.812 +L0 1/10 +3.866 +0.54 +0.164 +0.985 +0.49 +0.924 +1.12 +0.733 +PGD 0.05 +1.784 +0.727 +0.12 +0.986 +0.456 +0.849 +1.641 +0.717 +PGD 0.1 +4.598 +0.426 +0.288 +0.912 +0.962 +0.775 +4.779 +0.42 +23 + +Unrealistic Size +Perfect Scene +Unrealistic Lightning +Inconsistent LocationPublished as a conference paper at ICLR 2023 +Figure 13: Influence of inaccurate projection. The outline of the estimated depth of the target object +is blurred while using inaccurate projection in training. +Table 12: Defensive performance of the original and hardened Manydepth models. +Attacks +Original +L0+SelfSup +ABSE +δ ↑ +ABSE +δ ↑ +L0 1/20 +1.554 +0.78 +0.221 +0.996 +L0 1/10 +2.684 +0.662 +0.301 +0.994 +PGD 0.05 +7.112 +0.417 +1.270 +0.855 +PGD 0.1 +9.452 +0.339 +1.614 +0.83 +bounded and PGD attacks. The model hardened with our techniques is a lot more robust. The mean +depth estimation error is reduced by over 80%, which validates that our techniques are generic and +also work on state-of-the-art MDE models. +N +BROADER IMPACT AND LIMITATIONS +Our adversarial training method hardens the widely used self-supervised monocular depth estima- +tion networks and makes applications like autonomous driving more secure with regard to adver- +sarial attacks. Compared to original training, the hardening of such models with our method does +not require additional data or resources and the computational cost is affordable. The adversarial +attacks we studied are from existing works and we do not pose any additional threats. Some lim- +itations we could think of about our method are as follows. In our synthesis of different views, +we assume the physical-world object is a 2D image board instead of a 3D model to avoid using an +expensive scene rendering engine or high-fidelity simulator and to improve efficiency, which could +induce small errors in synthesis though. However, since most physical-world attacks are based on +adversarial patches attached to a flat surface, this 2D board assumption is realistic and practical. +Precise synthesis should consider lighting factors such as glare, reflections, shadows, etc., but how +to do that at a low-cost is an open problem and we leave it as our future work. In addition, although +our modeling of the relative positions and viewing angles of the camera and physical object does +not cover all real-world conditions, it considers the most common situations. There might be some +corner cases in which the adversarial attack could escape from our defense, but we still mitigate the +threats in the most common situations and improve the overall security a lot. +24 + +(a) Trained with inaccurate projection +(b) Trained with accurate projection \ No newline at end of file diff --git a/N9FRT4oBgHgl3EQfHDeZ/content/tmp_files/load_file.txt b/N9FRT4oBgHgl3EQfHDeZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3c176d3b1e1affd953848060061e2dc0a5b2a3c --- /dev/null +++ b/N9FRT4oBgHgl3EQfHDeZ/content/tmp_files/load_file.txt @@ -0,0 +1,1637 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf,len=1636 +page_content='Published as a conference paper at ICLR 2023 ADVERSARIAL TRAINING OF SELF-SUPERVISED MONOCULAR DEPTH ESTIMATION AGAINST PHYSICAL-WORLD ATTACKS Zhiyuan Cheng Purdue University cheng443@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='edu James Liang Rochester Institute of Technology jcl3689@rit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='edu Guanhong Tao Purdue University taog@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='edu Dongfang Liu∗ Rochester Institute of Technology dongfang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='liu@rit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='edu Xiangyu Zhang∗ Purdue University xyzhang@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='edu ABSTRACT Monocular Depth Estimation (MDE) is a critical component in applications such as autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' There are various attacks against MDE networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' These attacks, especially the physical ones, pose a great threat to the security of such sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Traditional adversarial training method requires ground-truth labels hence cannot be directly applied to self-supervised MDE that does not have ground- truth depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Some self-supervised model hardening techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', contrastive learning) ignore the domain knowledge of MDE and can hardly achieve optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In this work, we propose a novel adversarial training method for self-supervised MDE models based on view synthesis without using ground-truth depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We improve adversarial robustnessagainstphysical-worldattacksusingL0- norm-bounded perturbation in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We compare our method with supervised learning based and contrastive learning based methods that are tailored for MDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' ResultsontworepresentativeMDEnetworksshowthatweachievebetterrobustness against various adversarial attacks with nearly no benign performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 1 INTRODUCTION Monocular Depth Estimation (MDE) is a technique that estimates depth from a single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It enables 2D-to-3D projection by predicting the depth value for each pixel in a 2D image and serves as a very affordable replacement for the expensive Lidar sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It hence has a wide range of applications such as autonomous driving (Karpathy, 2020), visual SLAM (Wimbauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2021), and visual relocalization (von Stumberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In particular, self-supervised MDE gains fast-growing popularity in the industry (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', Tesla Autopilot (Karpathy, 2020)) because it does not require the ground-truth depth collected by Lidar during training while achieving comparable accu- racy with supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Exploiting vulnerabilities of deep neural networks, multiple digital- world (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020) and physical-world attacks (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2022) against MDE have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' They mainly use optimization-based methods to generate adversarial ex- amples to fool the MDE network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Due to the importance and broad usage of self-supervised MDE, these adversarial attacks have posed a great threat to the security of applications such as autonomous driving, which makes the defense and MDE model hardening an urgent need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Adversarial training (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2014) is the most popular and effective way to defend ad- versarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' However, it usually requires ground truth labels in training, making it not directly applicable to self-supervised MDE models with no depth ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Although contrastive learn- ing gains a lot of attention recently and has been used for self-supervised adversarial training (Ho & Nvasconcelos, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020), it does not considerthedomainknowledgeofdepth estimation and can hardly achieve optimal results (shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In addition, many existing adversarial training methods do not consider certain properties of physical-world attacks such as strong pertur- ∗Corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='13487v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='CV] 31 Jan 2023 Published as a conference paper at ICLR 2023 (a) Conceptual illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (b) Different Views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Figure 1: Self-supervised adversarial training of MDE with view synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' bations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Hence in this paper, we focus on addressing the problem of hardening self-supervised MDE models against physical-world attacks without requiring the ground-truth depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' A straightforward proposal to harden MDE models is to perturb 3D objects in various scenes and ensure the estimated depths remain correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' However, it is difficult to realize such adversarial train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' First, 3D perturbations are difficult to achieve in the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' While one can train the model in simulation, such training needs to be supported by a high-fidelity simulator and a powerful scene rendering engine that can precisely project 3D perturbations to 2D variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Second, since self-supervised MDE training does not have the ground-truth depth, even if realistic 3D perturba- tions could be achieved and used in training, the model may converge on incorrect (but robust) depth estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In this paper, we propose a new self-supervised adversarial training method for MDE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Figure 1a provides a conceptual illustration of our technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' A board A printed with the 2D image of a 3D object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', a car) is placed at a fixed location (next to the car at the top-right corner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We use two cameras (close to each other at the bottom) Ct and Cs to provide a stereo view of the board (images It and Is in Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Observe that there are fixed geometric relations between pixels in the two 2D views produced by the two respective cameras such that the image in one view can be transformed to yield the image from the other view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Intuitively, It can be acquired by shifting Is to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Note that when two cameras are not available, one can use two close-by frames in a video stream to form the two views as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' During adversarial training, camera Ct takes a picture It of the original 2D image board A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Similarly, camera Cs takes a picture Is of the board too (step \x82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The bounding box of the board A is recognized in It and the pixels in the bounding box corresponding to A are perturbed (step \x83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Note that these are 2D perturbations similar to those in traditional adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' At step \x84, the perturbed image It+perturbations is fed to the MDE model to make depth estimation, achieving a 3D projection of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Due to the perturbations, a vulnerable model generates distance errors as denoted by the red arrow between A and the projected 3D object in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' At step , we try to reconstruct It from Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The reconstruction is parameter- ized on the cameras’ relative pose transformations and the estimated distance of the object from the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Due to the distance error, the reconstructed image Is→t (shown in Figure 1b) is different from It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Observe that part of the car (the upper part inside the red circle) is distorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In comparison, Figure 1b also shows the reconstructed image Iben s→t without the perturbation, which is much more similar to It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The goal of our training (of the subject MDE model) is hence to reduce the differences betweenoriginalandreconstructedimages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='Theaboveprocessisconceptual,whosefaithfulrealization entails a substantial physical-world overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In Section 3, we describe how to avoid the majority of the physical-world cost through image synthesis and training on synthesized data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' While traditional adversarial training assumes bounded perturbations in L2 or L∞ norm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', mea- suring the overall perturbation magnitude on all pixels), physical-world attacks are usually un- bounded in those norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' They tend to be stronger attacks in order to be persistent with environ- mental condition variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' To harden MDE models against such attacks, we utilize a loss function that can effectively approximate the L0 norm (measuring the number of perturbed pixels regardless of their perturbation magnitude) while remaining differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Adversarial samples generated by minimizing this loss can effectively mimic physical attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We make the following contributions: We develop a new method to synthesize 2D images that follow physical-world constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', relative camera positions) and directly perturb such images in adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The physical world cost is hence minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 2 Distance Error Reconstructed View Is-t 3D Projection ① Loss ① 2D image A of the object +Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Noise View Is Depth Estimation View It + perturbation Camera Pose Trans CtIt Is Distorted Is-t jben s-tPublished as a conference paper at ICLR 2023 Our method utilizes the reconstruction consistency from one view to the other view to enable self-supervised adversarial training without the ground-truth depth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We generate L0-bounded perturbations with a differentiable loss and randomize the camera and objectsettingsduringsynthesistoeffectivelymimicphysical-worldattacksandimproverobustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We evaluate the method and compare it with a supervised learning baseline and a contrastive learning baseline, adapted from state-of-the-art adversarial contrastive learning (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Results show that our method achieves better robustness against various adversarial attacks with nearly no model performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The average depth estimation error of an adversarial object with 1/10 area of perturbation is reduced from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='08 m to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='25 m by our method, better than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='18 m by the supervised learning baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Moreover, the contrastive learning baseline degrades model performance a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Our physical-world experiments video is available at https: //youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='be/_b7E4yUFB-g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 2 RELATED WORKS Self-supervised MDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Due to the advantage of training without the depth ground truth, self- supervised MDE has gained much attention recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In such training, stereo image pairs and/or monocular videos are used as the training input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Basically, two images taken by camera(s) from adjacent poses are used in each optimization iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' A depth network and a pose network are used to estimate the depth map of one image and the transformation between the two camera poses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' With the depth map and pose transformation, it further calculates the pixel correspon- dence across the images and then tries to rearrange the pixels in one image to reconstruct the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The pose network and the depth network are updated simultaneously to minimize the reconstruction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (2016) first propose to use color consistency loss between stereo images in train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (2018) enable video-based training with two networks (one depth network and one pose network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Many following works improve the self-supervision with new loss terms (Godard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Yin & Shi, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Ramamonjisoa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020) or include temporal information (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Among them, Monodepth2 (Godard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2019) significantly improves the performance with several novel designs such as minimum photometric loss selection, masking out static pixels, and multi-scale depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Depthhints (Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2019) further improves it via additional depth suggestions obtained from stereo algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' While such unsupervised training is effective, how to improve its robustness against physical attack remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' MDE Attack and Defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (2020) use a deep feature annihilation loss to launch perturbation attack and patch attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (2020) design a universal attack with a multi- task strategy and Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (2020) generate targeted adversarial perturbation on images which can alter the depth map arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Hu & Okatani (2019) propose a defense method against pertur- bation attacks by masking out non-salient pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It requires another saliency prediction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' For physical-world attacks, Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (2022) generates a printable adversarial patch to make the vehicle disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' To the best of our knowledge, we are the first work focusing on improving the robustness of self-supervised MDE models against physical-world attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Adversarial Robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It is known that deep neural networks (DNN) are vulnerable to adversar- ial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Imperceptible input perturbations could lead to model misbehavior (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Moosavi-Dezfooli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Typically, adversarial training is used to improve the robustness of DNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It uses both benign and adversarial examples for training (Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Carlini & Wagner, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Tramèr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Adversarial training has been applied to many domains like image classification (Carlini & Wagner, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2018), object detec- tion (Zhang & Wang, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='a), and segmentation (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Hung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Arnab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2018) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' A common requirement for adversarial training is supervision because generating adversarial examples needs ground truth, and most tasks require labels for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Some semi-supervised adversarial learning methods (Carmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Alayrac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2019) use a small portion of labeled data to enhance robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Contrastive learning (Ho & Nvasconcelos, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020) is also used with adversarial examples either for better self-supervised learning or to improve robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In this work, we explore the adversarial training of MDE without using ground- truth depth and compare our method with contrastive learning-based and supervised learning-based methods that are specifically tailored for MDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' There are other defense techniques such as input transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' However, Athalye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (2018a) point out that these techniques largely rely on obfuscated gradients which may not lead to true robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In the scenarios of autonomous driving, 3 Published as a conference paper at ICLR 2023 Figure 2: (a) Bird view of the relative positions of the camera and the target object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (b) 3D coordinates of the four corners of the object in the camera frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (c) Projection from the physical-world object to the two views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' there are other works focusing on the security of Lidar or multi-sensor fusion-based systems (Tu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' They use sensor spoofing or adversarial shapes to fool the Lidar hardware or AI model, while in this work, we consider fully-vision-based autonomous driving systems in which MDE is the key component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 3 OUR DESIGN Our technique consists of a few components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The first one (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='1) is view synthesis that generates the two views It and Is of the object and is equivalent to step \x82 in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The second one (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='2) is robust adversarial perturbation that perturbs It to induce maximum distance errors (step \x83 in Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The third one (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='3) is the self-supervised adversarial training (steps \x84 and ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We will discuss the details of these components and then present the training pipeline at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='1 VIEW SYNTHESIS TO AVOID PHYSICAL WORLD SCENE MUTATION As mentioned in Section 1, conceptually we need to place an image board of the 3D object at some physical locations on streets and use two cameras to take pictures of the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In order to achieve robustness in training, we ought to vary the image of the object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', different cars), the position of the image board, the positions and angles of the cameras, and the street view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' This entails enormous overhead in the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Therefore, we propose a novel view synthesis technique that requires minimal physical world overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Instead, it directly synthesizes Is and It with an object in some street view, reflecting different settings of the aforementioned configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Specifically, we use one camera to take a picture It of a 2D image board A of the object, with the physical width and height W and H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As shown in Figure 2 (b), the four corners of the board have the 3D coordinates (xt 0, yt 0, zt 0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', and (xt 3, yt 3, zt 3), respectively, in the camera frame, namely, the coordinate system with the camera as the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The camera is placed at zc distance away from the board with an angle of α, as shown in Figure 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The size of the board is true to the rear of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' This is important for the later realistic synthesis step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' After that, we derive a projection function that can map a pixel in A to a pixel in It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The function is parameterized on W, H, zc, α, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' such that we can directly use it to synthesize a large number of It’s with different A’s by changing those parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' To acquire Is that is supposed to form a stereo view with It, we do not necessarily need another camera in the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Instead, we can use two neighboring video frames of some street view (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', from the KITTI dataset (Geiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2013)), denoted as Rt and Rs, to approximate a stereo pair taken by two close-by cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Note that the differences between the two images implicitly encode the relative positions of the two cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' A prominent benefit of such approximation is that a large number of camera placements and street views can be easily approximated by selecting various neighboring video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' This is consistent with existing works (Godard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We replace the area in It that does not correspond to A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', the background of the object, with the corresponding area in Rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Intuitively, we acquire a realistic It by stamping the object’s image to a background image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The synthesis respects physical world constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' A projection function parameterized by Rt and Rs can be derived to map a pixel in one camera’s view to a pixel in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Then, we project the part of It that corresponds to A using the projection function and stamp it on Rs, acquiring Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As such, the resulted view of A (in Is) is consistent with the camera pose denoted by Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It and Is are then used in model hardening (discussed later in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 4 W (ub, va) (us, v) (> (xt, yt, zt) α (xt,yt,z (ul,vt) (uz, v2) Zc 0,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='2) H< X (us;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='v) (us,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' vs) (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='zi) (x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' z2) (ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='vi) (uz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' v2) y A Is (a) (b) (c)Published as a conference paper at ICLR 2023 Formally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' if the center of the physical camera’s view aligns with the center of the image board A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' the correlation between a pixel (uA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' vA) in A and its 3D coordinate (xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' yt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' zt) is: � �� xt yt zt 1 � �� = � �� cos α 0 − sin α 0 0 1 0 0 sin α 0 cos α zc 0 0 0 1 � �� · � �� W/w 0 −W/2 0 H/h −H/2 0 0 0 0 0 1 � �� · � � uA vA 1 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (1) where w and h are the width and height of A in pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The other variables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', α, zc, W, H, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=') are defined in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The 3D coordinates can be further projected to pixels in It and Is as: �ut vt 1�⊤ = 1/zt · K · �xt yt zt 1�⊤ , [us vs 1]⊤ = 1/zs · K · [xs ys zs 1]⊤ , [xs ys zs 1]⊤ = Tt→s · �xt yt zt 1�⊤ , (2) where Tt→s is the camera pose transformation (CPT) that projects 3D coordinates in the phys- ical camera Ct’s coordinate system to coordinates in the other (virtual) camera Cs’s coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It is determined by Rs and Rt as mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' K is the camera intrinsic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Com- bining Equation 1 and Equation 2, we know the projections from pixel (uA, vA) of the object image to pixel (ut, vt) in It and to pixel (us, vs) in Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Let �ut vt 1�⊤ = P A→t zc,α (uA, vA) and [us vs 1]⊤ = P A→s zc,α,Tt→s(uA, vA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We synthesize It and Is as: It[u, v] = � A[uA, vA], [u v 1]⊤ = P A→t zc,α (uA, vA) Rt[u, v], otherwise , (3) Is[u, v] = � A[uA, vA], [u v 1]⊤ = P A→s zc,α,Tt→s(uA, vA) Rs[u, v], otherwise , (4) where Rt and Rs are the background images implicitly encoding the camera relative poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' A large number of It and Is are synthesized by varying Rt, Rs, zc, α, A, and used in hardening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The creation induces almost zero cost compared to creating a physical world dataset with similar diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='2 ROBUST ADVERSARIAL PERTURBATIONS We use an optimization based method to generate robust adversarial perturbations δ on the object image A composing the corresponding adversarial object A + δ and synthesize I′ t by replacing A with A + δ in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The synthesized I′ t is then used in adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We bound the perturbations with L0-norm, which is to constrain the number of perturbed pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Compared with digital-world attacks that use traditional L∞-norm or L2-norm-bounded perturbations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', FGSM (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2014), Deepfool (Moosavi-Dezfooli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2016), and PGD (Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2018)), physical-world attacks usually use adversarial patches (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2017) without restric- tions on the magnitude of perturbations in the patch area because stronger perturbations are needed to induce persistent model misbehavior in the presence of varying environmental conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', lighting conditions, viewing angles, distance and camera noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Hence, L0-norm is more suitable in physical-world attacks because it restricts the number of pixels to perturb without bounding the per- turbation magnitude of individual pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' However, the calculation of L0-norm is not differentiable by definition and hence not amenable for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We hence use a soft version of it as proposed in Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The main idea is to decompose the perturbations into positive and negative components and use the long-tail effects of tanh function, in the normalization term, to model the two ends of a pixel’s value change (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', zero perturbation or arbitrarily large perturbation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As such, a pixel tends to have very large perturbation or no perturbation at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' δ = maxp · (clip(bp, 0, 1) − clip(bn, 0, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (5) Lpixel = � h,w � max c �1 2(tanh(bp γ ) + 1) �� + � h,w � max c �1 2(tanh(bn γ ) + 1) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (6) Specifically, the perturbation is defined in Equation 5 and the normalization term is Lpixel in Equa- tion 6, where bp and bn are the positive and negative components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' clip() bounds the variable to a 5 Published as a conference paper at ICLR 2023 Figure 3: Pipeline of adversarial training of self-supervised monocular depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Solid lines denote data flow and dashed lines denote back propagation paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' range of [0,1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' h, w, c are the height, width and channels of image and γ is a scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We refer readers to Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (2022) for detailed explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Equation 7 gives the formal definition of our perturbation generation method, where Sp is a distri- bution of physical-world distance and view angles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', reflecting the relations between cameras and cars during real-world driving);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' SR is the set of background scenes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', various street views);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' D() is the MDE model which outputs the estimated depth map of given scenario and MSE() is the mean square error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' min bn,bp Ezc,α∼Sp,Rt∼SR � MSE � D (I′ t)−1 , 0 �� + Lpixel, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' L0(δ) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (7) Our adversarial goal is to make the target object further away, so we want to maximize the depth estimation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', minimize the reciprocal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Intuitively, we synthesize I′ t with random Rt and differ- ent α and zc of object A and use expectation of transformations (EoT) (Athalye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2018b) to improve physical-world robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We minimize the mean square error between zero and the re- ciprocal of synthesized scenario’s depth estimation in the adversarial loss term and use Lpixel as the normalization term of perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Parameter ϵ is a predefined L0-norm threshold of perturbations and it denotes the maximum ratio of pixels allowed to perturb (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', ϵ = 1/10 means 1/10 pixels can be perturbed at most).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='3 SELF-SUPERVISED MDE TRAINING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In each training iteration, we first synthesize It and Is as mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Perturbations are then generated to It to acquire I′ t following Equation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As illustrated in Figure 1a, we reconstruct a version of It from Is using the depth information derived from I′ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We call the resulted image Is→t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Intuitively, I′ t causes depth errors which distort the projection from Is to Is→t such that the latter appears different from It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Similar to how we project (uA, vA) to (ut, vt) or (us, vs) earlier in this section (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', Equation 2), we can project (us, vs) in Is to a pixel (us→t, vs→t) in Is→t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' This time, we use the depth information derived from I′ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Formally, the projection is defined as: �xs→t ys→t zs→t 1�⊤ = K−1 · DI′ t(us→t, vs→t) · �us→t vs→t 1�⊤ , [xs ys zs 1]⊤ = Tt→s · �xs→t ys→t zs→t 1�⊤ , [us vs 1]⊤ = 1/zs · K · [xs ys zs 1]⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (8) Intuitively, there are relations between 2D image pixels and 3D coordinates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', the first and the third formulas in Equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The 3D coordinates also have correlations decided by camera poses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', the second formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Observe that, the first 2D-to-3D relation is parameterized on DI′ t, the depth estimation of I′ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Let [us vs]⊤ = PDI′ t,Tt→s(us→t, vs→t) be the transformation function that projects a pixel in Is→t to a pixel in Is derived from Equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Is→t is synthesized as: Is→t[u, v] = Is[PDI′ t,Tt→s(u, v)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (9) Intuitively, it rearranges the pixels in Is to form Is→t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We then compare Is→t with It and minimize their difference to guide the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Our training pipeline is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' There are two trainable networks, the MDE model D and a camera transposing model TP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Recall that we need the camera pose transformation matrix Tt→s 6 → Data Flow → Back Propagation α,Zc,Rs IS A α,Zc, Rt D(It) pe(It,l +8Published as a conference paper at ICLR 2023 Table 1: Benign performance of original and hardened models on depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Monodepth2 DepthHints Models ABSE↓ RMSE↓ ABSR↓ SQR↓ δ ↑ ABSE↓ RMSE↓ ABSR↓ SQR↓ δ ↑ Original 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='125 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='631 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='807 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='877 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='194 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='756 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='928 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='526 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='213 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='701 For hardened models, A+B denotes generating adversarial perturbation with method A and training with method B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' between the two cameras’ coordinate systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We hence train the TP network that predicts Tt→s from a given pair of background images Rs and Rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We denote it as: Tt→s = TP(Rt, Rs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Observe that in Figure 3, from left to right, the pipeline takes the object image A and synthesizes images It and Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It and A are further used to derive adversarial sample I′ t, which is fed to the depth network D to acquire depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The depth information, the TP network’s output Tt→s, and Is are used to derive Is→t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Two outputs Is→t and It are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The training objective is hence as: min θD,θTP Lp = pe(It, Is→t), (10) which is to update the weight values of D and TP to minimize the photometric reconstruction error of the two outputs, denoted by pe().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Specific designs of pe() differ in literature but our model hardening technique is general to all self-supervised MDE methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 4 EVALUATION In this section, we evaluate the performance of our method in white-box, black-box, and physical- world attack scenarios, and discuss the ablations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Our code shall be available after acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='1 EXPERIMENTAL SETUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Networks and Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We use Monodepth2 (Godard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2019) and DepthHints (Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2019) as our subject networks to harden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' They are representative and popular self-supervised MDE models that are widely used as benchmarks in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Both models are trained on the KITTI dataset (Geiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2013) and our methods fine-tune the original models publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' There are no direct baselines available since no prior works have been focusing on harden- ing MDE models as far as we know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Hence we extend state-of-the-art contrastive learning-based and supervised learning-based adversarial training methods to MDE and use them as our baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' They do not require ground-truth depth, same as our self-supervised method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Details are in Appendix A Training Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In adversarial training, the ranges of distance zc and viewing angle α are sam- pled randomly from 5 to 10 meters and -30 to 30 degrees, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The view synthesis uses EoT (Athalye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We generate the adversarial perturbations with two methods: L0-norm- bounded with ϵ = 1/10 and L∞-norm-bounded (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', PGD (Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2018)) with ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The latter is for comparison purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We train with our self-supervised method and two baseline meth- ods based on contrastive learning and supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Hence, there are 6 approaches combining the 2 perturbation generation methods with the 3 training methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' With these approaches, we fine- tune the original model for 3 epochs on the KITTI dataset and produce 6 hardened models for each network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Other detailed configurations and the selection of 2D object images are in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We conduct various kinds of attacks to evaluate the robustness of different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' They are L0-norm-bounded attacks with ϵ = 1/20, 1/10, 1/5 and 1/3, L∞-norm-bounded (PGD) attacks with ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='2 (image data are normalized to [0,1]), and an adversarial patch attack in Mathew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Adversarial perturbation or patch is applied to an object image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The patch covers 1/10 of the object at the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Each attack is evaluated with 100 randomly selected background scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The object is placed at a distance range of 5 to 30 meters and a viewing angle range of -30 to 30 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We report the average attack performance over different background scenes, distances, and viewing angles for each attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In addition, we conduct the state-of-the-art physical-world attack (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2022) with the printed optimal patch and a real vehicle in driving scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Adversarial examples are in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Evaluation with more attacks are in Appendix J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We use the mean absolute error (ABSE), root mean square error (RMSE), relative absolute error (ABSR), relative square error (SQR), and the ratio of relative absolute error under 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='25 (δ) as 7 Published as a conference paper at ICLR 2023 Table 2: Defence performance of original and hardened models under attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Attacks Original L0+SelfSup (Ours) L0+Sup L0+Contras PGD+SelfSup PGD+Sup PGD+Contras ABSE↓ δ ↑ ABSE↓ δ ↑ ABSE↓ δ ↑ ABSE↓ δ ↑ ABSE↓ δ ↑ ABSE↓ δ ↑ ABSE↓ δ ↑ Monodepth2 L0 1/20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='95 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='93 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='98 Bold and underlining indicate the best and second best performance in each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Hardened models are named the same as Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' the evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' These metrics are widely used in evaluating depth estimation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Metric δ denotes the percentage of pixels of which the ratio between the estimated depth and ground- truth depth is smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It is the higher, the better and the others are the lower, the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The definition of each metric can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='2 MAIN RESULTS Benign Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Together with the original model, we have 7 models under test for each network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We evaluate the depth estimation performance on the KITTI dataset using the Eigen split and report the results in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As shown, self-supervised and supervised methods have little influence on the models’ depth estimation performance, which means these approaches can harden the model with nearly no benign performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In contrast, the contrastive learning- based approach performs the worst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The ABSE of estimated depth is over 1 m worse than the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The reason could be that contrastive learning itself does not consider the specific task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', MDE) but fine-tunes the encoder to filter out the adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Thus the benign performance is sacrificed during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The other two methods consider the depth estimation performance either by preserving the geometric relationship of 3D space in synthesized frames or by supervising the training with estimated depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' White-box Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We conduct various white-box attacks on each model to evaluate the robustness of hardened models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Specifically, for each model, we compare the estimated depth of the adversar- ial scene (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', I′ t) with that of the corresponding benign scene (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', It in Equation 3) and larger difference means worse defense performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Table 2 shows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As shown, all the hardened models have better robustness than the original models against all kinds of attacks, and it is generic on the two representative MED networks, which validates the effectiveness of our adversarial train- ing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Comparing different approaches, L0+SelfSup has the best performance in gen- eral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It reduces the ABSE caused by all-level L0-norm-bounded attacks from over 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='7 m to less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='6 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Specifically,theself-supervision-basedmethodoutperformsthecontrastivelearning-basedand the supervision-based methods regardless of the perturbation generation method used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It is because the self-supervision-based method follows the original training procedure that is carefully designed for the network and has been evaluated thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It is not surprising that models adversarially trainedwithL0-norm-boundedperturbation(ourmethod)achieve better robustness against L0-norm- bounded attacks and so do PGD-based methods, but more importantly, L0-norm-based training also has good defense performance against PGD attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The robustness of L0+SelfSup is only slightly worse than PGD+SelfSup on some PGD attacks and even better than it on stronger PGD attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' An explanation is that L0-norm does not restrict the magnitude of perturbation on each pixel, and stronger PGD attacks are closer to this setting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', high-magnitude perturbations) and can be well-defended using the L0-norm-based adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Monodepth2 is vulnerable to the patch attack, and this kind of attack can be well-defended by our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' L0+SelfSup also performs the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Depthhints itself is relatively robust to the patch attack, and our methods can fur- ther reduce the attack effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Our defense generalizes well to complex scenes including various road types, driving speed, and the density of surrounding objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Qualitative results are in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 8 Published as a conference paper at ICLR 2023 Figure 4: Physical-world attack and defence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Video: https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='be/ _b7E4yUFB-g Training Distance Range (m) Mean Absolute Error (m) 0 2 4 6 5 5~10 5~15 5~20 5~25 5~30 L0-Norm Attack 1/20 L0-Norm Attack 1/10 L0-Norm Attack 1/5 L0-Norm Attack 1/3 (a) L0-norm attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Training Distance Range (m) Mean Absolute Error (m) 0 2 4 6 5 5~10 5~15 5~20 5~25 5~30 PGD Performance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='05 PGD Performance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='1 PGD Performance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='2 Physical World ATK (b) PGD and patch attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Figure 5: Robustness with different training distance ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Table 3: Defence performance of original and hardened models under black-box attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Target Original L0+SelfSup (Ours) L0+Sup L0+Contras Source ABSE δ ↑ ABSE δ ↑ ABSE δ ↑ ABSE δ ↑ Original 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='71 L0+SelfSup 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='95 L0+Sup 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='89 L0+Contras 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='98 Bold indicates the best performance in each row or column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Black-box Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We also vali- date our methods against black-box attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We use the original Mon- odepth2 model and the models fine- tuned with L0-norm-bounded pertur- bations and the three training methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We perform L0-norm-bounded attacks on each model with ϵ = 1/10 and ap- ply the generated adversarial object to other models evaluating the black-box attack performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The first column in Table 3 denotes the source models and other columns are the target models’ defense performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Looking at each column, adversarial examples generated from L0+SelfSup have the worst attack performance, which indicates the low transferability of ad- versarial examples generated from the model trained with our self-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' From each row, we can observe that L0+SelfSup has the best defense performance against adversarial ex- amples generated from each source model, which further validates the robustness of L0+SelfSup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In summary, the self-supervised training method can produce safer and more robust models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Their adversarial examples have low transferability and they defend against black-box attacks well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Physical-world Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Our evaluation with the state-of-the-art physical-world MDE at- tack (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2022) validates the effectiveness of our method in various real-world lighting conditions, driving operations, road types, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The experimental settings are the same as Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Figure 4 shows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The first row is the real-world adversarial scene, in which a car is moving with an adversarial patch attached to the rear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The second row is the depth pre- dicted by the original Monodepth2 model and the third row is predicted by our hardened model (L0+SelfSup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The “hole” in the output of the original model caused by the adversarial patch is fixed in our hardened model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It is known that the adversarial attacks designed for the physical world are still generated in the digital world and they have better digital-world attack performance than physical-world performance because additional environmental variations in the physical world de- grade the effectiveness of adversarial patterns (Braunegg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Hence, defending attacks in the digital world is more difficult and our success in digital-world defense in previous experiments has implied effectiveness in the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='3 ABLATIONS Distance Range in Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' While synthesizing views in training, the range of distance zc of the target object is randomly sampled from d1 to d2 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In this ablation study, we evaluate the effect of using different ranges of distance in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We use L0+SelfSup to fine-tune the original Monodepth2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The ranges of distance we use in training are [5, 5], [5, 10], [5, 15], [5, 20], [5, 25] and [5, 30] (Unit: meter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Note that, for a fair comparison, the range of distance we use in model evaluation is always from 5 to 30 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The results are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As shown, the model trained with a distance range of 5-10 meters has the best robustness and a larger or smaller distance range could lead to worse performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It is because further distances lead to smaller objects on the image and fewer pixels are affected by the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Thus the attack performance is worse at further distances and training with these adversarial examples is not the most effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' If the distance in training is too small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 5 meters), the model cannot defend various scales of attack patterns and 9 Real-world Scene Original Model HardenedModePublished as a conference paper at ICLR 2023 cannot generalize well to further distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In our experiments, the range of 5-10 meters makes a good balance between training effectiveness and generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Other ablation studies about viewing angles range in training are in Appendix E, transferability to unseen target objects is in Appendix F, comparing training from scratch and fine-tuning is in Appendix G and the performance of method combinations can be found in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 5 CONCLUSION We tackle the problem of hardening self-supervised Monocular Depth Estimation (MDE) mod- els against physical-world attacks without using the depth ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We propose a self- supervised adversarial training method using view synthesis considering camera poses and use L0- norm-bounded perturbation generation in training to improve physical-world attacks robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Compared with traditional supervised learning-based and contrastive learning-based methods, our method achieves better robustness against various adversarial attacks in both the digital world and the physical world with nearly no benign performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 10 Published as a conference paper at ICLR 2023 6 REPRODUCIBILITY STATEMENT To help readers reproduce our results, we have described the implementation details in our experi- mental setup (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='1 and Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In the supplementary materials, we attached our source code and instructions to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We will open our source code right after acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The dataset we use is publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We also attached the videos of our physical-world attack in the 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D3vo: Deep depth, deep pose and deep uncertainty for monocular visual odometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In CVPR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Mang Ye, Xu Zhang, Pong C Yuen, and Shih-Fu Chang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Unsupervised embedding learning via invariant and spreading instance feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In CVPR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Zhichao Yin and Jianping Shi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Geonet: Unsupervised learning of dense depth, optical flow and camera pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In CVPR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 13 Published as a conference paper at ICLR 2023 Haichao Zhang and Jianyu Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Towards adversarially robust object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In ICCV, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Ziqi Zhang, Xinge Zhu, Yingwei Li, Xiangqun Chen, and Yao Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Adversarial attacks on monoc- ular depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' arXiv preprint arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='10315, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Kaichen Zhou, Lanqing Hong, Changhao Chen, Hang Xu, Chaoqiang Ye, Qingyong Hu, and Zhen- guo Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Devnet: Self-supervised monocular depth learning via density volume construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In ECCV, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Yuliang Zou, Zelun Luo, and Jia-Bin Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Df-net: Unsupervised joint learning of depth and flow using cross-task consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In ECCV, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Yuliang Zou, Pan Ji, Quoc-Huy Tran, Jia-Bin Huang, and Manmohan Chandraker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Learning monoc- ular visual odometry via self-supervised long-term modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In ECCV, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 14 Published as a conference paper at ICLR 2023 Appendix Adversarial Training of Self-supervised Monocular Depth Estimation against Physical-World Attacks This document provides more details about our work and additional experimental settings and result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We organize the content of our appendix as follows: Section A: the baseline methods we tailored for MDE specifically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Section B: more details about the training configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Section C: the formal definition of the metrics we used in our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Section D: the adversarial attack examples and qualitative results of defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Section E: the effect of different ranges of viewing angles in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Section F: the transferability evaluation of our hardened models to other target objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Section G: the difference between fine-tuning and training from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Section H: the model performance of combining different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Section I: comparing the supervised baseline trained with pseudo-depth labels and that trained with ground-truth depth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Section J: the robustness against more attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Section K: human evaluation of the quality of our synthesized images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Section L: the influence of inaccurate projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Section M: extension of our method to indoor scenes and more advanced MDE networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Section N: the broader impact and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' A BASELINES Adversarial Contrastive Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Contrastive learning is a widely used technique specifically tailored to self-supervised learning scenarios (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Misra & Maaten, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It has been used with adversarial exam- ples either to improve the robustness of models against adversarial attacks (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020) or to enhance the performance of contrastive learning itself (Ho & Nvasconcelos, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In this work, we extend a state-of-the-art contrastive learning-based adversarial training method (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020) to harden MDE models against physical attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We use it as a baseline to compare with our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Figure 6: (a) Adversarial contrastive learning of model encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The color-augmented benign (It) and adver- sarial (I′ t) examples are fed to the depth model encoder (the grey block) and one embedding (e′) is then fed to a prediction multi-layer perceptron (MLP) that transcribes the embedding to another embedding p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We maxi- mize the similarity of the output (p′) with the other embedding (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Backpropagation is only calculated along one side to update the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (b) Supervised adversarial training with the estimated depth as the pseudo- ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We use the output of the original model on benign examples as the ground truth to supervise the training of the subject model with adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The solid lines denote data flow and the dashed lines denote back propagation paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 15 Adversarial Image(It) Benign Image(It) (a) Adversarial Contrastive Learning (a) Stop Grad Depth CNN Depth CNN e MLP Sim (b) Supervised Adversarial Learning D(lt) D(It) MSE (b)Published as a conference paper at ICLR 2023 Figure 7: The various target objects in the transferability evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Similar to Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (2020), the positive pairs in our contrastive learning are the benign examples (It) and the corresponding adversarial examples (I′ t), and we further augment those examples by changing the color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Different from Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (2020), we do not need negative pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Instead, we use a learning method proposed in SimSiam (Chen & He, 2021) that only requires positive pairs and can achieve competitive performance with smaller batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Figure 6 (a) shows the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The key point is to maximize the similarity between the embeddings of the benign and adversarial examples so that their depth map outputs from the decoder network are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The parameters of the subject MDE model’s encoder and the prediction MLP network are updated iteratively in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We use color augmentation instead of other transformations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', resizing and rotation) because the embeddings should be similar among positive samples and the change of color would not affect the depth map output (but other transformations would).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Other settings such as the MLP network structure are the same as SimSiam (Chen & He, 2021) and we refer readers to it for detailed explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Supervised Adversarial Learning with Estimated Depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Since we do not have depth ground truth in the self-supervised scenario, one alternative way to do adversarial training is to use the estimated depth by the original model with inputs of benign samples as the pseudo ground truth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', pseudo labels) and perform supervised adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As shown in Figure 6 (b), we use mean square error (MSE) as the loss function to update MDE model parameters and minimize the difference between the model output of adversarial samples and the pseudo ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Using the pseudo ground truth predicted by an existing model is proved to be a simple and effective method in the field of semi-supervised learning (SSL) (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2013) and it has been used in adversarial training (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2021) and self-supervised MDE (Petrovai & Nedevschi, 2022) to boost model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Particularly, in the field of MDE, using pseudo-ground truth is good enough compared with using the real ground truth (Petrovai & Nedevschi, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Same as our supervised baseline, Petrovai & Nedevschi (2022) uses the depth estimated by an existing MDE model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', pseudo depth labels) to supervise the following MDE model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Results show that the pseudo-supervised model has similar or better performance than the reference model trained with ground-truth depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We also conduct experiments comparing the performance of supervised baseline trained with pseudo depth labels and ground-truth depth labels, which proves that a pseudo- supervised baseline is not a weak choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The results can be found in Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' B TRAINING CONFIGURATIONS We train our model with one GPU (Nvidia RTX A6000) that has a memory of 48G and the CPU is Intel Xeon Silver 4214R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' For each model, doing adversarial training from scratch takes around 70 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It includes 20 epochs of training on the KITTI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The fine-tuning of 3 epochs takes about 10 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The input resolution of our MDE model is 1024*320 and the original monodepth2 and depthhints models we used for fine-tuning are the official versions trained with both stereo images and videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In our hardening, we use stereo images with fixed camera pose transformation Tt→s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In perturbation generation, we use 10 steps and a step size of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='5 · ϵ/10 in L2 and L∞-bounded attacks to ensure that we can reach the boundary of the ϵ-ball from any starting point within it (Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2018) and a batch size of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In MDE training, the batch size is 32, and the learning rate is 1e-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We use Adam as the optimizer and other training setups are the same as the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As for the selection of 2D images of objects, as shown in Figure 2 (a) and Figure 2 (b), we have assumptions about the initial relative positions between the target object and the camera (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', the 3D coordinates of the center of the object is (0, 0, zc) in the camera’s coordinate system and the 16 (a) BMW SUV Black (b) Toyota Sedan Blue (c) Subaru Sedan White (d) Volvo SUV Grey (e) Traffic BarrierPublished as a conference paper at ICLR 2023 Figure 8: Examples of adversarial attacks in our robustness evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' viewing angle α of the camera is 0 degree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Hence, for a more realistic and high-quality synthesis, the camera should look at the center of the target object at the same height while taking the 2D image of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The width w and height h of the 2D image of the object should be proportional to the physical size W and H of it: w/W = h/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Moreover, when we prepared the 2D image of the object, we also prepared a corresponding mask to “cut out” the main body of the object for projection and we take the object together with its shadow to preserve reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Examples of object masks can be found in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We train models with L0 and L∞-bounded (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', PGD) perturbations in our evaluation but not L2 norm because Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (2018) has demonstrated that models hardened with L∞-bounded pertur- bations are also robust against L2-bounded attacks and our experiments in Appendix J also validate the robustness of our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In addition, physical-world attacks with adversarial patches have more resemblance to L0-bounded attacks that only restrict the ratio of perturbed pixels rather than the magnitude of the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' C EVALUATION METRICS The evaluation metrics we used in our evaluation are defined as follows, where we use X = {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', xn} to denote the estimated depth map and Y = {y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', yn} to denote the refer- ence depth map and I() is the indicator function that evaluates to 1 only when the condition is satisfied and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' ABSE = 1 n n � i=1 |xi − yi| (11) RMSE = � � � � 1 n n � i=1 (xi − yi)2 (12) ABSR = 1 n n � i=1 (|yi − xi| yi ) (13) SQR = 1 n n � i=1 (yi − xi)2 yi (14) 17 (a) Lo-norm bounded perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (ε = 1/10) (b) Lo-norm bounded perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='1) (c) Unbounded adversarial patch attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' (d) Optimal physical-world patch attackPublished as a conference paper at ICLR 2023 Figure 9: Qualitative results of the defensive performance of our hardened model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' δ = 1 n n � i=1 I(max{xi yi , yi xi } < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='25) (15) The mean absolute error (ABSE) and root mean square error (RMSE) are common metrics and are easy to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Intuitively, the relative absolute error (ABSR) is the mean ratio between the error and the ground truth value, and the relative square error (SQR) is the mean ratio between the square of error and the ground truth value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' δ denotes the percentage of pixels of which the ratio between the estimated depth and ground-truth depth is smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' D ADVERSARIAL ATTACK EXAMPLES Figure 8 gives examples of the three kinds of adversarial attacks we conducted in our robustness evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The first column is the original object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' the second column is the adversarial one and the third column is the adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We scale the adversarial perturbations of the L∞- norm-bounded attack for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' L0-norm restricts the number of perturbed pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' L∞-norm restricts the magnitude of perturbation of each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Adversarial patch attack perturbs pixels within the patch area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Figure 9 shows the qualitative results of our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The vertical axis denotes different street views and the horizontal axis denotes different models and attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Here we compare the perfor- 18 Original Model Hardened Model Lo Attack 1/10 Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Patch Attack Lo Attack 1/10 Ady.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Patch Attack Scene Depth Error Scene Depth ErrorScene Depth Error Scene Depth ErrorPublished as a conference paper at ICLR 2023 ABSE 0 1 2 3 4 L0-Norm L0-Norm L0-Norm 1/5 L0-Norm 1/3 PGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='05 PGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='1 PGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='2 Patch Attack Fine Tune From Scratch Figure 10: Robustness performance of fine-tuning and training from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Table 4: Benign performance of models trained with methods combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' ABSE RMSE ABSR SQR δ ↑ Original 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='125 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='631 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='807 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='877 SelfSup+Con 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='172 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='847 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='875 SelfSup+Sup 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='161 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='819 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='836 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='875 Con+Sup 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='408 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='849 All 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='171 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='875 Table 5: Ablations on the range of angles in adversarial training with L0+SelfSup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 0◦ −10◦ − 10◦ −20◦ − 20◦ −30◦ − 30◦ −40◦ − 40◦ Attacks ABSE ABSE ABSE ABSE ABSE L0 1/10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='260 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='251 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='249 L0 1/5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='351 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='347 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='350 PGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='662 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='543 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='544 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='539 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='536 PGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='587 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='472 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='465 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='467 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='470 mance of the original Monodepth2 model with our model hardened by L0+SelfSup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' For each street view, model, and attack, the first row is the adversarial scene (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', the scene with the adversarial object), the second row is the corresponding depth estimation and the third row is the depth estima- tion error caused by the adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As shown, our method mitigates the effectiveness of attacks significantly and the attacks can hardly cause any adversarial depth estimation error on our hardened model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In addition, our method works on different scenes including complex ones that have multiple pedestrians or vehicles at different speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It is because we do not have assump- tions about the scene geometry or surrounding objects in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Both the scene synthesis and depth estimation are single-image-based and the scene geometry or moving speed will not affect the quality of synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' E VIEWING ANGLES RANGE IN TRAINING Randomizing the degree of synthesis will make the model more robust in the physical-world settings because the camera’s viewing angle is not fixed toward the target object in practice, and the range we use (-30 degrees to 30 degrees) covers the most common situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Ablations of the range of angles during adversarial training do not have as much effect as changing the range of distance because distance has a dominant influence on the size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', number of pixels) of the adversarial object on the synthesized scene image (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', the further object looks smaller) and the number of adversarial pixels affects the attack performance a lot (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We conduct experiments with different viewing angle ranges, and other settings are the same as the ablation study of distance ranges in Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Table 5 shows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As shown, ranges of viewing angles have less influence on the defensive performance, and using a fixed setting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 0◦) still performs the worst, which is consistent with our claims above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' F TRANSFER TO OTHER TARGET OBJECTS The target object under attack may not be used in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In this experiment, we evaluate how the adversarially trained models perform on unseen target objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We evaluate with four vehicles and one traffic barrier using the original and the three hardened Monodepth2 models and we perform L0-norm-bounded attack on each object with ϵ = 1/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Figure 7 shows the examples of target 19 Published as a conference paper at ICLR 2023 Table 6: Defence performance of attacks on various target objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Models Original L0+SelfSup (Ours) L0+Sup L0+Contras Objects ABSE ↓ δ ↑ ABSE ↓ δ ↑ ABSE ↓ δ ↑ ABSE ↓ δ ↑ BMW SUV Black 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='988 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='767 Bold indicates the best performance in each row and underlining means the second best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' objects that we used in our transferability evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Figure 7(a) is the object we used in adversarial training and others are target objects under attack in the robustness evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Table 6 shows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The first column is the target objects under attack and the following columns are the perfor- mance of different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The first object (BMW SUV Black) is an object used in training and our hardened models are most robust on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The others are unseen objects during training and the hardened models can still mitigate the attack effect a lot compared with the original model, which validates the transferability of our hardened models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' L0+SelfSup still has the best performance in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' G TRAINING FROM SCRATCH We also compare the robustness performance of training from scratch and fine-tuning an existing model with our self-supervised adversarial training method using L0-norm-bounded perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We evaluate on Monodepth2 and Figure 10 shows the ABSE result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As shown, training from scratch can provide more robustness, especially for higher-level attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As for the benign performance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', depth estimation performance), the ABSE of the fine-tuned model is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='16 while the ABSE of the model trained from scratch is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='468, meaning that the model trained from scratch has slightly worse than benign performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' H ADVERSARIAL TRAINING METHODS COMBINATION We introduced three adversarial training methods in Section 3 and evaluated them separately in our main experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In this experiment, we explore the method combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Specifically, we combine different loss terms together as a total loss in adversarial training and there are four combinations in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The perturbation in training is L0-norm-bounded with ϵ = 1/10 and other evaluation setups are the same as our main experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The depth estimation performance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', clean performance) of each model is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Results show that models trained with the self-supervised method have equivalent performance as the original model and Con+Sup performs slightly worse than the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Table 7 shows the robustness performance under attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As shown, combining self-supervised learning with contrastive learning could achieve better robustness than self-supervised learning itself, and combining all three methods further improves the robustness under some attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' I SUPERVISED BASELINE WITH GROUND TRUTH DEPTH Although in the scope of this paper, we discuss self-supervised scenarios and assume the ground- truth depth is not available in training, we still conduct experiments comparing the defensive perfor- mance of our supervised baseline trained with pseudo-ground truth and that trained with ground- truth depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We use Monodepth2 as the subject model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The results in Table 8 show that us- ing ground-truth depth (L0+Sup(GT)) has a similar performance to using pseudo ground truth (L0+Sup(Pseudo)), and they are still worse than our self-supervised approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Hence, whether to use ground-truth depth is not the bottleneck, and our pseudo-supervised baseline is not a weak choice, which also has significant defensive performance against adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 20 Published as a conference paper at ICLR 2023 Table 7: Defence performance of models trained with methods combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Attacks Original SelfSup+Con SelfSup+Sup Con+Sup All ABSE δ ↑ ABSE δ ↑ ABSE δ ↑ ABSE δ ↑ ABSE δ ↑ Monodepth2 L0 1/20 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='277 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='995 Bold indicates the best performance in each row and underlining means the second best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Table 8: Performance of the supervised baseline trained with pseudo depth label (L0+Sup(Pseudo)) and ground-truth depth label (L0+Sup(GT)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Attacks Original L0+SelfSup (Ours) L0+Sup(Pseudo) L0+Sup(GT) ABSE ↓ δ ↑ ABSE ↓ δ ↑ ABSE ↓ δ ↑ ABSE ↓ δ ↑ L0 1/10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='75 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='69 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='32 J ROBUSTNESS AGAINST MORE ATTACKS In addition to the attacks evaluated in our main paper (PGD attacks, l0 attacks, adversarial patch at- tack and physical-world optimal patch attack), we have also evaluated the robustness of our models against more attacks: the L2-bounded PGD attack (Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2018), APGD attacks in AutoAt- tack (Croce & Hein, 2020), Square attack (Andriushchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020)(a query-based black-box attack), Gaussian Blur (Rauber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2020) and AdvLight (Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The subject network is Mondepth2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Results can be found in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As shown, Our model is still more robust than the original model in all cases, and our method outperforms all others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Attacks with an L∞-bound of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='1 can only cause less than 1 m depth estimation error on our hardened models while more than 10 m on the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The black-box attack does not have a good performance in the MDE task and only causes a mean depth estimation error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='924 m with 5000 queries to the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Although the Gaussian Blur and AdvLight are more stealthy and natural attacks, they are not as as effective as other methods in the task of MDE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', per-pixel-based regression task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' K QUALITY OF THE VIEW SYNTHESIS To demonstrate the quality of our view synthesis, we show more examples of the synthesized images in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Is and It are two adjacent views of the same scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Different target objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', the white sedan, black SUV and traffic barrier) are synthesized into the views with various distances zc and background scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' To evaluate the veracity of these images, we use Amazon Mechanical Turk to conduct human evaluations of our synthesized images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Participants are requested to evaluate the quality of the synthesized objects from 4 distinct perspectives: size, consistency of location, lighting, and overall quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The score ranges from 1 to 10 for each metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We use the score of 1 to indicate scenes synthesized with random projections and the score of 10 for real scenes taken by stereo cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Examples of perfect (score of 10) and unrealistic (score of 1) scenes for each metric are provided as references (See Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The results of the experiment with 100 participants can be found in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In each row, we show the number of participants who gave a score within the corresponding range and summarize the average in the last row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As demonstrated, We got an average score of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='7 regarding the size, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='21 regarding the location, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='43 regarding the lighting, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='74 regarding the overall quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Compared with the real scene, our synthesized scene is slightly inferior but still much better than random projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' More importantly, it works well to harden the model against physical-world attacks at a low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 21 Published as a conference paper at ICLR 2023 Table 9: Defensive performance of original and hardened models under more attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Attacks Original L0+SelfSup (Ours) L0+Sup L0+Contras ABSE δ ↑ ABSE δ ↑ ABSE δ ↑ ABSE δ ↑ L2-PGD ϵ = 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='403 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='294 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='996 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='741 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='919 L2-PGD ϵ = 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='522 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='597 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='984 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='516 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='479 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='437 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='734 L2-PGD ϵ = 24 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='913 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='387 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='7 APGD ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='739 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='976 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='614 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='793 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='859 APGD ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='945 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='603 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='793 Square Attack ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='1 N=5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='422 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='973 Gaussian Blur 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='288 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='997 AdvLight 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='513 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='988 Figure 11: More examples of view synthesis with different background scenes, target objects and distance zc of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Object mask is used to remove background of the 2D object image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' L INFLUENCE OF INACCURATE PROJECTIONS As stated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='1, we project a 2D image of the target object to two adjacent views considering the camera pose transformation Tt→s between Cs and Ct (Equation 3 and 4), then we enable the self-supervised training of MDE network by reconstructing It from Is using the estimated depth DI′ t (Equation 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In this section, we explore what influence the inaccurate projections of the two camera views would have on training results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In this experiment, we only consider half of the camera pose transformation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', Tt→s/2) while projecting the object to Is instead of the true value Tt→s, which leads to a wrong Is intentionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' For example, the distance between the stereo cameras in the KITTI dataset is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='54 m, but we use 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='27 m to synthesize Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It is still correctly synthesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We use L0-bounded perturbation with ϵ = 1/10 and self-supervised training to harden the Monodepth2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Figure 13 shows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As shown, the outline of the estimated depth of the target object 22 It Is Zc = 5 White Zc = 10 Zc = 15 It Is Toyota Sedan Object Mask Zc = 5 Zc = 10 Zc = 15 Blue It Is Volvo SUV Object Mask Zc = 5 Zc = 10 Zc = 15 Black It Is Traffic Object Mask Zc = 5 Zc = 10 Zc = 15 Barrier Figure 1l: More examples of view synthesisPublished as a conference paper at ICLR 2023 Figure 12: The reference images used in our human study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Table 10: Human evaluations of the quality of our synthesized images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We show the number of participants who gave a score in the corresponding range in each row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Score Ranges Size Location Lightning Overall 1-2 0 0 1 0 5-6 4 6 6 3 7-8 16 24 23 18 9-10 42 48 40 45 Total 100 100 100 100 Average Score 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='21 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='43 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='74 is blurred when the MDE model is trained with inaccurate projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' It remains clear for the model trained with accurate projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' M EXTENSION TO INDOOR SCENES AND ADVANCED NETWORK Although we focus on outdoor scenarios like autonomous driving, a domain where self-supervised MDE is widely adopted (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', Tesla Autopilot), our technique hardens the MDE networks and does not have any assumptions about indoor or outdoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Actually, both Monodepth2 and Depth- Hints models have been proven to be directly applicable on indoor scenes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', NYU-Depth-v2 Dataset) without any retraining (Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2022), which implies the indoor- scene capability of our hardened models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We further conduct additional experiments with indoor scenes in the NYU-Depth-v2 Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We launch adversarial attacks in a square area at the center of each scene to mimic a physical patch in the scene and evaluate the defensive performance of the original and hardened Monodepth2 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Results are shown in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As shown, our hardened models reduce the depth estimation error caused by the attack significantly and the model trained with our self-supervised method has the best defensive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Hence, the robustness of our hardened model still holds for indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Monodepth2 and DepthHints are the two popular and representative self-supervised MDE networks, and they are widely used as baselines in the literature, so we use them as our subject networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' How- ever, we do not have any assumptions about the MDE network structure, and our method should also work on state-of-the-art MDE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Hence we conduct additional experiments with Many- depth (Watson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' We use our L0-bounded perturbation with self-supervised adversarial training to harden the original Manydepth model, and Table 12 shows the defensive performance of the original and hardened models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' As shown, the original models are still vulnerable to both L0- Table 11: Defensive performance on the indoor scenes with the NYU-Depth-v2 Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Attacks Original L0+SelfSup (Ours) L0+Sup L0+Contras ABSE δ ↑ ABSE δ ↑ ABSE δ ↑ ABSE δ ↑ L0 1/20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='772 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='42 23 Unrealistic Size Perfect Scene Unrealistic Lightning Inconsistent LocationPublished as a conference paper at ICLR 2023 Figure 13: Influence of inaccurate projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The outline of the estimated depth of the target object is blurred while using inaccurate projection in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Table 12: Defensive performance of the original and hardened Manydepth models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Attacks Original L0+SelfSup ABSE δ ↑ ABSE δ ↑ L0 1/20 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='452 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='339 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='614 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content='83 bounded and PGD attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The model hardened with our techniques is a lot more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The mean depth estimation error is reduced by over 80%, which validates that our techniques are generic and also work on state-of-the-art MDE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' N BROADER IMPACT AND LIMITATIONS Our adversarial training method hardens the widely used self-supervised monocular depth estima- tion networks and makes applications like autonomous driving more secure with regard to adver- sarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Compared to original training, the hardening of such models with our method does not require additional data or resources and the computational cost is affordable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' The adversarial attacks we studied are from existing works and we do not pose any additional threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Some lim- itations we could think of about our method are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In our synthesis of different views, we assume the physical-world object is a 2D image board instead of a 3D model to avoid using an expensive scene rendering engine or high-fidelity simulator and to improve efficiency, which could induce small errors in synthesis though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' However, since most physical-world attacks are based on adversarial patches attached to a flat surface, this 2D board assumption is realistic and practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' Precise synthesis should consider lighting factors such as glare, reflections, shadows, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=', but how to do that at a low-cost is an open problem and we leave it as our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' In addition, although our modeling of the relative positions and viewing angles of the camera and physical object does not cover all real-world conditions, it considers the most common situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' There might be some corner cases in which the adversarial attack could escape from our defense, but we still mitigate the threats in the most common situations and improve the overall security a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} +page_content=' 24 (a) Trained with inaccurate projection (b) Trained with accurate projection' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FRT4oBgHgl3EQfHDeZ/content/2301.13487v1.pdf'} diff --git a/NtAzT4oBgHgl3EQfzP6J/content/tmp_files/2301.01766v1.pdf.txt b/NtAzT4oBgHgl3EQfzP6J/content/tmp_files/2301.01766v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..577ca3f4b047bc1723a1fada87a9c6c66ee52177 --- /dev/null +++ b/NtAzT4oBgHgl3EQfzP6J/content/tmp_files/2301.01766v1.pdf.txt @@ -0,0 +1,6001 @@ +Learning Gaussian Mixtures Using the +Wasserstein-Fisher-Rao Gradient Flow +Yuling Yan∗† +Kaizheng Wang∗‡ +Philippe Rigollet§ +January 5, 2023 +Abstract +Gaussian mixture models form a flexible and expressive parametric family of distributions that has +found applications in a wide variety of applications. Unfortunately, fitting these models to data is a no- +toriously hard problem from a computational perspective. Currently, only moment-based methods enjoy +theoretical guarantees while likelihood-based methods are dominated by heuristics such as Expectation- +Maximization that are known to fail in simple examples. In this work, we propose a new algorithm +to compute the nonparametric maximum likelihood estimator (NPMLE) in a Gaussian mixture model. +Our method is based on gradient descent over the space of probability measures equipped with the +Wasserstein-Fisher-Rao geometry for which we establish convergence guarantees. In practice, it can be +approximated using an interacting particle system where the weight and location of particles are updated +alternately. We conduct extensive numerical experiments to confirm the effectiveness of the proposed +algorithm compared not only to classical benchmarks but also to similar gradient descent algorithms +with respect to simpler geometries. In particular, these simulations illustrate the benefit of updating +both weight and location of the interacting particles. +Keywords: +Gaussian mixture model, nonparametric MLE, Wasserstein-Fisher-Rao geometry, optimal +transport, Wasserstein gradient flows, overparameterization +1 +Introduction +Owing to their flexibility and versatility, mixture models have emerged as central objects of statistical +modeling since their introduction by Pearson in the nineteenth century. However, this flexibility often comes +at computational cost: such models are often hard to fit and, until quite recently, the computational aspects +of mixture models have been overlooked. Still today, theory and practice diverge, with the former largely +focuses on the method of moments while the latter is dominated by variational approaches, chiefly maximum +likelihood. The goal of this work is to reduce this gap by developing new algorithms for maximum likelihood +estimation that are supported by theoretical guarantees. +Consider i.i.d. samples {Xi}1≤i≤N ∈ Rd generated from an isotropic Gaussian1 mixture ρ⋆ ∗ N(0, Id) +with density function +(ρ⋆ ∗ φ) (x) = +� +Rd φ (x − y) ρ⋆ (dy) , +where ρ⋆ is a mixing distribution over Rd, and φ(x) = (2π)−d/2 exp(−∥x∥2 +2/2) is the density function of the +isotropic Gaussian distribution N(0, Id). The goal is to learn the Gaussian mixture ρ⋆ ∗ N(0, Id) from N +samples. +∗The first two authors contributed equally. +†Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544, USA; Email: +yulingy@princeton.edu. +‡Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA; Email: +kaizheng.wang@columbia.edu. +§Department of Mathematics, MIT, Cambridge, MA 02139, USA; Email: rigollet@math.mit.edu. +1All of this work extends to general mixtures. See the Appendix for a general treatment. +1 +arXiv:2301.01766v1 [math.ST] 4 Jan 2023 + +The negative log-likelihood for this problem is defined as +ℓN (ρ) = − 1 +N +N +� +i=1 +log [(ρ ∗ φ) (Xi)] . +While ℓN itself is trivially a convex functional of ρ, the class of measures over which it is minimized is often +not. This is for example the case of finite mixture models where ρ is restricted to be a measure supported on +at most k atoms. This lack of convexity in the constraint is the main source of computational difficulty for +this problem. To overcome this limitation, Kiefer and Wolfowitz (1956) proposed the nonparametric maxi- +mum likelihood estimator (NPMLE) which prescribes to minimize ℓN over the set P(Rd) of all probability +distributions over Rd: +�ρ ∈ argmin +ρ∈P(Rd) +ℓN (ρ) . +(1.1) +While P(Rd) is convex, it is infinite-dimensional. In fact, NPMLE can be seen as an extreme instance of +overparameterization, a phenomenon that is currently challenging conventional statistical wisdom in deep +learning theory Allen-Zhu et al. (2019); Chizat and Bach (2018). +In fact, the solution of (1.1) enjoys +interesting structural properties when d = 1. More specifically, in this case, the optimization problem (1.1) +admits a unique solution �ρ (Jewell, 1982; Lindsay and Roeder, 1993) which furthermore is supported on +at most N atoms (Lindsay, 1983) in general. This upper bound was improved to OP(log N) by Polyanskiy +and Wu (2020) when ρ⋆ is sub-Gaussian. Quite strikingly, little is known about the structure of NPMLE in +dimension d ≥ 2. In fact, to the best of our knowledge, Theorem 1 below is the first to establish existence +of a solution to (1.1) in any dimension. +Classical statistical results provide Hellinger risk bounds for �ρ ∗ φ as an estimator of ρ⋆ ∗ φ (Dicker +and Zhao, 2016; Saha and Guntuboyina, 2020; Zhang, 2009). These rates are commensurate with minimax +optimality even over the larger class of C∞ density functions up to logarithmic factors. +Despite notable contributions, the computational aspects of NPMLE are still vastly under-explored. Most +of these contributions employ a discretization scheme by setting a fine grid in advance and solve (1.1) with +the additional constraint that ρ is supported on the grid (Jiang and Zhang, 2009; Koenker and Mizera, +2014; Lindsay, 1983; Zhang et al., 2022). While well understood theoretically, those methods suffer from the +curse of dimensionality, and their complexity scales exponentially with the dimension d. To overcome this +limitation, Zhang et al. (2022) proposed a heuristic that alternately updates the weights and the support +using the Expectation-Maximization (EM) algorithm but it does not come with theoretical guarantees. +Another notable contribution is the support reduction algorithm of Groeneboom et al. (2008) which is also +computationally inefficient since it requires to compute the minimizer of a nonconvex function in Rd in each +iteration. +In this work, we propose to solve (1.1) using gradient descent in the space of probability measures endowed +with the Wasserstein-Fisher-Rao (WFR) geometry (Chizat et al., 2018; Gallouët and Monsaingeon, 2017; +Kondratyev et al., 2016; Liero et al., 2018). More specifically, we introduce the WFR gradient flow of the +negative log-likelihood ℓN and show that it converges to the NPMLE under mild conditions. In turn, we +implement this WFR gradient flow using Euler discretization in time and particle discretization in space, thus +resulting in a system of weighted interacting particles. In essence, the resulting Algorithm 1 alternatively +updates locations and weights, like the EM algorithm described above but the updates coming from the +WFR gradient flow are inherently different. +As the name indicates, the WFR geometry is a composite of the Wasserstein geometry (Ambrosio et al., +2008; Otto, 2001) and the Fisher-Rao geometry (Bauer et al., 2016). The former component governs updates +of the support while the latter governs the weight updates. Either of these geometries leads to its own gradient +descent algorithm but our numerical results indicate that their combination is the key to achieving fast +convergence; see Section 4. In fact, we show that the fixed-location EM algorithm from prior literature (see, +e.g. Jiang and Zhang, 2009) implements gradient descent with respect to the Fisher-Rao geometry. As a +byproduct of our analysis, we also show that it converges to the NPMLE in the infinite-particle regime under +certain conditions. +2 + +2 +Nonparametric Maximum Likelihood Estimator (NPMLE) +In this section, we examine optimality conditions for the optimization problem (1.1). To that end, denote +by δℓN(ρ) the first variation of ℓN at a measure ρ and observe2 that it is given by +δℓN (ρ) : x �→ − 1 +N +N +� +i=1 +φ (x − Xi) +(ρ ∗ φ) (Xi). +(2.1) +The following theorem shows the existence as well as the optimality condition of NPMLE in general +dimension. The proof is deferred to Appendix B.1. +Theorem 1. The following properties hold for NPMLE: +1. (Existence) The minimizer of the optimization problem (1.1) exists. +2. (Optimality condition) A distribution �ρ ∈ P(Rd) is an NPMLE if and only if (i) δℓN(�ρ)(x) ≥ −1 holds +for all x ∈ Rd, and (ii) δℓN(�ρ)(x) = −1 for �ρ-a.e. x. +Remark 1. We show in Appendix B.1 that for any ρ ∈ P(Rd), +� +δℓN(ρ)dρ = −1 always holds. As a result, +the optimality condition (ii) in Theorem 1 is implied by (i), which means that δℓN(�ρ)(x) ≥ −1 for all x ∈ R +alone is already the necessary and sufficient condition for �ρ to be the NPMLE. However, we keep both +conditions in the theorem as each of them reveal important structural information of NPMLE. +Theorem 1 asserts the existence of NPMLE, but its uniqueness when d ≥ 2 is still an open problem. +Although the uniqueness is not settled, the convergence theory in this paper is still valid: in this case +NPMLE refers to any minimizer of (1.1). +Notation. +We use P(Rd) to denote the space of probability measures over Rd, P2(Rd) to denote the space +of probability measures over Rd with finite second moments, and Pac(Rd) to denote the space of probability +measures that are absolutely continuous with respect to the Lebesgue measure on Rd. Let ∆m−1 be the +m − 1 dimensional probability simplex. For any ρ ∈ P(Rd), supp(ρ) denotes its support set, i.e. the smallest +closed set C ⊆ Rd such that ρ(C) = 1. For any mapping T : Rd → Rd and any distribution ρ ∈ P(Rd), +let T#ρ be the pushforward (or image measure) of ρ by T, which is defined by T#ρ(A) = ρ(T −1(A)) +for any Borel set A in Rd. +For any x ∈ Rd, we use δx to denote the Dirac mass at point x. +For two +probability measures µ, ν ∈ P(Rd), we use µ ≪ ν to denote that µ is absolutely continuous with respect +to ν. For a sequence {ρn}∞ +n=0 in P(Rd) and ρ ∈ P(Rd), we write ρn +w→ ρ if ρn weakly converges to ρ, +i.e. +� +Rd f(x)ρn(dx) → +� +Rd f(x)ρ(dx) holds for every bounded continuous function f : Rd → R. Let C∞ +c (Rd) +be the set of smooth functions with compact support in Rd. We say that (ρt)t≥0 is a distributional solution +to the partial differential equation (PDE) ∂tρt = −div(ρtvt) + ρtαt where vt : Rd → Rd and αt : Rd → R, if +for any ϕ ∈ C∞ +c (Rd) it holds that +d +dt +� +Rd ϕ (x) ρt (dx) = +� +Rd [⟨∇ϕ (x) , vt (x)⟩ + ϕ(x)αt(x)] ρt (dx) . +Finally we use the shorthand ODE to refer to ordinary differential equations. +3 +Wasserstein-Fisher-Rao gradient descent +Gradient flows over probability measures are a useful tool in the development of sampling algorithms where +the goal is to produce samples from a target measure. A classical example arises when π is, for example, a +Bayesian posterior known only up to normalizing constant. In this context, Wasserstein-Fisher-Rao (WFR) +gradient flows have recently emerged as a strong alternative to vanilla Wasserstein gradient flows. Indeed, +they provide the backbone of the birth-death sampling algorithm of Lu et al. (2019a) as well as the particle- +based method proposed in Lu et al. (2022). We refer the reader to the recent manuscript of Chewi (2022) +for more details on sampling and the role of Wasserstein gradient flows in this context. +2Explicit calculations follow from standard arguments in calculus of variations and are deferred to Appendix C.1. +3 + +The likelihood maximization problem of interest in the present paper differs from sampling questions +because its aims at optimizing a different objective. Nevertheless, it remains an optimization problem and +the machinery of gradient flows over the space of probability measures may be deployed in this context. To +the best of our knowledge, this paper present the first attempt at such a deployment. More specifically, our +main algorithm to solve (1.1) relies on a specific discretization of the Wasserstein-Fisher-Rao gradient flow. +We begin with a short introduction to gradient flows over metric spaces of probability measures that can be +safely skipped by experts. +3.1 +Gradient flows over metric spaces of probability measures +Gradient flows over metric spaces of probability measures is a central topic of the calculus of variations that +has found applications in variety of fields ranging from analysis and geometry to probability and statistics. +We briefly discuss the main idea behind this powerful tool and refer the reader to the formidable book +of Ambrosio et al. (2008) for more details about this deep question. +Recall that our goal is to derive a gradient flow for the functional ℓN over the space of probability +measures. The nature of this gradient flow is simply a curve (ρt)t≥0 such that ∂tρt = −∇ℓN(ρt) for a notion +of gradient ∇ to be defined. In their seminal work, Jordan et al. (1998) were able to define a gradient flow +over the Wasserstein space by analogy to the Euclidean gradient flow without the need to actually define a +gradient. We follow their approach and define the gradient flow of ℓN with respect to a suitable geometry +with geodesic distance d(·, ·) over the space of probability measures as +∂tρt = lim +η→0 +ρη +t − ρt +η +, +where +ρη +t := arg min +ρ∈P(Rd) +�� +Rd δℓN (ρt) d (ρ − ρt) + 1 +2η d2 (ρ, ρt) +� +Given a distance, the existence of a limiting absolutely continuous curve (ρt)t≥0 is an important and central +question that we omit in this overview. +Our main focus in this work is the Wasserstein-Fisher-Rao distance which is a composite of the Fisher-Rao +distance and the (quadratic) Wasserstein distance. We now introduce these three distances. Recall that a +geodesic distance between two points measures the shortest curve that links these two points. The difference +between these three distances is governed by the differential structure put on probability distributions, which +roughly corresponds to type of curves that are allowed. The length of these curves is then measured using +a Riemannian metric which in all cases is rather straightforward so we focus our discussion on curves. In +turn, these curves and their lengths define a geometry on the space of probability measures. +Fisher-Rao distance. +The Fisher-Rao distance is linked to reaction equations of the form +∂tρt = ρt +� +αt − +� +αtdρt +� +, +(3.1) +where αt(x) ∈ R is a scalar that governs how much mass is created at x ∈ Rd and time t. It is easy to see that +these dynamics preserve the total mass 1 of probability distributions. Among all such curves that link ρ0 +to ρ1, the Fisher-Rao geodesic is the one that minimizes the total length. More specifically, the Fisher-Rao +distance dFR is defined as (Bauer et al., 2016), +d2 +FR (ρ0, ρ1) = inf +� � 1 +0 +� �� +αt − +� +αtdρt +�2� +dρtdt : (ρt, αt)t∈[0,1] solves +∂tρt = ρt +� +αt − +� +αtdρt +�� +. +(3.2) +While this will not be useful for our problem, it is worth noting that +d2 +FR (ρ0, ρ1) = 4 +� ��� +� +dρ0 +dλ − +� +dρ1 +dλ +��� +2 +dλ, +4 + +where λ is any positive measure such that ρ0, ρ1 ≪ λ (e.g., λ = ρ0 + ρ1.). Hence, up to a constant factor, +the Fisher-Rao distance is simply the Hellinger distance that is well-known to statisticians. +Wasserstein distance. +Given two probability measures ρ0, ρ1 ∈ P(Rd), the (quadratic) Wasserstein dis- +tance between ρ0 and ρ1 is defined as (Villani, 2009) +d2 +W (ρ0, ρ1) = +inf +π∈Π(ρ0,ρ1) +� +∥x − y∥2 +2 π (dx, dy) , +where the infimum is taken over all couplings of ρ0 and ρ1. +It also admits a geodesic distance interpretation by means of the Benamou-Brenier formula: +d2 +W (ρ0, ρ1) = inf +� � 1 +0 +� +∥vt∥2 dρtdt : (ρt, vt)t∈[0,1] solves ∂tρt = −div (ρtvt) +� +. +(3.3) +Here admissible curves are given by the continuity equation +∂tρt = −div (ρtvt) , +(3.4) +which describes the evolution of a density of particles in Rd evolving according to time-dependent vector +field vt : Rd → Rd. +Wasserstein-Fisher-Rao distance. +The geometry underlying the Wasserstein-Fisher-Rao distance is +built using curves that satisfy the following evolution equation: +∂tρt = −div (ρtvt) + ρt +� +αt − +� +αtdρt +� +. +Note that the right-hand side is precisely the sum of the right-hand sides for the reaction equation (3.1) +and the continuity equation (3.4) that govern the Fisher-Rao and the Wasserstein geometry respectively. As +such it is a composite of the two geometries. +In turn, the Wasserstein-Fisher-Rao distance dWFR is defined as (Chizat et al., 2018; Gallouët and Mon- +saingeon, 2017; Kondratyev et al., 2016; Liero et al., 2018) +d2 +WFR (ρ0, ρ1) = inf +� � 1 +0 +� � +∥vt∥2 + +� +αt − +� +αtdρt +�2� +dρtdt : (ρt, vt, αt)t∈[0,1] solves +∂tρt = −div (ρtvt) + ρt +� +αt − +� +αtdρt +�� +. +(3.5) +Equipped with this distance, we are in a position to define the WFR gradient flow and its time discretization, +the WFR gradient descent. +3.2 +Wasserstein-Fisher-Rao gradient descent +In this section we introduce our main algorithm: Wasserstein-Fisher-Rao Gradient Descent. +The gradient flow {ρt}t≥0 of the negative log-likelihood ℓN(ρ) in P2(Rd) with respect to the Wasserstein- +Fisher-Rao distance dWFR(·, ·) is given by +∂tρt = − [1 + δℓN (ρt)] ρt + div (ρt∇δℓN (ρt)) . +(3.6) +The formal derivation of (3.6) is based on the Riemannian structure underlying the Wasserstein-Fisher-Rao +metric can be found in Appendix C.4. +The WFR gradient flow is not readily implementable for because of two obstacles. First, it is described +in continuous time. Second, it requires the manipulation of full probability measures ρt on Rd which are +infinite dimensional objects. +5 + +To overcome the first obstacle, we employ a straightforward time-discretization scheme to obtain a WFR +gradient descent. +This algorithm produces a sequence of probability measures {ρn}n≥0. +It makes the +following two steps alternately: +d�ρn +dρn += 1 − η [1 + δℓN (ρn)] +(Fisher-Rao gradient update) +(3.7a) +ρn+1 = [id − η∇δℓN (�ρn)]# �ρn +(Wasserstein gradient update) +(3.7b) +for n = 0, 1, . . ., where η > 0 is the step size. Note that each corresponds to a summand on the right- +hand side of (3.6) In fact, (3.7a) is one step of gradient descent update with respect to the Fisher-Rao +geometry, and (3.7b) corresponds to a gradient step with respect to the Wasserstein geometry. It is also +worth mentioning that (3.7) is related to the splitting scheme in Gallouët and Monsaingeon (2017): (3.7) can +be viewed as forward Euler scheme (explicit scheme) for numerical approximation of (3.6), while Gallouët +and Monsaingeon (2017) uses backward Euler scheme (implicit scheme). The implementation of the latter +requires optimization over probability measures in each iterate, which is a difficult problem. +The following theorem shows that Wasserstein-Fisher-Rao gradient descent converges when initialized +from a distribution that puts weight on the entire space. The proof can be found in Appendix D. In fact, +a similar result for the continuous-time Wasserstein-Fisher-Rao gradient flow can be derived at the cost of +additional technical considerations. +Theorem 2 (Convergence to NPMLE). Suppose that the initialization ρ0 ∈ P(Rd) satisfies supp(ρ0) = +Rd. Consider the Wasserstein-Fisher-Rao gradient descent {ρn}n≥0 defined in (3.7). There exists η0 > 0 +determined by the samples {Xi}1≤i≤N, such that if 0 < η ≤ η0 and ρn +w→ �ρ when n → ∞, then �ρ is the +NPMLE. +The convergence result in Theorem 2 is conditional: we only show that if WFR gradient descent converges +weakly to some limiting distribution �ρ, then �ρ is an NPMLE. This is similar to the convergence theory +established by Chizat and Bach (2018) in their study of the training dynamics for shallow neural networks. +This limitation is due to a lack of geodesic convexity which prevents us from establishing unconditional +global convergence guarantees. +To overcome the second obstacle, we also need to discretize ρn in space. Thanks to the Wasserstein +update (3.7b), we need not use a fixed grid as in previous algorithms for NPMLE. Instead, observe that if +we initialize WFR gradient descent at a distribution supported on m atoms, then ρn remains supported on +m atoms for all n. In this case, ρn describes the evolution of m interacting particles with weights. +More concretely, consider the initialization +ρ0 = +m +� +l=1 +ω(l) +0 δµ(l) +0 , +where µ(1) +0 , . . . , µ(m) +0 +∈ Rd and ω0 = (ω(1) +0 , . . . , ω(m) +0 +) ∈ ∆m−1, +where the location of particles are independently sampled from the data points {Xi}1≤i≤N. The following +theorem gives a precise characterization of the Wasserstein-Fisher-Rao gradient flow initialized from ρ0. The +proof is deferred to Appendix E.3. +Theorem 3 (Particle Wasserstein-Fisher-Rao gradient flow). The system of coupled ODE +˙µ(j) +t += 1 +N +N +� +i=1 +φ +� +Xi − µ(j) +t +� +�m +l=1 ω(j) +t φ +� +Xi − µ(l) +t +� +� +Xi − µ(j) +t +� +, +(3.8a) +˙ω(j) +t += +� +� 1 +N +N +� +i=1 +φ +� +Xi − µ(j) +t +� +�m +l=1 ω(j) +t φ +� +Xi − µ(l) +t +� − 1 +� +� ω(j) +t , +(3.8b) +with initialization µ(1) +0 , . . . µ(m) +0 +i.i.d. +∼ Uniform({Xi}1≤i≤N) and ω0 = [ω(j) +0 ]1≤j≤m ∈ ∆m−1 has unique solution +on any time interval [0, T]. Moreover, the flow (ρt)t≥0 defined as +ρt := +m +� +l=1 +ω(l) +t δµ(l) +t +(3.9) +6 + +is the Wasserstein-Fisher-Rao gradient flow, i.e. a distributional solution to the PDE (3.6). +In practice, we can obtain a time discretization of the Wasserstein-Fisher-Rao gradient flow by discretizing +the ODE system (3.8), which gives the Wasserstein-Fisher-Rao gradient descent algorithm as summarized +in Algorithm 1. +Algorithm 1 Wasserstein-Fisher-Rao gradient descent. +Input: data {Xi}1≤i≤n, number of particles m, step size η > 0, maximum number of iterations t0. +Initialization: draw µ(1) +0 , . . . , µ(m) +0 +i.i.d. +∼ Unif({Xi}1≤i≤n) and ω(1) +0 += · · · = ω(m) +0 += 1/m. +Updates: for t = 0, 1, . . . , t0 do +µ(j) +t+1 = µ(j) +t ++ η 1 +N +N +� +i=1 +φ +� +Xi − µ(j) +t +� +�m +l=1 ω(j) +t φ +� +Xi − µ(l) +t +� +� +Xi − µ(j) +t +� +, +ω(j) +t+1 = ω(j) +t ++ η +� +� 1 +N +N +� +i=1 +φ +� +Xi − µ(j) +t+1 +� +�m +l=1 ω(j) +t φ +� +Xi − µ(l) +t+1 +� − 1 +� +� ω(j) +t , +for all j = 1, . . . , m. Here φ(x) = (2π)−d/2 exp(−∥x∥2 +2/2) is the probability density function of N(0, Id). +Output ρ = �m +j=1 ω(j) +t0 δµ(j) +t0 as the (approximate) NMPLE. +The proof techniques employed to establish Theorem 2 do not cover Algorithm 1 unfortunately since +the initial measure is not supported on the whole space. Nevertheless, since the NPMLE is known to be +supported on a small number of atoms (Polyanskiy and Wu, 2020) in certain cases, it is likely that taking m +large enough will be sufficient to establish convergence results. This conjecture is supported by our numerical +experiments in Section 4. +3.3 +Surrogate geometries +As discussed above the Wasserstein-Fisher-Rao (WFR) geometry is obtained as a composite of the Wasser- +stein geometry and the Fisher-Rao geometry. In turn, WFR gradient descent alternates between a Fisher-Rao +gradient step and a Wasserstein gradient step. This observation raises a burning question: is the composite +nature of WFR gradient descent necessary to obtain good performance of are either of these two build- +ing blocks alone sufficient? In this subsection, we explore properties and limitations of these two natural +alternatives. +Fisher-Rao gradient descent. +We show in Appendix C.2.1 that the Fisher-Rao gradient flow (ρt)t≥0 of +the function ℓN(ρ) is defined by the following PDE: +∂tρt = − [1 + δℓN (ρt)] ρt. +(3.11) +By time discretization, one readily obtains the Fisher-Rao gradient descent updates {ρn}n≥0: +dρn+1 +dρn += 1 − γ [1 + δℓN (ρn)] , +n = 0, 1, . . . , +(3.12) +where γ > 0 is the step size. Although Fisher-Rao gradient flow/descent is derived in an abstract way using +Riemannian geometry, it has intimate connection with some well-known algorithms. +1. Fisher-Rao gradient flow as proximal gradient flow. The Fisher-Rao gradient flow (3.11) can be viewed +as the continuous-time limit of the proximal gradient algorithm under the Fisher-Rao metric: +∂tρt = lim +η→0+ +ρη +t − ρt +η +, +where +ρη +t := arg min +ρ∈P(Rd) +�� +Rd δℓN (ρt) d (ρ − ρt) + 1 +2η d2 +FR (ρ, ρt) +� +. +(3.13) +7 + +2. Fisher-Rao gradient flow as mirror flow. The Fisher-Rao gradient flow (3.11) can also be viewed as the +continuous-time limit of mirror descent algorithm for ℓN(ρ) with Kullback-Leibler (KL) divergence +∂tρt = lim +η→0+ +ρη +t − ρt +η +, +where +ρη +t := arg min +ρ≪ρt +�� +Rd δℓN (ρt) d (ρ − ρt) + 1 +η KL (ρ ∥ ρt) +� +. +(3.14) +3. Fisher-Rao gradient descent as fixed-location EM algorithm. When ρ0 is discrete, Fisher-Rao gradient +descent (3.12) with step size γ = 1 coincides with the fixed-location EM algorithm for Gaussian mixture +model in e.g. Jiang and Zhang (2009). +Formal justifications of the above three connections can be found in Appendix C.2.2. The theorem below +shows that when ρ0 is diffused, the Fisher-Rao gradient descent enjoys appealing convergence property. The +proof is deferred to Appendix D. +Theorem 4 (Convergence to NPMLE). Suppose that the initialization ρ0 ∈ P(Rd) satisfies supp(ρ0) = Rd. +Consider the Fisher-Rao gradient descent {ρn}n≥0 defined in (3.12). +There exists η0 determined by the +samples {Xi}1≤i≤N, such that if 0 < η ≤ η0 and ρn +w→ �ρ when n → ∞, then �ρ is the NPMLE. +We can also show a similar result for the continuous-time Fisher-Rao gradient flow (ρt)t≥0 (i.e. a distri- +butional solution to the PDE (3.11)): if ρt +w→ �ρ as t → ∞, then �ρ is an NPMLE. +Theorem 4 provides convergence guarantees for Fisher-Rao gradient flow/descent when initialized from +a well-spread distribution. In practice, however, we can only initialize from a discrete distribution with m +particles +ρ0 = +m +� +l=1 +ω(l) +0 δµ(l), where µ(1), . . . , µ(m) ∈ Rd and ω0 = (ω(1) +0 , . . . , ω(m) +0 +) ∈ ∆m−1. +The following theorem characterizes the Fisher-Rao gradient flow when initialized from a discrete distribution +ρ0 with m mass points. +Theorem 5 (Particle Fisher-Rao gradient flow). The ODE system +˙ω(j) +t += −ω(j) +t +� +1 − 1 +N +N +� +i=1 +φ +� +Xi − µ(j)� +�m +l=1 ω(l) +t φ +� +Xi − µ(l)� +� +, +1 ≤ j ≤ m +(3.15) +with initialization ω0 = [ω(j) +0 ]1≤j≤m ∈ ∆m−1 has unique solution on any time interval [0, T]. Moreover, the +flow (ρt)t≥0 defined as +ρt := +m +� +l=1 +ω(l) +t δµ(l) +(3.16) +is the Fisher-Rao gradient flow, i.e. (3.16) is a distributional solution to the PDE (3.11). +The proof is deferred to Appendix E.1. For the purpose of implementation, we can further discretize the +gradient flow (3.16) with respect to time and obtain the Fisher-Rao gradient descent; see Algorithm 2. +A quick inspection of the pseudo-cpde presented in Algorithm 2 reveals a fatal flaw: the locations of the +particles are fixed at their initialization and only the weights of the particles are updated. This introduces +a systematic approximation error that may scale exponentially with the dimension d. +This diagnosic is +supported by our numerical experiments in Section 4. +Wasserstein gradient descent. +Wasserstein gradient flows have received significant attention recently +both from a theoretical perspective (Ambrosio et al., 2008; Santambrogio, 2017) and as a useful tool in a +variety of applications ranging from sampling (Chewi et al., 2020), to variation inference (Lambert et al., +2022), as well the theory of neural networks (Chizat and Bach, 2018; Sander et al., 2022). This practical +success is largely enabled by the fact that Wasserstein gradient flows can be implemented using interacting +particle systems. +In fact, akin to prior work on shallow neural networks (Chizat and Bach, 2018; Mei et al., 2018), the +Wasserstein gradient flow of the negative log-likelihood precisely describes the dynamics of gradient descent +8 + +Algorithm 2 Fisher-Rao gradient descent. +Input: data {Xi}1≤i≤n, number of particles m, step sizes η > 0, maximum number of iterations t0. +Initialization: draw µ(1), . . . , µ(m) i.i.d. +∼ Uniform({Xi}1≤i≤n) and ω(1) +0 += · · · = ω(m) +0 += 1/m. +Updates: for t = 0, 1, . . . , t0 do +ω(j) +t+1 = ω(j) +t ++ η +� +1 +N +N +� +i=1 +φ +� +Xi − µ(j)� +�m +l=1 ω(j) +t φ +� +Xi − µ(l)� − 1 +� +ω(j) +t , +for all j = 1, . . . , m. Here φ(x) = (2π)−d/2 exp(−∥x∥2 +2/2) is the probability density function of N(0, Id). +Output ρ = �m +j=1 ω(j) +t0 δµ(j) as the (approximate) NMPLE. +on the location parameters of the fitted mixture. Interested readers are referred to Theorem 8 in Appendix +F for a rigorous statement. +In Appendix C.3 we derive the gradient flow of ℓN(ρ) under the Wasserstein geometry, which evolves +according to the PDE +∂tρt = div (ρt∇δℓN (ρt)) +(3.18) +The corresponding discrete time algorithm is +ρt+1 = [id − η∇δℓN (ρt)]# ρt, +t = 0, 1, . . . +for some step size η > 0. When initialized from a discrete distribution +ρ0 = 1 +m +m +� +l=1 +δµ(l) +0 , where µ(1) +0 , . . . , µ(m) +0 +∈ Rd, +the following theorem gives a concise characterization of the Wasserstein gradient flow. The proof can be +found in Appendix E.2. +Theorem 6 (Particle Wasserstein gradient flow). The ODE system +˙µ(j) +t += 1 +N +N +� +i=1 +φ +� +Xi − µ(j) +t +� +�m +l=1 ω(j) +t φ +� +Xi − µ(l) +t +� +� +Xi − µ(j) +t +� +, +(3.19) +with initialization µ(1) +0 , . . . µ(m) +0 +i.i.d. +∼ +Uniform({Xi}1≤i≤N) has unique solution on any time interval [0, T]. +Moreover, the flow (ρt)t≥0 defined as +ρt := 1 +m +m +� +l=1 +δµ(l) +t +(3.20) +is the Wasserstein gradient flow, i.e. +(3.20) is a distributional solution to the PDE (3.18). +By time discretization, we also have the Wasserstein gradient descent algorithm; see Algorithm 3. +Unlike the Fisher-Rao gradient descent algorithm, the approximation error of the Wasserstein flow may +be mitigated since it allows particles to evolve in space. Nevertheless, this movement can take a long time to +move mass from one part of the space at initialization to a distant part of the space. Instead, Wasserstein- +Fisher-Rao gradient descent allows for particles to not only evolve in space but also have changing weights, +thus greatly improving the performance compared to vanilla Wasserstein gradient descent. This superiority +is, again, demonstrated in numerical experiments. In particular, owing to Theorem 8, these experiments +indicate that WFR gradient descent dominates the classical gradient descent on the location parameters of +the mixing distribution. +The reader will notice here the absence of convergence results analogous to Theorem 7 for WFR and +Theorem 4 for Fisher-Rao gradient descent. Unfortunately, we were not able to derive such convergence +results. The dynamics of the Wasserstein gradient flow are quite intricate and Appendix F is devoted to +establishing partial results that shed light on them. +9 + +Algorithm 3 Wasserstein gradient descent. +Input: data {Xi}1≤i≤n, number of particles m, step sizes η > 0, maximum number of iterations t0. +Initialization: draw µ(1) +0 , . . . , µ(m) +0 +i.i.d. +∼ Unif({Xi}1≤i≤n). +Updates: for t = 0, 1, . . . , t0 do +µ(j) +t+1 = µ(j) +t ++ η 1 +N +N +� +i=1 +φ +� +Xi − µ(j) +t +� +m−1 �m +l=1 φ +� +Xi − µ(l) +t +� +� +Xi − µ(j) +t +� +, +for all j = 1, . . . , m. Here φ(x) = (2π)−d/2 exp(−∥x∥2 +2/2) is the probability density function of N(0, Id). +Output ρ = m−1 �m +j=1 δµ(j) +t0 as the (approximate) NMPLE. +(a) bad local minima +(b) global minima +-5 +0 +5 +10 +15 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +-5 +0 +5 +10 +15 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +Figure 1: The density plots of ρ ∗ N(0, 1) (learned Gaussian mixture) and ρ⋆ ∗ N(0, 1) (true Gaussian +mixture), where ρ = 1 +3δµ1 + 1 +3δµ2 + 1 +3δµ3, ρ⋆ = ρd and (µ1, µ2, µ3) is the output returned by EM and GD +algorithms. Figure 1(a) shows the learned Gaussian mixture when (µ1, µ2, µ3) is a bad local minimum of ℓ, +and Figure 1(b) corresponds to the case when (µ1, µ2, µ3) is a global minimum of ℓ. +4 +Numerical experiments +In this section, we conduct a series of numerical experiments to validate and complement our theory. We +consider two mixing distributions in Rd: the first one is a continuous isotropic Gaussian distribution +ρc = N (0, Id) , +and the other one is a discrete distribution motivated by Jin et al. (2016) +ρd = +�1 +3δ−1 + 1 +3δ1 + 1 +3δ10 +� +⊗ (δ0)⊗(d−1) . +The second mixing distribution ρd is a product distribution with its first margin being a uniform distribution +over {−1, 1, 10} and the rest d−1 margins being a degenerate distribution taking a constant zero. According +to Jin et al. (2016), classical EM and gradient descent algorithm fail to learn the location of each component +of this Gaussian mixture even with infinite samples and known weights. +4.1 +Instability of classical algorithms +In this section, we compare the two classical algorithms for learning Gaussian mixture when the mixing +distribution ρ⋆ = ρd: (i) expectation–maximization (EM) algorithm, and (ii) the gradient descent (GD) +10 + +algorithm, with Wasserstein-Fisher-Rao gradient descent algorithm (cf. Algorithm 1) proposed in this paper. +For the first two algorithms, we assume that the number of mixture components k = 3 and the weights +ω⋆ +1 = ω⋆ +2 = ω⋆ +3 = 1/3 are known a priori, and implement the standard EM and GD algorithms for solving +the MLE +min +µ1,µ2,µ3 ℓ (µ1, µ2, µ3) := − 1 +N +N +� +i=1 +log +�1 +3 +3 +� +j=1 +1 +(2π)d/2 exp +� +−1 +2 ∥Xi − µj∥2 +2 +� � +. +The updating rule for EM algorithm is given by +µt+1 +j += +�N +i=1 ωt +i,jXi +�N +i=1 ωt +i,j +where +ωt +i,j = +ω⋆ +j φ +� +Xi; µt +j, Id +� +�3 +l=1 ω⋆ +l φ (Xi; µt +l, Id) +∀ i ∈ [N] . +(4.1) +for all j = 1, 2, 3 and t ≥ 0, with random initialization from the samples µ0 +1, µ0 +2, µ0 +3 +i.i.d. +∼ Uniform({Xi}1≤i≤N). +The GD algorithm coincides with the Wasserstein gradient descent (cf. Algorithm 3) with m = 3 particles. +We consider the one-dimensional setting (i.e. d = 1) for simplicity of visualization, and this turns out +to be enough for showing the instability of EM and GD. We generate N = 1500 samples {Xi}1≤i≤N from +ρ⋆ ∗ N(0, 1). Then we fix these samples and run 100 independent trials of the three algorithms. For EM, +we run 200 iterations. For GD and Wasserstein-Fisher-Rao gradient descent, we set all the step sizes to +be η = 0.1 and run t0 = 1000 iterations. Both EM and GD have two possible outputs: (i) the first one is +µ1 ≈ µ2 ≈ 10 and µ3 ≈ 0 (up to permutation), which is a bad local minimum of ℓ; (ii) the second one is +µ1 ≈ −1, µ2 ≈ 1 and µ3 ≈ 10, which is the global minimum of ℓ. Figure 1 displays the learned Gaussian +mixtures correspond to these two outputs, and it is clear that both algorighms fail to learn the true Gaussian +mixture when they converge to the bad local minima. In the 100 independent trials, EM converges to the +bad local minimum for 23 times, while GD converges to the bad local minimum for 32 times, both exhibiting +instability vis-à-vis random initialization. In contrast, as we will show in Section 4.3, Wasserstein-Fisher-Rao +gradient descent with number of particles m = 500 converges to NPMLE stably and learns the Gaussian +mixture as in Figure 4(c). +4.2 +Superiority of Wasserstein-Fisher-Rao gradient descent +In this section we compare the empirical performance of three gradient descent algorithms studied in this +paper: (i) Fisher-Rao gradient descent (cf. Algorithm 2), (ii) Wasserstein-Fisher-Rao gradient descent (cf. Al- +gorithm 1) and (iii) Wasserstein gradient descent (cf. Algorithm 3). We set the sample size N = 1500, the +dimension d = 10, maximum number of iterations t0 = 1000, the step size η = 10−1 for Algorithm 2 and +Algorithm 3, and η = 10−2 for Algorithm 1. Figure 2(a) displays the negative log-likelihood ℓN(ρ) (with +one standard deviation error bars) vs. the number of particles m over 20 independent trials for the three al- +gorithms. Unlike the previous experiment, we generate fresh samples {Xi}1≤i≤N in each independent trial. +As we can see, the loss decreases as we use more particles, and Wasserstein-Fisher-Rao gradient descent +achieves the smallest loss uniformly for all m. It can also be observed that the marginal benefit of increasing +the number of particles becomes negligible for Wasserstein-Fisher-Rao gradient descent when m > 500. Sim- +ilarly, Figure 2(b) depicts the negative log-likelihood ℓN(ρt) (with one standard deviation error bars) vs. the +iteration count t over 20 independent trials for the three algorithms. We can see that Wasserstein-Fisher-Rao +gradient descent again achieves the smallest loss uniformly in all iteration, confirming again the superiority +of the algorithm. +4.3 +Certifying the optimality condition for NPMLE in one dimension +The convergence guarantees in this paper only cover the infinite-particle limit of Wasserstein-Fisher-Rao +gradient descent (cf. Algorithm 1). In this section, we provide numerical evidence that Algorithm 1 converges +to (approximate) NPMLE when we use a large number of particles. +Recall from Theorem 1 and the following remark that ρ ∈ P(Rd) is NPMLE if and only if δℓN(ρ)(x) ≥ −1 +holds for all x ∈ Rd. We focus on the one-dimensional setting (i.e. d = 1) since it is computationally affordable +to check the function value of δℓN(ρ)(x) in one dimension (e.g. over a fine grid). For a discrete distribution +11 + +(a) training error for discrete ρ⋆ +(b) test error for discrete ρ⋆ +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +14.7 +14.8 +14.9 +15 +15.1 +15.2 +15.3 +15.4 +15.5 +15.6 +15.7 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +15.1 +15.2 +15.3 +15.4 +15.5 +15.6 +15.7 +15.8 +(c) training error for continuous ρ⋆ +(d) test error for continuous ρ⋆ +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +16 +16.5 +17 +17.5 +18 +18.5 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +17.9 +18 +18.1 +18.2 +18.3 +18.4 +18.5 +18.6 +Figure 2: Training or testing error (with error bars) of the three algorithms vs. the number of particles. The +training and testing errors are evaluated using ℓN(ρ) and ℓ∞(ρ) respectively. Figure (a) an (b) display the +training and testing errors when the mixing distribution ρ⋆ = ρd is discrete, while Figure (c) and (d) show +the training and testing errors when the mixing distribution ρ⋆ = ρc is continuous. The results are reported +over 20 independent trials for N = 1500, d = 10, and t0 = 1000. +ρ = �m +j=1 ωjδµj, we define the following suboptimality gaps: +gap (ρ) := sup +x∈R +max {−1 − δℓN (ρ) (x) , 0} , +� +gap(ρ) := +max +x∈grid(ρ) max {−1 − δℓN (ρ) (x) , 0} . +It is clear that when the first optimality gap gap(ρ) = 0, Theorem 1 asserts that ρ is the NPMLE. However +gap(ρ) is in general difficult to compute, and a practical scheme is to approximatly evaluate the supremum +over R by the maximum over a fine grid grid(ρ) ⊆ R, which gives the second optimality gap � +gap. In our +experiments, we take grid(ρ) to be a 0.01-net over [min1≤j≤m µj − 1, max1≤j≤m µj + 1]. +We set the sample size N = 1500, the dimension d = 1 and run Wasserstein-Fisher-Rao gradient descent +(cf. Algorithm 1) with number of particles m = 500, step sizes η1 = η2 = 10−1. +Figure 4 illustrates +the suboptimality gap � +gap in (4.3) (with one standard deviation error bars) vs. the iteration count over 20 +independent trials as well as the density plot of the output of the algorithm convolved with N(0, 1). Roughly +speaking, both suboptimality gaps decreases inversely proportional to the iteration counts. Lastly, Figure 5 +depicts the first variation δℓN(ρ), which clearly shows that the optimality condition is approximately satisfied +with high precision; see the caption of Figure 5 for more details. +12 + +(a) training error for discrete ρ⋆ +(b) test error for discrete ρ⋆ +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +14.7 +14.8 +14.9 +15 +15.1 +15.2 +15.3 +15.4 +15.5 +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +15.1 +15.2 +15.3 +15.4 +15.5 +15.6 +15.7 +15.8 +15.9 +(c) training error for continuous ρ⋆ +(d) test error for continuous ρ⋆ +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +16 +16.1 +16.2 +16.3 +16.4 +16.5 +16.6 +16.7 +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +1000 +17.94 +17.96 +17.98 +18 +18.02 +18.04 +18.06 +18.08 +18.1 +18.12 +18.14 +Figure 3: Training or testing error (with error bars) of the three algorithms vs. the iteration count. The +training and the testing errors are evaluated using ℓN(ρt) and ℓ∞(ρt) respectively. Figure (a) an (b) display +training and testing errors when the mixing distribution ρ⋆ = ρd is discrete, while Figure (c) and (d) show +training and testing errors when the mixing distribution ρ⋆ = ρc is continuous. The results are reported over +20 independent trials for N = 1500, d = 10, and m = 500. +5 +Discussion +The current paper proposes to solve the NPMLE for Gaussian mixtures using an interacting particle system +driven by Wasserstein-Fisher-Rao gradient descent. In the infinite-particle limit, we show that the proposed +algorithm converges to NPMLE under certain conditions. +In practice, we conduct extensive numerical +experiments to illustrate the capability of the proposed algorithm in exactly computing NPMLE using a +finite (or even small) number of particles, and also to demonstrate the superiority of the proposed algorithm +compared to other algorithms. Moving forward, there are numerous possible extensions that merit future +investigation. For example, our convergence theory only holds when the Wasserstein-Fisher-Rao gradient +descent is initialized from a distribution supported on the whole space; it would be of interest to extend the +current analysis to the more practical scenario where the algorithm is initialized from a discrete distribution +with a finite number of particles. Another interesting direction is to develop algorithms for learning Gaussian +mixtures beyond the isotropic case (i.e. without assuming that the covariance matrices are identity) using, +for example, Wasserstein-Fisher-Rao gradient flow over the Bures-Wasserstein space (Lambert et al., 2022). +13 + +(a) optimality gap for discrete ρ⋆ +(b) optimality gap for continuous ρ⋆ +102 +103 +104 +10-5 +10-4 +10-3 +10-2 +102 +103 +104 +10-4 +10-3 +10-2 +(c) density plot for discrete ρ⋆ +(d) density plot for continuous ρ⋆ +-5 +0 +5 +10 +15 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +Figure 4: Figures (a)-(b) display sub-optimality gaps (cf. (4.3)) vs. iteration count for Wasserstein-Fisher- +Rao gradient descent, with discrete mixing distribution ρ⋆ = ρd in Figure (a) and ρ⋆ = ρc in Figure (b). The +error bars are computed over 20 independent trials. Figures (c)-(d) are density plots of ρ and ρ⋆ convolved +with standard Gaussian, with discrete mixing distribution ρ⋆ = ρd in (c) and continuous mixing distribution +ρ⋆ = ρc in (d). The results are reported for N = 1500, d = 1 and m = 500. +Acknowledgements +The authors thank Donghao Wang for a helpful discussion. +Y. Yan is supported in part by Charlotte +Elizabeth Procter Honorific Fellowship from Princeton University. Part of this work was done during Y. Yan’s +visit to MIT in Fall 2022. K. Wang is supported by an NSF grant DMS-2210907 and a start-up grant at +Columbia University. P. Rigollet is supported by NSF grants IIS-1838071, DMS-2022448, and CCF-2106377. +14 + +(a) first variation δℓN(ρ) for discrete ρ⋆ = 1 +3δ−1 + 1 +3δ1 + 1 +3δ10 +-2 +0 +2 +4 +6 +8 +10 +12 +-1 +-0.8 +-0.6 +-0.4 +-0.2 +(b) first variation δℓN(ρ) for continuous ρ⋆ = N(0, 1) +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +-1 +-0.98 +-0.96 +-0.94 +-0.92 +-0.9 +-0.88 +Figure 5: The first variation δℓN(ρ) (in red line) for discrete ρ⋆ = ρd in the upper panel and continuous +ρ⋆ = ρc in the lower panel. The blue dots are {(µj, δℓN(ρ)(µj)) : 1 ≤ j ≤ m}, where the size of these dots +are proportional to the weights {ωj}1≤j≤m. The green line is a horizontal line y = −1. The two subfigures +in the upper panel zoom in the two regions −1.5 ≤ x ≤ 1.5 and 9.8 ≤ x ≤ 10.2, and the three subfigures in +the lower panel zoom in the three regions −4 ≤ x ≤ −3.4, −2 ≤ x ≤ 3 and 3.8 ≤ x ≤ 4.1. +A +Preliminaries +In the main text, we focused on the Gaussian mixture model where φ(x) = (2π)−d/2 exp(−∥x∥2 +2/2). In fact, +the algorithms and theorems in the current paper are also valid for a more general class of probability density +φ. In the appendices, we only assume that φ satisfies the following regularity assumption. +Assumption 1 (Regularity). Assume that the density φ(x) > 0 for any x ∈ Rd. +Furthermore, φ ∈ +Cmax{d,2}(Rd), lim∥x∥2→∞ φ(x) = 0, supx∈Rd ∥∇φ(x)∥2 < ∞ and supx∈Rd ∥∇2φ(x)∥2 < ∞. +It is clear that the Gaussian kernel φ(x) = (2π)−d/2 exp(−∥x∥2 +2/2) satisfies Assumption 1. We also define +the following sets and quantites that will be useful throughout the proof. +Definition 1. Let Ω = conv({Xi}1≤i≤N). For r ≥ 0, define Ωr = {x ∈ Rd : dist(x, Ω) ≤ r}, ¯φ(r) = +sup∥x∥2≥r φ(x) and φ(r) = inf∥x∥2≤r φ(x). +The following lemma shows that NPMLE is compactly supported. The proof can be found in Appendix +B.2. +15 + +Lemma 1 (Compact support of NPMLE). Let Assumption 1 hold and �ρ be any optimal solution to (1.1). +Define R1 = inf{r ≥ 0 : ¯φ(r) ≤ φ[diam(Ω)]/2} and +R = inf +� +r ≥ 0 : ¯φ(r) ≤ +¯φ(R1)φ(R1 + diam(Ω)) +8¯φ(0) +� +. +Then, we have supp(�ρ) ⊆ ΩR. +B +Proof of structual results for NPMLE +In this section, we prove the two structural results for NPMLE, namely Theorem 1 and Lemma 1 under +Assumption 1. +B.1 +Proof of Theorem 1 +Part 1: existence of NPMLE. +Note that the loss function ℓN is lower bounded +ℓN (ρ) = − 1 +N +N +� +i=1 +log [(ρ ∗ φ) (Xi)] ≥ − log ∥φ∥∞ , +(B.1) +where the last inequality follows from ρ∗φ(x) = +� +Rd φ (y − x) ρ (dy) ≤ ∥φ∥∞ for any x ∈ Rd. Therefore there +exists a sequence of probability distribution {ρn} such that +ℓN (ρn) ≤ +inf +ρ∈P(Rd) ℓN (ρ) + 1 +n. +(B.2) +Now we argue that {ρn} is tight. To that end, we show that there exists r > 0 such that for any ε > 0, it +holds ρn(Ωr) ≥ 1 − ε for n large enough. +For any n and r > 0, define +ρn,r := ρn (Ωr) · ρn|Ωr + ρn (Ωc +r) · Unif (Ω) , +where ρn|Ωr(·) = ρn(·|Ωr) is the conditional distribution of ρn given Ωr, and Unif(Ω) is the uniform distri- +bution on Ω. We have +ℓN (ρn) − ℓN (ρn,r) = − 1 +N +N +� +i=1 +log [(ρn ∗ φ) (Xi)] + 1 +N +N +� +i=1 +log [(ρn,r ∗ φ) (Xi)] = 1 +N +N +� +i=1 +log +�(ρn,r ∗ φ) (Xi) +(ρn ∗ φ) (Xi) +� +. +Note that for each i ∈ [N] +log +�(ρn,r ∗ φ) (Xi) +(ρn ∗ φ) (Xi) +� += log +�� +Ωr φ (Xi − y) ρn (dy) + ρn (Ωc +r) +� +Ω φ (Xi − y) dy +� +Ωr φ (Xi − y) ρn (dy) + +� +Ωc +r φ (Xi − y) ρn (dy) +� += log +� +1 + +ρn (Ωc +r) +� +Ω φ (Xi − y) dy − +� +Ωcr φ (Xi − y) ρn (dy) +(ρn ∗ φ) (Xi) +� +≥ log +� +1 + ρn (Ωc +r) +� +φ (diam (Ω)) − φ (r) +� +∥φ∥∞ +� +. +We can choose r > 0 to be sufficiently large so that φ(r) ≤ φ(diam(Ω))/2, and therefore for each i ∈ [N] +ℓN (ρn) − ℓN (ρn,r) ≥ log +� +1 + ρn (Ωc +r) φ (diam (Ω)) +2 ∥φ∥∞ +� +. +In view of (B.2), we know that +ℓN (ρn) − ℓN (ρn,r) ≤ 1 +n. +16 + +Taking the above two inequalities collectively give +ρn (Ωc +r) ≤ 2 ∥φ∥∞ +� +exp +� 1 +n +� +− 1 +� +φ (diam (Ω)) +. +Therefore we have +ρn (Ωc +r) ≤ ε +as long as +n ≥ nε := +� +1/ log +� +1 + εφ (diam (Ω)) +2 ∥φ∥∞ +�� +, +which implies that {ρn} is tight. We conclude using Prokhorov’s theorem: there exists a subsequence {ρnk} +and �ρ ∈ P(Rd) such that ρnk converges weakly to �ρ which must be a minimizer of (1.1). +Part 2: optimality condition. +First of all, it is straightforward to check that for any ρ ∈ M(Rd) +� +Rd δℓN (ρ) (x) ρ (dx) = − 1 +N +N +� +i=1 +� +φ (x − Xi) ρ (dx) +(ρ ∗ φ) (Xi) += − 1 +N +N +� +i=1 +(ρ ∗ φ) (Xi) +(ρ ∗ φ) (Xi) = −1. +If �ρ ∈ M(Rd) is the optimal solution to (1.1), then for any x ∈ Rd and any ε ∈ [0, 1] we have +ℓN (�ρ) ≤ ℓN ((1 − ε) �ρ + εδx) = ℓN (ρ + ε (δx − �ρ)) . +As a result we have +δℓN (�ρ) (x) + 1 = +� +Rd δℓN (�ρ) d (δx − �ρ) = lim +ε→0 +1 +ε [ℓN (�ρ + ε (δx − �ρ)) − ℓN (�ρ)] ≥ 0. +Since x is arbitrary, this implies that δℓN(�ρ)(x) ≥ −1 for any x ∈ Rd. Combine this with +� +Rd δℓN(�ρ)d�ρ = −1 +readily gives δℓN(�ρ)(x) = −1 for �ρ-a.e. x. +Conversely, if �ρ ∈ M(Rd) satisfies δℓN(�ρ)(x) ≥ −1 for all x ∈ Rd, then for any ρ ∈ M(Rd), it holds +0 ≤ +� +Rd δℓN (�ρ) dρ + 1 = +� +Rd δℓN (�ρ) d (ρ − �ρ) = lim +ε→0 +1 +ε [ℓN (�ρ + ε (ρ − �ρ)) − ℓN (�ρ)] ≤ ℓN (ρ) − ℓN (�ρ) +where the last inequality follows from convexity of the functional ρ �→ ℓN(ρ). The above display yields +that that �ρ is a global minimizer of (1.1). +B.2 +Proof of Lemma 1 +By Theorem 1, δℓN(�ρ) = −1 over supp(�ρ). We will show that |δℓN(�ρ)(y)| < 1/2 when y is too far away from +Ω, and then conclude that supp(�ρ) must stay close to Ω. To that end, we present some useful estimates in +the following lemma that will also be useful later. +Lemma 2. Let Assumption 1 hold. For any ρ ∈ P(Rd), x ∈ Ω and r ≥ 0, we have +ρ(Ωr)φ(r + diam(Ω)) ≤ (ρ ∗ φ)(x) ≤ ρ(Ωc +r)¯φ(r) + ρ(Ωr)¯φ(0) ≤ ¯φ(r) + ρ(Ωr)¯φ(0). +As a result, +− log +� +¯φ(r) + ρ(Ωr)¯φ(0) +� +≤ ℓN(ρ) ≤ − log +� +ρ(Ωr)φ(r + diam(Ω)) +� +. +Take any R ≥ 0 such that ¯φ(R) ≤ e−ℓN(ρ)/2. For any µ ∈ P(Rd) with ℓN(µ) ≤ ℓN(ρ), we have +µ(ΩR) ≥ e−ℓN(ρ)/[2¯φ(0)], +inf +x∈Ω(µ ∗ φ)(x) ≥ e−ℓN(ρ)φ(R + diam(Ω))/[2¯φ(0)], +sup +y∈Ωcr +|δℓN(µ)(y)| ≤ +2eℓN(ρ) ¯φ(0) +φ(R + diam(Ω)) · ¯φ(r), +∀r ≥ 0. +17 + +The proof of Lemma 2 is deferred to the end of this section. We come back to proving Lemma 1. Choose +any ρ0 ∈ P(Rd) supported on Ω. By Lemma 2, we have +ℓN(ρ0) ≤ − log +� +φ[diam(Ω)] +� +, +∀r ≥ 0. +Take R1 = inf{r ≥ 0 : ¯φ(r) ≤ φ[diam(Ω)]/2}. By the continuity of ¯φ, +¯φ(R1) ≤ φ[diam(Ω)]/2 ≤ e−ℓN(ρ0)/2. +Lemma 2 implies that +sup +y∈Ωcr +|δℓN(�ρ)(y)| ≤ +2eℓN(ρ0) ¯φ(0) +φ(R1 + diam(Ω)) · ¯φ(r) ≤ +4¯φ(0) +¯φ(R1)φ(R1 + diam(Ω)) · ¯φ(r), +∀r ≥ 0. +Let +R = inf +� +r ≥ 0 : ¯φ(r) ≤ +¯φ(R1)φ(R1 + diam(Ω)) +8¯φ(0) +� +. +The continuity of ¯φ leads to ¯φ(R) ≤ ¯φ(R1)φ(R1 +diam(Ω))/[8¯φ(0)] and thus supy∈Ωc +R |δℓN(�ρ)(y)| ≤ 1/2 < 1. +As a result, supp(�ρ) ∩ Ωc +R = ∅ and supp(�ρ) ⊆ ΩR. +Proof of Lemma 2. Note that φ(x − y) ≤ ¯φ(0) for any x, y ∈ Rd. If x ∈ Ω and y ∈ Ωc +r, then ∥x − y∥2 ≥ r +and thus φ(x − y) ≤ ¯φ(r). Hence, +(ρ ∗ φ)(x) ≤ +� +Ωr +φ(x − y)ρ(dy) + +� +Ωcr +φ(x − y)ρ(dy) ≤ ¯φ(0)ρ(Ωr) + ¯φ(r)ρ(Ωc +r). +If x ∈ Ω and y ∈ Ωr, then ∥x − y∥2 ≥ r + diam(Ω) and thus φ(x − y) ≥ φ(r + diam(Ω)). Therefore, +(ρ ∗ φ)(x) ≥ +� +Ωr +φ(x − y)ρ(dy) ≥ φ(r + diam(Ω))ρ(Ωr). +The desired bounds on ℓN(·) become obvious. If µ ∈ P(Rd) and ℓN(µ) ≤ ℓN(ρ), then our estimates of ℓ +implies that +− log +� +¯φ(r) + µ(Ωr)¯φ(0) +� +≤ ℓN(µ) ≤ ℓN(ρ), +∀r ≥ 0. +Hence, ¯φ(r) + µ(Ωr)¯φ(0) ≥ e−ℓN(ρ). By Assumption 1, lim +r→∞ +¯φ(r) = 0. Take any R ≥ 0 such that ¯φ(R) ≤ +e−ℓN(ρ)/2. Then, +µ(ΩR) ≥ [e−ℓN(ρ) − ¯φ(r)]/¯φ(0) ≥ e−ℓN(ρ)/[2¯φ(0)], +inf +x∈Ω(µ ∗ φ)(x) ≥ µ(ΩR)φ(R + diam(Ω)) ≥ e−ℓN(ρ)φ(R + diam(Ω))/[2¯φ(0)]. +For any r ≥ 0, we have ∥X − y∥2 ≥ r whenever X ∈ supp(ν) and y ∈ Ωc +r. Then, +|δℓN(µ)(y)| = 1 +N +N +� +i=1 +φ(Xi − y) +(µ ∗ φ)(Xi) ≤ +¯φ(r) +infx∈Ω(µ ∗ φ)(x), +∀r ≥ 0, +y ∈ Ωc +r. +The proof is finished by combining this and the lower bound on infx∈Ω(µ ∗ φ)(x) we have established above. +18 + +C +Derivation of gradient flows over P2(Rd) +C.1 +First variation +Recall that the population and finite-sample loss functions are +ℓ∞ (ρ) = EX∼(ρ⋆∗φ) {log [(ρ ∗ φ) (X)]} , +ℓN (ρ) = − 1 +N +N +� +i=1 +log [(ρ ∗ φ) (Xi)] . +The first variation of ℓN is defined as nay measurable function δℓ(ρ) : Rd → R satisfying +lim +ε→0 +ℓ (ρ + εX) − ℓ (ρ) +ε += +� +δℓ (ρ) dX +for all signed measures X satisfying +� +dX = 0. In particular, it is easy to see that the first variation is +defined up to an additive constant. +By direct computation, we have +lim +ε→0 +ℓN (ρ + εX) − ℓN (ρ) +ε += − 1 +N +N +� +i=1 +lim +ε→0 +log [[(ρ + εX) ∗ φ] (Xi)] − log [(ρ ∗ φ) (Xi)] +ε += − 1 +N +N +� +i=1 +lim +ε→0 +1 +ε log +� +1 + ε(X ∗ φ) (Xi) +(ρ ∗ φ) (Xi) +� += − 1 +N +N +� +i=1 +(X ∗ φ) (Xi) +(ρ ∗ φ) (Xi) += − 1 +N +N +� +i=1 +� +φ (x − Xi) +(ρ ∗ φ) (Xi)X (dx) , +As a result, we have +δℓN (ρ) : x → − 1 +N +N +� +i=1 +φ (x − Xi) +(ρ ∗ φ) (Xi). +Similarly, we can also compute +lim +ε→0 +ℓ∞ (ρ + εX) − ℓ∞ (ρ) +ε += lim +ε→0 +1 +ε +� +− +� +log +�(ρ + εX) ∗ φ (x) +(ρ ∗ φ) (x) +� +(ρ⋆ ∗ φ) (x) dx +� += − +� (X ∗ φ) (x) +(ρ ∗ φ) (x) (ρ⋆ ∗ φ) (x) dx = − +� (ρ⋆ ∗ φ) (x) +(ρ ∗ φ) (x) +�� +φ (x − y) X (dy) +� +dx += − +� �� (ρ⋆ ∗ φ) (x) +(ρ ∗ φ) (x) φ (x − y) dx +� +X (dy) . +which gives +δℓ∞ (ρ) : x → − +� (ρ⋆ ∗ φ) (y) +(ρ ∗ φ) (y) φ (x − y) dy. +C.2 +Fisher-Rao gradient flow +C.2.1 +A formal derivation of gradient flow using Riemannian geometry +We first introduce a Riemannian structure over P2(Rd) underlying the Fisher-Rao metric. Define the tangent +space at ρ ∈ P2(Rd) as +TanFR +ρ P2(Rd) := +� +ζ : ζ = ρ +� +α − +� +αdρ +� +for some α satisfying +� +α2dρ < ∞ +� +. +We equip the tangent space T FR +ρ P2(Rd) with the following Riemannian metric tensor gFR +ρ (·, ·) : TanFR +ρ P2(Rd)× +TanFR +ρ P2(Rd) → R as +gFR +ρ (ζ1, ζ2) := +� ζ1 · ζ2 +ρ2 +dρ +19 + += +� +Rd +� +α1 (x) − +� +Rd α1dρ +� � +α2 (x) − +� +Rd α2dρ +� +ρ (dx) += +� +Rd α1 (x) α2 (x) ρ (dx) − +�� +Rd α1dρ +� �� +Rd α2dρ +� +, +for any ζ1 = ρ(α1 − +� +α1dρ) and ζ2 = ρ(α2 − +� +α2dρ). The metric induced by this Riemannian structure, +namely the Fisher-Rao metric dFR(·, ·), satisfies the following property: +d2 +FR (ρ0, ρ1) = inf +� � 1 +0 +� �� +αt − +� +αtdρt +�2� +dρtdt : (ρt, αt)t∈[0,1] solves ∂tρt = ρtαt +� +. +Then we follow Gallouët and Monsaingeon (2017); Lu et al. (2019b) to derive the Fisher-Rao gradient flow +with respect to the functional ℓN. Let (ρt)t≥0 be a C1 curve satisfying ρ0 = ρ with initial velocity +∂tρt|t=0 = ζ = ρ +� +α − +� +αdρ +� +. +The Fisher-Rao gradient of ℓN at ρ is defined as the function gradFRℓN (ρ) ∈ L2(ρ) such that +d +dtℓN (ρt) +��� +t=0 = gFR +ρ (gradFRℓN (ρ) , ζ) . +To compute it, observe that the right-hand side of the above identity is given by +d +dtℓN (ρt) +��� +t=0 = +� +δℓN(ρ) · ∂tρt +��� +t=0 += +� +δℓN (ρ) , ζ = +� +δℓN (ρ) +� +α − +� +αdρ +� +dρ += +� � +δℓN (ρ) − +� +δℓN (ρ) dρ +� � +α − +� +αdρ +� +dρ += gFR +ρ +� +ρ +� +δℓN (ρ) − +� +δℓN (ρ) dρ +� +, ζ +� +. +Therefore +gFR +ρ (gradFRℓN (ρ) · ζ) = gFR +ρ +� +ρ +� +δℓN (ρ) − +� +δℓN (ρ) dρ +� +, ζ +� +holds for any ζ ∈ TanFR +ρ P2(Rd), and as a result +gradFRℓN (ρ) = ρ +� +δℓN (ρ) − +� +δℓN (ρ) dρ +� += ρ [δℓN (ρ) + 1] , +where we have used the fact that +� +δℓN(ρ)dρ = −1. Hence, the gradient flow of ℓN with respect to the +Fisher-Rao metric dFR is given by +∂tρt = −gradFRℓN (ρt) = −ρt [δℓN (ρt) + 1] . +C.2.2 +Other perspectives of Fisher-Rao gradient flow +In this section, we formally illustrate the connection between Fisher-Rao gradient flow (3.11) with proximal +gradient descent and mirror descent. For simplicity, we focus on the case when ρt is continuous; the case +when ρt is discrete is similar. We also show the connection between the particle Fisher-Rao gradient descent +(2) and the EM algorithm. +20 + +Fisher-Rao gradient flow as proximal gradient flow. +Consider the proximal gradient update in (3.13). +Recall that for µ, ν ∈ Pac(Rd), the Fisher-Rao distance can be expressed as +d2 +FR (µ, ν) = 4 +� ��� +µ (x) − +� +ν (x) +��2dx. +Note that for the purpose of defining a gradient flow, the metric only matters up to its second-order local +expansion +d2 +FR (µ, ν) = +� ∆2(x) +ν (x) dx + higher-order terms, +where ∆ = µ − ν. Therefore we obtain an asymptotically (as η → 0) equivalent problem +ρη +t := arg min +ρ∈Pac(Rd) +�� +Rd δℓN (ρt) d (ρ − ρt) + 1 +2η +� [ρ (x) − ρt (x)]2 +ρt (x) +dx +� +. +The first-order optimality condition is +δℓN (ρt) (x) + 1 +η · ρ (x) − ρt (x) +ρt (x) += c +for some constant c ∈ R. This gives +ρη +t (x) = ρt (x) [1 + xη − ηδℓN (ρt) (x)] . +Since ρη +t is a probability density, we have +1 = +� +Rd ρη +t (x) dx = +� +Rd ρt (x) [1 + cη − ηδℓN (ρt) (x)] dx = 1 + cη − η, +where we use the fact that +� +Rd δℓN(ρ)dρ = −1 for any ρ ∈ P2(Rd). This gives c = −1. Therefore +ρη +t (x) = ρt (x) [1 − η − ηδℓN (ρt) (x)] , +and as a result, +∂tρt = lim +η→0+ +ρη +t − ρt +η += − [1 + δℓN (ρt)] , +which recovers the Fisher-Rao gradient flow (3.11). +Fisher-Rao gradient flow as mirror flow. +Recall that the mirror descent update is defined as +ρη +t := arg min +ρ∈Pac(Rd) +� +Rd δℓN (ρt) d (ρ − ρt) + 1 +η KL (ρ ∥ ρt) . +The first variation of f(·) := KL(· ∥ ρt) is given by +δf (ρ) (x) = log +� ρ (x) +ρt (x) +� +, +therefore the first-order optimality condition reads +δℓN (ρt) (x) + 1 +η log +� ρ (x) +ρt (x) +� += c +for some constant c > 0. This gives +ρ (x) +ρt (x) = exp {η [c − δℓN (ρt) (x)]} ∝ exp [−ηδℓN (ρt) (x)] . +21 + +Since +� +Rd ρη +t (x)dx = 1, we know that the closed-form solution is given by +ρη +t (x) = +ρt (x) exp [−ηδℓN (ρt) (x)] +� +ρt (y) exp [−ηδℓN (ρt) (y)] dy . +(C.1) +Then as η → 0, we can compute +ρη +t (x) = +ρt (x) +� +1 − ηδℓN (ρt) (x) + O +� +η2�� +� +ρt (y) [1 − ηδℓN (ρt) (y) + O (η2)] dy = ρt (x) +� +1 − ηδℓN (ρt) (x) + O +� +η2�� +1 + η + O (η2) += ρt (x) +� +1 − η [1 + δℓN (ρt) (x)] + O +� +η2�� +, +where we use the fact that +� +δℓN(ρ)dρ = −1 for any ρ ∈ P2(Rd). Therefore the continuous-time limit of +mirror descent is +∂tρt = lim +η→0+ +ρη +t (x) − ρt (x) +η += − [1 + δℓN (ρt) (x)] , +which recovers the Fisher-Rao gradient flow (3.11). +Fisher-Rao gradient descent as EM algorithm. +Now we consider fitting a m-component Gaussian +mixture model with unknown weights {ω(j)}1≤j≤m, known location parameters {µj}1≤j≤m and isotropic +covariance. Given the data {Xi}1≤i≤N, the MLE is given by +arg max +ω∈∆m−1ℓ (ω) = 1 +N +N +� +i=1 +log +� +� +m +� +j=1 +ω(j)φ (Xi − µj) +� +� . +The Expectation-Maximization algorithm for solving the above MLE is given as follows. We first introduce +the latent i.i.d. random variables {Ji}1≤i≤N distributed P(Ji = j) = ω(j) for 1 ≤ j ≤ m, then the distribution +of the observed samples is Xi|Ji = j ∼ N(µj, Id). The joint distribution of (Xi, Ji) is given by +pω (x, j) = φ (Xi − µj) ω(j), +and conditional on Xi = x, the conditional distribution of Ji is given by +pω (j|x) = +pω (x, j) +�m +l=1 pω (x, l) = +φ (Xi − µj) ω(j) +�m +l=1 φ (Xi − µl) ω(l) . +Given the current estimate ωt, the E-step amounts to computing +Q (ω|ωt) = 1 +N +N +� +i=1 +m +� +j=1 +pωt (j|Xi) log pω (Xi, j) += 1 +N +N +� +i=1 +m +� +j=1 +φ (Xi − µj) ω(j) +t +�m +l=1 φ (Xi − µl) ω(l) +t +log +� +φ (Xi − µj) ω(j)� +. +The M-step is to update +ωt+1 := arg max +ω∈∆m−1Q (ω|ωt) , +which is given by +ω(j) +t+1 = 1 +N +N +� +i=1 +φ (Xi − µj) ω(j) +t +�m +l=1 φ (Xi − µl) ω(l) +t +∀ 1 ≤ j ≤ m. +This is equivalent to Algorithm 2 with step size η = 1. +22 + +C.3 +Wasserstein gradient flow +We introduce the Riemannian structure over P2(Rd) underlying the quadratic Wasserstein distance. We +define the tangent space at ρ ∈ P2(Rd) to be +TanW +ρ P2(Rd) = +� +ζ : ζ = −div (ρ∇u) for some u satisfying +� +∥∇u∥2 +2dρ < ∞ +� +. +We equip this tangent space with the L2(ρ) metric, namely we define the Riemannian metric tensor gW +ρ (·, ·) : +TanW +ρ P2(Rd) × TanW +ρ P2(Rd) → R as +gW +ρ (ζ1, ζ2) := +� +Rd +� +Rd ⟨∇u1, ∇u2⟩ ρ (dx) +for any ζ1 = −div(ρ∇u1) and ζ2 = −div(ρ∇u2). The metric induced by this Riemannian structure recovers +the quadratic Wasserstein distance, namely +d2 +W (ρ0, ρ1) = inf +� � 1 +0 +� +∥vt∥2 +2 dρtdt : (ρt, vt)t∈[0,1] solves ∂tρt + div(ρtvt) = 0 +� += +inf +π∈Π(ρ0,ρ1) +� +∥x − y∥2 +2 π (dx, dy) , +where Π(ρ0, ρ1) is the set of couplings of ρ0 and ρ1. This is known as the Benamou-Brenier formula for +the Wasserstein distance. Then we derive the Wasserstein gradient flow with respect to the functional ℓN. +Interested readers are referred to Ambrosio et al. (2008) for detailed introduction to Wasserstein gradient +flow. Let (ρt)t≥0 be a C1 curve satisfying ρ0 = ρ with initial velocity +∂tρt|t=0 = ζ = −div (ρ∇u) . +Then it should hold that +d +dtℓN (ρt) +��� +t=0 = gW +ρ (gradWℓN (ρ) , ζ) . +The left hand side of the above equation equals to +d +dtℓN (ρt) +��� +t=0 = +� +δℓN (ρ) ζdx = − +� +δℓN (ρ) div (ρ∇u) dx += − +� +⟨∇δℓN (ρ) , ∇u⟩ dρ += gW +ρ (−div (∇δℓN (ρ) ρ) , ζ) . +Therefore +gW +ρ (gradWℓN (ρ) , ζ) = gW +ρ (−div (∇δℓN (ρ) ρ) , ζ) +holds for any ζ ∈ TanW +ρ P2(Rd), and as a result +gradWℓN (ρ) = −div (∇δℓN (ρ) ρ) . +This shows that the gradient flow of ℓN with respect to the quadratic Wasserstein distance dW is given by +∂tρt = −gradWℓN (ρt) = div (∇δℓN (ρt) ρt) . +C.4 +Wasserstein-Fisher-Rao gradient flow +We introduce the Riemannian structure over P2(Rd) underlying the Wasserstein-Fisher-Rao metric. Define +the tangent space at ρ ∈ P2(Rd) to be +TanWFR +ρ +P2(Rd) = +� +ζ : ζ = −div (ρ∇u) + ρ +� +α − +� +αdρ +� +for some u, α : Rd → R +23 + +satisfying +� +(α2 + ∥∇u∥2 +2)dρ < ∞ +� +. +We equip this tangent space with the Riemannian metric tensor gWFR +ρ +(·, ·) : TanWFR +ρ +P2(Rd)×TanWFR +ρ +P2(Rd) → +R defined as +gWFR +ρ +(ζ1, ζ2) := +� +Rd ⟨∇u1, ∇u2⟩ ρ (dx) + +� +Rd +� +α1 (x) − +� +Rd α1dρ +� � +α2 (x) − +� +Rd α2dρ +� +ρ (dx) += +� +Rd ⟨∇u1, ∇u2⟩ ρ (dx) + +� +Rd α1 (x) α2 (x) ρ (dx) − +�� +Rd α1dρ +� �� +Rd α2dρ +� +for any ζ1 = −div(ρ∇u1) + ρ(α1 − +� +α1dρ) and ζ2 = −div(ρ∇u2) + ρ(α2 − +� +α2dρ). The metric induced by +the above Riemannian structure, namely the Wasserstein-Fisher-Rao metric dWFR(·, ·), is defined as +d2 +WFR (ρ0, ρ1) = inf +� � 1 +0 +� � +∥vt∥2 + +� +αt − +� +αtdρt +�2� +dρtdt : (ρt, vt, αt)0≤t≤1 +solves ∂tρt = −div(ρtvt) + ρtαt +� +. +Then we follow Gallouët and Monsaingeon (2017); Lu et al. (2019b) to derive the Wasserstein-Fisher-Rao +gradient flow with respect to the functional ℓN. Let (ρt)t≥0 be a C1 curve satisfying ρ0 = ρ with initial +velocity +∂tρt|t=0 = ζ = −div (ρ∇u) + ρ +� +α − +� +αdρ +� +. +Then it should hold that +d +dtℓN (ρt) +��� +t=0 = gWFR +ρ +(gradWFRℓN (ρ) , ζ) . +The left hand side of the above equation equals to +d +dtℓN (ρt) +��� +t=0 = +� +δℓN (ρ) ζdx = +� +δℓN (ρ) +� +−div (ρ∇u) + ρ +� +α − +� +αdρ +�� +dx += − +� +⟨∇δℓN (ρ) , ∇u⟩ dρ + +� � +δℓN (ρ) − +� +δℓN (ρ) dρ +� � +α − +� +αdρ +� +dρ += gWFR +ρ +� +−div (∇δℓN (ρ) ρ) + ρ +� +δℓN (ρ) − +� +δℓN (ρ) dρ +� +, ζ +� +. +Therefore +gWFR +ρ +(gradWℓN (ρ) , ζ) = gWFR +ρ +� +−div (∇δℓN (ρ) ρ) + ρ +� +δℓN (ρ) − +� +δℓN (ρ) dρ +� +, ζ +� +holds for any ζ ∈ TanWFR +ρ +P2(Rd), and as a result +gradWFRℓN (ρ) = −div (∇δℓN (ρ) ρ) + ρ +� +δℓN (ρ) − +� +δℓN (ρ) dρ +� += −div (∇δℓN (ρ) ρ) + ρ [1 + δℓN (ρ)] , +where we have used the fact that +� +δℓN(ρ)dρ = −1. This shows that the gradient flow of ℓN with respect to +the Wasserstein-Fisher-Rao metric dWFR is given by +∂tρt = −gradWFRℓN (ρt) = div (∇δℓN (ρt) ρt) − ρt [1 + δℓN (ρt)] . +24 + +D +Convergence theory (Proof of Theorem 4 and 2) +In this section, we provide a systematic treatment to the convergence of Fisher-Rao gradient descent and +Wasserstein-Fisher-Rao gradient descent. Instead of proving Theorem 4 and 2 separately, we prove a more +general convergence result in Theorem 7, which admits Theorem 4 and 2 as its special cases. +To begin with, we define the Wasserstein gradient descent update. +Definition 2 (Wasserstein gradient descent). For any ρ ∈ P(Rd) and η ≥ 0, we define ρW,η = (id − +η∇δℓN(ρ))#ρ. Given an initial distribution ρ0 ∈ P(Rd), the Wasserstein gradient descent for solving (1.1) +is defined recursively by ρn+1 = ρW,η +n +, ∀n ≥ 0. +In words, ρW,η is the push forward of ρ by the mapping x �→ x − η∇δℓN(ρ)(x). For any p ≥ 1, define the +p-Wasserstein distance between ρ0 and ρ1 in P(Rd) as +Wp (ρ0, ρ1) = +� +inf +π∈Π(ρ0,ρ1) +� +∥x − y∥p +p π (dx, dy) +�1/p +. +The following lemma characterizes the decrease in loss function ℓN by running one step of Wasserstein +gradient descent, which can be lower bounded by the squared Wasserstein distance between the two iterates. +Lemma 3 (Wasserstein gradient descent). Let Assumption 1 hold. Choose any ρ0 ∈ P(Rd) and define +c0 = e−ℓN(ρ0)φ(inf{r ≥ 0 : ¯φ(r) ≤ e−ℓN(ρ0)/2} + diam(Ω)) +2¯φ(0) +. +Suppose that +0 ≤ η < +c0 +supx∈Rd ∥∇2φ(x)∥2 + supx∈Rd ∥∇φ(x)∥2 +2/c0 +. +Define ρn+1 = ρW,η +n +for n ≥ 0. Then, for any n ≥ 0, +ℓN(ρn+1) − ℓN(ρn) ≤ −η +2EY ∼ρn∥∇δℓN(ρn)(Y )∥2 +2 ≤ − 1 +2η W 2 +2 (ρn+1, ρn). +In addition, if supp(ρ0) = Rd, then supp(ρn) = Rd for all n ≥ 0. +Proof. See Appendix D.1. +Then we define the Fisher-Rao gradient descent update. +Definition 3 (Fisher-Rao gradient descent). For any ρ ∈ P(Rd) and γ ∈ [0, 1], we define ρFR,γ ∈ P(Rd) +through +dρFR,γ +dρ += 1 − γ [δℓN(ρ) + 1] . +Given an initial distribution ρ0 ∈ P(Rd), the Fisher-Rao gradient descent for solving (1.1) is defined recur- +sively by ρn+1 = ρFR,γ +n +, ∀n ≥ 0. +It is easily seen that ρFR,γ = (1 − γ)ρ + γρFR,1 and +dρFR,1 +dρ +(x) = −δℓN(ρ)(x) = 1 +N +N +� +i=1 +φ(Xi − x) +(ρ ∗ φ)(Xi), +∀x ∈ Rd. +From Appendix C.2.1 we know that Fisher-Rao gradient descent with γ = 1 can be viewed as fixed-location +EM algorithm. The following lemma shows that by running one step of Fisher-Rao gradient descent, the +decrease in loss function can be lower bounded by the KL divergence between the two iterates. +Lemma 4 (Fisher-Rao gradient descent). Choose any ρ ∈ P(Rd) and γ ∈ [0, 1]. We have +ℓN(µ) ≤ ℓN(ρ) + KL(ρFR,1∥µ) − KL(ρFR,1∥ρ), +∀ µ ≪ ρ, +and +ℓN(ρFR,γ) − ℓN(ρ) ≤ −KL(ρFR,γ∥ρ). +25 + +Proof. See Appendix D.3. +Finally we define Wasserstein-Fisher-Rao gradient descent, which can be viewed as iteratively applying +one step of Wasserstein gradient descent (cf. Definition 2) and one step of Fisher-Rao gradient descent +(cf. Definition 3). +Definition 4 (Wasserstein-Fisher-Rao gradient descent). Let ρ0 ∈ P(Rd), η ≥ 0 and γ ∈ [0, 1]. +The +Wasserstein-Fisher-Rao gradient descent is defined through �ρn = ρW,η +n +and ρn+1 = �ρFR,γ +n +for all n ≥ 0. +The Wasserstein gradient descent and Fisher-Rao gradient descent are special cases of Wasserstein- +Fisher-Rao gradient descent with γ = 0 and η = 0, respectively. +The following theorem shows that if +Wasserstein-Fisher-Rao gradient descent converges weakly to a limit distribution, then this weak limit is the +NPMLE. +Theorem 7. Suppose that Assumption 1 holds. +Let {ρn}∞ +n=0 be the iterates of Wasserstein-Fisher-Rao +gradient descent, and +c0 = e−ℓN(ρ0)φ(inf{r ≥ 0 : ¯φ(r) ≤ e−ℓN(ρ0)/2} + diam(Ω)) +2¯φ(0) +. +If supp(ρ0) = Rd, γ ∈ (0, 1], and +0 ≤ η < +c0 +supx∈Rd ∥∇2φ(x)∥2 + supx∈Rd ∥∇φ(x)∥2 +2/c0 +, +then for all n ≥ 0, ℓN(ρn+1) ≤ ℓN(ρn) holds, and supp(ρn) = Rd. Furthermore, if {ρn}∞ +n=0 converges weakly +to ρ∞ ∈ P(Rd), then this limit ρ∞ is NPMLE, i.e. an optimal solution to (1.1). +Proof. See Appendix D.4. +Theorem 7 directly implies that Theorem 2 holds. By taking η = 0, this also implies that Theorem +4 holds. +Note that Theorem 7 requires γ > 0, therefore it does not provide convergence guarantee for +Wasserstein gradient descent. +Here are some key ideas for showing the optimality of the weak limit ρ∞. If ρ∞ is not an optimal solution, +then Theorem 1 implies that δℓN(ρ∞)(x0) < −1 − ε holds for some x0 ∈ Rd and ε > 0. We can find some +appropriate y ∈ (−1 − ε, −1 − ε/2) and study the sublevel set ¯S = {x ∈ Rd : δℓN(ρ∞) ≤ y}. Note that for n +large, we have ∇δℓN(ρn) ≈ ∇δℓN(ρ∞). We analyze the Wasserstein step and the Fisher-Rao step separately: +• For the Wasserstein step we show that for any x ∈ ¯S, the gradient descent step x−η∇δℓN(ρn)(x) remains +in ¯S; this follows from the definition of the gradient step and the fact that the step size is chosen small +enough. A crucial step therein is to choose y so that the gradient ∇δℓN(ρ∞) does not vanish on the +level set {x ∈ Rd : δℓN(ρ∞) = y}. The existence of such a y follows readily from Sard’s lemma. Since +�ρn = (id − ∇δℓN(ρn))#ρn, we get �ρn( ¯S) ≥ ρn( ¯S). +• For the Fisher-Rao step, we first establish that δℓN(�ρn)(x) < −1 − ε/4 for all x ∈ ¯S. Recalling that +dρn+1 +d�ρn +(·) = (1 − γ) + γ · [−δℓ(�ρn)(·)], +it readily yieldsρn+1( ¯S) ≥ (1 + ε/4)�ρn( ¯S). +Putting both steps together, we get that there exists N such that ρn+1( ¯S) ≥ (1+ε/4)ρn( ¯S) holds all n > N. +Note that this geometric improvement is entirely driven by the Fisher-Rao part of the proof. To conclude, +we use supp(ρN) = Rd to get �ρN( ¯S) > 0, and thus limn→∞ �ρn( ¯S) = ∞, which leads to a contradiction so +that ρ∞ must be an optimal solution to (1.1). +26 + +D.1 +Proof of Lemma 3 +We invoke a descent lemma Wasserstein gradient descent, whose proof is deferred to Appendix D.2. Such a +lemma is standard in convex optimization optimization (see, e.g., Bubeck, 2015, eq. (3.5)). It has appeared +for optimization over the Wasserstein space in Salim et al. (2020) under for functionals that are convex along +generalized geodesics, an assumption that does not hold for the negative log-likelihood ℓN. +Lemma 5 (A descent lemma). Let Assumption 1 hold. Choose any ρ ∈ P(Rd). Define c = infx∈Ω(ρ∗φ)(x), +G = supx∈Rd ∥∇φ(x)∥2 and H = supx∈Rd ∥∇2φ(x)∥2. +• We have +ℓN(ρW,η) − ℓN(ρ) ≤ −η +� +1 − η +2c +� +H + G2 +c +�� +EY ∼ρ∥∇δℓN(ρ)(Y )∥2 +2. +In addition, we have supx∈Rd ∥∇2δℓN(ρ)(x)∥2 ≤ H/c. +• If 0 ≤ η < c/H and supp(ρ) = Rd, then supp(ρW,η) = Rd. +We now come back to Lemma 3. Let R = inf{r ≥ 0 : ¯φ(r) ≤ e−ℓN(ρ0)/2}. Lemma 2 implies that for any +µ ∈ P(Rd) with ℓN(µ) ≤ ℓN(ρ0), we have +inf +x∈Ω(µ ∗ φ)(x) ≥ e−ℓN(ρ0)φ(R + diam(Ω))/[2¯φ(0)] = c0. +In particular, infx∈Ω(ρ0 ∗ φ)(x) ≥ c0. +When η ≤ c0/(H + G2/c0), Lemma 5 and the definition ρ1 = +(id − η∇δℓN(ρ0))#ρ0 together yield +ℓN(ρ1) − ℓN(ρ0) ≤ −η +2EY ∼ρ0∥∇δℓN(ρ0)(Y )∥2 +2 ≤ − 1 +2η W 2 +2 (ρ1, ρ0) ≤ 0. +Also, if supp(ρ0) = Rd, then supp(ρ1) = Rd. From ℓN(ρ1) ≤ ℓN(ρ0) we obtain that infx∈Ω(ρ1 ∗ φ)(x) ≥ c0. +Then, the proof is completed by induction. +D.2 +Proof of Lemma 5 +We prove the two results in Lemma 5 in sequence. +Part 1. +Let ν = N −1 �N +i=1 δXi be the empirical data distribution, and let h(x) = − log x for x > 0. Then +we can wirte +ℓN(ρ) = − 1 +N +N +� +i=1 +log [ρ ∗ φ (Xi)] = EX∼ν [h ((ρ ∗ φ)(X))] = EX∼ν [h (EY ∼ρ [φ(X − Y )])] +as well as +ℓN(ρW,η) = EX∼ν +� +h +� +EY ∼ρW,η [φ(X − Y )] +�� += EX∼ν [h (EY ∼ρ [φ (X − Y + η∇δℓN(ρ)(Y ))])] . +Define a(x) = EY ∼ρ[φ(x−Y +η∇δℓN(ρ)(Y ))] and b(x) = EY ∼ρ[φ(x−Y )] = (ρ∗φ)(x) for any x ∈ Rd. Then +we can write +ℓN(ρ) = EX∼ν [h (b(X))] , +ℓN(ρW,η) = EX∼ν [h (a(X))] . +Note that h′(x) = −x−1 < 0, h′′(x) = x−2 > 0 and h′′′(x) = −2x−3 < 0. For any a, b > 0, by Taylor’s +theorem +h(a) ≤ h(b) + h′(b)(a − b) + h′′(b) +2 +(a − b)2 = h(b) − a − b +b ++ (a − b)2 +2b2 +. +Taking the above two equations collectively gives +ℓN(ρW,η) − ℓN(ρ) = EX∼ν [h (a(X)) − h (b(X))] +27 + +≤ −EX∼ν +�a(X) − b(X) +b(X) +� +� +�� +� +=:α1 ++ 1 +2EX∼ν +�[a(X) − b(X)]2 +b(X)2 +� +� +�� +� +=:α2 +. +(D.1) +Now derive upper bounds for α1 and α2 respectively. +• To control α1, let G = supx∈Rd ∥∇φ(x)∥2 and observe that +|φ (x − y + η∇δℓN(ρ)(y)) − φ(x − y)| ≤ Gη ∥∇δℓN(ρ)(y)∥2 +for all x, y ∈ Rd, therefore +|a (x) − b (x)| ≤ GηEY ∼ρ [∥∇δℓN (ρ) (Y )∥2] +(D.2) +holds for all x ∈ Rd. Then we have +α1 +(i) +≤ +G2η2E2 +Y ∼ρ [∥∇δℓN (ρ) (Y )∥] +c2 +(ii) +≤ G2η2 +c2 +EY ∼ρ +� +∥∇δℓN (ρ) (Y )∥2 +2 +� +, +(D.3) +where (i) follows from (D.2) and the fact +c = inf +x∈Ω(ρ ∗ φ)(x) = inf +x∈Ω b(x) +(D.4) +and (ii) follows from Jensen’s inequality. +• Regarding α2, let H = supx∈Rd ∥∇2φ(x)∥2 and we have +φ (x − y + η∇δℓN(ρ)(y)) − φ(x − y) ≥ ⟨∇φ(x − y), η∇δℓN(ρ)(y)⟩ − H +2 η2∥∇δℓN(ρ)(y)∥2 +2, +for any x, y ∈ Rd, and therefore +a(x) − b(x) ≥ EY ∼ρ [⟨∇φ(x − Y ), η∇δℓN(ρ)(Y )⟩] − H +2 η2EY ∼ρ +� +∥∇δℓN(ρ)(Y )∥2 +2 +� +(D.5) +for any x ∈ Rd. Since b(x) > 0, we have +α2 +(i) +≤ −EX∼ν +� +� +EY ∼ρ [⟨∇φ(X − Y ), η∇δℓN(ρ)(Y )⟩] − H +2 η2EY ∼ρ +� +∥∇δℓN(ρ)(Y )∥2 +2 +� +b (X) +� +� += −ηEY ∼ρ +�� +EX∼ν +�∇φ(X − Y ) +b(X) +� +, ∇δℓN(ρ)(Y ) +�� ++ Hη2 +2 +EX∼ν +� +1 +b(X) +� +EY ∼ρ +� +∥∇δℓN(ρ)(Y )∥2 +2 +� +(ii) += −ηEY ∼ρ +� +∥∇δℓN(ρ)(Y )∥2 +2 +� ++ Hη2 +2 +EX∼ν +� +1 +b(X) +� +EY ∼ρ +� +∥∇δℓN(ρ)(Y )∥2 +2 +� +(iii) +≤ +� +−η + Hη2 +2c +� +EY ∼ρ +� +∥∇δℓN(ρ)(Y )∥2 +2 +� +. +(D.6) +Here (i) utilizes (D.5); (ii) holds since +δℓN(ρ)(y) = −EX∼ν +�φ(X − y) +b(X) +� +, +∇δℓN(ρ)(y) = EX∼ν +�∇φ(X − y) +b(X) +� +, +(D.7) +and therefore +EY ∼ρ +�� +EX∼ν +�∇φ(X − Y ) +b(X) +� +, ∇δℓN(ρ)(Y ) +�� += EY ∼ρ +� +∥∇δℓN(ρ)(Y )∥2 +2 +� +; +and (iii) follows from (D.4). +28 + +Taking (D.1), (D.3) and (D.6) collectively gives +ℓN(ρW,η) − ℓN(ρ) ≤ α1 + α2 ≤ +� +− η + Hη2 +2c ++ G2η2 +2c2 +� +EY ∼ρ +� +∥∇δℓN(ρ)(Y )∥2 +2 +� += −η +� +1 − η +2c +� +H + G2 +c +�� +EY ∼ρ +� +∥∇δℓN(ρ)(Y )∥2 +2 +� +. +Finally, we learn from (D.7) and (D.4) that +��∇2δℓN(ρ)(x) +�� +2 = +����EX∼ν +�∇2φ(X − x) +b (X) +����� +2 +≤ supx∈Rd ∥∇2φ(x)∥2 +infx∈Rd b (x) += H +c . +Part 2. +When η < c/H, the mapping x �→ η∇δℓN(ρ)(x) is a contraction. Let ϕ(x) = x − η∇δℓN(ρ)(x). +By Lemma 6, ϕ : Rd → Rd is a bijection and ϕ−1 is Lipschitz. The second-order differentiability of δℓN(ρ) +implies the differentiability of ϕ and thus ϕ−1. If supp(ρ) = Rd, then supp(ρW,η) = supp(ϕ#ρ) = Rd. +D.3 +Proof of Lemma 4 +Let ν = N −1 �N +i=1 δXi be the empirical data distribution. For any µ ≪ ρ, we have +ℓN(µ) = −EX∼ν [log ((µ ∗ φ)(X))] = −EX∼ν [log (EY ∼µ [φ(X − Y )])] += −EX∼ν +� +log +� +EY ∼ρ +�dµ +dρ (Y ) · φ(X − Y ) +��� += −EX∼ν +� +log +� +EY ∼ρ +�dµ +dρ (Y ) · (ρ ∗ φ)(X) · φ(X − Y ) +(ρ ∗ φ)(X) +��� +. +For any x ∈ Rd, we can define a new probability measure ρFR,1 +x +∈ P(Rd) through +dρFR,1 +x +dρ +(·) = φ(x − ·) +(ρ ∗ φ)(x), +(D.8) +and we can check that ρFR,1 +x +is indeed a probability measure since +� +Rd dρFR,1 +x += +� +Rd +φ(x − y) +(ρ ∗ φ)(x)ρ (dx) = 1. +Then we have, by the convexity of t �→ − log t and Jensen’s inequality, +ℓN(µ) +(i) += −EX∼ν +� +log +� +EY ∼ρFR,1 +X +�dµ +dρ (Y ) · (ρ ∗ φ)(X) +��� +(ii) +≤ −EX∼ν +� +EY ∼ρFR,1 +X +� +log +�dµ +dρ (Y ) · (ρ ∗ φ)(X) +��� +(iii) += −EX∼ν +� +EY ∼ρ +� +log +�dµ +dρ (Y ) · (ρ ∗ φ)(X) +� +· φ(X − Y ) +(ρ ∗ φ)(X) +�� += − EX∼ν,Y ∼ρ +� +log +�dµ +dρ (Y ) +� +· φ(X − Y ) +(ρ ∗ φ)(X) +� +� +�� +� +=:β1 +− EX∼ν,Y ∼ρ +� +log ((ρ ∗ φ)(X)) · φ(X − Y ) +(ρ ∗ φ)(X) +� +� +�� +� +=:β2 +. +(D.9) +Here (i) and (iii) utilizes (D.8), and (ii) follows from Jensen’s inequality and the convexity of t �→ − log t. +Then we study the two terms β1 and β2 respectively. Regarding β1, we have +β1 = EY ∼ρ +� +log +�dµ +dρ (Y ) +� +EX∼ν +� φ(X − Y ) +(ρ ∗ φ)(X) +�� += EY ∼ρ +� +log +�dµ +dρ (Y ) +� +· dρFR,1 +dρ +(Y ) +� += EY ∼ρFR,1 +� +log +�dµ +dρ (Y ) +�� +29 + += EY ∼ρFR,1 +� +log +� +dµ +dρFR,1 (Y ) +�� ++ EY ∼ρFR,1 +� +log +�dρFR,1 +dρ +(Y ) +�� += −KL +� +ρFR,1 ∥ µ +� ++ KL +� +ρFR,1 ∥ ρ +� +. +(D.10) +Regarding β2, we have +β2 = EX∼ν +� +log ((ρ ∗ φ)(X)) EY ∼ρ +� φ(X − Y ) +(ρ ∗ φ)(X) +�� += EX∼ν [log ((ρ ∗ φ)(X))] = −ℓN (ρ) . +(D.11) +Taking (D.9), (D.10) and (D.11) collectively yields +ℓN(µ) ≤ ℓN(ρ) + KL +� +ρFR,1 ∥ µ +� +− KL +� +ρFR,1 ∥ ρ +� +, +∀ µ ≪ ρ. +By taking µ = ρFR,1, we get +ℓN +� +ρFR,1� +≤ ℓN (ρ) − KL +� +ρFR,1 ∥ ρ +� +. +(D.12) +Recall that for any γ ∈ (0, 1) we have ρFR,γ = (1 − γ)ρ + γρFR,1. Therefore +ℓN +� +ρFR,γ� (i) +≤ (1 − γ)ℓN(ρ) + γℓN(ρFR,1) +(ii) +≤ ℓN(ρ) − γKL +� +ρFR,1 ∥ ρ +� +, +(D.13) +where (i) holds since ℓN(ρ) is ℓ2-convex in ρ, and (ii) follows from (D.12). By the ℓ2-convexity of KL(· ∥ ρ), +we have +KL +� +ρFR,γ ∥ ρ +� +≤ (1 − γ)KL (ρ ∥ ρ) + γKL +� +ρFR,1 ∥ ρ +� += γKL +� +ρFR,1 ∥ ρ +� +. +(D.14) +Combine (D.13) and (D.14) to achieve +ℓN +� +ρFR,γ� +≤ ℓN(ρ) − KL +� +ρFR,γ ∥ ρ +� +. +D.4 +Proof of Theorem 7 +Suppose that for some n ≥ 0, ℓN(ρn) ≤ ℓN(ρ0) holds, and supp(ρn) = Rd. Define +cn = e−ℓN(ρn)φ(inf{r ≥ 0 : ¯φ(r) ≤ e−ℓN(ρn)/2} + diam(Ω)) +2¯φ(0) +. +Then, cn ≥ c0 and thus +0 ≤ η < +cn +supx∈Rd ∥∇2φ(x)∥2 + supx∈Rd ∥∇φ(x)∥2 +2/cn +. +Theorem 3 and the upper bound on η immediately gives ℓ(�ρn) ≤ ℓ(ρn) and supp(�ρn) = Rd. +Let ν = +N −1 �N +i=1 δXi be the empirical data distribution. From Lemma 4, the updating rule +dρn+1 +d�ρn +(·) = (1 − γ) + γEX∼ν +� φ(X − ·) +(�ρn ∗ φ)(X) +� +, +(D.15) +and the positivity of φ, we see that ℓ(ρn+1) ≤ ℓ(�ρn) and supp(ρn+1) = Rd. Therefore we can use induction +to show that, for all n ≥ 0, the inequality ℓ(ρn+1) ≤ ℓ(�ρn) ≤ ℓ(ρn) holds, and supp(ρn) = Rd. In view of +(B.1), both sequences {ℓ(ρn)}∞ +n=0 and {ℓ(�ρn)}∞ +n=0 converge and have the same limit. Consequently, +ℓ(ρn) − ℓ(�ρn) → 0. +(D.16) +Now, suppose that {ρn}∞ +n=0 converges weakly to some ρ∞ ∈ P(Rd). +Claim 1. {δℓN(ρn)}∞ +n=0 converges uniformly to δℓN(ρ∞) over compact sets. +30 + +Proof. It is easily seen that +sup +x∈Rd ∥∇δℓN(ρn)(x)∥2 = sup +x∈Rd +����EX∼ν +� ∇φ(X − x) +(ρn ∗ φ)(X) +����� +2 +≤ supx∈Rd ∥∇φ(x)∥2 +infx∈Ω(ρn ∗ φ)(x) . +Assumption 1 forces supx∈Rd ∥∇φ(x)∥2 < ∞. By Lemma (2) and the fact that ℓ(ρn) ≤ ℓ(ρ0), we have +inf +x∈Ω(ρn ∗ φ)(x) ≥ c0, +∀n ≥ 0. +(D.17) +Hence, {δℓN(ρn)}∞ +n=0 are uniformly equicontinuous. Therefore, it suffices to prove that {δℓN(ρn)}∞ +n=0 con- +verges pointwise to δℓN(ρ∞). +Since φ is bounded and continuous (cf. Assumption 1), (ρn ∗ φ)(x) → (ρ∞ ∗ φ)(x) holds for every x ∈ Rd. +Recall that +δℓ(ρ)(x) = −EX∼ν +� φ(X − x) +(ρ ∗ φ)(X) +� +, +∀ρ ∈ P(Rd). +(D.18) +Based on the boundedness of φ and the lower bound (D.17), we use the bounded convergence theorem to +derive δℓN(ρn)(x) → δℓN(ρ∞)(x) for every x ∈ Rd. This proves the claim. +Claim 2. {∇δℓN(ρn)}∞ +n=0 converges uniformly to ∇δℓN(ρ∞) over compact sets. +Proof. The proof is similar to that of Claim 1 and is thus omitted. +Claim 3. {δℓN(�ρn)}∞ +n=0 converges uniformly to δℓN(ρ∞) over compact sets. +Proof. Lemma 3 implies that W 2 +2 (�ρn, ρn) ≤ 2η[ℓN(ρn) − ℓN(�ρn)]. By (D.16), W2(�ρn, ρn) → 0 and thus +W1(�ρn, ρn) → 0. Since φ is Lipschitz, we have +sup +x∈Rd |(�ρn ∗ φ)(x) − (ρn ∗ φ)(x)| → 0. +From the above uniform bound, (D.17), and (D.18) we obtain that +sup +x∈Rd |δℓN(�ρn)(x) − δℓN(ρn)(x)| → 0. +Then, the proof is completed by applying Claim 1. +We now come back to Theorem (7). +Suppose that ρ∞ is not an optimal solution. +Then in view of +Theorem (1), there exists ε > 0 such that δℓN(ρ∞)(x) < −1 − ε for some x ∈ Rd.Similar to showing the +pointwise convergence of δℓN(ρn) to δℓN(ρ∞) in Claim (1), we can use the bounded convergence theorem to +show that ℓ(ρn) → ℓ(ρ∞). Hence, ℓ(ρ∞) ≤ ℓ(ρ0). Lemma (2) immdeiately implies that +inf +x∈Ω(ρ∞ ∗ φ)(x) ≥ c0, +∀n ≥ 0, +(D.19) +and lim∥x∥2→∞ δℓ(ρ∞)(x) = 0. Therefore, the function δℓ(ρ∞) achieves its minimum value at some x0 ∈ Rd. +We have δℓN(ρ∞)(x0) < −1 − ε. +For notational simplicity, let fn(x) = δℓN(ρn)(x) and f(x) = δℓN(ρ∞)(x). By Assumption 1, f ∈ Cd(Rd). +The second part of Lemma (7) guarantees the existence of some y ∈ (−1 − ε, −1 − ε/2) such that +S(y) = +� +x ∈ Rd : f(x) = y +� +is compact and infx∈S(y) ∥∇f(x)∥2 > 0. Denote by ξ = infx∈S(y) ∥∇f(x)∥2 and ¯S = {x ∈ Rd : f(x) ≤ y}. +The sublevel set ¯S is compact. According to the fact that f(x0) < y and the continuity of f, ¯S has positive +Lebesgue measure. Therefore we have +ρn( ¯S) > 0, +∀ n ≥ 0. +(D.20) +31 + +• We first show that �ρn( ¯S) ≥ ρn( ¯S) holds for sufficently large n. By Claim 2, there exists N > 0 such that +sup +x∈ ¯S +∥∇fn(x) − ∇f(x)∥2 ≤ ξ/11, +∀ n > N. +By the assumed upper bound on η and the estimate (D.19), we have +η ≤ +c0 +supx∈Rd ∥∇2φ(x)∥2 +≤ +1 +supx∈Rd ∥∇2f(x)∥2 +. +On top of the above, the first part of Lemma (D.23) implies that +{x − η∇fn(x) : x ∈ ¯S} ⊆ ¯S, +∀ n > N. +Since �ρn = (id − η∇fn)#ρn, we have +�ρn( ¯S) ≥ ρn( ¯S), +∀ n > N. +(D.21) +• Then we prove that ρn+1( ¯S) > (1 + ε/4)�ρn( ¯S) for sufficiently large n. By Claim 3, there exists N ′ > 0 +such that +sup +x∈ ¯S +|δℓ(�ρn)(x) − f(x)| ≤ ε/4, +∀ n > N ′. +Consequently, +sup +x∈ ¯S +δℓ(�ρn)(x) ≤ sup +x∈ ¯S +f(x) + ε/4 ≤ −1 − ε/4, +∀ n > N ′. +The expression (D.15) implies that +dρn+1 +d�ρn +(x) = (1 − γ) + γ · [−δℓ(�ρn)(x)] ≥ 1 + γε/4, +∀ n > N ′, x ∈ ¯S. +Therefore we have +ρn+1( ¯S) ≥ (1 + γε/4)�ρn( ¯S), +∀n > N ′. +(D.22) +Taking (D.21) and (D.22) collectively yields +ρn+1( ¯S) ≥ (1 + γε/4)ρn( ¯S), +∀n > max{N, N ′}. +This combined with (D.20) immediately leads to limn→∞ ρn( ¯S) = ∞, which contradicts ρn(Rd) = 1 for all +n ≥ 0. Therefore ρ∞ must be an optimal solution to (1.1), i.e. ρ∞ is the NPMLE. +D.5 +Technical lemmas +Here is a standard result about the Lipschitz perturbation of identity mapping. +Lemma 6. Let B be a Banach space with norm ∥ · ∥. If f : +B → B has Lipschitz constant c < 1, then +ϕ : B → B, x �→ x + f(x) is bijective and ϕ−1 is (1 − c)−1-Lipschitz. +Proof. Choose any y ∈ B. The mapping ψ : x �→ y − f(x) has Lipschitz constant c < 1 (in terms of the +norm ∥ · ∥). By the Banach fixed-point theorem, there exists a unique z such that z = ψ(z), which implies +that y = ϕ(z). Hence ϕ is bijective and ϕ−1 : B → B is well-defined. +For any y1, y2 ∈ B, define xi = ϕ−1(yi). Then, yi = xi + f(xi) and +∥y2 − y1∥2 = ∥ϕ(x2) − ϕ(x1)∥2 ≥ ∥x2 − x1∥2 − ∥f(x2) − f(x1)∥2 ≥ (1 − c)∥x2 − x1∥2, +∥ϕ−1(y2) − ϕ−1(y1)∥2 = ∥x2 − x1∥2 ≤ (1 − c)−1∥y2 − y1∥2. +This proves the Lipschitz smoothness of ϕ−1. +32 + +Below we show that one step of approximate gradient descent cannot expand certain sub-level sets of a +function. +Lemma 7. Assume that f ∈ C2(Rd) has a finite minimum value y0. Define S(y) = {x ∈ Rd : f(x) = y} +for y ∈ R. We have the followings. +• Suppose that supx∈Rd ∥∇2f(x)∥2 ≤ L < ∞. Choose any η ∈ [0, 1/L] and y > y0. Define ¯S = {x ∈ Rd : +f(x) ≤ y}. If g : Rd → Rd satisfies +sup +x∈ ¯S +∥g(x) − ∇f(x)∥2 ≤ 1 +11 +inf +x∈S(y) ∥∇f(x)∥2, +then +{x − ηg(x) : x ∈ ¯S} ⊆ ¯S. +• Suppose that inf∥x∥2≥R f(x) > y0 holds for some R and in addition, f ∈ Cd(Rd). Then, for any ε > 0, +there exists y ∈ (y0, y0 + ε) such that S(y) is compact and +inf +x∈S(y) ∥∇f(x)∥2 > 0. +Proof. To prove the first part, we choose any x ∈ ¯S and we will show that x − ηg(x) ∈ ¯S. +Let δ = +supz∈ ¯S ∥g(z) − ∇f(z)∥2. By supx∈Rd ∥∇2f(x)∥2 ≤ L and 0 ≤ η ≤ 1/L, +f(x − ηg(x)) ≤ f(x) + ⟨∇f(x), −ηg(x)⟩ + L +2 ∥ηg(x)∥2 +2 +≤ f(x) + ⟨∇f(x), −η∇f(x) − η[g(x) − ∇f(x)]⟩ + Lη2 +2 [∥∇f(x)∥2 + ∥g(x) − ∇f(x)∥2]2 +≤ f(x) − η∥∇f(x)∥2 +2 + ηδ∥∇f(x)∥2 + Lη2 +2 [∥∇f(x)∥2 +2 + 2δ∥∇f(x)∥2 + δ2] += f(x) + η∥∇f(x)∥2 +2 +� +− +� +1 − Lη +2 +� ++ (1 + Lη) +δ +∥∇f(x)∥2 ++ Lη +2 +� +δ +∥∇f(x)∥2 +�2� +≤ f(x) + η∥∇f(x)∥2 +2 +� +− 1 +2 + +2δ +∥∇f(x)∥2 ++ 1 +2 +� +δ +∥∇f(x)∥2 +�2� +. +(D.23) +Define ξ = infz∈S(y) ∥∇f(z)∥2. If ξ = 0, then δ = 0. The bound (D.23) yields +f(x − ηg(x)) ≤ f(x) − η +2∥∇f(x)∥2 +2 ≤ f(x) ≤ y +and thus x − ηg(x) ∈ ¯S. From now on we assume that ξ > 0. +Case 1. +When ∥∇f(x)∥2 > ( +√ +5 + 2)δ, we have +δ +∥∇f(x)∥2 +< +1 +√ +5 + 2 = +√ +5 − 2. +By (D.23), +f(x − ηg(x)) − f(x) ≤ η∥∇f(x)∥2 +2 +�1 +2 +� +δ +∥∇f(x)∥2 ++ 2 +�2 +− 5 +2 +� +≤ 0. +Hence, f(x − ηg(x)) ≤ f(x) ≤ y and x − ηg(x) ∈ ¯S. +Case 2. +When ∥∇f(x)∥2 ≤ ( +√ +5 + 2)δ, we use δ ≤ ξ/11 and +√ +5 < 5/2 to get +∥∇f(x)∥2 ≤ ( +√ +5 + 2)δ ≤ +√ +5 + 2 +11 +ξ < 9ξ +22 < ξ +2. +33 + +Therefore, x /∈ S(y). As x ∈ ¯S, we must have f(x) < y. +Note that for any z ∈ S(y), we have ∥∇f(z)∥2 ≥ ξ > 0. By supx∈Rd ∥∇2f(x)∥2 ≤ L, we have +∥z − x∥2 ≥ ∥∇f(z) − ∇f(x)∥2 +L +≥ ∥∇f(z)∥2 − ∥∇f(x)∥2 +L +> ξ − ξ/2 +L += ξ +2L. +Therefore, infz∈S(y) ∥z − x∥2 ≥ +ξ +2L. We have +B ∩ S(y) = ∅, +where +B = +� +x′ ∈ Rd : ∥x′ − x∥2 < ξ +2L +� +. +(D.24) +We claim that B ⊆ ¯S. If this is not true, then f(x′) > y holds for some x′ ∈ B. Since f(x) < y and f +is continuous, there exists t ∈ (0, 1) such that f((1 − t)x + tx′) = y, i.e. (1 − t)x + tx′ ∈ S(y). The fact +(1 − t)x + tx′ ∈ B leads to B ∩ S(y) ̸= ∅, which contradicts (D.24). +On the other hand, we have +∥[x − ηg(x)] − x∥2 = η∥g(x)∥2 ≤ η∥∇f(x)∥2 + ηδ ≤ ( +√ +5 + 3)δ +L +≤ ( +√ +5 + 3)ξ +11L +< ξ +2L. +Therefore, x − ηg(x) ∈ B ⊆ ¯S. This proves the first part. +We now come to the second part. Let y1 = inf∥x∥2≥R f(x). Thanks to the continuity of f, the image +set {f(x) : x ∈ Rd} contains the interval (y0, y1). Since f : Rd → R is Cd, Sard’s lemma asserts that the +set of critical values, i.e. the image set of critical points {f(x) : +∇f(x) = 0}, has Lebesgue measure 0. +Consequently, for any ε > 0, there exists a regular value y ∈ (y0, min{y0 + ε, y1}), i.e. ∇f(x) ̸= 0 so long as +f(x) = y. Because of y < y1, S(y) ⊆ {x ∈ Rd : ∥x∥2 ≥ R} must be compact. The continuity of ∇f implies +that infx∈S(y) ∥∇f(x)∥2 > 0. +E +Well-posedness of particle gradient flows +E.1 +Fisher-Rao gradient flow (Proof of Theorem 5) +In this section, we will show that for the ODE system (3.15) +˙ω(j) +t += −ω(j) +t +� +1 − 1 +N +N +� +i=1 +φ +� +Xi − µ(j)� +�m +l=1 ω(l) +t φ +� +Xi − µ(l)� +� +, +∀ t ≥ 0, j ∈ [m] +with initial value ω0 ∈ ∆m−1, the solution exists, is unique, and (ρt)t≥0 where ρt := �m +l=1 ω(l) +t δµ(l) is a +Fisher-Rao gradient flow in the sense of (3.11). +First of all, we will use Picard-Lindelöf theorem to prove existence and uniqueness of the solution. Define +a function f : Rm → Rm as +f (y) = [fj (y)]1≤j≤m , +fj (y) = −yj +� +1 − 1 +N +N +� +i=1 +φ +� +Xi − µ(j)� +�m +l=1 ylφ +� +Xi − µ(l)� +� +. +Then we can rewrite the ODE system as ˙ωt = f(ωt). For any y ∈ Rm such that ∥y − ω0∥2 ≤ ε for some +δ > 0 to be specified later, we know that +m +� +l=1 +ylφ +� +Xi − µ(l)� +≥ +m +� +l=1 +ω(l) +0 φ +� +Xi − µ(l)� +− +m +� +l=1 +� +ω(l) +0 − yl +� +φ +� +Xi − µ(l)� +(i) +≥ +min +i∈[N],l∈[m] φ +� +Xi − µ(l)� +− ∥y − ω0∥2 +m +� +l=1 +φ2 � +Xi − µ(l)� +(ii) +≥ +min +i∈[N],l∈[m] φ +� +Xi − µ(l)� +− +εm +(2π)d +34 + +for any i ∈ [N], where (i) follows from ω(l) +0 +∈ ∆m−1 and Cauchy-Schwarz inequality, while (ii) holds since +∥φ∥∞ ≤ (2π)−d/2. Therefore by taking +ε := (2π)d +2m +min +i∈[N],l∈[m] φ +� +Xi − µ(l)� +, +we know that for any ∥y − ω0∥2 ≤ ε it holds that +m +� +l=1 +ylφ +� +Xi − µ(l)� +≥ 1 +2 +min +i∈[N],l∈[m] φ +� +Xi − µ(l)� +≜ δ. +(E.1) +Therefore we can check that +���[∇fj (y)]j +��� = +�����− +� +1 − 1 +N +N +� +i=1 +φ +� +Xi − µ(j)� +�m +l=1 ylφ +� +Xi − µ(l)� +� +− +� +1 +N +N +� +i=1 +yjφ2 � +Xi − µ(j)� +��m +l=1 ylφ +� +Xi − µ(l)��2 +������ +≤ 1 + ∥φ∥∞ +δ ++ (1 + ε) ∥φ∥2 +∞ +δ2 += 1 + +1 +(2π)d/2 δ ++ +1 + ε +(2π)d δ2 , +and for l ̸= j +��[∇fj (y)]l +�� = +�����−yj +� +1 +N +N +� +i=1 +φ +� +Xi − µ(j)� +φ +� +Xi − µ(l)� +��m +l=1 ylφ +� +Xi − µ(l)��2 +������ ≤ (1 + ε) ∥φ∥2 +∞ +δ2 += +1 + ε +(2π)d δ2 . +As a result, we know that for any +max +y:∥y−ω0∥2≤ε ∥∇fj (y)∥2 ≤ √m +� +1 + +1 +(2π)d/2 δ ++ +1 + ε +(2π)d δ2 +� +≜ Clip, +(E.2) +and hence f(y) is CLip-Lipschitz continuous in {y : ∥y − ω0∥2 ≤ ε} where CLip := √mClip. In addition, it is +easy to show that +max +y:∥y−ω0∥2≤ε ∥fj (y)∥2 ≤ −yj + 1 +N +N +� +i=1 +yjφ +� +Xi − µ(j)� +�m +l=1 ylφ +� +Xi − µ(l)� ≤ 1 + ε ≜ M. +By Picard-Lindelöf theorem, there exists t0 > 0 such that the ODE has a unique solution on the time +interval [0, t0]. We first check that ω(j) +t +> 0 for any j ∈ [m] and t ∈ [0, t0]. If this is not true, define +t⋆ := min +� +t ∈ [0, t0] : ∃ j ∈ [m] s.t. ω(j) +t +≤ 0 +� +and suppose ω(j⋆) +t⋆ +≤ 0 for j⋆ ∈ [m]. Then we know that ω(j) +t +> 0 for any j ∈ [m] and 0 ≤ t ≤ t⋆, and hence +˙ω(j) +t +≥ −ω(j) +t +for all t ∈ [0, t⋆]. Then we can use Grönwall’s lemma to achieve ω(j) +t +≥ ω(j) +0 e−t for all t ∈ [0, t⋆], +and as a result ω(j⋆) +t⋆ +> 0, which is a contradiction. In addition, we can also check that +d +dt +m +� +j=1 +ω(j) +t += +m +� +j=1 +˙ω(j) +t += − +m +� +j=1 +ω(j) +t +� +1 − 1 +N +N +� +i=1 +φ +� +Xi − µ(j)� +�m +l=1 ω(l) +t φ +� +Xi − µ(l)� +� += 0 +for all t ∈ [0, t0]. As as a result ωt ∈ ∆m−1 for any 0 ≤ t ≤ t0. By repeating the same procedure as above +(notice that the above proof only depends on ω0 ∈ ∆m−1, and t0 only depends on universal constants CLip +and M and does not depend on ω0), we can show that the ODE has a unique solution on [t0, 2t0], [2t0, 3t0], +and so on. This shows the existence and uniqueness of the solution to the ODE system (3.15). +Next, we show that (ρt)t≥0 defined as ρt := �m +l=1 ω(l) +t δµ(l) solves (3.11). Note that ρt is a probability +measure since we have shown that ωt ∈ ∆m−1 for any t ≥ 0. For any test function ϕ(x) ∈ C∞ +c , we have +d +dt +� +Rd ϕ (x) ρt (dx) = d +dt +� +� +m +� +j=1 +ω(j) +t ϕ +� +µ(j) +t +� +� +� = +m +� +j=1 +˙ω(j) +t ϕ +� +µ(j) +t +� +35 + += − +m +� +j=1 +� +1 + δℓN (ρt) +� +µ(j) +t +�� +ω(j) +t ϕ +� +µ(j) +t +� += − +� +Rd [1 + δℓN (ρt) (x)] ϕ (x) ρt (dx) . +This proves that +∂tρt = − [δℓ (ρt) + 1] ρt +holds in the sense of distributions. +E.2 +Wasserstein gradient flow (Proof of Theorem 6) +In this section we will show that the ODE system (3.19) +˙µ(j) +t += 1 +N +N +� +i=1 +φ +� +Xi − µ(j) +t +� +m−1 �m +l=1 φ +� +Xi − µ(l) +t +� +� +Xi − µ(j) +t +� +, +∀ t ≥ 0, j ∈ [m] +has unique solution, and (ρt)t≥0 where ρt := m−1 �m +l=1 δµ(l) +t +is a Wasserstein gradient flow in the sense of +(3.18). +We will use Picard-Lindelöf theorem to prove existence and uniqueness of the solution. Define a suffi- +ciently large constant +R := max +1≤i≤N ∥Xi∥2 . +For each j ∈ [m], define a function f (j) : Rmd → Rd as +f (j) (z) = 1 +N +N +� +i=1 +φ (Xi − zj) +m−1 �m +l=1 φ (Xi − zl) (Xi − zj) , +where z = [zj]1≤j≤m ∈ Rmd and z1, . . . , zm ∈ Rd, and let f(z) = [f (j)(z)]1≤j≤m. Then we can write the +ODE system as ˙µt = f(µt) where µt = [µ(j) +t ]1≤j≤m. Denote by f (j)(z) = [f (j) +k (z)]1≤k≤d. For any z ∈ Rmd +satisfying maxj∈[m] ∥zj∥2 ≤ 2R, we have +min +1≤i≤N +1 +m +m +� +l=1 +φ (Xi − zl) ≥ +1 +(2π)d/2 exp +� +−9 +2R2 +� +≜ δ. +(E.3) +Then we can compute for l ̸= j +���∇zlf (j) +k +(z) +��� +2 = +�����∇zl +1 +N +N +� +i=1 +φ (Xi − zj) +m−1 �m +l=1 φ (Xi − zl)e⊤ +k (Xi − zj) +����� +2 += +�����− 1 +N +N +� +i=1 +φ (Xi − zj) m−1φ (Xi − zl) +[m−1 �m +l=1 φ (Xi − zl)]2 +e⊤ +k (Xi − zj) (Xi − zl) +����� +2 +(i) +≤ m−1 ∥φ∥2 +∞ +δ2 +1 +N +N +� +i=1 +��e⊤ +k (Xi − zj) +�� ∥Xi − zl∥2 +(ii) +≤ +9m−1 +δ2 (2π)d R2, +where (i) utilizes (E.3) and (ii) follows from maxi∈[N] ∥Xi∥2 ≤ R and maxj∈[m] ∥zj∥2 ≤ 2R. Similarly we +have +���∇zjf (j) +k +(z) +��� +2 = +�����∇zj +1 +N +N +� +i=1 +φ (Xi − zj) +m−1 �m +l=1 φ (Xi − zl)e⊤ +k (Xi − zj) +����� +2 +≤ +�����− 1 +N +N +� +i=1 +φ2 (Xi − zj) m−1 +[m−1 �m +l=1 φ (Xi − zl)]2 e⊤ +k (Xi − zj) (Xi − zj) +����� +2 +36 + ++ +����� +1 +N +N +� +i=1 +φ (Xi − zj) +m−1 �m +l=1 φ (Xi − zl) +� +e⊤ +k (Xi − zj) (Xi − zj) − ek +� +����� +2 +≤ 9 +� +m−1 +δ2 (2π)d + +1 +δ (2π)d/2 +� +R2 + ∥φ∥∞ +δ +. +As a result, we have +���∇zf (j) +k +(z) +��� +2 = +� +� +� +� +m +� +l=1 +���∇zlf (j) +k +(z) +��� +2 +2 ≤ +� +� +� +�(m + 1) +� +9m−1 +δ2 (2π)d R2 +�2 ++ 2 +1 +δ2 (2π)d ≜ Clip. +(E.4) +Therefore f (j)(z) is +√ +dCLip-Lipschitz continuous in {z : maxj∈[m] ∥zj∥2 ≤ 2R}, and hence f(z) is CLip- +Lipschitz continous where CLip := Clip +√ +md. In addition, it is straightforward to show that for any z ∈ Rmd +satisfying maxj∈[m] ∥zj∥2 ≤ 2R, +���f (j) (z) +��� +2 ≤ ∥φ∥∞ +δ +3R ≜ M +holds for all 1 ≤ j ≤ m. +Recall that µ(1) +0 , . . . , µ(m) +0 +are i.i.d. sampled from Uniform({Xi}1≤i≤N), therefore maxj∈[m] ∥µ(j) +0 ∥2 ≤ R. +Therefore we have +{z : ∥z − µ0∥2 ≤ R} ⊆ +� +z : max +j∈[m] ∥zj∥2 ≤ 2R +� +, +where µ0 = [µ(j) +0 ]1≤j≤m. Hence f(z) is +√ +mdCLip-Lipschitz continous in {z : ∥z − µ0∥2 ≤ R}. Then we can +use Picard-Lindelöf theorem to show that, there exists t0 > 0 such that the ODE has a unique solution on +the time interval [0, t0]. For any t ∈ [0, t0] and j ∈ [m], we can compute +d +dt∥µ(j) +t ∥2 +2 = 2⟨µ(j) +t , ˙µ(j) +t ⟩ = 2 +N +N +� +i=1 +φ +� +Xi − µ(j) +t +� +m−1 �m +l=1 φ +� +Xi − µ(l) +t +�µ(j)⊤ +t +� +Xi − µ(j) +t +� +≤ 2 +N +N +� +i=1 +φ +� +Xi − µ(j) +t +� +m−1 �m +l=1 φ +� +Xi − µ(l) +t +� +� +∥Xi∥2 ∥µ(j) +t ∥2 − ∥µ(j) +t ∥2 +2 +� +≤ 2 +N +N +� +i=1 +φ +� +Xi − µ(j) +t +� +m−1 �m +l=1 φ +� +Xi − µ(l) +t +� +� +R − ∥µ(j) +t ∥2 +� +∥µ(j) +t ∥2, +where we use Cauchy-Schwarz inequality in the penultimate step. This shows that +d +dt∥µ(j) +t ∥2 +2 < 0 as long +as ∥µ(j) +t0 ∥2 > R, and as a result maxj∈[m] ∥µ(j) +t0 ∥2 ≤ R. Then we can repeat the same analysis as above +(notice that the above proof only requires maxj∈[m] ∥µ(j) +0 ∥2 ≤ R, and t0 only depends on universal constants +CLip and M) to show that the ODE has a unique solution on [t0, 2t0], [2t0, 3t0], and so on. This shows the +existence and uniqueness of the solution to the ODE system (3.15). +Finally we check that (ρt)t≥0 defined as ρt := �m +l=1 m−1δµ(l) +t +solves (3.11). For any test function ϕ(x) ∈ +C∞ +c , we have +d +dt +� +Rd ϕ (x) ρt (dx) = d +dt +� +� 1 +m +m +� +j=1 +ϕ +� +µ(j) +t +� +� +� = 1 +m +m +� +j=1 +� +∇ϕ +� +µ(j) +t +� +, ˙µ(j) +t +� += 1 +m +m +� +j=1 +� +∇ϕ +� +µ(j) +t +� +, −∇δℓN (ρt) +� +µ(j) +t +�� += − +� +Rd ⟨∇ϕ (x) , ∇δℓN (ρt)⟩ ρt (dx) . +37 + +This proves that +∂tρt = div (ρt∇δℓ (ρt)) +holds in the sense of distributions. +E.3 +Wasserstein-Fisher-Rao gradient flow (Proof of Theorem 3) +In this section we will show that the ODE system (3.8) +˙µ(j) +t += 1 +N +N +� +i=1 +φ +� +Xi − µ(j) +t +� +m−1 �m +l=1 φ +� +Xi − µ(l) +t +� +� +Xi − µ(j) +t +� +, +˙ω(j) +t += +� +� 1 +N +N +� +i=1 +φ +� +Xi − µ(j) +t +� +�m +l=1 ω(j) +t φ +� +Xi − µ(l) +t +� − 1 +� +� ω(j) +t , +has unique solution, and (ρt)t≥0 where ρt := �m +l=1 ω(l) +t δµ(l) +t +is a Wasserstein-Fisher-Rao gradient flow in the +sense of (3.6). We will integrate the proof techniques used in the previous two sections. +We will again use Picard-Lindelöf theorem to prove existence and uniqueness of the solution. For each +j ∈ [m], define two functions f (j) : ∆m−1 × Rmd → Rd and g(j) : ∆m−1 × Rmd → R as +f (j) (y, z) = 1 +N +N +� +i=1 +φ (Xi − zj) +�m +l=1 ylφ (Xi − zl) (Xi − zj) , +g(j) (y, z) = − +� +1 − 1 +N +N +� +i=1 +φ (Xi − zj) +�m +l=1 ylφ (Xi − zl) +� +yj, +where y = [yj]1≤j≤m ∈ ∆m−1, z = [zj]1≤j≤m ∈ Rmd with z1, . . . , zm ∈ Rd. Let +f (y, z) := +� +�� +f (1) (y, z) +... +f (m) (y, z) +� +�� , +g (y, z) := +� +�� +g(1) (y, z) +... +g(m) (y, z) +� +�� , +and +h (y, z) = +� f (y, z) +g (y, z) +� +. +Then we can write the ODE system as +� ˙µt +˙ωt +� += h +�� µt +ωt +�� +, +where µt = [µ(j) +t ]1≤j≤m and ωt = [ω(j) +t ]1≤j≤m. Denote by f (j)(z) = [f (j) +k (z)]1≤k≤d. +For any z ∈ Rmd satisfying maxj∈[m] ∥zj∥2 ≤ 2R and any y ∈ Rm satisfying ∥y − ω0∥2 ≤ ε where +R := max +1≤i≤N ∥Xi∥2 , +ε := (2π)d/2 +2m +exp +� +−9 +2R2 +� +, +we have for any i ∈ [N] +m +� +l=1 +ylφ (Xi − zl) ≥ +m +� +l=1 +ω(l) +0 φ (Xi − zl) − +m +� +l=1 +� +ω(l) +0 − yl +� +φ (Xi − zl) +(i) +≥ +min +i∈[N],l∈[m] φ (Xi − zl) − ∥y − ω0∥2 +m +� +l=1 +φ2 � +Xi − µ(l)� +(ii) +≥ +1 +(2π)d/2 exp +� +−9 +2R2 +� +− +εm +(2π)d = +1 +2 (2π)d/2 exp +� +−9 +2R2 +� +≜ δ. +(E.5) +38 + +Here (i) follows from ω0 ∈ ∆m−1 and the Cauchy-Schwarz inequality, while (ii) and (iii) holds since ∥Xi − +zl∥2 ≤ 3R for any i ∈ [N] and l ∈ [m]. For any j ∈ [m], denote by f (j)(z) = [f (j) +k (z)]1≤k≤d. Similar to the +proof of (E.4), we can use (E.5) to show that +���∇zf (j) +k +(y, z) +��� +2 ≤ +� +� +� +�(m + 1) +� +9m−1 +δ2 (2π)d R2 +�2 ++ 2 +1 +δ2 (2π)d . +We also have +∇yf (j) +k +(y, z) = ∇y +1 +N +N +� +i=1 +φ (Xi − zj) +�m +l=1 ylφ (Xi − zl)e⊤ +k (Xi − zj) += − 1 +N +N +� +i=1 +φ (Xi − zj) +[�m +l=1 ylφ (Xi − zl)]2 e⊤ +k (Xi − zj) +� +�� +φ (Xi − z1) +... +φ (Xi − zm) +� +�� , +and as a result +���∇yf (j) +k +(y, z) +��� +2 ≤ +√m ∥φ∥2 +∞ +δ2 +max +i∈[N],j∈[m] ∥Xi − zj∥2 ≤ 3√mR +δ2 (2π)d . +In addition, we can compute +∇yg(j) (y, z) = −∇y +�� +1 − 1 +N +N +� +i=1 +φ (Xi − zj) +�m +l=1 ylφ (Xi − zl) +� +yj +� += − +� +1 − 1 +N +N +� +i=1 +φ (Xi − zj) +�m +l=1 ylφ (Xi − zl) +� +ej + 1 +N +N +� +i=1 +yjφ (Xi − zj) +[�m +l=1 ylφ (Xi − zl)]2 +� +�� +φ (Xi − z1) +... +φ (Xi − zm) +� +�� +and for each l ∈ [m] +∇zlg(j) (y, z) = −∇zl +�� +1 − 1 +N +N +� +i=1 +φ (Xi − zj) +�m +l=1 ylφ (Xi − zl) +� +yj +� += yj +1 +N +N +� +i=1 +� +∇zlφ (Xi − zj) +�m +l=1 ylφ (Xi − zl) − φ (Xi − zj) yl∇zlφ (Xi − zl) +[�m +l=1 ylφ (Xi − zl)]2 +� += 1 +N +N +� +i=1 +� +1 {l = j} +yjφ (Xi − zj) +�m +l=1 ylφ (Xi − zl) (Xi − zj) − yjφ (Xi − zj) ylφ (Xi − zl) +[�m +l=1 ylφ (Xi − zl)]2 +(Xi − zl) +� +. +As a result, we have +���∇yg(j) (y, z) +��� +2 ≤ +�����1 − 1 +N +N +� +i=1 +φ (Xi − zj) +�m +l=1 ylφ (Xi − zl) +����� + 1 +N +N +� +i=1 +����� +yjφ (Xi − zj) +[�m +l=1 ylφ (Xi − zl)]2 +����� +√m ∥φ∥∞ +≤ 1 + ∥φ∥∞ +δ ++ (1 + ε) √m +δ2 +∥φ∥2 +∞ = 1 + +1 +(2π)d/2 δ ++ (1 + ε) √m +(2π)d δ2 +and +���∇zlg(j) (y, z) +��� +2 ≤ 1 +N +N +� +i=1 +(1 + ε) ∥φ∥∞ +δ +∥Xi − zj∥2 + 1 +N +N +� +i=1 +(1 + ε)2 ∥φ∥2 +∞ +δ2 +∥Xi − zl∥2 +≤ 3 (1 + ε) R +(2π)d/2 δ ++ 3 (1 + ε)2 R ∥φ∥2 +∞ +(2π)d δ2 +. +39 + +Therefore for any k ∈ [d], j ∈ [m], z ∈ Rmd satisfying maxj∈[m] ∥zj∥2 ≤ 2R and y ∈ Rm such that +∥y − ω0∥2 ≤ ε, we have +���∇f (j) +k +(y, z) +��� +2 = +����∇zf (j) +k +(y, z) +��� +2 +2 + +���∇yf (j) +k +(y, z) +��� +2 +2 +≤ +� +� +� +�(m + 1) +� +9m−1 +δ2 (2π)d R2 +�2 ++ +2 +δ2 (2π)d + +� +3√mR +δ2 (2π)d +�2 +≜ Clip,f, +which suggests that f (j)(y, z) is +√ +dClip,f-Lipschitz continuous, and +���∇g(j) (y, z) +��� +2 = +� +� +� +���∇yg(j) (y, z) +��2 +2 + +m +� +l=1 +��∇zlg(j) (y, z) +��2 +2 +≤ +� +� +� +� +� +1 + +1 +(2π)d/2 δ ++ (1 + ε) √m +(2π)d δ2 +�2 ++ m +� +3 (1 + ε) R +(2π)d/2 δ ++ 3 (1 + ε)2 R ∥φ∥2 +∞ +(2π)d δ2 +�2 +≜ Clip,g. +which suggests that g(j)(y, z) is Clip,g-Lipschitz continuous. This allows us to conclude that h(y, z) is CLip- +continuous in {(y, z) : ∥y − ω0∥2 ≤ ε, maxj∈[m] ∥zj∥2 ≤ 2R}, where CLip = +� +mC2 +lip,g + mdC2 +lip,f. In addition, +it is easy to check that for any z ∈ Rmd satisfying maxj∈[m] ∥zj∥2 ≤ 2R and any y ∈ Rm satisfying +∥y − ω0∥2 ≤ ε, +max +j∈[m] +���f (j) (y, z) +��� +2 ≤ +3R +δ (2π)d/2 , +max +j∈[m] +���g(j) (y, z) +��� ≤ +� +1 + +1 +δ (2π)d/2 +� +(1 + ε) +and therefore +∥h (y, z)∥2 ≤ +� +� +� +�m +� +3R +δ (2π)d/2 +�2 ++ m +�� +1 + +1 +δ (2π)d/2 +� +(1 + ε) +�2 +≜ M. +Recall that µ(1) +0 , . . . , µ(m) +0 +are i.i.d. sampled from Uniform({Xi}1≤i≤N), therefore +(ω0, µ0) ∈ (y, z) : +� +(y, z) : ∥y − ω0∥2 ≤ ε, max +j∈[m] ∥zj∥2 ≤ 2R +� +where µ0 = [µ(j) +0 ]1≤j≤m. We are ready to apply Picard-Lindelöf theorem to show that there exists t0 > 0, +only depending on CLip and M, such that the ODE has a unique solution on the time interval [0, t0]. We can +use the same argument in the proof of Theorem 5 in Appendix E.1 to show that ωt ∈ ∆m−1 for all t ∈ [0, t0], +and can use the same argument in the proof of Theorem 6 in Appendix E.2 to show that maxj∈[m] ∥µ(j) +t ∥2 ≤ R +for all t ∈ [0, t0]. Then we can repeat the same analysis as above (notice that the above proof only requires +ω0 ∈ ∆m−1 and maxj∈[m] ∥µ(j) +0 ∥2 ≤ R, and t0 only depends on universal constants CLip and M) to show that +the ODE has a unique solution on [t0, 2t0], [2t0, 3t0], and so on. This shows the existence and uniqueness of +the solution to the ODE system (3.8). +Finally we check that (ρt)t≥0 defined as ρt := �m +l=1 ω(l) +t δµ(l) +t +solves (3.11). Note that ρt is a probability +measure since we have shown that ωt ∈ ∆m−1 for any t ≥ 0. For any test function ϕ(x) ∈ C∞ +c , we have +d +dt +� +Rd ϕ (x) ρt (dx) = d +dt +� +� +m +� +j=1 +ω(j) +t ϕ +� +µ(j) +t +� +� +� = +m +� +j=1 +� +˙ω(j) +t ϕ +� +µ(j) +t +� ++ ω(j) +t +� +∇ϕ +� +µ(j) +t +� +, ˙µ(j) +t +�� += − +m +� +j=1 +� +1 + δℓN (ρt) +� +µ(j) +t +�� +ω(j) +t ϕ +� +µ(j) +t +� ++ +m +� +j=1 +ω(j) +t +� +∇ϕ +� +µ(j) +t +� +, −∇δℓN (ρt) +� +µ(j) +t +�� +40 + += − +� +Rd [1 + δℓN (ρt) (x)] ϕ (x) ρt (dx) − +� +Rd ⟨∇ϕ (x) , ∇δℓN (ρt)⟩ ρt (dx) . +This proves that +∂tρt = div (ρt∇δℓ (ρt)) − [δℓ (ρt) + 1] ρt +holds in the sense of distributions. +F +Properties of Wasserstein gradient flow +In this section, we present some preliminary results on Wasserstein gradient flow for learning Gaussian +mixtures. We also discuss the implications of these results, as well as the technical difficulty of obtaining +more general results. +We first establish the connection between the Wasserstein gradient flow and the classical gradient flow +in the Euclidean space. Suppose we fit the data {Xi}1≤i≤N using a m-component Gaussian mixture model +1 +m +m +� +j=1 +N +� +µ(j), Id +� +, +where {µ(j)}1≤j≤m is the location of the m Gaussian components. The negative likelihood function is +ℓN,m +� +µ(1), . . . , µ(m)� +:= − 1 +N +N +� +i=1 +log +� +� 1 +m +m +� +j=1 +φ +� +Xi − µ(j)� +� +� . +(F.1) +The gradient flow for minimizing (F.1), denoted by (µt)t≥0 where µt = [µ(j) +t ]1≤j≤m, is given by the following +ODE system +˙µ(j) +t += −∇µ(j)ℓN,m +� +µ(1) +t , . . . , µ(m) +t +� +(F.2) +with initialization µ(1) +0 , . . . , µ(m) +0 +i.i.d. +∼ Uniform({Xi}1≤i≤N). The following theorem shows that the gradient +flow (F.2) captures the evolution of the location of particles in the Wasserstein gradient flow (3.18) initialized +from a discrete distribution +1 +m +�m +l=1 δµ(l) +0 . The proof is deferred to Appendix F.1. +Theorem 8. Consider the Euclidean gradient flow (µt)t≥0 in (F.2). Then the flow (ρt)t≥0 defined as +ρt := 1 +m +m +� +l=1 +δµ(l) +t +(F.3) +is the Wasserstein gradient flow, i.e. (F.3) is a distributional solution to the PDE (3.18). +Similar connection can also be established for the gradient descent algorithm for minimizing (F.1) and +the particle Wasserstein gradient descent (cf. Algorithm 3), which is omitted for brevity. +Then we focus on the infinite sample limit of Wasserstein gradent flow and analyze its convergence +property. The population level loss function is +ℓ∞ (ρ) = −EX∼ρ⋆∗φ {log [ρ ∗ φ (X)]} = KL (ρ⋆ ∗ φ ∥ ρ ∗ φ) + const. +(F.4) +In Appendix C.1 we have computed that +δℓ∞ (ρ) = − +� +Rd +ρ⋆ ∗ φ (y) +ρ ∗ φ (y) φ (x − y) dy. +(F.5) +We know that the Wasserstein gradient flow (ρt)t≥0 with respect to ℓ∞(ρ) is described by the following PDE: +∂tρt = div (ρt∇δℓ∞ (ρt)) +(F.6) +41 + +with ρ0 = ρ⋆ ∗ N(0, Id), which is the data distribution when we have infinite samples. This Wasserstein +gradient flow has the following particle interpretation: suppose at time t = 0 we initialize a particle x0 ∼ ρ0 +in the vector field (vt)t≥0 where vt = −∇δℓ∞(ρt), namely +˙xt = vt (xt) , +then xt ∼ ρt, namely the marginal distribution of (xt)t≥0 evolves according to the Wasserstein gradient flow. +The following theorem shows that, when the true mixing distribution ρ⋆ is a singleton (we assume +without loss of generality that ρ⋆ = δ0), Wasserstein gradient flow converges to ρ⋆. The proof can be found +in Appendix F.2. +Theorem 9. Consider the Wasserstein gradient flow in (F.6) with ρ⋆ = δ0. +For any ε < 1, we have +� +Rd ∥x∥2 +2ρt(dx) = O(ε) as long as +t ≥ exp (2d) ε−1−max{8, +√ +8d}. +Although Theorem 9 only focuses on the case when ρ⋆ is a singleton, the convergence result already +provides some intuition about the behavior of Wasserstein gradient flow in more general setting. Consider +a well-separated Gaussian mixture model with K components. +Assume that the mixing distribution is +ρ⋆ = �K +j=1 ω⋆ +j δµ⋆ +j , and the location of each Gaussian components, {µ⋆ +j}1≤j≤K, are well-separated. Since +the push-forward mapping vt = −∇δℓ∞(ρt) is localized (see F.5), there exists some T > 0 such that the +Wasserstein gradient flow (F.6) initialized from ρ⋆ ∗ N(0, Id) can be approximated, up to time T, by +ρt ≈ +K +� +j=1 +ω⋆ +j ρ(j) +t +∀ t ∈ [0, T] , +where for each j ∈ [K], ρ(j) +t +is the Wasserstein gradient flow ∂tρ(j) +t += div(ρ(j) +t ∇δℓ∞(ρ(j) +t )) with initialization +ρ(j) +0 += N(µ⋆ +j, Id). This suggests that Wasserstein gradient flow approximately converges to ρ⋆ since, by +Theorem 9, each ρ(j) +t +converges to δµ⋆ +j . However this observation also suggests that Wasserstein gradient +flow is not robust to weight mismatch. Consider initializing the Wasserstein gradient flow (F.6) with ρ0 = +�ρ ∗ N(0, 1), where �ρ = �K +j=1 �ωjδµ⋆ +j is a mixing distribution with correct support {µ⋆ +j}1≤j≤K but wrong +weights {�ωj}1≤j≤K ̸= {ω⋆ +j }1≤j≤K. Then we also have +ρt ≈ +K +� +j=1 +�ωjρ(j) +t +∀ t ∈ [0, T] , +which shows that ρt approximately converges to �ρ instead of ρ⋆ when 0 ≤ t ≤ T. Note that the time length T +that such approximations are valid can be arbitrarily large as long as the separation mini̸=j ∥µ⋆ +i −µ⋆ +j∥2 → ∞. +The above discussion suggests that using the correct initial weights are important for Wasserstein gradient +flow to converge to the true mixing distribution. +We also would like to compare the convergence rate in Theorem 9 to a benchmark provided by the Bures- +Wasserstein gradient flow. The Bures-Wasserstein gradient flow is defined on the space of non-degenerate +Gaussian distributions on Rd, denoted by BW(Rd) = Rd×Sd +++ (where we identify a non-degenerate Gaussian +distribution ν = N(µ, Σ) with (µ, Σ) ∈ Rd × Sd +++) equipped with the Wasserstein distance (3.3), which has +the following closed form expression +d2 +W (ν1, ν2) = ∥µ1 − µ2∥2 +2 + tr +� +Σ1 + Σ2 − 2 +� +Σ1/2 +1 +Σ2Σ1/2 +1 +�1/2� +when ν1 = N(µ1, Σ1) and ν2 = N(µ2, Σ2) are both non-degenerate Gaussians. +The Bures-Wasserstein +gradient flow (νt)t≥0 can be viewed as the Wasserstein gradient flow (ρt)t≥0 constrained to lie on BW(Rd). +We refer interested readers to Altschuler et al. (2021); Lambert et al. (2022) for more detailed discussion. +We can see from the proof of Theorem 9 that the push-forward mapping vt(x) of Wasserstein gradient flow +decays exponentially fast as ∥x∥2 → ∞, this will make the Wasserstein gradient flow (ρt)t≥0 becomes more +and more heavy-tailed . However the push-forward mapping of Bures-Wasserstein gradient flow is always +42 + +linear, and the Bures-Wasserstein gradient flow (νt)t≥0 is always Gaussian. For example, in Appendix F.3 +we can compute the push forward mapping explicitly for the two gradient flows at t = 0: +v0 (x) = −1 +3 +�4 +3 +�d/2 +exp +� +−∥x∥2 +2 +6 +� +x +(Wasserstein), +v0 (x) = x +4 +(Bures-Wasserstein). +Therefore it is natural to expect that Bures-Wasserstein gradient flow (νt)t≥0 initialized from ν0 = N(0, Id) +converges faster than Wasserstein gradient flow (ρt)t≥0 initialized from ρ0 = δ0. In Appendix F.3 we show +that the Bures-Wasserstein gradient flow (νt = N(µt, Σt))t≥0 is characterized by the following ODE: +µt = 0 +˙Σt = −2 (Σt + Id)−1 Σ2 +t (Σt + Id)−1 . +We also show that Σt is sandwiched between +1 +1 + 2tI ⪯ Σt ⪯ +2 +2 + tI, +and as a result +� +Rd ∥x∥2 +2νt(dx) = O(d/t). Since Bures-Wasserstein gradient flow is not converging exponen- +tially fast (we can see that the convergence rate is polynomial in t), we conjecture that Wasserstein gradient +flow does not enjoy exponential convergence as well. +Lastly, we numerically show in Figure 6 that the loss function ℓ∞(ρ) (cf. (F.4)) is not geodesically convex +(Ambrosio et al., 2008) even when ρ⋆ = δ0. We can also check that Polyak-Łojasiewicz (PL) inequality +∀ ρ : +∥∇Wℓ∞ (ρ)∥2 +ρ ≥ CPL [ℓ∞ (ρ) − ℓ∞ (ρ⋆)] +for some CPL > 0 +does not hold in general: consider ρ⋆ = 1 +2δ−1 + 1 +2δ1 and ρ = δ0, then it is straightforward to check that +∇Wℓ∞(ρ) = 0 but ℓ∞(ρ) > ℓ∞(ρ⋆). Therefore we cannot use standard proof technique (e.g. Ambrosio et al. +(2008) when the loss function is geodesically convex, or Chewi et al. (2020) when there is a PL inequality) +to show exponential convergence for the Wasserstein gradient flow (F.6). +F.1 +Proof of Theorem 8 +It is straightforward to compute the gradient of ℓN,m. For any j ∈ [m], we have +∇µ(j)ℓN,m +� +µ(1), . . . , µ(m)� += − 1 +N +N +� +i=1 +1 +�m +l=1 φ +� +Xi − µ(l)�∇µ(j)φ +� +Xi − µ(j)� += − 1 +N +N +� +i=1 +φ +� +Xi − µ(l)� +�m +l=1 φ +� +Xi − µ(l)� +� +Xi − µ(l)� +. +Therefore the Euclidean gradient flow (F.2) is given by +˙µ(j) +t += −∇µ(j)ℓN,m +� +µ(1) +t , . . . , µ(m) +t +� += 1 +N +N +� +i=1 +φ +� +Xi − µ(l) +t +� +�m +l=1 φ +� +Xi − µ(l) +t +� +� +Xi − µ(l) +t +� +. +Then we can invoke Theorem 6 to finish the proof. +F.2 +Proof of Theorem 9 +Step 1: characterizing the push-forward mapping. +First of all, it is straightforward to check that +(ρt)t≥0 is spherically symmetric for all t ≥ 0, namely ρt(dx) only depends on ∥x∥2. +The push-forward +mapping vt(x) : Rd → Rd at time t is +vt (x) = −∇δℓ∞ (ρt) (x) = ∇x +� +Rd +ρ⋆ ∗ φ (y) +ρt ∗ φ (y) φ (x − y) dy +43 + +(a) constant speed geodesic (ρt)0≤t≤1 joining ρ0 = N(0, 3) to ρ1 = N(0, 1) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +-5.4 +-5.3 +-5.2 +-5.1 +-5 +-4.9 +-4.8 +-4.7 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +-0.66 +-0.64 +-0.62 +-0.6 +-0.58 +-0.56 +-0.54 +-0.52 +-0.5 +(b) constant speed geodesic (ρt)0≤t≤1 joining ρ0 = N(0, 1) to ρ1 = δ0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +-5.5 +-5.48 +-5.46 +-5.44 +-5.42 +-5.4 +-5.38 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +-0.25 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +Figure 6: The loss function ℓ∞(ρt) or its derivative ℓ′ +∞(ρt) vs. t. In Figures (a), (ρt)0≤t≤1 is the constant +speed geodesic joining ρ0 = N(0, 3) to ρ1 = N(0, 1). In Figures (b), (ρt)0≤t≤1 is the constant speed geodesic +joining ρ0 = N(0, 1) to ρ1 = δ0. This shows that ℓ∞(ρ) is not globally geodesically convex, but might be +locally geodesically conex around ρ⋆ = δ0. += − +� +Rd +ρ⋆ ∗ φ (y) +ρt ∗ φ (y) (x − y) φ (x − y) dy += +� +Rd ∇y +ρ⋆ ∗ φ (y) +ρt ∗ φ (y) φ (y − x) dy += +� +Rd ht (y) φ (y − x) dy, +(F.7) +where the penultimate line follows from Stein’s lemma or Gaussian integration by parts, and ht(y) in the +last line is defined as +ht (y) := ∇ +φ (y) +ρt ∗ φ (y) = ∇y +1 +� +exp +� +− 1 +2 ∥z∥2 +2 + y⊤z +� +ρt (dz) += − +� +exp +� +− 1 +2 ∥z∥2 +2 + y⊤z +� +zρt (dz) +�� +exp +� +− 1 +2 ∥z∥2 +2 + y⊤z +� +ρt (dz) +�2 = −φ (y) +� +φ (y − z) zρt (dz) +�� +φ (y − z) ρt (dz) +�2 += −φ (y) +� +φ (y − z) zρt (dz) +[ρt ∗ φ (y)]2 += − +φ (y) +ρt ∗ φ (y) · +� +φ (y − z) zρt (dz) +ρt ∗ φ (y) +. +(F.8) +44 + +For any y ∈ Rd, we can compute +� +φ (y − z) zρt (dz) = +� +y⊤z>0 +φ (y − z) zρt (dz) + +� +y⊤z<0 +φ (y − z) zρt (dz) + +� +y⊤z=0 +φ (y − z) zρt (dz) +(i) += +� +y⊤z>0 +[φ (y − z) − φ (y + z)] zρt (dz) += φ (y) +� +y⊤z>0 +� +exp +� +y⊤z +� +− exp +� +−y⊤z +�� +z exp +� +−1 +2 ∥z∥2 +2 +� +ρt (dz) +(ii) += φ (y) +� +y⊤z>0 +� +exp +� +y⊤z +� +− exp +� +−y⊤z +�� y⊤z +∥y∥2 +2 +y exp +� +−1 +2 ∥z∥2 +2 +� +ρt (dz) += +� +y⊤z>0 +� +exp +� +y⊤z +� +− exp +� +−y⊤z +�� y⊤z +∥y∥2 +2 +exp +� +−1 +2 ∥z∥2 +2 +� +ρt (dz) +� +�� +� +=:at +· φ (y) y. +(F.9) +Here (i) and (ii) both follow from the spherical symmetry of ρt, and it is straightforward to check that the +integral in the last line does not depend on y due to the spherical symmetry of ρt, therefore at is a universal +constant that is independent of y. +Note that when y⊤z > 0, we have exp(y⊤z) − exp(−y⊤z) ≥ 2y⊤z, +therefore +at ≥ 2 +� +y⊤z>0 +� +y⊤z +�2 +∥y∥2 +2 +exp +� +−1 +2 ∥z∥2 +2 +� +ρt (dz) = +� +Rd +� +y⊤z +�2 +∥y∥2 +2 +exp +� +−1 +2 ∥z∥2 +2 +� +ρt (dz) . +Since at does not depend on y, we take y = ei for i ∈ [d] to achieve +at ≥ +� +Rd z2 +i exp +� +−1 +2 ∥z∥2 +2 +� +ρt (dz) , +∀ i ∈ [d]. +By taking average over d, we have +at ≥ 1 +d +� +Rd ∥z∥2 +2 exp +� +−1 +2 ∥z∥2 +2 +� +ρt (dz) = mt +d , +(F.10) +where we define +mt := +� +∥z∥2 +2 exp +� +−1 +2 ∥z∥2 +2 +� +ρt (dz) . +Taking (F.8), (F.9) and (F.10) collectively gives +ht (y) = −aty +� +φ (y) +ρt ∗ φ (y) +�2 +, +where +at ≥ mt +d . +Then we use (F.7) to characterize the push-forward mapping: +vt (x) = +� +y⊤x>0 +h (y) φ (y − x) dy + +� +y⊤x<0 +h (y) φ (y − x) dy + +� +y⊤x=0 +h (y) φ (y − x) dy +(i) += +� +y⊤x>0 +h (y) [φ (y − x) − φ (−y − x)] dy += −φ (x) at +� +y⊤x>0 +y +� +φ (y) +ρt ∗ φ (y) +�2 � +exp +� +y⊤x +� +− exp +� +−y⊤x +�� +exp +� +−1 +2 ∥y∥2 +2 +� +dy +(ii) += −φ (x) at +� +y⊤x>0 +x⊤y +∥x∥2 +2 +x +� +φ (y) +ρt ∗ φ (y) +�2 � +exp +� +y⊤x +� +− exp +� +−y⊤x +�� +exp +� +−1 +2 ∥y∥2 +2 +� +dy += −at +� +y⊤x>0 +x⊤y +∥x∥2 +2 +� +φ (y) +ρt ∗ φ (y) +�2 � +exp +� +y⊤x +� +− exp +� +−y⊤x +�� +exp +� +−1 +2 ∥y∥2 +2 +� +dy +� +�� +� +=:bt +· φ (x) x. +45 + +Similar to (F.9), here (i) and (ii) both follow from the spherical symmetry of ρt, and the integral in the last +line does not depend on x due to the spherical symmetry of ρt, as a result bt is a universal constant that is +independent of x. Note that when y⊤x > 0, we have exp(y⊤x) − exp(−y⊤x) ≥ 2y⊤x, therefore +bt ≥ 2 +� +y⊤x>0 +x⊤y +∥x∥2 +2 +� +φ (y) +ρt ∗ φ (y) +�2 +y⊤x exp +� +−1 +2 ∥y∥2 +2 +� +dy += +� +Rd +� +x⊤y +�2 +∥x∥2 +2 +� +φ (y) +ρt ∗ φ (y) +�2 +exp +� +−1 +2 ∥y∥2 +2 +� +dy. +Note that ∥ρt ∗ φ∥∞ ≤ ∥φ∥∞ ≤ (2π)−d/2, and as a result +bt ≥ +� � +x⊤y +�2 +∥x∥2 +2 +exp +� +−3 +2 ∥y∥2 +2 +� +dy = 1 +3 +�2π +3 +�d/2 +Therefore we have +vt (x) = −atbtφ (x) x = −ctφ (x) x +(F.11) +where +ct := atbt ≥ 1 +3d +�2π +3 +�d/2 +mt. +(F.12) +Step 2: showing the convergence of Wasserstein gradient flow. +Recall the particle interpretation +of Wasserstein gradient flow as follows: let x0 ∼ ρ0 = N(0, Id) and ˙xt = vt(xt), then for any t ≥ 0 we have +xt ∼ ρt. This allows us to compute +∂tE +� +∥xt∥2 +2 +� += 2E [⟨xt, ˙xt⟩] = 2E [⟨xt, vt (xt)⟩] +(i) += −2ctE +� +∥xt∥2 +2 φ (xt) +� +(ii) +≤ − 2 +3d +�2π +3 +�d/2 +mtE +� +∥xt∥2 +2 φ (xt) +� +(iii) += − 2 +3d +�4π2 +3 +�d/2 +E2 � +∥xt∥2 +2 φ (xt) +� +, +(F.13) +where (i) follows from (F.11), (ii) utilizes (F.12), and (iii) holds since +mt = +� +∥z∥2 +2 exp +� +−1 +2 ∥z∥2 +2 +� +ρt (dz) = (2π)d/2 E +� +∥xt∥2 +2 φ (xt) +� +. +For any τ > 0, by Cauchy-Schwarz inequality we have +E +� +∥x0∥2 +2 1 +� +∥x0∥2 +2 > d + τ +�� (i) +≤ +� +E ∥x0∥4 +2 +�1/2 � +P +� +∥x0∥2 +2 > d + τ +��1/2 +. +Note that ∥x0∥2 +2 ∼ χ2(d), therefore E∥x0∥4 +2 = var(∥x0∥2 +2) + (E∥x0∥2 +2)2 = 2d + d2. In addition, by the tail +probability bound for χ2 random variables (e.g. Wainwright (2019, equation (2.18))), we have +P +� +∥x0∥2 +2 > d + τ +� +≤ exp +� +− min +�τ 2 +8d, τ +8 +�� +. +Therefore we have +E +� +∥x0∥2 +2 1 +� +∥x0∥2 +2 > d + τ +�� +≤ +� +2d + d2 exp +� +− min +�τ 2 +8d, τ +8 +�� +≤ (d + 1) exp +� +− min +�τ 2 +8d, τ +8 +�� +≤ ε +46 + +as long as we choose +τ ≜ max +� +8 log d + 1 +ε +, +� +8d log d + 1 +ε +� +. +Since the push forward mapping vt(x) is always pointing towards zero (cf. +(F.11) and (F.12)), we know +that ∥xt∥2 is non-increasing in t. Therefore we have +E +� +∥xt∥2 +2 φ (xt) +� +≥ E +� +∥xt∥2 +2 φ (xt) 1 +� +∥x0∥2 +2 ≤ d + τ +�� +(i) +≥ (2π)−d/2 exp +� +−d + τ +2 +� +E +� +∥xt∥2 +2 1 +� +∥x0∥2 +2 ≤ d + τ +�� +≥ (2π)−d/2 exp +� +−d + τ +2 +� � +E +� +∥xt∥2 +2 +� +− E +� +∥xt∥2 +2 1 +� +∥x0∥2 +2 > d + τ +��� +(ii) +≥ (2π)−d/2 exp +� +−d + τ +2 +� � +E +� +∥xt∥2 +2 +� +− E +� +∥x0∥2 +2 1 +� +∥x0∥2 +2 > d + τ +��� +≥ (2π)−d/2 exp +� +−d + τ +2 +� � +E +� +∥xt∥2 +2 +� +− ε +� +, +(F.14) +where both (i) and (ii) follows from the fact that ∥xt∥2 is non-increasing. Taking (F.13) and (F.14) collectively +gives +∂tE +� +∥xt∥2 +2 +� +≤ − 2 +3d +�4π2 +3 +�d/2 +(2π)−d exp [− (d + τ)] +� +E +� +∥xt∥2 +2 +� +− ε +�2 += − 2 +3d +�1 +3 +�d/2 +exp [− (d + τ)] +� +E +� +∥xt∥2 +2 +� +− ε +�2 +. +Let f(t) = E[∥xt∥2 +2], we know that f(0) = d and +df +dt ≤ − 2 +3d +�1 +3 +�d/2 +exp [− (d + τ)] (f − ε)2 . +Solving this ordinary differential inequality gives +1 +f (t) − ε − +1 +f (0) − ε ≥ 2 +3d +�1 +3 +�d/2 +exp [− (d + τ)] t, +which is equivalent to +E +� +∥xt∥2 +2 +� +≤ ε + +� +2 +3d +�1 +3 +�d/2 +exp (−d − τ) t + +1 +d − ε +�−1 +. +Then we immediately know that E[∥xt∥2 +2] ≤ O(ε) as long as +t ≥ exp (2d) ε−1−max{8, +√ +8d}. +F.3 +Calculation for Bures-Wasserstein gradient flow +Define ℓ(µ, Σ) = ℓ∞(ρ) where we parameterize ρ = N(µ, Σ). Then we can compute +ℓ (µ, Σ) = − +� +log +� +(2π)−d/2 [det (Σ + Id)]−1/2 exp +� +−1 +2 (x − µ)⊤ (Σ + Id)−1 (x − µ) +�� +φ (x) dx + constant += 1 +2 log det (Σ + Id) + +� 1 +2 (x − µ)⊤ (Σ + Id)−1 (x − µ) φ (x) dx + constant +47 + += 1 +2 log det (Σ + Id) + 1 +2Ex∼N (0,I) +� +(x − µ)⊤ (Σ + Id)−1 (x − µ) +� ++ constant += 1 +2 log det (Σ + Id) + 1 +2tr +� +(Σ + Id)−1� ++ 1 +2µ⊤ (Σ + Id)−1 µ + constant. +Then we can compute the Euclidean gradient of ℓ(µ, Σ) as follows: +∇µℓ (µ, Σ) = (Σ + Id)−1 µ, +∇Σℓ (µ, Σ) = 1 +2 (Σ + Id)−1 − 1 +2 (Σ + Id)−2 − 1 +2 (Σ + Id)−1 µµ⊤ (Σ + Id)−1 += 1 +2 (Σ + Id)−1 � +Σ + Id − Id − µµ⊤� +(Σ + Id)−1 += 1 +2 (Σ + Id)−1 � +Σ − µµ⊤� +(Σ + Id)−1 . +According to Lambert et al. (2022, Appendix B.3), when initialized from (µ0, Σ0) = (0, Id), the Bures- +Wasserstein gradient flow can be described using the following ODE: +˙µt = − (Σt + Id)−1 µt +˙Σt = −Σt (Σt + Id)−1 � +Σt − µµ⊤� +(Σt + Id)−1 − (Σt + Id)−1 � +Σt − µµ⊤� +(Σt + Id)−1 Σt +with initial condition µ0 = 0 and Σ0 = Id. It is straightforward to check that µt = 0 for all t ≥ 0, and the +dynamic of Σt is governed by +˙Σt = −Σt (Σt + Id)−1 Σt (Σt + Id)−1 − (Σt + Id)−1 Σt (Σt + Id)−1 Σt += −Σt (Σt + Id)−1 + 2 (Σt + Id)−1 Σt (Σt + Id)−1 − (Σt + Id)−1 Σt += −2Id + 2 (Σt + Id)−1 + 2 (Σt + Id)−1 Σt (Σt + Id)−1 += −2 (Σt + Id)−1 Σ2 +t (Σt + Id)−1 +with initial condition Σ0 = Id. We can check that the off-diagonal entries of Σt are always zero, and its +diagonal entries are identical and evloves according to the following ODE +˙σt = −2 +σ2 +t +(σt + 1)2 +with initial condition σ0 = 1. It is straightforward to check that σt is monotonically decreasing and is always +non-negative, namely 0 ≤ σt ≤ 1 always holds. Therefore we have +−2σ2 +t ≤ ˙σt ≤ −1 +2σ2 +t . +This gives +1 +1 + 2t ≤ σt ≤ +2 +2 + t, +and therefore +1 +1 + 2tI ⪯ Σt ⪯ +2 +2 + tI, +which suggests that ρt converges to ρ⋆ at the speed of O(d/t). +When t = 0, we can compute the push forward mapping of Wasserstein gradient flow explicitly, which +intuitively explains why Wasserstein gradient flow does not converge exponentially fast. We first compute +∇ρ⋆ ∗ φ (y) +ρ0 ∗ φ (y) == ∇ +(det I)−d/2 exp +� +− 1 +2 ∥y∥2 +2 +� +(det 2I)−d/2 exp +� +− 1 +4 ∥y∥2 +2 +� = 2d/2∇ exp +� +−1 +4 ∥y∥2 +2 +� += −2d/2−1y exp +� +−1 +4 ∥y∥2 +2 +� +, +48 + +then the push forward mapping at t = 0 is given by x �→ v0(x) where +v0 (x) = +� � +∇y +ρ⋆ ∗ φ (y) +ρ0 ∗ φ (y) +� +φ (y − x) dy = −2d/2−1 +� +y exp +� +−1 +4 ∥y∥2 +2 +� +· +1 +(2π)d/2 exp +� +−1 +2 ∥x − y∥2 +2 +� +dy += − 2d/2−1 +(2π)d/2 +� +y exp +� +−1 +2 ∥x∥2 +2 + x⊤y − 3 +4 ∥y∥2 +2 +� +dy = − 2d/2−1 +(2π)d/2 +� +y exp +� +−1 +6 ∥x∥2 +2 − 3 +4 +����y − 2 +3x +���� +2 +2 +� +dy += −2d/2−1 +�2 +3 +�d/2 +exp +� +−1 +6 ∥x∥2 +2 +� � +1 +(2π)d/2 (2/3)d/2 y exp +� +−3 +4 +����y − 2 +3x +���� +2 +2 +� +dy += −1 +3 +�4 +3 +�d/2 +exp +� +−1 +6 ∥x∥2 +2 +� +x. +On the other hand, in view of Lambert et al. (2022, Appendix B.3), the Bures-Wasserstein gradient at time +t = 0 is given by +∇BWℓ∞ (ρ0) = +� ∇µℓ (µ0, Σ0) +2∇Σℓ (µ0, Σ0) +� += +� +0 +1 +4Id +� +, +and therefore the push forward mapping at t = 0 is given by x �→ x/4. +References +Allen-Zhu, Z., Li, Y., and Song, Z. (2019). A convergence theory for deep learning via over-parameterization. +ICML. +Altschuler, J., Chewi, S., Gerber, P. R., and Stromme, A. (2021). +Averaging on the bures-wasserstein +manifold: dimension-free convergence of gradient descent. Advances in Neural Information Processing +Systems, 34:22132–22145. +Ambrosio, L., Gigli, N., and Savaré, G. (2008). Gradient flows: in metric spaces and in the space of probability +measures. Springer Science & Business Media. +Bauer, M., Bruveris, M., and Michor, P. W. (2016). Uniqueness of the fisher–rao metric on the space of +smooth densities. Bulletin of the London Mathematical Society, 48(3):499–506. +Bubeck, S. (2015). Convex optimization: algorithms and complexity. 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High-dimensional statistics: A non-asymptotic viewpoint, volume 48. Cambridge +University Press. +Zhang, C.-H. (2009). Generalized maximum likelihood estimation of normal mixture densities. Statistica +Sinica, pages 1297–1318. +Zhang, Y., Cui, Y., Sen, B., and Toh, K.-C. (2022). On efficient and scalable computation of the nonpara- +metric maximum likelihood estimator in mixture models. arXiv preprint arXiv:2208.07514. +51 + diff --git a/NtAzT4oBgHgl3EQfzP6J/content/tmp_files/load_file.txt b/NtAzT4oBgHgl3EQfzP6J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..37807eac1423681f980b0328b39d261facbba55d --- /dev/null +++ b/NtAzT4oBgHgl3EQfzP6J/content/tmp_files/load_file.txt @@ -0,0 +1,2046 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf,len=2045 +page_content='Learning Gaussian Mixtures Using the Wasserstein-Fisher-Rao Gradient Flow Yuling Yan∗† Kaizheng Wang∗‡ Philippe Rigollet§ January 5, 2023 Abstract Gaussian mixture models form a flexible and expressive parametric family of distributions that has found applications in a wide variety of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Unfortunately, fitting these models to data is a no- toriously hard problem from a computational perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Currently, only moment-based methods enjoy theoretical guarantees while likelihood-based methods are dominated by heuristics such as Expectation- Maximization that are known to fail in simple examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In this work, we propose a new algorithm to compute the nonparametric maximum likelihood estimator (NPMLE) in a Gaussian mixture model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Our method is based on gradient descent over the space of probability measures equipped with the Wasserstein-Fisher-Rao geometry for which we establish convergence guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In practice, it can be approximated using an interacting particle system where the weight and location of particles are updated alternately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We conduct extensive numerical experiments to confirm the effectiveness of the proposed algorithm compared not only to classical benchmarks but also to similar gradient descent algorithms with respect to simpler geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In particular, these simulations illustrate the benefit of updating both weight and location of the interacting particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Keywords: Gaussian mixture model, nonparametric MLE, Wasserstein-Fisher-Rao geometry, optimal transport, Wasserstein gradient flows, overparameterization 1 Introduction Owing to their flexibility and versatility, mixture models have emerged as central objects of statistical modeling since their introduction by Pearson in the nineteenth century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' However, this flexibility often comes at computational cost: such models are often hard to fit and, until quite recently, the computational aspects of mixture models have been overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Still today, theory and practice diverge, with the former largely focuses on the method of moments while the latter is dominated by variational approaches, chiefly maximum likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The goal of this work is to reduce this gap by developing new algorithms for maximum likelihood estimation that are supported by theoretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Consider i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' samples {Xi}1≤i≤N ∈ Rd generated from an isotropic Gaussian1 mixture ρ⋆ ∗ N(0, Id) with density function (ρ⋆ ∗ φ) (x) = � Rd φ (x − y) ρ⋆ (dy) , where ρ⋆ is a mixing distribution over Rd, and φ(x) = (2π)−d/2 exp(−∥x∥2 2/2) is the density function of the isotropic Gaussian distribution N(0, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The goal is to learn the Gaussian mixture ρ⋆ ∗ N(0, Id) from N samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' ∗The first two authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' †Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Email: yulingy@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' ‡Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Email: kaizheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='wang@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' §Department of Mathematics, MIT, Cambridge, MA 02139, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Email: rigollet@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 1All of this work extends to general mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' See the Appendix for a general treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='01766v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='ST] 4 Jan 2023 The negative log-likelihood for this problem is defined as ℓN (ρ) = − 1 N N � i=1 log [(ρ ∗ φ) (Xi)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' While ℓN itself is trivially a convex functional of ρ, the class of measures over which it is minimized is often not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This is for example the case of finite mixture models where ρ is restricted to be a measure supported on at most k atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This lack of convexity in the constraint is the main source of computational difficulty for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' To overcome this limitation, Kiefer and Wolfowitz (1956) proposed the nonparametric maxi- mum likelihood estimator (NPMLE) which prescribes to minimize ℓN over the set P(Rd) of all probability distributions over Rd: �ρ ∈ argmin ρ∈P(Rd) ℓN (ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) While P(Rd) is convex, it is infinite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In fact, NPMLE can be seen as an extreme instance of overparameterization, a phenomenon that is currently challenging conventional statistical wisdom in deep learning theory Allen-Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Chizat and Bach (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In fact, the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) enjoys interesting structural properties when d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' More specifically, in this case, the optimization problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) admits a unique solution �ρ (Jewell, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lindsay and Roeder, 1993) which furthermore is supported on at most N atoms (Lindsay, 1983) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This upper bound was improved to OP(log N) by Polyanskiy and Wu (2020) when ρ⋆ is sub-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Quite strikingly, little is known about the structure of NPMLE in dimension d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In fact, to the best of our knowledge, Theorem 1 below is the first to establish existence of a solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) in any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Classical statistical results provide Hellinger risk bounds for �ρ ∗ φ as an estimator of ρ⋆ ∗ φ (Dicker and Zhao, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Saha and Guntuboyina, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Zhang, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' These rates are commensurate with minimax optimality even over the larger class of C∞ density functions up to logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Despite notable contributions, the computational aspects of NPMLE are still vastly under-explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Most of these contributions employ a discretization scheme by setting a fine grid in advance and solve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) with the additional constraint that ρ is supported on the grid (Jiang and Zhang, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Koenker and Mizera, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lindsay, 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' While well understood theoretically, those methods suffer from the curse of dimensionality, and their complexity scales exponentially with the dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' To overcome this limitation, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2022) proposed a heuristic that alternately updates the weights and the support using the Expectation-Maximization (EM) algorithm but it does not come with theoretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Another notable contribution is the support reduction algorithm of Groeneboom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2008) which is also computationally inefficient since it requires to compute the minimizer of a nonconvex function in Rd in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In this work, we propose to solve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) using gradient descent in the space of probability measures endowed with the Wasserstein-Fisher-Rao (WFR) geometry (Chizat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Gallouët and Monsaingeon, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Kondratyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Liero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' More specifically, we introduce the WFR gradient flow of the negative log-likelihood ℓN and show that it converges to the NPMLE under mild conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In turn, we implement this WFR gradient flow using Euler discretization in time and particle discretization in space, thus resulting in a system of weighted interacting particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In essence, the resulting Algorithm 1 alternatively updates locations and weights, like the EM algorithm described above but the updates coming from the WFR gradient flow are inherently different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' As the name indicates, the WFR geometry is a composite of the Wasserstein geometry (Ambrosio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Otto, 2001) and the Fisher-Rao geometry (Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The former component governs updates of the support while the latter governs the weight updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Either of these geometries leads to its own gradient descent algorithm but our numerical results indicate that their combination is the key to achieving fast convergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In fact, we show that the fixed-location EM algorithm from prior literature (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Jiang and Zhang, 2009) implements gradient descent with respect to the Fisher-Rao geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' As a byproduct of our analysis, we also show that it converges to the NPMLE in the infinite-particle regime under certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 2 2 Nonparametric Maximum Likelihood Estimator (NPMLE) In this section, we examine optimality conditions for the optimization problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' To that end, denote by δℓN(ρ) the first variation of ℓN at a measure ρ and observe2 that it is given by δℓN (ρ) : x �→ − 1 N N � i=1 φ (x − Xi) (ρ ∗ φ) (Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) The following theorem shows the existence as well as the optimality condition of NPMLE in general dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The proof is deferred to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The following properties hold for NPMLE: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (Existence) The minimizer of the optimization problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (Optimality condition) A distribution �ρ ∈ P(Rd) is an NPMLE if and only if (i) δℓN(�ρ)(x) ≥ −1 holds for all x ∈ Rd, and (ii) δℓN(�ρ)(x) = −1 for �ρ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We show in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 that for any ρ ∈ P(Rd), � δℓN(ρ)dρ = −1 always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' As a result, the optimality condition (ii) in Theorem 1 is implied by (i), which means that δℓN(�ρ)(x) ≥ −1 for all x ∈ R alone is already the necessary and sufficient condition for �ρ to be the NPMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' However, we keep both conditions in the theorem as each of them reveal important structural information of NPMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Theorem 1 asserts the existence of NPMLE, but its uniqueness when d ≥ 2 is still an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Although the uniqueness is not settled, the convergence theory in this paper is still valid: in this case NPMLE refers to any minimizer of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We use P(Rd) to denote the space of probability measures over Rd, P2(Rd) to denote the space of probability measures over Rd with finite second moments, and Pac(Rd) to denote the space of probability measures that are absolutely continuous with respect to the Lebesgue measure on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let ∆m−1 be the m − 1 dimensional probability simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any ρ ∈ P(Rd), supp(ρ) denotes its support set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' the smallest closed set C ⊆ Rd such that ρ(C) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any mapping T : Rd → Rd and any distribution ρ ∈ P(Rd), let T#ρ be the pushforward (or image measure) of ρ by T, which is defined by T#ρ(A) = ρ(T −1(A)) for any Borel set A in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any x ∈ Rd, we use δx to denote the Dirac mass at point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For two probability measures µ, ν ∈ P(Rd), we use µ ≪ ν to denote that µ is absolutely continuous with respect to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For a sequence {ρn}∞ n=0 in P(Rd) and ρ ∈ P(Rd), we write ρn w→ ρ if ρn weakly converges to ρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' � Rd f(x)ρn(dx) → � Rd f(x)ρ(dx) holds for every bounded continuous function f : Rd → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let C∞ c (Rd) be the set of smooth functions with compact support in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We say that (ρt)t≥0 is a distributional solution to the partial differential equation (PDE) ∂tρt = −div(ρtvt) + ρtαt where vt : Rd → Rd and αt : Rd → R, if for any ϕ ∈ C∞ c (Rd) it holds that d dt � Rd ϕ (x) ρt (dx) = � Rd [⟨∇ϕ (x) , vt (x)⟩ + ϕ(x)αt(x)] ρt (dx) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Finally we use the shorthand ODE to refer to ordinary differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 3 Wasserstein-Fisher-Rao gradient descent Gradient flows over probability measures are a useful tool in the development of sampling algorithms where the goal is to produce samples from a target measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' A classical example arises when π is, for example, a Bayesian posterior known only up to normalizing constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In this context, Wasserstein-Fisher-Rao (WFR) gradient flows have recently emerged as a strong alternative to vanilla Wasserstein gradient flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Indeed, they provide the backbone of the birth-death sampling algorithm of Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2019a) as well as the particle- based method proposed in Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We refer the reader to the recent manuscript of Chewi (2022) for more details on sampling and the role of Wasserstein gradient flows in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 2Explicit calculations follow from standard arguments in calculus of variations and are deferred to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 3 The likelihood maximization problem of interest in the present paper differs from sampling questions because its aims at optimizing a different objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Nevertheless, it remains an optimization problem and the machinery of gradient flows over the space of probability measures may be deployed in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' To the best of our knowledge, this paper present the first attempt at such a deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' More specifically, our main algorithm to solve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) relies on a specific discretization of the Wasserstein-Fisher-Rao gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We begin with a short introduction to gradient flows over metric spaces of probability measures that can be safely skipped by experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 Gradient flows over metric spaces of probability measures Gradient flows over metric spaces of probability measures is a central topic of the calculus of variations that has found applications in variety of fields ranging from analysis and geometry to probability and statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We briefly discuss the main idea behind this powerful tool and refer the reader to the formidable book of Ambrosio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2008) for more details about this deep question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Recall that our goal is to derive a gradient flow for the functional ℓN over the space of probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The nature of this gradient flow is simply a curve (ρt)t≥0 such that ∂tρt = −∇ℓN(ρt) for a notion of gradient ∇ to be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In their seminal work, Jordan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (1998) were able to define a gradient flow over the Wasserstein space by analogy to the Euclidean gradient flow without the need to actually define a gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We follow their approach and define the gradient flow of ℓN with respect to a suitable geometry with geodesic distance d(·, ·) over the space of probability measures as ∂tρt = lim η→0 ρη t − ρt η , where ρη t := arg min ρ∈P(Rd) �� Rd δℓN (ρt) d (ρ − ρt) + 1 2η d2 (ρ, ρt) � Given a distance, the existence of a limiting absolutely continuous curve (ρt)t≥0 is an important and central question that we omit in this overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Our main focus in this work is the Wasserstein-Fisher-Rao distance which is a composite of the Fisher-Rao distance and the (quadratic) Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We now introduce these three distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Recall that a geodesic distance between two points measures the shortest curve that links these two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The difference between these three distances is governed by the differential structure put on probability distributions, which roughly corresponds to type of curves that are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The length of these curves is then measured using a Riemannian metric which in all cases is rather straightforward so we focus our discussion on curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In turn, these curves and their lengths define a geometry on the space of probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Fisher-Rao distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The Fisher-Rao distance is linked to reaction equations of the form ∂tρt = ρt � αt − � αtdρt � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) where αt(x) ∈ R is a scalar that governs how much mass is created at x ∈ Rd and time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' It is easy to see that these dynamics preserve the total mass 1 of probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Among all such curves that link ρ0 to ρ1, the Fisher-Rao geodesic is the one that minimizes the total length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' More specifically, the Fisher-Rao distance dFR is defined as (Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2016), d2 FR (ρ0, ρ1) = inf � � 1 0 � �� αt − � αtdρt �2� dρtdt : (ρt, αt)t∈[0,1] solves ∂tρt = ρt � αt − � αtdρt �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2) While this will not be useful for our problem, it is worth noting that d2 FR (ρ0, ρ1) = 4 � ��� � dρ0 dλ − � dρ1 dλ ��� 2 dλ, 4 where λ is any positive measure such that ρ0, ρ1 ≪ λ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', λ = ρ0 + ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Hence, up to a constant factor, the Fisher-Rao distance is simply the Hellinger distance that is well-known to statisticians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Given two probability measures ρ0, ρ1 ∈ P(Rd), the (quadratic) Wasserstein dis- tance between ρ0 and ρ1 is defined as (Villani, 2009) d2 W (ρ0, ρ1) = inf π∈Π(ρ0,ρ1) � ∥x − y∥2 2 π (dx, dy) , where the infimum is taken over all couplings of ρ0 and ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' It also admits a geodesic distance interpretation by means of the Benamou-Brenier formula: d2 W (ρ0, ρ1) = inf � � 1 0 � ∥vt∥2 dρtdt : (ρt, vt)t∈[0,1] solves ∂tρt = −div (ρtvt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3) Here admissible curves are given by the continuity equation ∂tρt = −div (ρtvt) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4) which describes the evolution of a density of particles in Rd evolving according to time-dependent vector field vt : Rd → Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Wasserstein-Fisher-Rao distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The geometry underlying the Wasserstein-Fisher-Rao distance is built using curves that satisfy the following evolution equation: ∂tρt = −div (ρtvt) + ρt � αt − � αtdρt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that the right-hand side is precisely the sum of the right-hand sides for the reaction equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) and the continuity equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4) that govern the Fisher-Rao and the Wasserstein geometry respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' As such it is a composite of the two geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In turn, the Wasserstein-Fisher-Rao distance dWFR is defined as (Chizat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Gallouët and Mon- saingeon, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Kondratyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Liero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2018) d2 WFR (ρ0, ρ1) = inf � � 1 0 � � ∥vt∥2 + � αt − � αtdρt �2� dρtdt : (ρt, vt, αt)t∈[0,1] solves ∂tρt = −div (ρtvt) + ρt � αt − � αtdρt �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5) Equipped with this distance, we are in a position to define the WFR gradient flow and its time discretization, the WFR gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 Wasserstein-Fisher-Rao gradient descent In this section we introduce our main algorithm: Wasserstein-Fisher-Rao Gradient Descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The gradient flow {ρt}t≥0 of the negative log-likelihood ℓN(ρ) in P2(Rd) with respect to the Wasserstein- Fisher-Rao distance dWFR(·, ·) is given by ∂tρt = − [1 + δℓN (ρt)] ρt + div (ρt∇δℓN (ρt)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6) The formal derivation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6) is based on the Riemannian structure underlying the Wasserstein-Fisher-Rao metric can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The WFR gradient flow is not readily implementable for because of two obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' First, it is described in continuous time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Second, it requires the manipulation of full probability measures ρt on Rd which are infinite dimensional objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 5 To overcome the first obstacle, we employ a straightforward time-discretization scheme to obtain a WFR gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This algorithm produces a sequence of probability measures {ρn}n≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' It makes the following two steps alternately: d�ρn dρn = 1 − η [1 + δℓN (ρn)] (Fisher-Rao gradient update) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7a) ρn+1 = [id − η∇δℓN (�ρn)]# �ρn (Wasserstein gradient update) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7b) for n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', where η > 0 is the step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that each corresponds to a summand on the right- hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6) In fact, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7a) is one step of gradient descent update with respect to the Fisher-Rao geometry, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7b) corresponds to a gradient step with respect to the Wasserstein geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' It is also worth mentioning that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7) is related to the splitting scheme in Gallouët and Monsaingeon (2017): (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7) can be viewed as forward Euler scheme (explicit scheme) for numerical approximation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6), while Gallouët and Monsaingeon (2017) uses backward Euler scheme (implicit scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The implementation of the latter requires optimization over probability measures in each iterate, which is a difficult problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The following theorem shows that Wasserstein-Fisher-Rao gradient descent converges when initialized from a distribution that puts weight on the entire space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The proof can be found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In fact, a similar result for the continuous-time Wasserstein-Fisher-Rao gradient flow can be derived at the cost of additional technical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Theorem 2 (Convergence to NPMLE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Suppose that the initialization ρ0 ∈ P(Rd) satisfies supp(ρ0) = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Consider the Wasserstein-Fisher-Rao gradient descent {ρn}n≥0 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' There exists η0 > 0 determined by the samples {Xi}1≤i≤N, such that if 0 < η ≤ η0 and ρn w→ �ρ when n → ∞, then �ρ is the NPMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The convergence result in Theorem 2 is conditional: we only show that if WFR gradient descent converges weakly to some limiting distribution �ρ, then �ρ is an NPMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This is similar to the convergence theory established by Chizat and Bach (2018) in their study of the training dynamics for shallow neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This limitation is due to a lack of geodesic convexity which prevents us from establishing unconditional global convergence guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' To overcome the second obstacle, we also need to discretize ρn in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Thanks to the Wasserstein update (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7b), we need not use a fixed grid as in previous algorithms for NPMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Instead, observe that if we initialize WFR gradient descent at a distribution supported on m atoms, then ρn remains supported on m atoms for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In this case, ρn describes the evolution of m interacting particles with weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' More concretely, consider the initialization ρ0 = m � l=1 ω(l) 0 δµ(l) 0 , where µ(1) 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , µ(m) 0 ∈ Rd and ω0 = (ω(1) 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , ω(m) 0 ) ∈ ∆m−1, where the location of particles are independently sampled from the data points {Xi}1≤i≤N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The following theorem gives a precise characterization of the Wasserstein-Fisher-Rao gradient flow initialized from ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The proof is deferred to Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Theorem 3 (Particle Wasserstein-Fisher-Rao gradient flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The system of coupled ODE ˙µ(j) t = 1 N N � i=1 φ � Xi − µ(j) t � �m l=1 ω(j) t φ � Xi − µ(l) t � � Xi − µ(j) t � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8a) ˙ω(j) t = � � 1 N N � i=1 φ � Xi − µ(j) t � �m l=1 ω(j) t φ � Xi − µ(l) t � − 1 � � ω(j) t , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8b) with initialization µ(1) 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' µ(m) 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' ∼ Uniform({Xi}1≤i≤N) and ω0 = [ω(j) 0 ]1≤j≤m ∈ ∆m−1 has unique solution on any time interval [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Moreover, the flow (ρt)t≥0 defined as ρt := m � l=1 ω(l) t δµ(l) t (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='9) 6 is the Wasserstein-Fisher-Rao gradient flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' a distributional solution to the PDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In practice, we can obtain a time discretization of the Wasserstein-Fisher-Rao gradient flow by discretizing the ODE system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8), which gives the Wasserstein-Fisher-Rao gradient descent algorithm as summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Algorithm 1 Wasserstein-Fisher-Rao gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Input: data {Xi}1≤i≤n, number of particles m, step size η > 0, maximum number of iterations t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Initialization: draw µ(1) 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , µ(m) 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' ∼ Unif({Xi}1≤i≤n) and ω(1) 0 = · · · = ω(m) 0 = 1/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Updates: for t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , t0 do µ(j) t+1 = µ(j) t + η 1 N N � i=1 φ � Xi − µ(j) t � �m l=1 ω(j) t φ � Xi − µ(l) t � � Xi − µ(j) t � , ω(j) t+1 = ω(j) t + η � � 1 N N � i=1 φ � Xi − µ(j) t+1 � �m l=1 ω(j) t φ � Xi − µ(l) t+1 � − 1 � � ω(j) t , for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Here φ(x) = (2π)−d/2 exp(−∥x∥2 2/2) is the probability density function of N(0, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Output ρ = �m j=1 ω(j) t0 δµ(j) t0 as the (approximate) NMPLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The proof techniques employed to establish Theorem 2 do not cover Algorithm 1 unfortunately since the initial measure is not supported on the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Nevertheless, since the NPMLE is known to be supported on a small number of atoms (Polyanskiy and Wu, 2020) in certain cases, it is likely that taking m large enough will be sufficient to establish convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This conjecture is supported by our numerical experiments in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 Surrogate geometries As discussed above the Wasserstein-Fisher-Rao (WFR) geometry is obtained as a composite of the Wasser- stein geometry and the Fisher-Rao geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In turn, WFR gradient descent alternates between a Fisher-Rao gradient step and a Wasserstein gradient step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This observation raises a burning question: is the composite nature of WFR gradient descent necessary to obtain good performance of are either of these two build- ing blocks alone sufficient?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In this subsection, we explore properties and limitations of these two natural alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Fisher-Rao gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We show in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 that the Fisher-Rao gradient flow (ρt)t≥0 of the function ℓN(ρ) is defined by the following PDE: ∂tρt = − [1 + δℓN (ρt)] ρt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11) By time discretization, one readily obtains the Fisher-Rao gradient descent updates {ρn}n≥0: dρn+1 dρn = 1 − γ [1 + δℓN (ρn)] , n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='12) where γ > 0 is the step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Although Fisher-Rao gradient flow/descent is derived in an abstract way using Riemannian geometry, it has intimate connection with some well-known algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Fisher-Rao gradient flow as proximal gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The Fisher-Rao gradient flow (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11) can be viewed as the continuous-time limit of the proximal gradient algorithm under the Fisher-Rao metric: ∂tρt = lim η→0+ ρη t − ρt η , where ρη t := arg min ρ∈P(Rd) �� Rd δℓN (ρt) d (ρ − ρt) + 1 2η d2 FR (ρ, ρt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='13) 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Fisher-Rao gradient flow as mirror flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The Fisher-Rao gradient flow (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11) can also be viewed as the continuous-time limit of mirror descent algorithm for ℓN(ρ) with Kullback-Leibler (KL) divergence ∂tρt = lim η→0+ ρη t − ρt η , where ρη t := arg min ρ≪ρt �� Rd δℓN (ρt) d (ρ − ρt) + 1 η KL (ρ ∥ ρt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='14) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Fisher-Rao gradient descent as fixed-location EM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' When ρ0 is discrete, Fisher-Rao gradient descent (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='12) with step size γ = 1 coincides with the fixed-location EM algorithm for Gaussian mixture model in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Jiang and Zhang (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Formal justifications of the above three connections can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The theorem below shows that when ρ0 is diffused, the Fisher-Rao gradient descent enjoys appealing convergence property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The proof is deferred to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Theorem 4 (Convergence to NPMLE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Suppose that the initialization ρ0 ∈ P(Rd) satisfies supp(ρ0) = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Consider the Fisher-Rao gradient descent {ρn}n≥0 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' There exists η0 determined by the samples {Xi}1≤i≤N, such that if 0 < η ≤ η0 and ρn w→ �ρ when n → ∞, then �ρ is the NPMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We can also show a similar result for the continuous-time Fisher-Rao gradient flow (ρt)t≥0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' a distri- butional solution to the PDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11)): if ρt w→ �ρ as t → ∞, then �ρ is an NPMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Theorem 4 provides convergence guarantees for Fisher-Rao gradient flow/descent when initialized from a well-spread distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In practice, however, we can only initialize from a discrete distribution with m particles ρ0 = m � l=1 ω(l) 0 δµ(l), where µ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , µ(m) ∈ Rd and ω0 = (ω(1) 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , ω(m) 0 ) ∈ ∆m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The following theorem characterizes the Fisher-Rao gradient flow when initialized from a discrete distribution ρ0 with m mass points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Theorem 5 (Particle Fisher-Rao gradient flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The ODE system ˙ω(j) t = −ω(j) t � 1 − 1 N N � i=1 φ � Xi − µ(j)� �m l=1 ω(l) t φ � Xi − µ(l)� � , 1 ≤ j ≤ m (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='15) with initialization ω0 = [ω(j) 0 ]1≤j≤m ∈ ∆m−1 has unique solution on any time interval [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Moreover, the flow (ρt)t≥0 defined as ρt := m � l=1 ω(l) t δµ(l) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='16) is the Fisher-Rao gradient flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='16) is a distributional solution to the PDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The proof is deferred to Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For the purpose of implementation, we can further discretize the gradient flow (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='16) with respect to time and obtain the Fisher-Rao gradient descent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' see Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' A quick inspection of the pseudo-cpde presented in Algorithm 2 reveals a fatal flaw: the locations of the particles are fixed at their initialization and only the weights of the particles are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This introduces a systematic approximation error that may scale exponentially with the dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This diagnosic is supported by our numerical experiments in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Wasserstein gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Wasserstein gradient flows have received significant attention recently both from a theoretical perspective (Ambrosio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Santambrogio, 2017) and as a useful tool in a variety of applications ranging from sampling (Chewi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2020), to variation inference (Lambert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2022), as well the theory of neural networks (Chizat and Bach, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Sander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This practical success is largely enabled by the fact that Wasserstein gradient flows can be implemented using interacting particle systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In fact, akin to prior work on shallow neural networks (Chizat and Bach, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2018), the Wasserstein gradient flow of the negative log-likelihood precisely describes the dynamics of gradient descent 8 Algorithm 2 Fisher-Rao gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Input: data {Xi}1≤i≤n, number of particles m, step sizes η > 0, maximum number of iterations t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Initialization: draw µ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , µ(m) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' ∼ Uniform({Xi}1≤i≤n) and ω(1) 0 = · · · = ω(m) 0 = 1/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Updates: for t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , t0 do ω(j) t+1 = ω(j) t + η � 1 N N � i=1 φ � Xi − µ(j)� �m l=1 ω(j) t φ � Xi − µ(l)� − 1 � ω(j) t , for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Here φ(x) = (2π)−d/2 exp(−∥x∥2 2/2) is the probability density function of N(0, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Output ρ = �m j=1 ω(j) t0 δµ(j) as the (approximate) NMPLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' on the location parameters of the fitted mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Interested readers are referred to Theorem 8 in Appendix F for a rigorous statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 we derive the gradient flow of ℓN(ρ) under the Wasserstein geometry, which evolves according to the PDE ∂tρt = div (ρt∇δℓN (ρt)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='18) The corresponding discrete time algorithm is ρt+1 = [id − η∇δℓN (ρt)]# ρt, t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' for some step size η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' When initialized from a discrete distribution ρ0 = 1 m m � l=1 δµ(l) 0 , where µ(1) 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , µ(m) 0 ∈ Rd, the following theorem gives a concise characterization of the Wasserstein gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The proof can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Theorem 6 (Particle Wasserstein gradient flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The ODE system ˙µ(j) t = 1 N N � i=1 φ � Xi − µ(j) t � �m l=1 ω(j) t φ � Xi − µ(l) t � � Xi − µ(j) t � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='19) with initialization µ(1) 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' µ(m) 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' ∼ Uniform({Xi}1≤i≤N) has unique solution on any time interval [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Moreover, the flow (ρt)t≥0 defined as ρt := 1 m m � l=1 δµ(l) t (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='20) is the Wasserstein gradient flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='20) is a distributional solution to the PDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By time discretization, we also have the Wasserstein gradient descent algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' see Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Unlike the Fisher-Rao gradient descent algorithm, the approximation error of the Wasserstein flow may be mitigated since it allows particles to evolve in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Nevertheless, this movement can take a long time to move mass from one part of the space at initialization to a distant part of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Instead, Wasserstein- Fisher-Rao gradient descent allows for particles to not only evolve in space but also have changing weights, thus greatly improving the performance compared to vanilla Wasserstein gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This superiority is, again, demonstrated in numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In particular, owing to Theorem 8, these experiments indicate that WFR gradient descent dominates the classical gradient descent on the location parameters of the mixing distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The reader will notice here the absence of convergence results analogous to Theorem 7 for WFR and Theorem 4 for Fisher-Rao gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Unfortunately, we were not able to derive such convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The dynamics of the Wasserstein gradient flow are quite intricate and Appendix F is devoted to establishing partial results that shed light on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 9 Algorithm 3 Wasserstein gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Input: data {Xi}1≤i≤n, number of particles m, step sizes η > 0, maximum number of iterations t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Initialization: draw µ(1) 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , µ(m) 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' ∼ Unif({Xi}1≤i≤n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Updates: for t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , t0 do µ(j) t+1 = µ(j) t + η 1 N N � i=1 φ � Xi − µ(j) t � m−1 �m l=1 φ � Xi − µ(l) t � � Xi − µ(j) t � , for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Here φ(x) = (2π)−d/2 exp(−∥x∥2 2/2) is the probability density function of N(0, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Output ρ = m−1 �m j=1 δµ(j) t0 as the (approximate) NMPLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (a) bad local minima (b) global minima 5 0 5 10 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 5 0 5 10 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 Figure 1: The density plots of ρ ∗ N(0, 1) (learned Gaussian mixture) and ρ⋆ ∗ N(0, 1) (true Gaussian mixture), where ρ = 1 3δµ1 + 1 3δµ2 + 1 3δµ3, ρ⋆ = ρd and (µ1, µ2, µ3) is the output returned by EM and GD algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Figure 1(a) shows the learned Gaussian mixture when (µ1, µ2, µ3) is a bad local minimum of ℓ, and Figure 1(b) corresponds to the case when (µ1, µ2, µ3) is a global minimum of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 4 Numerical experiments In this section, we conduct a series of numerical experiments to validate and complement our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We consider two mixing distributions in Rd: the first one is a continuous isotropic Gaussian distribution ρc = N (0, Id) , and the other one is a discrete distribution motivated by Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2016) ρd = �1 3δ−1 + 1 3δ1 + 1 3δ10 � ⊗ (δ0)⊗(d−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The second mixing distribution ρd is a product distribution with its first margin being a uniform distribution over {−1, 1, 10} and the rest d−1 margins being a degenerate distribution taking a constant zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' According to Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2016), classical EM and gradient descent algorithm fail to learn the location of each component of this Gaussian mixture even with infinite samples and known weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 Instability of classical algorithms In this section, we compare the two classical algorithms for learning Gaussian mixture when the mixing distribution ρ⋆ = ρd: (i) expectation–maximization (EM) algorithm, and (ii) the gradient descent (GD) 10 algorithm, with Wasserstein-Fisher-Rao gradient descent algorithm (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Algorithm 1) proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For the first two algorithms, we assume that the number of mixture components k = 3 and the weights ω⋆ 1 = ω⋆ 2 = ω⋆ 3 = 1/3 are known a priori, and implement the standard EM and GD algorithms for solving the MLE min µ1,µ2,µ3 ℓ (µ1, µ2, µ3) := − 1 N N � i=1 log �1 3 3 � j=1 1 (2π)d/2 exp � −1 2 ∥Xi − µj∥2 2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The updating rule for EM algorithm is given by µt+1 j = �N i=1 ωt i,jXi �N i=1 ωt i,j where ωt i,j = ω⋆ j φ � Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' µt j, Id � �3 l=1 ω⋆ l φ (Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' µt l, Id) ∀ i ∈ [N] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) for all j = 1, 2, 3 and t ≥ 0, with random initialization from the samples µ0 1, µ0 2, µ0 3 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' ∼ Uniform({Xi}1≤i≤N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The GD algorithm coincides with the Wasserstein gradient descent (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Algorithm 3) with m = 3 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We consider the one-dimensional setting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' d = 1) for simplicity of visualization, and this turns out to be enough for showing the instability of EM and GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We generate N = 1500 samples {Xi}1≤i≤N from ρ⋆ ∗ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we fix these samples and run 100 independent trials of the three algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For EM, we run 200 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For GD and Wasserstein-Fisher-Rao gradient descent, we set all the step sizes to be η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 and run t0 = 1000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Both EM and GD have two possible outputs: (i) the first one is µ1 ≈ µ2 ≈ 10 and µ3 ≈ 0 (up to permutation), which is a bad local minimum of ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (ii) the second one is µ1 ≈ −1, µ2 ≈ 1 and µ3 ≈ 10, which is the global minimum of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Figure 1 displays the learned Gaussian mixtures correspond to these two outputs, and it is clear that both algorighms fail to learn the true Gaussian mixture when they converge to the bad local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In the 100 independent trials, EM converges to the bad local minimum for 23 times, while GD converges to the bad local minimum for 32 times, both exhibiting instability vis-à-vis random initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In contrast, as we will show in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3, Wasserstein-Fisher-Rao gradient descent with number of particles m = 500 converges to NPMLE stably and learns the Gaussian mixture as in Figure 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 Superiority of Wasserstein-Fisher-Rao gradient descent In this section we compare the empirical performance of three gradient descent algorithms studied in this paper: (i) Fisher-Rao gradient descent (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Algorithm 2), (ii) Wasserstein-Fisher-Rao gradient descent (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Al- gorithm 1) and (iii) Wasserstein gradient descent (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We set the sample size N = 1500, the dimension d = 10, maximum number of iterations t0 = 1000, the step size η = 10−1 for Algorithm 2 and Algorithm 3, and η = 10−2 for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Figure 2(a) displays the negative log-likelihood ℓN(ρ) (with one standard deviation error bars) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' the number of particles m over 20 independent trials for the three al- gorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Unlike the previous experiment, we generate fresh samples {Xi}1≤i≤N in each independent trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' As we can see, the loss decreases as we use more particles, and Wasserstein-Fisher-Rao gradient descent achieves the smallest loss uniformly for all m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' It can also be observed that the marginal benefit of increasing the number of particles becomes negligible for Wasserstein-Fisher-Rao gradient descent when m > 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Sim- ilarly, Figure 2(b) depicts the negative log-likelihood ℓN(ρt) (with one standard deviation error bars) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' the iteration count t over 20 independent trials for the three algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We can see that Wasserstein-Fisher-Rao gradient descent again achieves the smallest loss uniformly in all iteration, confirming again the superiority of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 Certifying the optimality condition for NPMLE in one dimension The convergence guarantees in this paper only cover the infinite-particle limit of Wasserstein-Fisher-Rao gradient descent (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In this section, we provide numerical evidence that Algorithm 1 converges to (approximate) NPMLE when we use a large number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Recall from Theorem 1 and the following remark that ρ ∈ P(Rd) is NPMLE if and only if δℓN(ρ)(x) ≥ −1 holds for all x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We focus on the one-dimensional setting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' d = 1) since it is computationally affordable to check the function value of δℓN(ρ)(x) in one dimension (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' over a fine grid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For a discrete distribution 11 (a) training error for discrete ρ⋆ (b) test error for discrete ρ⋆ 100 200 300 400 500 600 700 800 900 1000 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='9 15 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7 100 200 300 400 500 600 700 800 900 1000 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8 (c) training error for continuous ρ⋆ (d) test error for continuous ρ⋆ 100 200 300 400 500 600 700 800 900 1000 16 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5 17 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5 18 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5 100 200 300 400 500 600 700 800 900 1000 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='9 18 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6 Figure 2: Training or testing error (with error bars) of the three algorithms vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' the number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The training and testing errors are evaluated using ℓN(ρ) and ℓ∞(ρ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Figure (a) an (b) display the training and testing errors when the mixing distribution ρ⋆ = ρd is discrete, while Figure (c) and (d) show the training and testing errors when the mixing distribution ρ⋆ = ρc is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The results are reported over 20 independent trials for N = 1500, d = 10, and t0 = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' ρ = �m j=1 ωjδµj, we define the following suboptimality gaps: gap (ρ) := sup x∈R max {−1 − δℓN (ρ) (x) , 0} , � gap(ρ) := max x∈grid(ρ) max {−1 − δℓN (ρ) (x) , 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' It is clear that when the first optimality gap gap(ρ) = 0, Theorem 1 asserts that ρ is the NPMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' However gap(ρ) is in general difficult to compute, and a practical scheme is to approximatly evaluate the supremum over R by the maximum over a fine grid grid(ρ) ⊆ R, which gives the second optimality gap � gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In our experiments, we take grid(ρ) to be a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='01-net over [min1≤j≤m µj − 1, max1≤j≤m µj + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We set the sample size N = 1500, the dimension d = 1 and run Wasserstein-Fisher-Rao gradient descent (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Algorithm 1) with number of particles m = 500, step sizes η1 = η2 = 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Figure 4 illustrates the suboptimality gap � gap in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3) (with one standard deviation error bars) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' the iteration count over 20 independent trials as well as the density plot of the output of the algorithm convolved with N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Roughly speaking, both suboptimality gaps decreases inversely proportional to the iteration counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lastly, Figure 5 depicts the first variation δℓN(ρ), which clearly shows that the optimality condition is approximately satisfied with high precision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' see the caption of Figure 5 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 12 (a) training error for discrete ρ⋆ (b) test error for discrete ρ⋆ 0 100 200 300 400 500 600 700 800 900 1000 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='9 15 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5 0 100 200 300 400 500 600 700 800 900 1000 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='9 (c) training error for continuous ρ⋆ (d) test error for continuous ρ⋆ 0 100 200 300 400 500 600 700 800 900 1000 16 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7 0 100 200 300 400 500 600 700 800 900 1000 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='94 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='96 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='98 18 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='02 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='04 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='06 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='08 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='12 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='14 Figure 3: Training or testing error (with error bars) of the three algorithms vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' the iteration count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The training and the testing errors are evaluated using ℓN(ρt) and ℓ∞(ρt) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Figure (a) an (b) display training and testing errors when the mixing distribution ρ⋆ = ρd is discrete, while Figure (c) and (d) show training and testing errors when the mixing distribution ρ⋆ = ρc is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The results are reported over 20 independent trials for N = 1500, d = 10, and m = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 5 Discussion The current paper proposes to solve the NPMLE for Gaussian mixtures using an interacting particle system driven by Wasserstein-Fisher-Rao gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In the infinite-particle limit, we show that the proposed algorithm converges to NPMLE under certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In practice, we conduct extensive numerical experiments to illustrate the capability of the proposed algorithm in exactly computing NPMLE using a finite (or even small) number of particles, and also to demonstrate the superiority of the proposed algorithm compared to other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Moving forward, there are numerous possible extensions that merit future investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For example, our convergence theory only holds when the Wasserstein-Fisher-Rao gradient descent is initialized from a distribution supported on the whole space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' it would be of interest to extend the current analysis to the more practical scenario where the algorithm is initialized from a discrete distribution with a finite number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Another interesting direction is to develop algorithms for learning Gaussian mixtures beyond the isotropic case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' without assuming that the covariance matrices are identity) using, for example, Wasserstein-Fisher-Rao gradient flow over the Bures-Wasserstein space (Lambert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 13 (a) optimality gap for discrete ρ⋆ (b) optimality gap for continuous ρ⋆ 102 103 104 10-5 10-4 10-3 10-2 102 103 104 10-4 10-3 10-2 (c) density plot for discrete ρ⋆ (d) density plot for continuous ρ⋆ 5 0 5 10 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='02 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='18 5 4 3 2 1 0 1 2 3 4 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 Figure 4: Figures (a)-(b) display sub-optimality gaps (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3)) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' iteration count for Wasserstein-Fisher- Rao gradient descent, with discrete mixing distribution ρ⋆ = ρd in Figure (a) and ρ⋆ = ρc in Figure (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The error bars are computed over 20 independent trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Figures (c)-(d) are density plots of ρ and ρ⋆ convolved with standard Gaussian, with discrete mixing distribution ρ⋆ = ρd in (c) and continuous mixing distribution ρ⋆ = ρc in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The results are reported for N = 1500, d = 1 and m = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Acknowledgements The authors thank Donghao Wang for a helpful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Yan is supported in part by Charlotte Elizabeth Procter Honorific Fellowship from Princeton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Part of this work was done during Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Yan’s visit to MIT in Fall 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Wang is supported by an NSF grant DMS-2210907 and a start-up grant at Columbia University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Rigollet is supported by NSF grants IIS-1838071, DMS-2022448, and CCF-2106377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 14 (a) first variation δℓN(ρ) for discrete ρ⋆ = 1 3δ−1 + 1 3δ1 + 1 3δ10 2 0 2 4 6 8 10 12 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 (b) first variation δℓN(ρ) for continuous ρ⋆ = N(0, 1) 5 4 3 2 1 0 1 2 3 4 5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='88 Figure 5: The first variation δℓN(ρ) (in red line) for discrete ρ⋆ = ρd in the upper panel and continuous ρ⋆ = ρc in the lower panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The blue dots are {(µj, δℓN(ρ)(µj)) : 1 ≤ j ≤ m}, where the size of these dots are proportional to the weights {ωj}1≤j≤m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The green line is a horizontal line y = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The two subfigures in the upper panel zoom in the two regions −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5 ≤ x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8 ≤ x ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2, and the three subfigures in the lower panel zoom in the three regions −4 ≤ x ≤ −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4, −2 ≤ x ≤ 3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8 ≤ x ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' A Preliminaries In the main text, we focused on the Gaussian mixture model where φ(x) = (2π)−d/2 exp(−∥x∥2 2/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In fact, the algorithms and theorems in the current paper are also valid for a more general class of probability density φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In the appendices, we only assume that φ satisfies the following regularity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Assumption 1 (Regularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Assume that the density φ(x) > 0 for any x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Furthermore, φ ∈ Cmax{d,2}(Rd), lim∥x∥2→∞ φ(x) = 0, supx∈Rd ∥∇φ(x)∥2 < ∞ and supx∈Rd ∥∇2φ(x)∥2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' It is clear that the Gaussian kernel φ(x) = (2π)−d/2 exp(−∥x∥2 2/2) satisfies Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We also define the following sets and quantites that will be useful throughout the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let Ω = conv({Xi}1≤i≤N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For r ≥ 0, define Ωr = {x ∈ Rd : dist(x, Ω) ≤ r}, ¯φ(r) = sup∥x∥2≥r φ(x) and φ(r) = inf∥x∥2≤r φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The following lemma shows that NPMLE is compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The proof can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 15 Lemma 1 (Compact support of NPMLE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let Assumption 1 hold and �ρ be any optimal solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Define R1 = inf{r ≥ 0 : ¯φ(r) ≤ φ[diam(Ω)]/2} and R = inf � r ≥ 0 : ¯φ(r) ≤ ¯φ(R1)φ(R1 + diam(Ω)) 8¯φ(0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then, we have supp(�ρ) ⊆ ΩR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' B Proof of structual results for NPMLE In this section, we prove the two structural results for NPMLE, namely Theorem 1 and Lemma 1 under Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 Proof of Theorem 1 Part 1: existence of NPMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that the loss function ℓN is lower bounded ℓN (ρ) = − 1 N N � i=1 log [(ρ ∗ φ) (Xi)] ≥ − log ∥φ∥∞ , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) where the last inequality follows from ρ∗φ(x) = � Rd φ (y − x) ρ (dy) ≤ ∥φ∥∞ for any x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore there exists a sequence of probability distribution {ρn} such that ℓN (ρn) ≤ inf ρ∈P(Rd) ℓN (ρ) + 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2) Now we argue that {ρn} is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' To that end, we show that there exists r > 0 such that for any ε > 0, it holds ρn(Ωr) ≥ 1 − ε for n large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any n and r > 0, define ρn,r := ρn (Ωr) · ρn|Ωr + ρn (Ωc r) · Unif (Ω) , where ρn|Ωr(·) = ρn(·|Ωr) is the conditional distribution of ρn given Ωr, and Unif(Ω) is the uniform distri- bution on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We have ℓN (ρn) − ℓN (ρn,r) = − 1 N N � i=1 log [(ρn ∗ φ) (Xi)] + 1 N N � i=1 log [(ρn,r ∗ φ) (Xi)] = 1 N N � i=1 log �(ρn,r ∗ φ) (Xi) (ρn ∗ φ) (Xi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that for each i ∈ [N] log �(ρn,r ∗ φ) (Xi) (ρn ∗ φ) (Xi) � = log �� Ωr φ (Xi − y) ρn (dy) + ρn (Ωc r) � Ω φ (Xi − y) dy � Ωr φ (Xi − y) ρn (dy) + � Ωc r φ (Xi − y) ρn (dy) � = log � 1 + ρn (Ωc r) � Ω φ (Xi − y) dy − � Ωcr φ (Xi − y) ρn (dy) (ρn ∗ φ) (Xi) � ≥ log � 1 + ρn (Ωc r) � φ (diam (Ω)) − φ (r) � ∥φ∥∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We can choose r > 0 to be sufficiently large so that φ(r) ≤ φ(diam(Ω))/2, and therefore for each i ∈ [N] ℓN (ρn) − ℓN (ρn,r) ≥ log � 1 + ρn (Ωc r) φ (diam (Ω)) 2 ∥φ∥∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In view of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2), we know that ℓN (ρn) − ℓN (ρn,r) ≤ 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 16 Taking the above two inequalities collectively give ρn (Ωc r) ≤ 2 ∥φ∥∞ � exp � 1 n � − 1 � φ (diam (Ω)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore we have ρn (Ωc r) ≤ ε as long as n ≥ nε := � 1/ log � 1 + εφ (diam (Ω)) 2 ∥φ∥∞ �� , which implies that {ρn} is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We conclude using Prokhorov’s theorem: there exists a subsequence {ρnk} and �ρ ∈ P(Rd) such that ρnk converges weakly to �ρ which must be a minimizer of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Part 2: optimality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' First of all, it is straightforward to check that for any ρ ∈ M(Rd) � Rd δℓN (ρ) (x) ρ (dx) = − 1 N N � i=1 � φ (x − Xi) ρ (dx) (ρ ∗ φ) (Xi) = − 1 N N � i=1 (ρ ∗ φ) (Xi) (ρ ∗ φ) (Xi) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' If �ρ ∈ M(Rd) is the optimal solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1), then for any x ∈ Rd and any ε ∈ [0, 1] we have ℓN (�ρ) ≤ ℓN ((1 − ε) �ρ + εδx) = ℓN (ρ + ε (δx − �ρ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' As a result we have δℓN (�ρ) (x) + 1 = � Rd δℓN (�ρ) d (δx − �ρ) = lim ε→0 1 ε [ℓN (�ρ + ε (δx − �ρ)) − ℓN (�ρ)] ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Since x is arbitrary, this implies that δℓN(�ρ)(x) ≥ −1 for any x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Combine this with � Rd δℓN(�ρ)d�ρ = −1 readily gives δℓN(�ρ)(x) = −1 for �ρ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Conversely, if �ρ ∈ M(Rd) satisfies δℓN(�ρ)(x) ≥ −1 for all x ∈ Rd, then for any ρ ∈ M(Rd), it holds 0 ≤ � Rd δℓN (�ρ) dρ + 1 = � Rd δℓN (�ρ) d (ρ − �ρ) = lim ε→0 1 ε [ℓN (�ρ + ε (ρ − �ρ)) − ℓN (�ρ)] ≤ ℓN (ρ) − ℓN (�ρ) where the last inequality follows from convexity of the functional ρ �→ ℓN(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The above display yields that that �ρ is a global minimizer of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 Proof of Lemma 1 By Theorem 1, δℓN(�ρ) = −1 over supp(�ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We will show that |δℓN(�ρ)(y)| < 1/2 when y is too far away from Ω, and then conclude that supp(�ρ) must stay close to Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' To that end, we present some useful estimates in the following lemma that will also be useful later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let Assumption 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any ρ ∈ P(Rd), x ∈ Ω and r ≥ 0, we have ρ(Ωr)φ(r + diam(Ω)) ≤ (ρ ∗ φ)(x) ≤ ρ(Ωc r)¯φ(r) + ρ(Ωr)¯φ(0) ≤ ¯φ(r) + ρ(Ωr)¯φ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' As a result, − log � ¯φ(r) + ρ(Ωr)¯φ(0) � ≤ ℓN(ρ) ≤ − log � ρ(Ωr)φ(r + diam(Ω)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Take any R ≥ 0 such that ¯φ(R) ≤ e−ℓN(ρ)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any µ ∈ P(Rd) with ℓN(µ) ≤ ℓN(ρ), we have µ(ΩR) ≥ e−ℓN(ρ)/[2¯φ(0)], inf x∈Ω(µ ∗ φ)(x) ≥ e−ℓN(ρ)φ(R + diam(Ω))/[2¯φ(0)], sup y∈Ωcr |δℓN(µ)(y)| ≤ 2eℓN(ρ) ¯φ(0) φ(R + diam(Ω)) · ¯φ(r), ∀r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 17 The proof of Lemma 2 is deferred to the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We come back to proving Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Choose any ρ0 ∈ P(Rd) supported on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By Lemma 2, we have ℓN(ρ0) ≤ − log � φ[diam(Ω)] � , ∀r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Take R1 = inf{r ≥ 0 : ¯φ(r) ≤ φ[diam(Ω)]/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By the continuity of ¯φ, ¯φ(R1) ≤ φ[diam(Ω)]/2 ≤ e−ℓN(ρ0)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lemma 2 implies that sup y∈Ωcr |δℓN(�ρ)(y)| ≤ 2eℓN(ρ0) ¯φ(0) φ(R1 + diam(Ω)) · ¯φ(r) ≤ 4¯φ(0) ¯φ(R1)φ(R1 + diam(Ω)) · ¯φ(r), ∀r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let R = inf � r ≥ 0 : ¯φ(r) ≤ ¯φ(R1)φ(R1 + diam(Ω)) 8¯φ(0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The continuity of ¯φ leads to ¯φ(R) ≤ ¯φ(R1)φ(R1 +diam(Ω))/[8¯φ(0)] and thus supy∈Ωc R |δℓN(�ρ)(y)| ≤ 1/2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' As a result, supp(�ρ) ∩ Ωc R = ∅ and supp(�ρ) ⊆ ΩR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that φ(x − y) ≤ ¯φ(0) for any x, y ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' If x ∈ Ω and y ∈ Ωc r, then ∥x − y∥2 ≥ r and thus φ(x − y) ≤ ¯φ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Hence, (ρ ∗ φ)(x) ≤ � Ωr φ(x − y)ρ(dy) + � Ωcr φ(x − y)ρ(dy) ≤ ¯φ(0)ρ(Ωr) + ¯φ(r)ρ(Ωc r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' If x ∈ Ω and y ∈ Ωr, then ∥x − y∥2 ≥ r + diam(Ω) and thus φ(x − y) ≥ φ(r + diam(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore, (ρ ∗ φ)(x) ≥ � Ωr φ(x − y)ρ(dy) ≥ φ(r + diam(Ω))ρ(Ωr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The desired bounds on ℓN(·) become obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' If µ ∈ P(Rd) and ℓN(µ) ≤ ℓN(ρ), then our estimates of ℓ implies that − log � ¯φ(r) + µ(Ωr)¯φ(0) � ≤ ℓN(µ) ≤ ℓN(ρ), ∀r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Hence, ¯φ(r) + µ(Ωr)¯φ(0) ≥ e−ℓN(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By Assumption 1, lim r→∞ ¯φ(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Take any R ≥ 0 such that ¯φ(R) ≤ e−ℓN(ρ)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then, µ(ΩR) ≥ [e−ℓN(ρ) − ¯φ(r)]/¯φ(0) ≥ e−ℓN(ρ)/[2¯φ(0)], inf x∈Ω(µ ∗ φ)(x) ≥ µ(ΩR)φ(R + diam(Ω)) ≥ e−ℓN(ρ)φ(R + diam(Ω))/[2¯φ(0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any r ≥ 0, we have ∥X − y∥2 ≥ r whenever X ∈ supp(ν) and y ∈ Ωc r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then, |δℓN(µ)(y)| = 1 N N � i=1 φ(Xi − y) (µ ∗ φ)(Xi) ≤ ¯φ(r) infx∈Ω(µ ∗ φ)(x), ∀r ≥ 0, y ∈ Ωc r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The proof is finished by combining this and the lower bound on infx∈Ω(µ ∗ φ)(x) we have established above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 18 C Derivation of gradient flows over P2(Rd) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 First variation Recall that the population and finite-sample loss functions are ℓ∞ (ρ) = EX∼(ρ⋆∗φ) {log [(ρ ∗ φ) (X)]} , ℓN (ρ) = − 1 N N � i=1 log [(ρ ∗ φ) (Xi)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The first variation of ℓN is defined as nay measurable function δℓ(ρ) : Rd → R satisfying lim ε→0 ℓ (ρ + εX) − ℓ (ρ) ε = � δℓ (ρ) dX for all signed measures X satisfying � dX = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In particular, it is easy to see that the first variation is defined up to an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By direct computation, we have lim ε→0 ℓN (ρ + εX) − ℓN (ρ) ε = − 1 N N � i=1 lim ε→0 log [[(ρ + εX) ∗ φ] (Xi)] − log [(ρ ∗ φ) (Xi)] ε = − 1 N N � i=1 lim ε→0 1 ε log � 1 + ε(X ∗ φ) (Xi) (ρ ∗ φ) (Xi) � = − 1 N N � i=1 (X ∗ φ) (Xi) (ρ ∗ φ) (Xi) = − 1 N N � i=1 � φ (x − Xi) (ρ ∗ φ) (Xi)X (dx) , As a result, we have δℓN (ρ) : x → − 1 N N � i=1 φ (x − Xi) (ρ ∗ φ) (Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Similarly, we can also compute lim ε→0 ℓ∞ (ρ + εX) − ℓ∞ (ρ) ε = lim ε→0 1 ε � − � log �(ρ + εX) ∗ φ (x) (ρ ∗ φ) (x) � (ρ⋆ ∗ φ) (x) dx � = − � (X ∗ φ) (x) (ρ ∗ φ) (x) (ρ⋆ ∗ φ) (x) dx = − � (ρ⋆ ∗ φ) (x) (ρ ∗ φ) (x) �� φ (x − y) X (dy) � dx = − � �� (ρ⋆ ∗ φ) (x) (ρ ∗ φ) (x) φ (x − y) dx � X (dy) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' which gives δℓ∞ (ρ) : x → − � (ρ⋆ ∗ φ) (y) (ρ ∗ φ) (y) φ (x − y) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 Fisher-Rao gradient flow C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 A formal derivation of gradient flow using Riemannian geometry We first introduce a Riemannian structure over P2(Rd) underlying the Fisher-Rao metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Define the tangent space at ρ ∈ P2(Rd) as TanFR ρ P2(Rd) := � ζ : ζ = ρ � α − � αdρ � for some α satisfying � α2dρ < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We equip the tangent space T FR ρ P2(Rd) with the following Riemannian metric tensor gFR ρ (·, ·) : TanFR ρ P2(Rd)× TanFR ρ P2(Rd) → R as gFR ρ (ζ1, ζ2) := � ζ1 · ζ2 ρ2 dρ 19 = � Rd � α1 (x) − � Rd α1dρ � � α2 (x) − � Rd α2dρ � ρ (dx) = � Rd α1 (x) α2 (x) ρ (dx) − �� Rd α1dρ � �� Rd α2dρ � , for any ζ1 = ρ(α1 − � α1dρ) and ζ2 = ρ(α2 − � α2dρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The metric induced by this Riemannian structure, namely the Fisher-Rao metric dFR(·, ·), satisfies the following property: d2 FR (ρ0, ρ1) = inf � � 1 0 � �� αt − � αtdρt �2� dρtdt : (ρt, αt)t∈[0,1] solves ∂tρt = ρtαt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we follow Gallouët and Monsaingeon (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2019b) to derive the Fisher-Rao gradient flow with respect to the functional ℓN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let (ρt)t≥0 be a C1 curve satisfying ρ0 = ρ with initial velocity ∂tρt|t=0 = ζ = ρ � α − � αdρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The Fisher-Rao gradient of ℓN at ρ is defined as the function gradFRℓN (ρ) ∈ L2(ρ) such that d dtℓN (ρt) ��� t=0 = gFR ρ (gradFRℓN (ρ) , ζ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' To compute it, observe that the right-hand side of the above identity is given by d dtℓN (ρt) ��� t=0 = � δℓN(ρ) · ∂tρt ��� t=0 = � δℓN (ρ) , ζ = � δℓN (ρ) � α − � αdρ � dρ = � � δℓN (ρ) − � δℓN (ρ) dρ � � α − � αdρ � dρ = gFR ρ � ρ � δℓN (ρ) − � δℓN (ρ) dρ � , ζ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore gFR ρ (gradFRℓN (ρ) · ζ) = gFR ρ � ρ � δℓN (ρ) − � δℓN (ρ) dρ � , ζ � holds for any ζ ∈ TanFR ρ P2(Rd), and as a result gradFRℓN (ρ) = ρ � δℓN (ρ) − � δℓN (ρ) dρ � = ρ [δℓN (ρ) + 1] , where we have used the fact that � δℓN(ρ)dρ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Hence, the gradient flow of ℓN with respect to the Fisher-Rao metric dFR is given by ∂tρt = −gradFRℓN (ρt) = −ρt [δℓN (ρt) + 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 Other perspectives of Fisher-Rao gradient flow In this section, we formally illustrate the connection between Fisher-Rao gradient flow (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11) with proximal gradient descent and mirror descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For simplicity, we focus on the case when ρt is continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' the case when ρt is discrete is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We also show the connection between the particle Fisher-Rao gradient descent (2) and the EM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 20 Fisher-Rao gradient flow as proximal gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Consider the proximal gradient update in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Recall that for µ, ν ∈ Pac(Rd), the Fisher-Rao distance can be expressed as d2 FR (µ, ν) = 4 � ��� µ (x) − � ν (x) ��2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that for the purpose of defining a gradient flow, the metric only matters up to its second-order local expansion d2 FR (µ, ν) = � ∆2(x) ν (x) dx + higher-order terms, where ∆ = µ − ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore we obtain an asymptotically (as η → 0) equivalent problem ρη t := arg min ρ∈Pac(Rd) �� Rd δℓN (ρt) d (ρ − ρt) + 1 2η � [ρ (x) − ρt (x)]2 ρt (x) dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The first-order optimality condition is δℓN (ρt) (x) + 1 η · ρ (x) − ρt (x) ρt (x) = c for some constant c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This gives ρη t (x) = ρt (x) [1 + xη − ηδℓN (ρt) (x)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Since ρη t is a probability density, we have 1 = � Rd ρη t (x) dx = � Rd ρt (x) [1 + cη − ηδℓN (ρt) (x)] dx = 1 + cη − η, where we use the fact that � Rd δℓN(ρ)dρ = −1 for any ρ ∈ P2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This gives c = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore ρη t (x) = ρt (x) [1 − η − ηδℓN (ρt) (x)] , and as a result, ∂tρt = lim η→0+ ρη t − ρt η = − [1 + δℓN (ρt)] , which recovers the Fisher-Rao gradient flow (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Fisher-Rao gradient flow as mirror flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Recall that the mirror descent update is defined as ρη t := arg min ρ∈Pac(Rd) � Rd δℓN (ρt) d (ρ − ρt) + 1 η KL (ρ ∥ ρt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The first variation of f(·) := KL(· ∥ ρt) is given by δf (ρ) (x) = log � ρ (x) ρt (x) � , therefore the first-order optimality condition reads δℓN (ρt) (x) + 1 η log � ρ (x) ρt (x) � = c for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This gives ρ (x) ρt (x) = exp {η [c − δℓN (ρt) (x)]} ∝ exp [−ηδℓN (ρt) (x)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 21 Since � Rd ρη t (x)dx = 1, we know that the closed-form solution is given by ρη t (x) = ρt (x) exp [−ηδℓN (ρt) (x)] � ρt (y) exp [−ηδℓN (ρt) (y)] dy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) Then as η → 0, we can compute ρη t (x) = ρt (x) � 1 − ηδℓN (ρt) (x) + O � η2�� � ρt (y) [1 − ηδℓN (ρt) (y) + O (η2)] dy = ρt (x) � 1 − ηδℓN (ρt) (x) + O � η2�� 1 + η + O (η2) = ρt (x) � 1 − η [1 + δℓN (ρt) (x)] + O � η2�� , where we use the fact that � δℓN(ρ)dρ = −1 for any ρ ∈ P2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore the continuous-time limit of mirror descent is ∂tρt = lim η→0+ ρη t (x) − ρt (x) η = − [1 + δℓN (ρt) (x)] , which recovers the Fisher-Rao gradient flow (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Fisher-Rao gradient descent as EM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Now we consider fitting a m-component Gaussian mixture model with unknown weights {ω(j)}1≤j≤m, known location parameters {µj}1≤j≤m and isotropic covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Given the data {Xi}1≤i≤N, the MLE is given by arg max ω∈∆m−1ℓ (ω) = 1 N N � i=1 log � � m � j=1 ω(j)φ (Xi − µj) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The Expectation-Maximization algorithm for solving the above MLE is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We first introduce the latent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' random variables {Ji}1≤i≤N distributed P(Ji = j) = ω(j) for 1 ≤ j ≤ m, then the distribution of the observed samples is Xi|Ji = j ∼ N(µj, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The joint distribution of (Xi, Ji) is given by pω (x, j) = φ (Xi − µj) ω(j), and conditional on Xi = x, the conditional distribution of Ji is given by pω (j|x) = pω (x, j) �m l=1 pω (x, l) = φ (Xi − µj) ω(j) �m l=1 φ (Xi − µl) ω(l) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Given the current estimate ωt, the E-step amounts to computing Q (ω|ωt) = 1 N N � i=1 m � j=1 pωt (j|Xi) log pω (Xi, j) = 1 N N � i=1 m � j=1 φ (Xi − µj) ω(j) t �m l=1 φ (Xi − µl) ω(l) t log � φ (Xi − µj) ω(j)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The M-step is to update ωt+1 := arg max ω∈∆m−1Q (ω|ωt) , which is given by ω(j) t+1 = 1 N N � i=1 φ (Xi − µj) ω(j) t �m l=1 φ (Xi − µl) ω(l) t ∀ 1 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This is equivalent to Algorithm 2 with step size η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 22 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 Wasserstein gradient flow We introduce the Riemannian structure over P2(Rd) underlying the quadratic Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We define the tangent space at ρ ∈ P2(Rd) to be TanW ρ P2(Rd) = � ζ : ζ = −div (ρ∇u) for some u satisfying � ∥∇u∥2 2dρ < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We equip this tangent space with the L2(ρ) metric, namely we define the Riemannian metric tensor gW ρ (·, ·) : TanW ρ P2(Rd) × TanW ρ P2(Rd) → R as gW ρ (ζ1, ζ2) := � Rd � Rd ⟨∇u1, ∇u2⟩ ρ (dx) for any ζ1 = −div(ρ∇u1) and ζ2 = −div(ρ∇u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The metric induced by this Riemannian structure recovers the quadratic Wasserstein distance, namely d2 W (ρ0, ρ1) = inf � � 1 0 � ∥vt∥2 2 dρtdt : (ρt, vt)t∈[0,1] solves ∂tρt + div(ρtvt) = 0 � = inf π∈Π(ρ0,ρ1) � ∥x − y∥2 2 π (dx, dy) , where Π(ρ0, ρ1) is the set of couplings of ρ0 and ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This is known as the Benamou-Brenier formula for the Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we derive the Wasserstein gradient flow with respect to the functional ℓN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Interested readers are referred to Ambrosio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2008) for detailed introduction to Wasserstein gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let (ρt)t≥0 be a C1 curve satisfying ρ0 = ρ with initial velocity ∂tρt|t=0 = ζ = −div (ρ∇u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then it should hold that d dtℓN (ρt) ��� t=0 = gW ρ (gradWℓN (ρ) , ζ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The left hand side of the above equation equals to d dtℓN (ρt) ��� t=0 = � δℓN (ρ) ζdx = − � δℓN (ρ) div (ρ∇u) dx = − � ⟨∇δℓN (ρ) , ∇u⟩ dρ = gW ρ (−div (∇δℓN (ρ) ρ) , ζ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore gW ρ (gradWℓN (ρ) , ζ) = gW ρ (−div (∇δℓN (ρ) ρ) , ζ) holds for any ζ ∈ TanW ρ P2(Rd), and as a result gradWℓN (ρ) = −div (∇δℓN (ρ) ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This shows that the gradient flow of ℓN with respect to the quadratic Wasserstein distance dW is given by ∂tρt = −gradWℓN (ρt) = div (∇δℓN (ρt) ρt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4 Wasserstein-Fisher-Rao gradient flow We introduce the Riemannian structure over P2(Rd) underlying the Wasserstein-Fisher-Rao metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Define the tangent space at ρ ∈ P2(Rd) to be TanWFR ρ P2(Rd) = � ζ : ζ = −div (ρ∇u) + ρ � α − � αdρ � for some u, α : Rd → R 23 satisfying � (α2 + ∥∇u∥2 2)dρ < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We equip this tangent space with the Riemannian metric tensor gWFR ρ (·, ·) : TanWFR ρ P2(Rd)×TanWFR ρ P2(Rd) → R defined as gWFR ρ (ζ1, ζ2) := � Rd ⟨∇u1, ∇u2⟩ ρ (dx) + � Rd � α1 (x) − � Rd α1dρ � � α2 (x) − � Rd α2dρ � ρ (dx) = � Rd ⟨∇u1, ∇u2⟩ ρ (dx) + � Rd α1 (x) α2 (x) ρ (dx) − �� Rd α1dρ � �� Rd α2dρ � for any ζ1 = −div(ρ∇u1) + ρ(α1 − � α1dρ) and ζ2 = −div(ρ∇u2) + ρ(α2 − � α2dρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The metric induced by the above Riemannian structure, namely the Wasserstein-Fisher-Rao metric dWFR(·, ·), is defined as d2 WFR (ρ0, ρ1) = inf � � 1 0 � � ∥vt∥2 + � αt − � αtdρt �2� dρtdt : (ρt, vt, αt)0≤t≤1 solves ∂tρt = −div(ρtvt) + ρtαt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we follow Gallouët and Monsaingeon (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2019b) to derive the Wasserstein-Fisher-Rao gradient flow with respect to the functional ℓN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let (ρt)t≥0 be a C1 curve satisfying ρ0 = ρ with initial velocity ∂tρt|t=0 = ζ = −div (ρ∇u) + ρ � α − � αdρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then it should hold that d dtℓN (ρt) ��� t=0 = gWFR ρ (gradWFRℓN (ρ) , ζ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The left hand side of the above equation equals to d dtℓN (ρt) ��� t=0 = � δℓN (ρ) ζdx = � δℓN (ρ) � −div (ρ∇u) + ρ � α − � αdρ �� dx = − � ⟨∇δℓN (ρ) , ∇u⟩ dρ + � � δℓN (ρ) − � δℓN (ρ) dρ � � α − � αdρ � dρ = gWFR ρ � −div (∇δℓN (ρ) ρ) + ρ � δℓN (ρ) − � δℓN (ρ) dρ � , ζ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore gWFR ρ (gradWℓN (ρ) , ζ) = gWFR ρ � −div (∇δℓN (ρ) ρ) + ρ � δℓN (ρ) − � δℓN (ρ) dρ � , ζ � holds for any ζ ∈ TanWFR ρ P2(Rd), and as a result gradWFRℓN (ρ) = −div (∇δℓN (ρ) ρ) + ρ � δℓN (ρ) − � δℓN (ρ) dρ � = −div (∇δℓN (ρ) ρ) + ρ [1 + δℓN (ρ)] , where we have used the fact that � δℓN(ρ)dρ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This shows that the gradient flow of ℓN with respect to the Wasserstein-Fisher-Rao metric dWFR is given by ∂tρt = −gradWFRℓN (ρt) = div (∇δℓN (ρt) ρt) − ρt [1 + δℓN (ρt)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 24 D Convergence theory (Proof of Theorem 4 and 2) In this section, we provide a systematic treatment to the convergence of Fisher-Rao gradient descent and Wasserstein-Fisher-Rao gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Instead of proving Theorem 4 and 2 separately, we prove a more general convergence result in Theorem 7, which admits Theorem 4 and 2 as its special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' To begin with, we define the Wasserstein gradient descent update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Definition 2 (Wasserstein gradient descent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any ρ ∈ P(Rd) and η ≥ 0, we define ρW,η = (id − η∇δℓN(ρ))#ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Given an initial distribution ρ0 ∈ P(Rd), the Wasserstein gradient descent for solving (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) is defined recursively by ρn+1 = ρW,η n , ∀n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In words, ρW,η is the push forward of ρ by the mapping x �→ x − η∇δℓN(ρ)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any p ≥ 1, define the p-Wasserstein distance between ρ0 and ρ1 in P(Rd) as Wp (ρ0, ρ1) = � inf π∈Π(ρ0,ρ1) � ∥x − y∥p p π (dx, dy) �1/p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The following lemma characterizes the decrease in loss function ℓN by running one step of Wasserstein gradient descent, which can be lower bounded by the squared Wasserstein distance between the two iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lemma 3 (Wasserstein gradient descent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let Assumption 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Choose any ρ0 ∈ P(Rd) and define c0 = e−ℓN(ρ0)φ(inf{r ≥ 0 : ¯φ(r) ≤ e−ℓN(ρ0)/2} + diam(Ω)) 2¯φ(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Suppose that 0 ≤ η < c0 supx∈Rd ∥∇2φ(x)∥2 + supx∈Rd ∥∇φ(x)∥2 2/c0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Define ρn+1 = ρW,η n for n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then, for any n ≥ 0, ℓN(ρn+1) − ℓN(ρn) ≤ −η 2EY ∼ρn∥∇δℓN(ρn)(Y )∥2 2 ≤ − 1 2η W 2 2 (ρn+1, ρn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In addition, if supp(ρ0) = Rd, then supp(ρn) = Rd for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we define the Fisher-Rao gradient descent update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Definition 3 (Fisher-Rao gradient descent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any ρ ∈ P(Rd) and γ ∈ [0, 1], we define ρFR,γ ∈ P(Rd) through dρFR,γ dρ = 1 − γ [δℓN(ρ) + 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Given an initial distribution ρ0 ∈ P(Rd), the Fisher-Rao gradient descent for solving (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) is defined recur- sively by ρn+1 = ρFR,γ n , ∀n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' It is easily seen that ρFR,γ = (1 − γ)ρ + γρFR,1 and dρFR,1 dρ (x) = −δℓN(ρ)(x) = 1 N N � i=1 φ(Xi − x) (ρ ∗ φ)(Xi), ∀x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' From Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 we know that Fisher-Rao gradient descent with γ = 1 can be viewed as fixed-location EM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The following lemma shows that by running one step of Fisher-Rao gradient descent, the decrease in loss function can be lower bounded by the KL divergence between the two iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lemma 4 (Fisher-Rao gradient descent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Choose any ρ ∈ P(Rd) and γ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We have ℓN(µ) ≤ ℓN(ρ) + KL(ρFR,1∥µ) − KL(ρFR,1∥ρ), ∀ µ ≪ ρ, and ℓN(ρFR,γ) − ℓN(ρ) ≤ −KL(ρFR,γ∥ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Finally we define Wasserstein-Fisher-Rao gradient descent, which can be viewed as iteratively applying one step of Wasserstein gradient descent (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Definition 2) and one step of Fisher-Rao gradient descent (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Definition 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Definition 4 (Wasserstein-Fisher-Rao gradient descent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let ρ0 ∈ P(Rd), η ≥ 0 and γ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The Wasserstein-Fisher-Rao gradient descent is defined through �ρn = ρW,η n and ρn+1 = �ρFR,γ n for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The Wasserstein gradient descent and Fisher-Rao gradient descent are special cases of Wasserstein- Fisher-Rao gradient descent with γ = 0 and η = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The following theorem shows that if Wasserstein-Fisher-Rao gradient descent converges weakly to a limit distribution, then this weak limit is the NPMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Suppose that Assumption 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let {ρn}∞ n=0 be the iterates of Wasserstein-Fisher-Rao gradient descent, and c0 = e−ℓN(ρ0)φ(inf{r ≥ 0 : ¯φ(r) ≤ e−ℓN(ρ0)/2} + diam(Ω)) 2¯φ(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' If supp(ρ0) = Rd, γ ∈ (0, 1], and 0 ≤ η < c0 supx∈Rd ∥∇2φ(x)∥2 + supx∈Rd ∥∇φ(x)∥2 2/c0 , then for all n ≥ 0, ℓN(ρn+1) ≤ ℓN(ρn) holds, and supp(ρn) = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Furthermore, if {ρn}∞ n=0 converges weakly to ρ∞ ∈ P(Rd), then this limit ρ∞ is NPMLE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' an optimal solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Theorem 7 directly implies that Theorem 2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By taking η = 0, this also implies that Theorem 4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that Theorem 7 requires γ > 0, therefore it does not provide convergence guarantee for Wasserstein gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Here are some key ideas for showing the optimality of the weak limit ρ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' If ρ∞ is not an optimal solution, then Theorem 1 implies that δℓN(ρ∞)(x0) < −1 − ε holds for some x0 ∈ Rd and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We can find some appropriate y ∈ (−1 − ε, −1 − ε/2) and study the sublevel set ¯S = {x ∈ Rd : δℓN(ρ∞) ≤ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that for n large, we have ∇δℓN(ρn) ≈ ∇δℓN(ρ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We analyze the Wasserstein step and the Fisher-Rao step separately: For the Wasserstein step we show that for any x ∈ ¯S, the gradient descent step x−η∇δℓN(ρn)(x) remains in ¯S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' this follows from the definition of the gradient step and the fact that the step size is chosen small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' A crucial step therein is to choose y so that the gradient ∇δℓN(ρ∞) does not vanish on the level set {x ∈ Rd : δℓN(ρ∞) = y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The existence of such a y follows readily from Sard’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Since �ρn = (id − ∇δℓN(ρn))#ρn, we get �ρn( ¯S) ≥ ρn( ¯S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For the Fisher-Rao step, we first establish that δℓN(�ρn)(x) < −1 − ε/4 for all x ∈ ¯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Recalling that dρn+1 d�ρn (·) = (1 − γ) + γ · [−δℓ(�ρn)(·)], it readily yieldsρn+1( ¯S) ≥ (1 + ε/4)�ρn( ¯S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Putting both steps together, we get that there exists N such that ρn+1( ¯S) ≥ (1+ε/4)ρn( ¯S) holds all n > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that this geometric improvement is entirely driven by the Fisher-Rao part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' To conclude, we use supp(ρN) = Rd to get �ρN( ¯S) > 0, and thus limn→∞ �ρn( ¯S) = ∞, which leads to a contradiction so that ρ∞ must be an optimal solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 Proof of Lemma 3 We invoke a descent lemma Wasserstein gradient descent, whose proof is deferred to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Such a lemma is standard in convex optimization optimization (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', Bubeck, 2015, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' It has appeared for optimization over the Wasserstein space in Salim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2020) under for functionals that are convex along generalized geodesics, an assumption that does not hold for the negative log-likelihood ℓN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lemma 5 (A descent lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let Assumption 1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Choose any ρ ∈ P(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Define c = infx∈Ω(ρ∗φ)(x), G = supx∈Rd ∥∇φ(x)∥2 and H = supx∈Rd ∥∇2φ(x)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We have ℓN(ρW,η) − ℓN(ρ) ≤ −η � 1 − η 2c � H + G2 c �� EY ∼ρ∥∇δℓN(ρ)(Y )∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In addition, we have supx∈Rd ∥∇2δℓN(ρ)(x)∥2 ≤ H/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' If 0 ≤ η < c/H and supp(ρ) = Rd, then supp(ρW,η) = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We now come back to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let R = inf{r ≥ 0 : ¯φ(r) ≤ e−ℓN(ρ0)/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lemma 2 implies that for any µ ∈ P(Rd) with ℓN(µ) ≤ ℓN(ρ0), we have inf x∈Ω(µ ∗ φ)(x) ≥ e−ℓN(ρ0)φ(R + diam(Ω))/[2¯φ(0)] = c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In particular, infx∈Ω(ρ0 ∗ φ)(x) ≥ c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' When η ≤ c0/(H + G2/c0), Lemma 5 and the definition ρ1 = (id − η∇δℓN(ρ0))#ρ0 together yield ℓN(ρ1) − ℓN(ρ0) ≤ −η 2EY ∼ρ0∥∇δℓN(ρ0)(Y )∥2 2 ≤ − 1 2η W 2 2 (ρ1, ρ0) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Also, if supp(ρ0) = Rd, then supp(ρ1) = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' From ℓN(ρ1) ≤ ℓN(ρ0) we obtain that infx∈Ω(ρ1 ∗ φ)(x) ≥ c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then, the proof is completed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 Proof of Lemma 5 We prove the two results in Lemma 5 in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let ν = N −1 �N i=1 δXi be the empirical data distribution, and let h(x) = − log x for x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we can wirte ℓN(ρ) = − 1 N N � i=1 log [ρ ∗ φ (Xi)] = EX∼ν [h ((ρ ∗ φ)(X))] = EX∼ν [h (EY ∼ρ [φ(X − Y )])] as well as ℓN(ρW,η) = EX∼ν � h � EY ∼ρW,η [φ(X − Y )] �� = EX∼ν [h (EY ∼ρ [φ (X − Y + η∇δℓN(ρ)(Y ))])] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Define a(x) = EY ∼ρ[φ(x−Y +η∇δℓN(ρ)(Y ))] and b(x) = EY ∼ρ[φ(x−Y )] = (ρ∗φ)(x) for any x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we can write ℓN(ρ) = EX∼ν [h (b(X))] , ℓN(ρW,η) = EX∼ν [h (a(X))] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that h′(x) = −x−1 < 0, h′′(x) = x−2 > 0 and h′′′(x) = −2x−3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any a, b > 0, by Taylor’s theorem h(a) ≤ h(b) + h′(b)(a − b) + h′′(b) 2 (a − b)2 = h(b) − a − b b + (a − b)2 2b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Taking the above two equations collectively gives ℓN(ρW,η) − ℓN(ρ) = EX∼ν [h (a(X)) − h (b(X))] 27 ≤ −EX∼ν �a(X) − b(X) b(X) � � �� � =:α1 + 1 2EX∼ν �[a(X) − b(X)]2 b(X)2 � � �� � =:α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) Now derive upper bounds for α1 and α2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' To control α1, let G = supx∈Rd ∥∇φ(x)∥2 and observe that |φ (x − y + η∇δℓN(ρ)(y)) − φ(x − y)| ≤ Gη ∥∇δℓN(ρ)(y)∥2 for all x, y ∈ Rd, therefore |a (x) − b (x)| ≤ GηEY ∼ρ [∥∇δℓN (ρ) (Y )∥2] (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2) holds for all x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we have α1 (i) ≤ G2η2E2 Y ∼ρ [∥∇δℓN (ρ) (Y )∥] c2 (ii) ≤ G2η2 c2 EY ∼ρ � ∥∇δℓN (ρ) (Y )∥2 2 � , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3) where (i) follows from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2) and the fact c = inf x∈Ω(ρ ∗ φ)(x) = inf x∈Ω b(x) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4) and (ii) follows from Jensen’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Regarding α2, let H = supx∈Rd ∥∇2φ(x)∥2 and we have φ (x − y + η∇δℓN(ρ)(y)) − φ(x − y) ≥ ⟨∇φ(x − y), η∇δℓN(ρ)(y)⟩ − H 2 η2∥∇δℓN(ρ)(y)∥2 2, for any x, y ∈ Rd, and therefore a(x) − b(x) ≥ EY ∼ρ [⟨∇φ(x − Y ), η∇δℓN(ρ)(Y )⟩] − H 2 η2EY ∼ρ � ∥∇δℓN(ρ)(Y )∥2 2 � (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5) for any x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Since b(x) > 0, we have α2 (i) ≤ −EX∼ν � � EY ∼ρ [⟨∇φ(X − Y ), η∇δℓN(ρ)(Y )⟩] − H 2 η2EY ∼ρ � ∥∇δℓN(ρ)(Y )∥2 2 � b (X) � � = −ηEY ∼ρ �� EX∼ν �∇φ(X − Y ) b(X) � , ∇δℓN(ρ)(Y ) �� + Hη2 2 EX∼ν � 1 b(X) � EY ∼ρ � ∥∇δℓN(ρ)(Y )∥2 2 � (ii) = −ηEY ∼ρ � ∥∇δℓN(ρ)(Y )∥2 2 � + Hη2 2 EX∼ν � 1 b(X) � EY ∼ρ � ∥∇δℓN(ρ)(Y )∥2 2 � (iii) ≤ � −η + Hη2 2c � EY ∼ρ � ∥∇δℓN(ρ)(Y )∥2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6) Here (i) utilizes (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (ii) holds since δℓN(ρ)(y) = −EX∼ν �φ(X − y) b(X) � , ∇δℓN(ρ)(y) = EX∼ν �∇φ(X − y) b(X) � , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7) and therefore EY ∼ρ �� EX∼ν �∇φ(X − Y ) b(X) � , ∇δℓN(ρ)(Y ) �� = EY ∼ρ � ∥∇δℓN(ρ)(Y )∥2 2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' and (iii) follows from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 28 Taking (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6) collectively gives ℓN(ρW,η) − ℓN(ρ) ≤ α1 + α2 ≤ � − η + Hη2 2c + G2η2 2c2 � EY ∼ρ � ∥∇δℓN(ρ)(Y )∥2 2 � = −η � 1 − η 2c � H + G2 c �� EY ∼ρ � ∥∇δℓN(ρ)(Y )∥2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Finally, we learn from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4) that ��∇2δℓN(ρ)(x) �� 2 = ����EX∼ν �∇2φ(X − x) b (X) ����� 2 ≤ supx∈Rd ∥∇2φ(x)∥2 infx∈Rd b (x) = H c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Part 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' When η < c/H, the mapping x �→ η∇δℓN(ρ)(x) is a contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let ϕ(x) = x − η∇δℓN(ρ)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By Lemma 6, ϕ : Rd → Rd is a bijection and ϕ−1 is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The second-order differentiability of δℓN(ρ) implies the differentiability of ϕ and thus ϕ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' If supp(ρ) = Rd, then supp(ρW,η) = supp(ϕ#ρ) = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 Proof of Lemma 4 Let ν = N −1 �N i=1 δXi be the empirical data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any µ ≪ ρ, we have ℓN(µ) = −EX∼ν [log ((µ ∗ φ)(X))] = −EX∼ν [log (EY ∼µ [φ(X − Y )])] = −EX∼ν � log � EY ∼ρ �dµ dρ (Y ) · φ(X − Y ) ��� = −EX∼ν � log � EY ∼ρ �dµ dρ (Y ) · (ρ ∗ φ)(X) · φ(X − Y ) (ρ ∗ φ)(X) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any x ∈ Rd, we can define a new probability measure ρFR,1 x ∈ P(Rd) through dρFR,1 x dρ (·) = φ(x − ·) (ρ ∗ φ)(x), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8) and we can check that ρFR,1 x is indeed a probability measure since � Rd dρFR,1 x = � Rd φ(x − y) (ρ ∗ φ)(x)ρ (dx) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we have, by the convexity of t �→ − log t and Jensen’s inequality, ℓN(µ) (i) = −EX∼ν � log � EY ∼ρFR,1 X �dµ dρ (Y ) · (ρ ∗ φ)(X) ��� (ii) ≤ −EX∼ν � EY ∼ρFR,1 X � log �dµ dρ (Y ) · (ρ ∗ φ)(X) ��� (iii) = −EX∼ν � EY ∼ρ � log �dµ dρ (Y ) · (ρ ∗ φ)(X) � φ(X − Y ) (ρ ∗ φ)(X) �� = − EX∼ν,Y ∼ρ � log �dµ dρ (Y ) � φ(X − Y ) (ρ ∗ φ)(X) � � �� � =:β1 − EX∼ν,Y ∼ρ � log ((ρ ∗ φ)(X)) · φ(X − Y ) (ρ ∗ φ)(X) � � �� � =:β2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='9) Here (i) and (iii) utilizes (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8), and (ii) follows from Jensen’s inequality and the convexity of t �→ − log t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we study the two terms β1 and β2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Regarding β1, we have β1 = EY ∼ρ � log �dµ dρ (Y ) � EX∼ν � φ(X − Y ) (ρ ∗ φ)(X) �� = EY ∼ρ � log �dµ dρ (Y ) � dρFR,1 dρ (Y ) � = EY ∼ρFR,1 � log �dµ dρ (Y ) �� 29 = EY ∼ρFR,1 � log � dµ dρFR,1 (Y ) �� + EY ∼ρFR,1 � log �dρFR,1 dρ (Y ) �� = −KL � ρFR,1 ∥ µ � + KL � ρFR,1 ∥ ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='10) Regarding β2, we have β2 = EX∼ν � log ((ρ ∗ φ)(X)) EY ∼ρ � φ(X − Y ) (ρ ∗ φ)(X) �� = EX∼ν [log ((ρ ∗ φ)(X))] = −ℓN (ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11) Taking (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='9), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='10) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11) collectively yields ℓN(µ) ≤ ℓN(ρ) + KL � ρFR,1 ∥ µ � − KL � ρFR,1 ∥ ρ � , ∀ µ ≪ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By taking µ = ρFR,1, we get ℓN � ρFR,1� ≤ ℓN (ρ) − KL � ρFR,1 ∥ ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='12) Recall that for any γ ∈ (0, 1) we have ρFR,γ = (1 − γ)ρ + γρFR,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore ℓN � ρFR,γ� (i) ≤ (1 − γ)ℓN(ρ) + γℓN(ρFR,1) (ii) ≤ ℓN(ρ) − γKL � ρFR,1 ∥ ρ � , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='13) where (i) holds since ℓN(ρ) is ℓ2-convex in ρ, and (ii) follows from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By the ℓ2-convexity of KL(· ∥ ρ), we have KL � ρFR,γ ∥ ρ � ≤ (1 − γ)KL (ρ ∥ ρ) + γKL � ρFR,1 ∥ ρ � = γKL � ρFR,1 ∥ ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='14) Combine (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='13) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='14) to achieve ℓN � ρFR,γ� ≤ ℓN(ρ) − KL � ρFR,γ ∥ ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4 Proof of Theorem 7 Suppose that for some n ≥ 0, ℓN(ρn) ≤ ℓN(ρ0) holds, and supp(ρn) = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Define cn = e−ℓN(ρn)φ(inf{r ≥ 0 : ¯φ(r) ≤ e−ℓN(ρn)/2} + diam(Ω)) 2¯φ(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then, cn ≥ c0 and thus 0 ≤ η < cn supx∈Rd ∥∇2φ(x)∥2 + supx∈Rd ∥∇φ(x)∥2 2/cn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Theorem 3 and the upper bound on η immediately gives ℓ(�ρn) ≤ ℓ(ρn) and supp(�ρn) = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let ν = N −1 �N i=1 δXi be the empirical data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' From Lemma 4, the updating rule dρn+1 d�ρn (·) = (1 − γ) + γEX∼ν � φ(X − ·) (�ρn ∗ φ)(X) � , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='15) and the positivity of φ, we see that ℓ(ρn+1) ≤ ℓ(�ρn) and supp(ρn+1) = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore we can use induction to show that, for all n ≥ 0, the inequality ℓ(ρn+1) ≤ ℓ(�ρn) ≤ ℓ(ρn) holds, and supp(ρn) = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In view of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1), both sequences {ℓ(ρn)}∞ n=0 and {ℓ(�ρn)}∞ n=0 converge and have the same limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Consequently, ℓ(ρn) − ℓ(�ρn) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='16) Now, suppose that {ρn}∞ n=0 converges weakly to some ρ∞ ∈ P(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' {δℓN(ρn)}∞ n=0 converges uniformly to δℓN(ρ∞) over compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 30 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' It is easily seen that sup x∈Rd ∥∇δℓN(ρn)(x)∥2 = sup x∈Rd ����EX∼ν � ∇φ(X − x) (ρn ∗ φ)(X) ����� 2 ≤ supx∈Rd ∥∇φ(x)∥2 infx∈Ω(ρn ∗ φ)(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Assumption 1 forces supx∈Rd ∥∇φ(x)∥2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By Lemma (2) and the fact that ℓ(ρn) ≤ ℓ(ρ0), we have inf x∈Ω(ρn ∗ φ)(x) ≥ c0, ∀n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='17) Hence, {δℓN(ρn)}∞ n=0 are uniformly equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore, it suffices to prove that {δℓN(ρn)}∞ n=0 con- verges pointwise to δℓN(ρ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Since φ is bounded and continuous (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Assumption 1), (ρn ∗ φ)(x) → (ρ∞ ∗ φ)(x) holds for every x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Recall that δℓ(ρ)(x) = −EX∼ν � φ(X − x) (ρ ∗ φ)(X) � , ∀ρ ∈ P(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='18) Based on the boundedness of φ and the lower bound (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='17), we use the bounded convergence theorem to derive δℓN(ρn)(x) → δℓN(ρ∞)(x) for every x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' {∇δℓN(ρn)}∞ n=0 converges uniformly to ∇δℓN(ρ∞) over compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The proof is similar to that of Claim 1 and is thus omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' {δℓN(�ρn)}∞ n=0 converges uniformly to δℓN(ρ∞) over compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lemma 3 implies that W 2 2 (�ρn, ρn) ≤ 2η[ℓN(ρn) − ℓN(�ρn)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='16), W2(�ρn, ρn) → 0 and thus W1(�ρn, ρn) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Since φ is Lipschitz, we have sup x∈Rd |(�ρn ∗ φ)(x) − (ρn ∗ φ)(x)| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' From the above uniform bound, (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='17), and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='18) we obtain that sup x∈Rd |δℓN(�ρn)(x) − δℓN(ρn)(x)| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then, the proof is completed by applying Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We now come back to Theorem (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Suppose that ρ∞ is not an optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then in view of Theorem (1), there exists ε > 0 such that δℓN(ρ∞)(x) < −1 − ε for some x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='Similar to showing the pointwise convergence of δℓN(ρn) to δℓN(ρ∞) in Claim (1), we can use the bounded convergence theorem to show that ℓ(ρn) → ℓ(ρ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Hence, ℓ(ρ∞) ≤ ℓ(ρ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lemma (2) immdeiately implies that inf x∈Ω(ρ∞ ∗ φ)(x) ≥ c0, ∀n ≥ 0, (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='19) and lim∥x∥2→∞ δℓ(ρ∞)(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore, the function δℓ(ρ∞) achieves its minimum value at some x0 ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We have δℓN(ρ∞)(x0) < −1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For notational simplicity, let fn(x) = δℓN(ρn)(x) and f(x) = δℓN(ρ∞)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By Assumption 1, f ∈ Cd(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The second part of Lemma (7) guarantees the existence of some y ∈ (−1 − ε, −1 − ε/2) such that S(y) = � x ∈ Rd : f(x) = y � is compact and infx∈S(y) ∥∇f(x)∥2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Denote by ξ = infx∈S(y) ∥∇f(x)∥2 and ¯S = {x ∈ Rd : f(x) ≤ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The sublevel set ¯S is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' According to the fact that f(x0) < y and the continuity of f, ¯S has positive Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore we have ρn( ¯S) > 0, ∀ n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='20) 31 We first show that �ρn( ¯S) ≥ ρn( ¯S) holds for sufficently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By Claim 2, there exists N > 0 such that sup x∈ ¯S ∥∇fn(x) − ∇f(x)∥2 ≤ ξ/11, ∀ n > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By the assumed upper bound on η and the estimate (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='19), we have η ≤ c0 supx∈Rd ∥∇2φ(x)∥2 ≤ 1 supx∈Rd ∥∇2f(x)∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' On top of the above, the first part of Lemma (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='23) implies that {x − η∇fn(x) : x ∈ ¯S} ⊆ ¯S, ∀ n > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Since �ρn = (id − η∇fn)#ρn, we have �ρn( ¯S) ≥ ρn( ¯S), ∀ n > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='21) Then we prove that ρn+1( ¯S) > (1 + ε/4)�ρn( ¯S) for sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By Claim 3, there exists N ′ > 0 such that sup x∈ ¯S |δℓ(�ρn)(x) − f(x)| ≤ ε/4, ∀ n > N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Consequently, sup x∈ ¯S δℓ(�ρn)(x) ≤ sup x∈ ¯S f(x) + ε/4 ≤ −1 − ε/4, ∀ n > N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The expression (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='15) implies that dρn+1 d�ρn (x) = (1 − γ) + γ · [−δℓ(�ρn)(x)] ≥ 1 + γε/4, ∀ n > N ′, x ∈ ¯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore we have ρn+1( ¯S) ≥ (1 + γε/4)�ρn( ¯S), ∀n > N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='22) Taking (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='21) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='22) collectively yields ρn+1( ¯S) ≥ (1 + γε/4)ρn( ¯S), ∀n > max{N, N ′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This combined with (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='20) immediately leads to limn→∞ ρn( ¯S) = ∞, which contradicts ρn(Rd) = 1 for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore ρ∞ must be an optimal solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' ρ∞ is the NPMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5 Technical lemmas Here is a standard result about the Lipschitz perturbation of identity mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let B be a Banach space with norm ∥ · ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' If f : B → B has Lipschitz constant c < 1, then ϕ : B → B, x �→ x + f(x) is bijective and ϕ−1 is (1 − c)−1-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Choose any y ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The mapping ψ : x �→ y − f(x) has Lipschitz constant c < 1 (in terms of the norm ∥ · ∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By the Banach fixed-point theorem, there exists a unique z such that z = ψ(z), which implies that y = ϕ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Hence ϕ is bijective and ϕ−1 : B → B is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any y1, y2 ∈ B, define xi = ϕ−1(yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then, yi = xi + f(xi) and ∥y2 − y1∥2 = ∥ϕ(x2) − ϕ(x1)∥2 ≥ ∥x2 − x1∥2 − ∥f(x2) − f(x1)∥2 ≥ (1 − c)∥x2 − x1∥2, ∥ϕ−1(y2) − ϕ−1(y1)∥2 = ∥x2 − x1∥2 ≤ (1 − c)−1∥y2 − y1∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This proves the Lipschitz smoothness of ϕ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 32 Below we show that one step of approximate gradient descent cannot expand certain sub-level sets of a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Assume that f ∈ C2(Rd) has a finite minimum value y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Define S(y) = {x ∈ Rd : f(x) = y} for y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We have the followings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Suppose that supx∈Rd ∥∇2f(x)∥2 ≤ L < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Choose any η ∈ [0, 1/L] and y > y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Define ¯S = {x ∈ Rd : f(x) ≤ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' If g : Rd → Rd satisfies sup x∈ ¯S ∥g(x) − ∇f(x)∥2 ≤ 1 11 inf x∈S(y) ∥∇f(x)∥2, then {x − ηg(x) : x ∈ ¯S} ⊆ ¯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Suppose that inf∥x∥2≥R f(x) > y0 holds for some R and in addition, f ∈ Cd(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then, for any ε > 0, there exists y ∈ (y0, y0 + ε) such that S(y) is compact and inf x∈S(y) ∥∇f(x)∥2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' To prove the first part, we choose any x ∈ ¯S and we will show that x − ηg(x) ∈ ¯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let δ = supz∈ ¯S ∥g(z) − ∇f(z)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By supx∈Rd ∥∇2f(x)∥2 ≤ L and 0 ≤ η ≤ 1/L, f(x − ηg(x)) ≤ f(x) + ⟨∇f(x), −ηg(x)⟩ + L 2 ∥ηg(x)∥2 2 ≤ f(x) + ⟨∇f(x), −η∇f(x) − η[g(x) − ∇f(x)]⟩ + Lη2 2 [∥∇f(x)∥2 + ∥g(x) − ∇f(x)∥2]2 ≤ f(x) − η∥∇f(x)∥2 2 + ηδ∥∇f(x)∥2 + Lη2 2 [∥∇f(x)∥2 2 + 2δ∥∇f(x)∥2 + δ2] = f(x) + η∥∇f(x)∥2 2 � − � 1 − Lη 2 � + (1 + Lη) δ ∥∇f(x)∥2 + Lη 2 � δ ∥∇f(x)∥2 �2� ≤ f(x) + η∥∇f(x)∥2 2 � − 1 2 + 2δ ∥∇f(x)∥2 + 1 2 � δ ∥∇f(x)∥2 �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='23) Define ξ = infz∈S(y) ∥∇f(z)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' If ξ = 0, then δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The bound (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='23) yields f(x − ηg(x)) ≤ f(x) − η 2∥∇f(x)∥2 2 ≤ f(x) ≤ y and thus x − ηg(x) ∈ ¯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' From now on we assume that ξ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' When ∥∇f(x)∥2 > ( √ 5 + 2)δ, we have δ ∥∇f(x)∥2 < 1 √ 5 + 2 = √ 5 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='23), f(x − ηg(x)) − f(x) ≤ η∥∇f(x)∥2 2 �1 2 � δ ∥∇f(x)∥2 + 2 �2 − 5 2 � ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Hence, f(x − ηg(x)) ≤ f(x) ≤ y and x − ηg(x) ∈ ¯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' When ∥∇f(x)∥2 ≤ ( √ 5 + 2)δ, we use δ ≤ ξ/11 and √ 5 < 5/2 to get ∥∇f(x)∥2 ≤ ( √ 5 + 2)δ ≤ √ 5 + 2 11 ξ < 9ξ 22 < ξ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 33 Therefore, x /∈ S(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' As x ∈ ¯S, we must have f(x) < y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that for any z ∈ S(y), we have ∥∇f(z)∥2 ≥ ξ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By supx∈Rd ∥∇2f(x)∥2 ≤ L, we have ∥z − x∥2 ≥ ∥∇f(z) − ∇f(x)∥2 L ≥ ∥∇f(z)∥2 − ∥∇f(x)∥2 L > ξ − ξ/2 L = ξ 2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore, infz∈S(y) ∥z − x∥2 ≥ ξ 2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We have B ∩ S(y) = ∅, where B = � x′ ∈ Rd : ∥x′ − x∥2 < ξ 2L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='24) We claim that B ⊆ ¯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' If this is not true, then f(x′) > y holds for some x′ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Since f(x) < y and f is continuous, there exists t ∈ (0, 1) such that f((1 − t)x + tx′) = y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (1 − t)x + tx′ ∈ S(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The fact (1 − t)x + tx′ ∈ B leads to B ∩ S(y) ̸= ∅, which contradicts (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' On the other hand, we have ∥[x − ηg(x)] − x∥2 = η∥g(x)∥2 ≤ η∥∇f(x)∥2 + ηδ ≤ ( √ 5 + 3)δ L ≤ ( √ 5 + 3)ξ 11L < ξ 2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore, x − ηg(x) ∈ B ⊆ ¯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This proves the first part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We now come to the second part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let y1 = inf∥x∥2≥R f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Thanks to the continuity of f, the image set {f(x) : x ∈ Rd} contains the interval (y0, y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Since f : Rd → R is Cd, Sard’s lemma asserts that the set of critical values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' the image set of critical points {f(x) : ∇f(x) = 0}, has Lebesgue measure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Consequently, for any ε > 0, there exists a regular value y ∈ (y0, min{y0 + ε, y1}), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' ∇f(x) ̸= 0 so long as f(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Because of y < y1, S(y) ⊆ {x ∈ Rd : ∥x∥2 ≥ R} must be compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The continuity of ∇f implies that infx∈S(y) ∥∇f(x)∥2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' E Well-posedness of particle gradient flows E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 Fisher-Rao gradient flow (Proof of Theorem 5) In this section, we will show that for the ODE system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='15) ˙ω(j) t = −ω(j) t � 1 − 1 N N � i=1 φ � Xi − µ(j)� �m l=1 ω(l) t φ � Xi − µ(l)� � , ∀ t ≥ 0, j ∈ [m] with initial value ω0 ∈ ∆m−1, the solution exists, is unique, and (ρt)t≥0 where ρt := �m l=1 ω(l) t δµ(l) is a Fisher-Rao gradient flow in the sense of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' First of all, we will use Picard-Lindelöf theorem to prove existence and uniqueness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Define a function f : Rm → Rm as f (y) = [fj (y)]1≤j≤m , fj (y) = −yj � 1 − 1 N N � i=1 φ � Xi − µ(j)� �m l=1 ylφ � Xi − µ(l)� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we can rewrite the ODE system as ˙ωt = f(ωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any y ∈ Rm such that ∥y − ω0∥2 ≤ ε for some δ > 0 to be specified later, we know that m � l=1 ylφ � Xi − µ(l)� ≥ m � l=1 ω(l) 0 φ � Xi − µ(l)� − m � l=1 � ω(l) 0 − yl � φ � Xi − µ(l)� (i) ≥ min i∈[N],l∈[m] φ � Xi − µ(l)� − ∥y − ω0∥2 m � l=1 φ2 � Xi − µ(l)� (ii) ≥ min i∈[N],l∈[m] φ � Xi − µ(l)� − εm (2π)d 34 for any i ∈ [N], where (i) follows from ω(l) 0 ∈ ∆m−1 and Cauchy-Schwarz inequality, while (ii) holds since ∥φ∥∞ ≤ (2π)−d/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore by taking ε := (2π)d 2m min i∈[N],l∈[m] φ � Xi − µ(l)� , we know that for any ∥y − ω0∥2 ≤ ε it holds that m � l=1 ylφ � Xi − µ(l)� ≥ 1 2 min i∈[N],l∈[m] φ � Xi − µ(l)� ≜ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) Therefore we can check that ���[∇fj (y)]j ��� = �����− � 1 − 1 N N � i=1 φ � Xi − µ(j)� �m l=1 ylφ � Xi − µ(l)� � − � 1 N N � i=1 yjφ2 � Xi − µ(j)� ��m l=1 ylφ � Xi − µ(l)��2 ������ ≤ 1 + ∥φ∥∞ δ + (1 + ε) ∥φ∥2 ∞ δ2 = 1 + 1 (2π)d/2 δ + 1 + ε (2π)d δ2 , and for l ̸= j ��[∇fj (y)]l �� = �����−yj � 1 N N � i=1 φ � Xi − µ(j)� φ � Xi − µ(l)� ��m l=1 ylφ � Xi − µ(l)��2 ������ ≤ (1 + ε) ∥φ∥2 ∞ δ2 = 1 + ε (2π)d δ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' As a result, we know that for any max y:∥y−ω0∥2≤ε ∥∇fj (y)∥2 ≤ √m � 1 + 1 (2π)d/2 δ + 1 + ε (2π)d δ2 � ≜ Clip, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2) and hence f(y) is CLip-Lipschitz continuous in {y : ∥y − ω0∥2 ≤ ε} where CLip := √mClip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In addition, it is easy to show that max y:∥y−ω0∥2≤ε ∥fj (y)∥2 ≤ −yj + 1 N N � i=1 yjφ � Xi − µ(j)� �m l=1 ylφ � Xi − µ(l)� ≤ 1 + ε ≜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By Picard-Lindelöf theorem, there exists t0 > 0 such that the ODE has a unique solution on the time interval [0, t0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We first check that ω(j) t > 0 for any j ∈ [m] and t ∈ [0, t0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' If this is not true, define t⋆ := min � t ∈ [0, t0] : ∃ j ∈ [m] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' ω(j) t ≤ 0 � and suppose ω(j⋆) t⋆ ≤ 0 for j⋆ ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we know that ω(j) t > 0 for any j ∈ [m] and 0 ≤ t ≤ t⋆, and hence ˙ω(j) t ≥ −ω(j) t for all t ∈ [0, t⋆].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we can use Grönwall’s lemma to achieve ω(j) t ≥ ω(j) 0 e−t for all t ∈ [0, t⋆], and as a result ω(j⋆) t⋆ > 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In addition, we can also check that d dt m � j=1 ω(j) t = m � j=1 ˙ω(j) t = − m � j=1 ω(j) t � 1 − 1 N N � i=1 φ � Xi − µ(j)� �m l=1 ω(l) t φ � Xi − µ(l)� � = 0 for all t ∈ [0, t0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' As as a result ωt ∈ ∆m−1 for any 0 ≤ t ≤ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By repeating the same procedure as above (notice that the above proof only depends on ω0 ∈ ∆m−1, and t0 only depends on universal constants CLip and M and does not depend on ω0), we can show that the ODE has a unique solution on [t0, 2t0], [2t0, 3t0], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This shows the existence and uniqueness of the solution to the ODE system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Next, we show that (ρt)t≥0 defined as ρt := �m l=1 ω(l) t δµ(l) solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that ρt is a probability measure since we have shown that ωt ∈ ∆m−1 for any t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any test function ϕ(x) ∈ C∞ c , we have d dt � Rd ϕ (x) ρt (dx) = d dt � � m � j=1 ω(j) t ϕ � µ(j) t � � � = m � j=1 ˙ω(j) t ϕ � µ(j) t � 35 = − m � j=1 � 1 + δℓN (ρt) � µ(j) t �� ω(j) t ϕ � µ(j) t � = − � Rd [1 + δℓN (ρt) (x)] ϕ (x) ρt (dx) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This proves that ∂tρt = − [δℓ (ρt) + 1] ρt holds in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 Wasserstein gradient flow (Proof of Theorem 6) In this section we will show that the ODE system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='19) ˙µ(j) t = 1 N N � i=1 φ � Xi − µ(j) t � m−1 �m l=1 φ � Xi − µ(l) t � � Xi − µ(j) t � , ∀ t ≥ 0, j ∈ [m] has unique solution, and (ρt)t≥0 where ρt := m−1 �m l=1 δµ(l) t is a Wasserstein gradient flow in the sense of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We will use Picard-Lindelöf theorem to prove existence and uniqueness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Define a suffi- ciently large constant R := max 1≤i≤N ∥Xi∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For each j ∈ [m], define a function f (j) : Rmd → Rd as f (j) (z) = 1 N N � i=1 φ (Xi − zj) m−1 �m l=1 φ (Xi − zl) (Xi − zj) , where z = [zj]1≤j≤m ∈ Rmd and z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , zm ∈ Rd, and let f(z) = [f (j)(z)]1≤j≤m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we can write the ODE system as ˙µt = f(µt) where µt = [µ(j) t ]1≤j≤m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Denote by f (j)(z) = [f (j) k (z)]1≤k≤d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any z ∈ Rmd satisfying maxj∈[m] ∥zj∥2 ≤ 2R, we have min 1≤i≤N 1 m m � l=1 φ (Xi − zl) ≥ 1 (2π)d/2 exp � −9 2R2 � ≜ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3) Then we can compute for l ̸= j ���∇zlf (j) k (z) ��� 2 = �����∇zl 1 N N � i=1 φ (Xi − zj) m−1 �m l=1 φ (Xi − zl)e⊤ k (Xi − zj) ����� 2 = �����− 1 N N � i=1 φ (Xi − zj) m−1φ (Xi − zl) [m−1 �m l=1 φ (Xi − zl)]2 e⊤ k (Xi − zj) (Xi − zl) ����� 2 (i) ≤ m−1 ∥φ∥2 ∞ δ2 1 N N � i=1 ��e⊤ k (Xi − zj) �� ∥Xi − zl∥2 (ii) ≤ 9m−1 δ2 (2π)d R2, where (i) utilizes (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3) and (ii) follows from maxi∈[N] ∥Xi∥2 ≤ R and maxj∈[m] ∥zj∥2 ≤ 2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Similarly we have ���∇zjf (j) k (z) ��� 2 = �����∇zj 1 N N � i=1 φ (Xi − zj) m−1 �m l=1 φ (Xi − zl)e⊤ k (Xi − zj) ����� 2 ≤ �����− 1 N N � i=1 φ2 (Xi − zj) m−1 [m−1 �m l=1 φ (Xi − zl)]2 e⊤ k (Xi − zj) (Xi − zj) ����� 2 36 + ����� 1 N N � i=1 φ (Xi − zj) m−1 �m l=1 φ (Xi − zl) � e⊤ k (Xi − zj) (Xi − zj) − ek � ����� 2 ≤ 9 � m−1 δ2 (2π)d + 1 δ (2π)d/2 � R2 + ∥φ∥∞ δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' As a result, we have ���∇zf (j) k (z) ��� 2 = � � � � m � l=1 ���∇zlf (j) k (z) ��� 2 2 ≤ � � � �(m + 1) � 9m−1 δ2 (2π)d R2 �2 + 2 1 δ2 (2π)d ≜ Clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4) Therefore f (j)(z) is √ dCLip-Lipschitz continuous in {z : maxj∈[m] ∥zj∥2 ≤ 2R}, and hence f(z) is CLip- Lipschitz continous where CLip := Clip √ md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In addition, it is straightforward to show that for any z ∈ Rmd satisfying maxj∈[m] ∥zj∥2 ≤ 2R, ���f (j) (z) ��� 2 ≤ ∥φ∥∞ δ 3R ≜ M holds for all 1 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Recall that µ(1) 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , µ(m) 0 are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' sampled from Uniform({Xi}1≤i≤N), therefore maxj∈[m] ∥µ(j) 0 ∥2 ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore we have {z : ∥z − µ0∥2 ≤ R} ⊆ � z : max j∈[m] ∥zj∥2 ≤ 2R � , where µ0 = [µ(j) 0 ]1≤j≤m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Hence f(z) is √ mdCLip-Lipschitz continous in {z : ∥z − µ0∥2 ≤ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we can use Picard-Lindelöf theorem to show that, there exists t0 > 0 such that the ODE has a unique solution on the time interval [0, t0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any t ∈ [0, t0] and j ∈ [m], we can compute d dt∥µ(j) t ∥2 2 = 2⟨µ(j) t , ˙µ(j) t ⟩ = 2 N N � i=1 φ � Xi − µ(j) t � m−1 �m l=1 φ � Xi − µ(l) t �µ(j)⊤ t � Xi − µ(j) t � ≤ 2 N N � i=1 φ � Xi − µ(j) t � m−1 �m l=1 φ � Xi − µ(l) t � � ∥Xi∥2 ∥µ(j) t ∥2 − ∥µ(j) t ∥2 2 � ≤ 2 N N � i=1 φ � Xi − µ(j) t � m−1 �m l=1 φ � Xi − µ(l) t � � R − ∥µ(j) t ∥2 � ∥µ(j) t ∥2, where we use Cauchy-Schwarz inequality in the penultimate step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This shows that d dt∥µ(j) t ∥2 2 < 0 as long as ∥µ(j) t0 ∥2 > R, and as a result maxj∈[m] ∥µ(j) t0 ∥2 ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we can repeat the same analysis as above (notice that the above proof only requires maxj∈[m] ∥µ(j) 0 ∥2 ≤ R, and t0 only depends on universal constants CLip and M) to show that the ODE has a unique solution on [t0, 2t0], [2t0, 3t0], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This shows the existence and uniqueness of the solution to the ODE system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Finally we check that (ρt)t≥0 defined as ρt := �m l=1 m−1δµ(l) t solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any test function ϕ(x) ∈ C∞ c , we have d dt � Rd ϕ (x) ρt (dx) = d dt � � 1 m m � j=1 ϕ � µ(j) t � � � = 1 m m � j=1 � ∇ϕ � µ(j) t � , ˙µ(j) t � = 1 m m � j=1 � ∇ϕ � µ(j) t � , −∇δℓN (ρt) � µ(j) t �� = − � Rd ⟨∇ϕ (x) , ∇δℓN (ρt)⟩ ρt (dx) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 37 This proves that ∂tρt = div (ρt∇δℓ (ρt)) holds in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 Wasserstein-Fisher-Rao gradient flow (Proof of Theorem 3) In this section we will show that the ODE system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8) ˙µ(j) t = 1 N N � i=1 φ � Xi − µ(j) t � m−1 �m l=1 φ � Xi − µ(l) t � � Xi − µ(j) t � , ˙ω(j) t = � � 1 N N � i=1 φ � Xi − µ(j) t � �m l=1 ω(j) t φ � Xi − µ(l) t � − 1 � � ω(j) t , has unique solution, and (ρt)t≥0 where ρt := �m l=1 ω(l) t δµ(l) t is a Wasserstein-Fisher-Rao gradient flow in the sense of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We will integrate the proof techniques used in the previous two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We will again use Picard-Lindelöf theorem to prove existence and uniqueness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For each j ∈ [m], define two functions f (j) : ∆m−1 × Rmd → Rd and g(j) : ∆m−1 × Rmd → R as f (j) (y, z) = 1 N N � i=1 φ (Xi − zj) �m l=1 ylφ (Xi − zl) (Xi − zj) , g(j) (y, z) = − � 1 − 1 N N � i=1 φ (Xi − zj) �m l=1 ylφ (Xi − zl) � yj, where y = [yj]1≤j≤m ∈ ∆m−1, z = [zj]1≤j≤m ∈ Rmd with z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , zm ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let f (y, z) := � �� f (1) (y, z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' f (m) (y, z) � �� , g (y, z) := � �� g(1) (y, z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' g(m) (y, z) � �� , and h (y, z) = � f (y, z) g (y, z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we can write the ODE system as � ˙µt ˙ωt � = h �� µt ωt �� , where µt = [µ(j) t ]1≤j≤m and ωt = [ω(j) t ]1≤j≤m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Denote by f (j)(z) = [f (j) k (z)]1≤k≤d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any z ∈ Rmd satisfying maxj∈[m] ∥zj∥2 ≤ 2R and any y ∈ Rm satisfying ∥y − ω0∥2 ≤ ε where R := max 1≤i≤N ∥Xi∥2 , ε := (2π)d/2 2m exp � −9 2R2 � , we have for any i ∈ [N] m � l=1 ylφ (Xi − zl) ≥ m � l=1 ω(l) 0 φ (Xi − zl) − m � l=1 � ω(l) 0 − yl � φ (Xi − zl) (i) ≥ min i∈[N],l∈[m] φ (Xi − zl) − ∥y − ω0∥2 m � l=1 φ2 � Xi − µ(l)� (ii) ≥ 1 (2π)d/2 exp � −9 2R2 � − εm (2π)d = 1 2 (2π)d/2 exp � −9 2R2 � ≜ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5) 38 Here (i) follows from ω0 ∈ ∆m−1 and the Cauchy-Schwarz inequality, while (ii) and (iii) holds since ∥Xi − zl∥2 ≤ 3R for any i ∈ [N] and l ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any j ∈ [m], denote by f (j)(z) = [f (j) k (z)]1≤k≤d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Similar to the proof of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4), we can use (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5) to show that ���∇zf (j) k (y, z) ��� 2 ≤ � � � �(m + 1) � 9m−1 δ2 (2π)d R2 �2 + 2 1 δ2 (2π)d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We also have ∇yf (j) k (y, z) = ∇y 1 N N � i=1 φ (Xi − zj) �m l=1 ylφ (Xi − zl)e⊤ k (Xi − zj) = − 1 N N � i=1 φ (Xi − zj) [�m l=1 ylφ (Xi − zl)]2 e⊤ k (Xi − zj) � �� φ (Xi − z1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' φ (Xi − zm) � �� , and as a result ���∇yf (j) k (y, z) ��� 2 ≤ √m ∥φ∥2 ∞ δ2 max i∈[N],j∈[m] ∥Xi − zj∥2 ≤ 3√mR δ2 (2π)d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In addition, we can compute ∇yg(j) (y, z) = −∇y �� 1 − 1 N N � i=1 φ (Xi − zj) �m l=1 ylφ (Xi − zl) � yj � = − � 1 − 1 N N � i=1 φ (Xi − zj) �m l=1 ylφ (Xi − zl) � ej + 1 N N � i=1 yjφ (Xi − zj) [�m l=1 ylφ (Xi − zl)]2 � �� φ (Xi − z1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' φ (Xi − zm) � �� and for each l ∈ [m] ∇zlg(j) (y, z) = −∇zl �� 1 − 1 N N � i=1 φ (Xi − zj) �m l=1 ylφ (Xi − zl) � yj � = yj 1 N N � i=1 � ∇zlφ (Xi − zj) �m l=1 ylφ (Xi − zl) − φ (Xi − zj) yl∇zlφ (Xi − zl) [�m l=1 ylφ (Xi − zl)]2 � = 1 N N � i=1 � 1 {l = j} yjφ (Xi − zj) �m l=1 ylφ (Xi − zl) (Xi − zj) − yjφ (Xi − zj) ylφ (Xi − zl) [�m l=1 ylφ (Xi − zl)]2 (Xi − zl) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' As a result, we have ���∇yg(j) (y, z) ��� 2 ≤ �����1 − 1 N N � i=1 φ (Xi − zj) �m l=1 ylφ (Xi − zl) ����� + 1 N N � i=1 ����� yjφ (Xi − zj) [�m l=1 ylφ (Xi − zl)]2 ����� √m ∥φ∥∞ ≤ 1 + ∥φ∥∞ δ + (1 + ε) √m δ2 ∥φ∥2 ∞ = 1 + 1 (2π)d/2 δ + (1 + ε) √m (2π)d δ2 and ���∇zlg(j) (y, z) ��� 2 ≤ 1 N N � i=1 (1 + ε) ∥φ∥∞ δ ∥Xi − zj∥2 + 1 N N � i=1 (1 + ε)2 ∥φ∥2 ∞ δ2 ∥Xi − zl∥2 ≤ 3 (1 + ε) R (2π)d/2 δ + 3 (1 + ε)2 R ∥φ∥2 ∞ (2π)d δ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 39 Therefore for any k ∈ [d],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' j ∈ [m],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' z ∈ Rmd satisfying maxj∈[m] ∥zj∥2 ≤ 2R and y ∈ Rm such that ∥y − ω0∥2 ≤ ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' we have ���∇f (j) k (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' z) ��� 2 = ����∇zf (j) k (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' z) ��� 2 2 + ���∇yf (j) k (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' z) ��� 2 2 ≤ � � � �(m + 1) � 9m−1 δ2 (2π)d R2 �2 + 2 δ2 (2π)d + � 3√mR δ2 (2π)d �2 ≜ Clip,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' which suggests that f (j)(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' z) is √ dClip,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='f-Lipschitz continuous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' and ���∇g(j) (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' z) ��� 2 = � � � ���∇yg(j) (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' z) ��2 2 + m � l=1 ��∇zlg(j) (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' z) ��2 2 ≤ � � � � � 1 + 1 (2π)d/2 δ + (1 + ε) √m (2π)d δ2 �2 + m � 3 (1 + ε) R (2π)d/2 δ + 3 (1 + ε)2 R ∥φ∥2 ∞ (2π)d δ2 �2 ≜ Clip,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' which suggests that g(j)(y, z) is Clip,g-Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This allows us to conclude that h(y, z) is CLip- continuous in {(y, z) : ∥y − ω0∥2 ≤ ε, maxj∈[m] ∥zj∥2 ≤ 2R}, where CLip = � mC2 lip,g + mdC2 lip,f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In addition, it is easy to check that for any z ∈ Rmd satisfying maxj∈[m] ∥zj∥2 ≤ 2R and any y ∈ Rm satisfying ∥y − ω0∥2 ≤ ε, max j∈[m] ���f (j) (y, z) ��� 2 ≤ 3R δ (2π)d/2 , max j∈[m] ���g(j) (y, z) ��� ≤ � 1 + 1 δ (2π)d/2 � (1 + ε) and therefore ∥h (y, z)∥2 ≤ � � � �m � 3R δ (2π)d/2 �2 + m �� 1 + 1 δ (2π)d/2 � (1 + ε) �2 ≜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Recall that µ(1) 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , µ(m) 0 are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' sampled from Uniform({Xi}1≤i≤N), therefore (ω0, µ0) ∈ (y, z) : � (y, z) : ∥y − ω0∥2 ≤ ε, max j∈[m] ∥zj∥2 ≤ 2R � where µ0 = [µ(j) 0 ]1≤j≤m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We are ready to apply Picard-Lindelöf theorem to show that there exists t0 > 0, only depending on CLip and M, such that the ODE has a unique solution on the time interval [0, t0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We can use the same argument in the proof of Theorem 5 in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 to show that ωt ∈ ∆m−1 for all t ∈ [0, t0], and can use the same argument in the proof of Theorem 6 in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 to show that maxj∈[m] ∥µ(j) t ∥2 ≤ R for all t ∈ [0, t0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we can repeat the same analysis as above (notice that the above proof only requires ω0 ∈ ∆m−1 and maxj∈[m] ∥µ(j) 0 ∥2 ≤ R, and t0 only depends on universal constants CLip and M) to show that the ODE has a unique solution on [t0, 2t0], [2t0, 3t0], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This shows the existence and uniqueness of the solution to the ODE system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Finally we check that (ρt)t≥0 defined as ρt := �m l=1 ω(l) t δµ(l) t solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that ρt is a probability measure since we have shown that ωt ∈ ∆m−1 for any t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any test function ϕ(x) ∈ C∞ c , we have d dt � Rd ϕ (x) ρt (dx) = d dt � � m � j=1 ω(j) t ϕ � µ(j) t � � � = m � j=1 � ˙ω(j) t ϕ � µ(j) t � + ω(j) t � ∇ϕ � µ(j) t � , ˙µ(j) t �� = − m � j=1 � 1 + δℓN (ρt) � µ(j) t �� ω(j) t ϕ � µ(j) t � + m � j=1 ω(j) t � ∇ϕ � µ(j) t � , −∇δℓN (ρt) � µ(j) t �� 40 = − � Rd [1 + δℓN (ρt) (x)] ϕ (x) ρt (dx) − � Rd ⟨∇ϕ (x) , ∇δℓN (ρt)⟩ ρt (dx) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This proves that ∂tρt = div (ρt∇δℓ (ρt)) − [δℓ (ρt) + 1] ρt holds in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' F Properties of Wasserstein gradient flow In this section, we present some preliminary results on Wasserstein gradient flow for learning Gaussian mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We also discuss the implications of these results, as well as the technical difficulty of obtaining more general results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We first establish the connection between the Wasserstein gradient flow and the classical gradient flow in the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Suppose we fit the data {Xi}1≤i≤N using a m-component Gaussian mixture model 1 m m � j=1 N � µ(j), Id � , where {µ(j)}1≤j≤m is the location of the m Gaussian components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The negative likelihood function is ℓN,m � µ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , µ(m)� := − 1 N N � i=1 log � � 1 m m � j=1 φ � Xi − µ(j)� � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) The gradient flow for minimizing (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1), denoted by (µt)t≥0 where µt = [µ(j) t ]1≤j≤m, is given by the following ODE system ˙µ(j) t = −∇µ(j)ℓN,m � µ(1) t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , µ(m) t � (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2) with initialization µ(1) 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , µ(m) 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' ∼ Uniform({Xi}1≤i≤N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The following theorem shows that the gradient flow (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2) captures the evolution of the location of particles in the Wasserstein gradient flow (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='18) initialized from a discrete distribution 1 m �m l=1 δµ(l) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The proof is deferred to Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Consider the Euclidean gradient flow (µt)t≥0 in (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then the flow (ρt)t≥0 defined as ρt := 1 m m � l=1 δµ(l) t (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3) is the Wasserstein gradient flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3) is a distributional solution to the PDE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Similar connection can also be established for the gradient descent algorithm for minimizing (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1) and the particle Wasserstein gradient descent (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Algorithm 3), which is omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we focus on the infinite sample limit of Wasserstein gradent flow and analyze its convergence property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The population level loss function is ℓ∞ (ρ) = −EX∼ρ⋆∗φ {log [ρ ∗ φ (X)]} = KL (ρ⋆ ∗ φ ∥ ρ ∗ φ) + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4) In Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 we have computed that δℓ∞ (ρ) = − � Rd ρ⋆ ∗ φ (y) ρ ∗ φ (y) φ (x − y) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5) We know that the Wasserstein gradient flow (ρt)t≥0 with respect to ℓ∞(ρ) is described by the following PDE: ∂tρt = div (ρt∇δℓ∞ (ρt)) (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6) 41 with ρ0 = ρ⋆ ∗ N(0, Id), which is the data distribution when we have infinite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This Wasserstein gradient flow has the following particle interpretation: suppose at time t = 0 we initialize a particle x0 ∼ ρ0 in the vector field (vt)t≥0 where vt = −∇δℓ∞(ρt), namely ˙xt = vt (xt) , then xt ∼ ρt, namely the marginal distribution of (xt)t≥0 evolves according to the Wasserstein gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The following theorem shows that, when the true mixing distribution ρ⋆ is a singleton (we assume without loss of generality that ρ⋆ = δ0), Wasserstein gradient flow converges to ρ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The proof can be found in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Consider the Wasserstein gradient flow in (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6) with ρ⋆ = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any ε < 1, we have � Rd ∥x∥2 2ρt(dx) = O(ε) as long as t ≥ exp (2d) ε−1−max{8, √ 8d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Although Theorem 9 only focuses on the case when ρ⋆ is a singleton, the convergence result already provides some intuition about the behavior of Wasserstein gradient flow in more general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Consider a well-separated Gaussian mixture model with K components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Assume that the mixing distribution is ρ⋆ = �K j=1 ω⋆ j δµ⋆ j , and the location of each Gaussian components, {µ⋆ j}1≤j≤K, are well-separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Since the push-forward mapping vt = −∇δℓ∞(ρt) is localized (see F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='5), there exists some T > 0 such that the Wasserstein gradient flow (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6) initialized from ρ⋆ ∗ N(0, Id) can be approximated, up to time T, by ρt ≈ K � j=1 ω⋆ j ρ(j) t ∀ t ∈ [0, T] , where for each j ∈ [K], ρ(j) t is the Wasserstein gradient flow ∂tρ(j) t = div(ρ(j) t ∇δℓ∞(ρ(j) t )) with initialization ρ(j) 0 = N(µ⋆ j, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This suggests that Wasserstein gradient flow approximately converges to ρ⋆ since, by Theorem 9, each ρ(j) t converges to δµ⋆ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' However this observation also suggests that Wasserstein gradient flow is not robust to weight mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Consider initializing the Wasserstein gradient flow (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6) with ρ0 = �ρ ∗ N(0, 1), where �ρ = �K j=1 �ωjδµ⋆ j is a mixing distribution with correct support {µ⋆ j}1≤j≤K but wrong weights {�ωj}1≤j≤K ̸= {ω⋆ j }1≤j≤K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we also have ρt ≈ K � j=1 �ωjρ(j) t ∀ t ∈ [0, T] , which shows that ρt approximately converges to �ρ instead of ρ⋆ when 0 ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that the time length T that such approximations are valid can be arbitrarily large as long as the separation mini̸=j ∥µ⋆ i −µ⋆ j∥2 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The above discussion suggests that using the correct initial weights are important for Wasserstein gradient flow to converge to the true mixing distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We also would like to compare the convergence rate in Theorem 9 to a benchmark provided by the Bures- Wasserstein gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The Bures-Wasserstein gradient flow is defined on the space of non-degenerate Gaussian distributions on Rd, denoted by BW(Rd) = Rd×Sd ++ (where we identify a non-degenerate Gaussian distribution ν = N(µ, Σ) with (µ, Σ) ∈ Rd × Sd ++) equipped with the Wasserstein distance (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3), which has the following closed form expression d2 W (ν1, ν2) = ∥µ1 − µ2∥2 2 + tr � Σ1 + Σ2 − 2 � Σ1/2 1 Σ2Σ1/2 1 �1/2� when ν1 = N(µ1, Σ1) and ν2 = N(µ2, Σ2) are both non-degenerate Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The Bures-Wasserstein gradient flow (νt)t≥0 can be viewed as the Wasserstein gradient flow (ρt)t≥0 constrained to lie on BW(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We refer interested readers to Altschuler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lambert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2022) for more detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We can see from the proof of Theorem 9 that the push-forward mapping vt(x) of Wasserstein gradient flow decays exponentially fast as ∥x∥2 → ∞, this will make the Wasserstein gradient flow (ρt)t≥0 becomes more and more heavy-tailed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' However the push-forward mapping of Bures-Wasserstein gradient flow is always 42 linear, and the Bures-Wasserstein gradient flow (νt)t≥0 is always Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For example, in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 we can compute the push forward mapping explicitly for the two gradient flows at t = 0: v0 (x) = −1 3 �4 3 �d/2 exp � −∥x∥2 2 6 � x (Wasserstein), v0 (x) = x 4 (Bures-Wasserstein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore it is natural to expect that Bures-Wasserstein gradient flow (νt)t≥0 initialized from ν0 = N(0, Id) converges faster than Wasserstein gradient flow (ρt)t≥0 initialized from ρ0 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 we show that the Bures-Wasserstein gradient flow (νt = N(µt, Σt))t≥0 is characterized by the following ODE: µt = 0 ˙Σt = −2 (Σt + Id)−1 Σ2 t (Σt + Id)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We also show that Σt is sandwiched between 1 1 + 2tI ⪯ Σt ⪯ 2 2 + tI, and as a result � Rd ∥x∥2 2νt(dx) = O(d/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Since Bures-Wasserstein gradient flow is not converging exponen- tially fast (we can see that the convergence rate is polynomial in t), we conjecture that Wasserstein gradient flow does not enjoy exponential convergence as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Lastly, we numerically show in Figure 6 that the loss function ℓ∞(ρ) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4)) is not geodesically convex (Ambrosio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=', 2008) even when ρ⋆ = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We can also check that Polyak-Łojasiewicz (PL) inequality ∀ ρ : ∥∇Wℓ∞ (ρ)∥2 ρ ≥ CPL [ℓ∞ (ρ) − ℓ∞ (ρ⋆)] for some CPL > 0 does not hold in general: consider ρ⋆ = 1 2δ−1 + 1 2δ1 and ρ = δ0, then it is straightforward to check that ∇Wℓ∞(ρ) = 0 but ℓ∞(ρ) > ℓ∞(ρ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore we cannot use standard proof technique (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Ambrosio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2008) when the loss function is geodesically convex, or Chewi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2020) when there is a PL inequality) to show exponential convergence for the Wasserstein gradient flow (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 Proof of Theorem 8 It is straightforward to compute the gradient of ℓN,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any j ∈ [m], we have ∇µ(j)ℓN,m � µ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , µ(m)� = − 1 N N � i=1 1 �m l=1 φ � Xi − µ(l)�∇µ(j)φ � Xi − µ(j)� = − 1 N N � i=1 φ � Xi − µ(l)� �m l=1 φ � Xi − µ(l)� � Xi − µ(l)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore the Euclidean gradient flow (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2) is given by ˙µ(j) t = −∇µ(j)ℓN,m � µ(1) t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' , µ(m) t � = 1 N N � i=1 φ � Xi − µ(l) t � �m l=1 φ � Xi − µ(l) t � � Xi − µ(l) t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we can invoke Theorem 6 to finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 Proof of Theorem 9 Step 1: characterizing the push-forward mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' First of all, it is straightforward to check that (ρt)t≥0 is spherically symmetric for all t ≥ 0, namely ρt(dx) only depends on ∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' The push-forward mapping vt(x) : Rd → Rd at time t is vt (x) = −∇δℓ∞ (ρt) (x) = ∇x � Rd ρ⋆ ∗ φ (y) ρt ∗ φ (y) φ (x − y) dy 43 (a) constant speed geodesic (ρt)0≤t≤1 joining ρ0 = N(0, 3) to ρ1 = N(0, 1) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='05 Figure 6: The loss function ℓ∞(ρt) or its derivative ℓ′ ∞(ρt) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In Figures (a), (ρt)0≤t≤1 is the constant speed geodesic joining ρ0 = N(0, 3) to ρ1 = N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In Figures (b), (ρt)0≤t≤1 is the constant speed geodesic joining ρ0 = N(0, 1) to ρ1 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This shows that ℓ∞(ρ) is not globally geodesically convex, but might be locally geodesically conex around ρ⋆ = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' = − � Rd ρ⋆ ∗ φ (y) ρt ∗ φ (y) (x − y) φ (x − y) dy = � Rd ∇y ρ⋆ ∗ φ (y) ρt ∗ φ (y) φ (y − x) dy = � Rd ht (y) φ (y − x) dy, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7) where the penultimate line follows from Stein’s lemma or Gaussian integration by parts, and ht(y) in the last line is defined as ht (y) := ∇ φ (y) ρt ∗ φ (y) = ∇y 1 � exp � − 1 2 ∥z∥2 2 + y⊤z � ρt (dz) = − � exp � − 1 2 ∥z∥2 2 + y⊤z � zρt (dz) �� exp � − 1 2 ∥z∥2 2 + y⊤z � ρt (dz) �2 = −φ (y) � φ (y − z) zρt (dz) �� φ (y − z) ρt (dz) �2 = −φ (y) � φ (y − z) zρt (dz) [ρt ∗ φ (y)]2 = − φ (y) ρt ∗ φ (y) · � φ (y − z) zρt (dz) ρt ∗ φ (y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8) 44 For any y ∈ Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' we can compute ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='φ (y − z) zρt (dz) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤z>0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='φ (y − z) zρt (dz) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤z<0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='φ (y − z) zρt (dz) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤z=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='φ (y − z) zρt (dz) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤z>0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='[φ (y − z) − φ (y + z)] zρt (dz) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='= φ (y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤z>0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='− exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='−y⊤z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='z exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ∥z∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='ρt (dz) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='(ii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='= φ (y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤z>0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤z ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤z>0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤z ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ∥z∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='ρt (dz) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='=:at ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='φ (y) y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='9) Here (i) and (ii) both follow from the spherical symmetry of ρt, and it is straightforward to check that the integral in the last line does not depend on y due to the spherical symmetry of ρt, therefore at is a universal constant that is independent of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that when y⊤z > 0, we have exp(y⊤z) − exp(−y⊤z) ≥ 2y⊤z, therefore at ≥ 2 � y⊤z>0 � y⊤z �2 ∥y∥2 2 exp � −1 2 ∥z∥2 2 � ρt (dz) = � Rd � y⊤z �2 ∥y∥2 2 exp � −1 2 ∥z∥2 2 � ρt (dz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Since at does not depend on y, we take y = ei for i ∈ [d] to achieve at ≥ � Rd z2 i exp � −1 2 ∥z∥2 2 � ρt (dz) , ∀ i ∈ [d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' By taking average over d, we have at ≥ 1 d � Rd ∥z∥2 2 exp � −1 2 ∥z∥2 2 � ρt (dz) = mt d , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='10) where we define mt := � ∥z∥2 2 exp � −1 2 ∥z∥2 2 � ρt (dz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Taking (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='8), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='9) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='10) collectively gives ht (y) = −aty � φ (y) ρt ∗ φ (y) �2 , where at ≥ mt d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we use (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='7) to characterize the push-forward mapping: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='vt (x) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤x>0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='h (y) φ (y − x) dy + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤x<0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='h (y) φ (y − x) dy + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤x=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='h (y) φ (y − x) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤x>0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='h (y) [φ (y − x) − φ (−y − x)] dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='= −φ (x) at ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤x>0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='φ (y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='ρt ∗ φ (y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='�2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤x ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ∥y∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='(ii) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='= −φ (x) at ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='φ (y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='ρt ∗ φ (y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='�2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='− exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='−y⊤x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ∥y∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='= −at ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y⊤x>0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='x⊤y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='∥x∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='φ (y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='ρt ∗ φ (y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='�2 � ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ∥y∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='=:bt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='φ (x) x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' 45 Similar to (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='9), here (i) and (ii) both follow from the spherical symmetry of ρt, and the integral in the last line does not depend on x due to the spherical symmetry of ρt, as a result bt is a universal constant that is independent of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that when y⊤x > 0, we have exp(y⊤x) − exp(−y⊤x) ≥ 2y⊤x, therefore bt ≥ 2 � y⊤x>0 x⊤y ∥x∥2 2 � φ (y) ρt ∗ φ (y) �2 y⊤x exp � −1 2 ∥y∥2 2 � dy = � Rd � x⊤y �2 ∥x∥2 2 � φ (y) ρt ∗ φ (y) �2 exp � −1 2 ∥y∥2 2 � dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that ∥ρt ∗ φ∥∞ ≤ ∥φ∥∞ ≤ (2π)−d/2, and as a result bt ≥ � � x⊤y �2 ∥x∥2 2 exp � −3 2 ∥y∥2 2 � dy = 1 3 �2π 3 �d/2 Therefore we have vt (x) = −atbtφ (x) x = −ctφ (x) x (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11) where ct := atbt ≥ 1 3d �2π 3 �d/2 mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='12) Step 2: showing the convergence of Wasserstein gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Recall the particle interpretation of Wasserstein gradient flow as follows: let x0 ∼ ρ0 = N(0, Id) and ˙xt = vt(xt), then for any t ≥ 0 we have xt ∼ ρt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This allows us to compute ∂tE � ∥xt∥2 2 � = 2E [⟨xt, ˙xt⟩] = 2E [⟨xt, vt (xt)⟩] (i) = −2ctE � ∥xt∥2 2 φ (xt) � (ii) ≤ − 2 3d �2π 3 �d/2 mtE � ∥xt∥2 2 φ (xt) � (iii) = − 2 3d �4π2 3 �d/2 E2 � ∥xt∥2 2 φ (xt) � , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='13) where (i) follows from (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11), (ii) utilizes (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='12), and (iii) holds since mt = � ∥z∥2 2 exp � −1 2 ∥z∥2 2 � ρt (dz) = (2π)d/2 E � ∥xt∥2 2 φ (xt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' For any τ > 0, by Cauchy-Schwarz inequality we have E � ∥x0∥2 2 1 � ∥x0∥2 2 > d + τ �� (i) ≤ � E ∥x0∥4 2 �1/2 � P � ∥x0∥2 2 > d + τ ��1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Note that ∥x0∥2 2 ∼ χ2(d), therefore E∥x0∥4 2 = var(∥x0∥2 2) + (E∥x0∥2 2)2 = 2d + d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' In addition, by the tail probability bound for χ2 random variables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Wainwright (2019, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='18))), we have P � ∥x0∥2 2 > d + τ � ≤ exp � − min �τ 2 8d, τ 8 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore we have E � ∥x0∥2 2 1 � ∥x0∥2 2 > d + τ �� ≤ � 2d + d2 exp � − min �τ 2 8d, τ 8 �� ≤ (d + 1) exp � − min �τ 2 8d, τ 8 �� ≤ ε 46 as long as we choose τ ≜ max � 8 log d + 1 ε , � 8d log d + 1 ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Since the push forward mapping vt(x) is always pointing towards zero (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='11) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='12)), we know that ∥xt∥2 is non-increasing in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore we have E � ∥xt∥2 2 φ (xt) � ≥ E � ∥xt∥2 2 φ (xt) 1 � ∥x0∥2 2 ≤ d + τ �� (i) ≥ (2π)−d/2 exp � −d + τ 2 � E � ∥xt∥2 2 1 � ∥x0∥2 2 ≤ d + τ �� ≥ (2π)−d/2 exp � −d + τ 2 � � E � ∥xt∥2 2 � − E � ∥xt∥2 2 1 � ∥x0∥2 2 > d + τ ��� (ii) ≥ (2π)−d/2 exp � −d + τ 2 � � E � ∥xt∥2 2 � − E � ∥x0∥2 2 1 � ∥x0∥2 2 > d + τ ��� ≥ (2π)−d/2 exp � −d + τ 2 � � E � ∥xt∥2 2 � − ε � , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='14) where both (i) and (ii) follows from the fact that ∥xt∥2 is non-increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Taking (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='13) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='14) collectively gives ∂tE � ∥xt∥2 2 � ≤ − 2 3d �4π2 3 �d/2 (2π)−d exp [− (d + τ)] � E � ∥xt∥2 2 � − ε �2 = − 2 3d �1 3 �d/2 exp [− (d + τ)] � E � ∥xt∥2 2 � − ε �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Let f(t) = E[∥xt∥2 2], we know that f(0) = d and df dt ≤ − 2 3d �1 3 �d/2 exp [− (d + τ)] (f − ε)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Solving this ordinary differential inequality gives 1 f (t) − ε − 1 f (0) − ε ≥ 2 3d �1 3 �d/2 exp [− (d + τ)] t, which is equivalent to E � ∥xt∥2 2 � ≤ ε + � 2 3d �1 3 �d/2 exp (−d − τ) t + 1 d − ε �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we immediately know that E[∥xt∥2 2] ≤ O(ε) as long as t ≥ exp (2d) ε−1−max{8, √ 8d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 Calculation for Bures-Wasserstein gradient flow Define ℓ(µ, Σ) = ℓ∞(ρ) where we parameterize ρ = N(µ, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we can compute ℓ (µ, Σ) = − � log � (2π)−d/2 [det (Σ + Id)]−1/2 exp � −1 2 (x − µ)⊤ (Σ + Id)−1 (x − µ) �� φ (x) dx + constant = 1 2 log det (Σ + Id) + � 1 2 (x − µ)⊤ (Σ + Id)−1 (x − µ) φ (x) dx + constant 47 = 1 2 log det (Σ + Id) + 1 2Ex∼N (0,I) � (x − µ)⊤ (Σ + Id)−1 (x − µ) � + constant = 1 2 log det (Σ + Id) + 1 2tr � (Σ + Id)−1� + 1 2µ⊤ (Σ + Id)−1 µ + constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Then we can compute the Euclidean gradient of ℓ(µ, Σ) as follows: ∇µℓ (µ, Σ) = (Σ + Id)−1 µ, ∇Σℓ (µ, Σ) = 1 2 (Σ + Id)−1 − 1 2 (Σ + Id)−2 − 1 2 (Σ + Id)−1 µµ⊤ (Σ + Id)−1 = 1 2 (Σ + Id)−1 � Σ + Id − Id − µµ⊤� (Σ + Id)−1 = 1 2 (Σ + Id)−1 � Σ − µµ⊤� (Σ + Id)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' According to Lambert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2022, Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3), when initialized from (µ0, Σ0) = (0, Id), the Bures- Wasserstein gradient flow can be described using the following ODE: ˙µt = − (Σt + Id)−1 µt ˙Σt = −Σt (Σt + Id)−1 � Σt − µµ⊤� (Σt + Id)−1 − (Σt + Id)−1 � Σt − µµ⊤� (Σt + Id)−1 Σt with initial condition µ0 = 0 and Σ0 = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' It is straightforward to check that µt = 0 for all t ≥ 0, and the dynamic of Σt is governed by ˙Σt = −Σt (Σt + Id)−1 Σt (Σt + Id)−1 − (Σt + Id)−1 Σt (Σt + Id)−1 Σt = −Σt (Σt + Id)−1 + 2 (Σt + Id)−1 Σt (Σt + Id)−1 − (Σt + Id)−1 Σt = −2Id + 2 (Σt + Id)−1 + 2 (Σt + Id)−1 Σt (Σt + Id)−1 = −2 (Σt + Id)−1 Σ2 t (Σt + Id)−1 with initial condition Σ0 = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We can check that the off-diagonal entries of Σt are always zero, and its diagonal entries are identical and evloves according to the following ODE ˙σt = −2 σ2 t (σt + 1)2 with initial condition σ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' It is straightforward to check that σt is monotonically decreasing and is always non-negative, namely 0 ≤ σt ≤ 1 always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' Therefore we have −2σ2 t ≤ ˙σt ≤ −1 2σ2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' This gives 1 1 + 2t ≤ σt ≤ 2 2 + t, and therefore 1 1 + 2tI ⪯ Σt ⪯ 2 2 + tI, which suggests that ρt converges to ρ⋆ at the speed of O(d/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' When t = 0, we can compute the push forward mapping of Wasserstein gradient flow explicitly, which intuitively explains why Wasserstein gradient flow does not converge exponentially fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' We first compute ∇ρ⋆ ∗ φ (y) ρ0 ∗ φ (y) == ∇ (det I)−d/2 exp � − 1 2 ∥y∥2 2 � (det 2I)−d/2 exp � − 1 4 ∥y∥2 2 � = 2d/2∇ exp � −1 4 ∥y∥2 2 � = −2d/2−1y exp � −1 4 ∥y∥2 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='then the push forward mapping at t = 0 is given by x �→ v0(x) where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='v0 (x) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='∇y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='ρ⋆ ∗ φ (y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='ρ0 ∗ φ (y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='φ (y − x) dy = −2d/2−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4 ∥y∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='(2π)d/2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ∥x − y∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='= − 2d/2−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='(2π)d/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ∥x∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 + x⊤y − 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4 ∥y∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='dy = − 2d/2−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='(2π)d/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='y exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6 ∥x∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 − 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='����y − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='= −2d/2−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='�d/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6 ∥x∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='(2π)d/2 (2/3)d/2 y exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='����y − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='= −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='�4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='�d/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='6 ∥x∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' On the other hand, in view of Lambert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content=' (2022, Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtAzT4oBgHgl3EQfzP6J/content/2301.01766v1.pdf'} +page_content='3), the 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In this paper we describe the mathematical foundations of a new approach to semi- +supervised Machine Learning. Using techniques of Symbolic Computation and Computer Algebra, +we apply the concept of persistent homology to obtain a new semi-supervised learning method. +INTRODUCTION +Machine Learning and Deep Learning methods have become the state-of-the-art approach for +solving data classification tasks. In order to use those methods, it is necessary to acquire and +label a considerable amount of data; however, this is not straightforward in some fields, since data +annotation is time consuming and may require expert knowledge. This challenge can be tackled +by means of semi-supervised learning methods that take advantage of both labelled and unlabelled +data. In our team we have applied this Machine Learning paradigm in various applied projects +(e.g. [3]). In this paper, we present a new semi-supervised learning method based on techniques +from Topological Data Analysis. In particular, we have used a homological approach that consists +of studying the persistence diagrams associated with data from binary classification tasks using +the bottleneck and Wasserstein distances. In addition, we have carried out a thorough analysis +of the developed method using 5 structured datasets. The results show that the semi-supervised +method developed in this work outperforms both the results obtained with models trained with +only manually labelled data, and those obtained with classical semi-supervised learning methods, +improving the models by up to a 16%. +1. CONCEPTUAL PRESENTATION +Our method falls within the discipline of Topological Data Analysis (hereinafter TDA), a field +devoted to extracting topological and geometrical information from data. And the problem under- +taken is motivated by the challenge of obtaining enough annotated data to apply Machine Learning +techniques. To that end, a family of methods that has been successfully applied in the literature is +semi-supervised learning. Semi-supervised learning methods provide a means of using unlabelled +data to improve models’ performance when we have access to a large corpus of data that is difficult +to annotate. Traditional semi-supervised learning algorithms, such as Label Spreading [4] and La- +bel Propagation [5], focus on the distance among the data points to annotate unlabelled data points; +i.e. on the metric and density characteristics of the data in a dataset. However, there are contexts +where metric approaches could be misleading. As shown in Figure 1, distances are not the right +discriminators in complex situations and, therefore other ideas are needed. Our inspiration comes +from the Manifold Hypothesis [2], which explores when high dimensional data could tend to lie in +low dimensional manifolds. Roughly speaking, our method works under the hypothesis that each +This work was partially supported by the projects PID2020-115225RB-I00 and PID2020-116641GB-I00, funded by +MCIN/AEI/10.13039/501100011033 and by “European Union NextGenerationEU/PRTR”. +1 +arXiv:2301.11658v1 [cs.LG] 27 Jan 2023 + +2 +ADRIÁN INÉS, CÉSAR DOMÍNGUEZ, JÓNATHAN HERAS, GADEA MATA AND JULIO RUBIO +Unlabeled +Class 0 +Class 1 +Supervised +Self training +Label Spreading +Label Propagation +FIGURE 1. Example with two “connected manifolds” +class in the dataset lies on a manifold. In particular, homological information should be respected +when we add an unlabelled point to one of the classes. Our method is therefore as follows: given +two sets of data points A and B, corresponding to the points labelled with class 1 and class 2, re- +spectively, we assume there are two manifolds associated with each set, MA and MB respectively; +now, given an unlabelled data point x, if x belongs to class 1, for instance, then A ∪ {x} would +lie on a manifold more similar to MA than the manifold corresponding to B ∪ {x} with respect to +MB. +All the code developed for this project and also the conducted experiments are available at the +project webpage https://github.com/adines/TTASSL. +2. DESCRIPTION OF THE METHOD +In this section, we describe the semi-supervised learning algorithm that we have designed to +tackle binary classification tasks. We start with a set X1 of points from class 1, a set X2 of points +from class 2, and a set X of unlabelled points. The objective of our algorithms is to annotate the +elements of X by using topological properties of X1 and X2. We assume some familiarity with +notions employed in TDA such as Vietoris-Rips filtration (we denote the Vietoris-Rips filtration +associated with a set X by VX), persistence diagrams (we denote the persistence diagram associated +with a filtration F by P(F)), and the bottleneck and Wasserstein distances (denoted by dB and dW +respectively). For a detailed introduction to these topics see [6]. +Our semi-supervised learning algorithm takes as input the sets X1 and X2, a point x ∈ X, a +threshold value t, and a flag that indicates whether the bottleneck or the Wasserstein distance should +be used. We denote the chosen distance as d. The output produced by our algorithm is whether the +point x belongs to X1, X2 or none of them. In order to decide the output of the algorithm, our +hypothesis is that if a point belongs to X1, analogously for X2, then when adding the point to the + +SEMI-SUPERVISED MACHINE LEARNING: A HOMOLOGICAL APPROACH +3 +manifold on which X1 lies, the topological variation will be minimal; whereas if the point does not +belong to X1, the variation will be greater. In particular, we proceed as follows: +(1) Construct the Vietoris-Rips filtrations VX1, VX2, VX1∪{x} and VX2∪{x}; +(2) Construct the persistence diagrams P(VX1), P(VX2), P(VX1∪{x}) and P(VX2∪{x}); +(3) Compute the distances d(P(VX1), P(VX1∪{x})) and d(P(VX2), P(VX2∪{x})), from now +on d1 and d2 respectively; +(4) If both d1 and d2 are greater than the threshold t, return none; otherwise, return the set +associated with the minimum of the distances d1 and d2. +The algorithm above is diagrammatically described in Figure 2, and it is applied to all the points +of the set of unlabelled points X. +distance 0.1285 +distance 0.4958 +FIGURE 2. Example of the application of our method using the bottleneck distance, and +using 0.6 as threshold value. +3. EVALUATION +Table 1 presents the results with 5 different datasets taken from the UCI Machine Learning +Repository [1], training the models with two machine learning algorithms, which are Support Vec- +tor Machines (SVM in the table) and Random Forest (RF), and comparing our method with three +classical semi-supervised learning techniques (namely, Label Propagation [5], Label Spreading [4], + +1.0 +0.8 +0.6 +0.4 +0.2 +Ho +0.0 +Hi +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Birth1.0 ++ ++ ++ +0 ++ ++ +× 1 +0.8 ++ +0.6 ++ +0.4 +X ++ ++ +0.2 +X +0.0 ++ ++ +0.2 +-0.4 +X +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.01.0 ++ ++ ++ +0 ++ ++ +X +1 +0.8 +C +unlabelled ++ +0.6 ++ +0.4 +X ++ ++ +0.2 +0.0 +. ++ +0.2 +-0.4 +X +X +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.01.0 ++ ++ +0 ++ ++ +0.8 ++ +0.6 ++ +0.4 ++ ++ +0.2 +0.0 ++ +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.001.0 ++ ++ +0 ++ +0.8 ++ +0.6 ++ +0.4 ++ ++ +0.2 +0.0 ++ ++ +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.000.5 +0.4 +0.3 +0.2 +0.1 - +Ho +0.0 +Hi +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Birth0.5 +0.4 +. +0.3 - +. +0.2 +0.1 - +Ho +0.0 +Hi +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Birthx +0.4 - +0.2 +0.0 +X +X +0.2 +X +X +-0.4 +0.0 +0.5 +1.0 +1.5 +2.00.4 +0.2 +X +X +0.0 +X +X +0.2 +X +-0.4 +X +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.00.5 +0.4 - +0.3 +Death +0.2 - +0.1 +Ho +0.0 +H1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Birth4 +ADRIÁN INÉS, CÉSAR DOMÍNGUEZ, JÓNATHAN HERAS, GADEA MATA AND JULIO RUBIO +TABLE 1. Accuracy results for the SVM and RF classifiers trained with data annotated +for each of the annotation methods (classical and homological) together with the results +obtained with the initial data (base) in the 5 structured datasets. The best result for each +dataset is highlighted in bold face. +Banknote +Breast Cancer +Ionosphere +Prima Indian +Sonar +Mean (std) +Method +SVM +RF +SVM +RF +SVM +RF +SVM +RF +SVM +RF +SVM +RF +Base +97.0 +88.6 +89.3 +96.1 +80.0 +93.3 +65.7 +60.8 +61.3 +64.5 +78.7(15.2) +80.7(16.7) +Label Propagation +97.4 +93.2 +90.3 +89.3 +86.7 +86.7 +64.3 +68.5 +58.1 +54.8 +79.3(17.1) +78.5(16.3) +Label Spreading +97.4 +93.2 +90.3 +89.3 +86.7 +86.7 +64.3 +68.5 +58.1 +54.8 +79.3(17.1) +78.5(16.3) +Self Training classifier +95.1 +93.6 +35.9 +35.9 +85.0 +86.7 +66.4 +66.4 +58.1 +67.7 +68.1(23.2) +70.1(22.4) +Bottleneck threshold 0.8 +99.2 +92.4 +93.2 +91.3 +78.3 +95.0 +63.6 +64.3 +61.3 +64.5 +79.1(17.0) +81.5(15.6) +Bottleneck threshold 0.6 +99.2 +91.3 +89.3 +90.3 +75.0 +88.3 +59.4 +63.6 +48.4 +45.2 +74.3(20.9) +75.7(20.6) +Bottleneck threshold 0.4 +97.4 +90.5 +87.4 +85.4 +78.3 +86.7 +63.6 +62.9 +45.2 +45.2 +74.4(20.5) +74.1(19.5) +Bottleneck threshold 0.2 +97.4 +90.5 +87.4 +85.4 +78.3 +86.7 +63.6 +62.9 +45.2 +45.2 +74.4(20.5) +74.1(19.5) +Bottleneck threshold 0.0 +97.4 +90.5 +87.4 +85.4 +78.3 +86.7 +63.6 +62.9 +45.2 +45.2 +77.1(22.6) +74.1(19.5) +Wasserstein threshold 0.8 +97.4 +89.8 +92.2 +88.4 +80.0 +95.0 +68.5 +67.8 +61.3 +64.5 +79.9(15.3) +81.1(13.9) +Wasserstein threshold 0.6 +99.2 +93.6 +89.3 +87.4 +70.0 +91.7 +61.5 +61.5 +74.2 +61.3 +78.9(15.2) +79.1(16.3) +Wasserstein threshold 0.4 +97.0 +96.2 +87.4 +87.4 +76.7 +81.7 +60.8 +62.9 +71.0 +71.0 +78.6(14.1) +79.8(13.2) +Wasserstein threshold 0.2 +97.0 +96.2 +87.4 +87.4 +76.7 +81.7 +60.8 +62.9 +71.0 +71.0 +78.6(14.1) +79.8(13.2) +Wasserstein threshold 0.0 +97.0 +96.2 +87.4 +87.4 +76.7 +81.7 +60.8 +62.9 +71.0 +71.0 +78.6(14.1) +79.8(13.2) +and Self Training) to annotate the unlabelled data. From these results, we can extract several con- +clusions: our method improves the base results in 8 out of the 10 models and obtains better results +than the classical semi-supervised learning techniques in 8 out of the 10 models. +4. CONCLUSIONS AND FURTHER WORK +In this paper, we have studied the application of Topological Data Analysis techniques to the +semi-supervised learning setting. The results show that our method can create classification mod- +els that achieve better results than those obtained when using classical semi-supervised learning +methods. We plan to extend our work in different ways. First of all, the proposed method can be +expanded to multi-class classification tasks, and, an iterative version of the algorithm can be easily +developed. In addition, we plan to design new semi-supervised learning algorithms based on other +notions from TDA, taking further advantage of the Manifold Hypothesis. +REFERENCES +[1] D. Dua and C. Graff. UCI Machine Learning Repository. http://archive.ics.uci.edu/ml. 2017 +[2] C. Fefferman, S. Mitter and H. Narayanan. Testing the Manifold Hypothesis. Journal of the American Mathematical +Society. Vol. 29 (4), 983–1049. 2016 +[3] A. Inés, C. Domínguez, J. Heras, E. Mata, and V. Pascual. Biomedical image classification made easier thanks to +transfer and semi-supervised learning. Computer Methods and Programs in Biomedicine. Vol. 198, 105782. 2021 +[4] D. Zhou, O. Bousquet, T. N. Lal, J. Weston and B. Schölkopf. Learning with local and global consistency. Advances +in Neural Information Processing Systems 16, 321–328. 2004 +[5] X. Zhu and Z. Ghahramani. Learning from Labeled and Unlabeled Data with Label Propagation. Tech. Report. 2002 +[6] A. Zomorodian. Topological data analysis. Advances in Applied and Computational Topology. Vol. 70, 1–39. 2012 +Departamento de Matemáticas y Computación. Universidad de La Rioja +Email address: {adrian.ines, cesar.dominguez, jonathan.heras}@unirioja.es +Email address: {gadea.mata, julio.rubio}@unirioja.es + diff --git a/QtFJT4oBgHgl3EQf3C0d/content/tmp_files/load_file.txt b/QtFJT4oBgHgl3EQf3C0d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b379d4d8d0cc4285a4e569f516df35370d09dca5 --- /dev/null +++ b/QtFJT4oBgHgl3EQf3C0d/content/tmp_files/load_file.txt @@ -0,0 +1,466 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf,len=465 +page_content='SEMI-SUPERVISED MACHINE LEARNING: A HOMOLOGICAL APPROACH ADRIÁN INÉS, CÉSAR DOMÍNGUEZ, JÓNATHAN HERAS, GADEA MATA AND JULIO RUBIO ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' In this paper we describe the mathematical foundations of a new approach to semi- supervised Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Using techniques of Symbolic Computation and Computer Algebra, we apply the concept of persistent homology to obtain a new semi-supervised learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' INTRODUCTION Machine Learning and Deep Learning methods have become the state-of-the-art approach for solving data classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' In order to use those methods, it is necessary to acquire and label a considerable amount of data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' however, this is not straightforward in some fields, since data annotation is time consuming and may require expert knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' This challenge can be tackled by means of semi-supervised learning methods that take advantage of both labelled and unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' In our team we have applied this Machine Learning paradigm in various applied projects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' In this paper, we present a new semi-supervised learning method based on techniques from Topological Data Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' In particular, we have used a homological approach that consists of studying the persistence diagrams associated with data from binary classification tasks using the bottleneck and Wasserstein distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' In addition, we have carried out a thorough analysis of the developed method using 5 structured datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' The results show that the semi-supervised method developed in this work outperforms both the results obtained with models trained with only manually labelled data, and those obtained with classical semi-supervised learning methods, improving the models by up to a 16%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' CONCEPTUAL PRESENTATION Our method falls within the discipline of Topological Data Analysis (hereinafter TDA), a field devoted to extracting topological and geometrical information from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' And the problem under- taken is motivated by the challenge of obtaining enough annotated data to apply Machine Learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' To that end, a family of methods that has been successfully applied in the literature is semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Semi-supervised learning methods provide a means of using unlabelled data to improve models’ performance when we have access to a large corpus of data that is difficult to annotate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Traditional semi-supervised learning algorithms, such as Label Spreading [4] and La- bel Propagation [5], focus on the distance among the data points to annotate unlabelled data points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' on the metric and density characteristics of the data in a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' However, there are contexts where metric approaches could be misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' As shown in Figure 1, distances are not the right discriminators in complex situations and, therefore other ideas are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Our inspiration comes from the Manifold Hypothesis [2], which explores when high dimensional data could tend to lie in low dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Roughly speaking, our method works under the hypothesis that each This work was partially supported by the projects PID2020-115225RB-I00 and PID2020-116641GB-I00, funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='13039/501100011033 and by “European Union NextGenerationEU/PRTR”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='11658v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='LG] 27 Jan 2023 2 ADRIÁN INÉS, CÉSAR DOMÍNGUEZ, JÓNATHAN HERAS, GADEA MATA AND JULIO RUBIO Unlabeled Class 0 Class 1 Supervised Self training Label Spreading Label Propagation FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Example with two “connected manifolds” class in the dataset lies on a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' In particular, homological information should be respected when we add an unlabelled point to one of the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Our method is therefore as follows: given two sets of data points A and B, corresponding to the points labelled with class 1 and class 2, re- spectively, we assume there are two manifolds associated with each set, MA and MB respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' now, given an unlabelled data point x, if x belongs to class 1, for instance, then A ∪ {x} would lie on a manifold more similar to MA than the manifold corresponding to B ∪ {x} with respect to MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' All the code developed for this project and also the conducted experiments are available at the project webpage https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='com/adines/TTASSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' DESCRIPTION OF THE METHOD In this section, we describe the semi-supervised learning algorithm that we have designed to tackle binary classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' We start with a set X1 of points from class 1, a set X2 of points from class 2, and a set X of unlabelled points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' The objective of our algorithms is to annotate the elements of X by using topological properties of X1 and X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' We assume some familiarity with notions employed in TDA such as Vietoris-Rips filtration (we denote the Vietoris-Rips filtration associated with a set X by VX), persistence diagrams (we denote the persistence diagram associated with a filtration F by P(F)), and the bottleneck and Wasserstein distances (denoted by dB and dW respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' For a detailed introduction to these topics see [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Our semi-supervised learning algorithm takes as input the sets X1 and X2, a point x ∈ X, a threshold value t, and a flag that indicates whether the bottleneck or the Wasserstein distance should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' We denote the chosen distance as d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' The output produced by our algorithm is whether the point x belongs to X1, X2 or none of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' In order to decide the output of the algorithm, our hypothesis is that if a point belongs to X1, analogously for X2, then when adding the point to the SEMI-SUPERVISED MACHINE LEARNING: A HOMOLOGICAL APPROACH 3 manifold on which X1 lies, the topological variation will be minimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' whereas if the point does not belong to X1, the variation will be greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' In particular, we proceed as follows: (1) Construct the Vietoris-Rips filtrations VX1, VX2, VX1∪{x} and VX2∪{x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' (2) Construct the persistence diagrams P(VX1), P(VX2), P(VX1∪{x}) and P(VX2∪{x});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' (3) Compute the distances d(P(VX1), P(VX1∪{x})) and d(P(VX2), P(VX2∪{x})), from now on d1 and d2 respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' (4) If both d1 and d2 are greater than the threshold t, return none;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' otherwise, return the set associated with the minimum of the distances d1 and d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' The algorithm above is diagrammatically described in Figure 2, and it is applied to all the points of the set of unlabelled points X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='1285 distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='4958 FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Example of the application of our method using the bottleneck distance, and using 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='6 as threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' EVALUATION Table 1 presents the results with 5 different datasets taken from the UCI Machine Learning Repository [1], training the models with two machine learning algorithms, which are Support Vec- tor Machines (SVM in the table) and Random Forest (RF), and comparing our method with three classical semi-supervised learning techniques (namely, Label Propagation [5], Label Spreading [4], 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='8 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='2) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='7(16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='7) Label Propagation 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='3 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='7 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='7 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='1 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='3(17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='1) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='5(16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='3) Label Spreading 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='3 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='3 86.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='1) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='5(16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='3) Self Training classifier 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='6 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='9 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='9 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='7 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='1 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='7 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='1(23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='2) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='1(22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='4) Bottleneck threshold 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='2 91.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='1(17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='0) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='5(15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='6) Bottleneck threshold 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='6 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='3 89.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='6(14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='1) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='8(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='2) and Self Training) to annotate the unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' From these results, we can extract several con- clusions: our method improves the base results in 8 out of the 10 models and obtains better results than the classical semi-supervised learning techniques in 8 out of the 10 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' CONCLUSIONS AND FURTHER WORK In this paper, we have studied the application of Topological Data Analysis techniques to the semi-supervised learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' The results show that our method can create classification mod- els that achieve better results than those obtained when using classical semi-supervised learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' We plan to extend our work in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' First of all, the proposed method can be expanded to multi-class classification tasks, and, an iterative version of the algorithm can be easily developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' In addition, we plan to design new semi-supervised learning algorithms based on other notions from TDA, taking further advantage of the Manifold Hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' REFERENCES [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Dua and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Graff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' UCI Machine Learning Repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' http://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='edu/ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' 2017 [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Fefferman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Mitter and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Narayanan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Testing the Manifold Hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Journal of the American Mathematical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' 29 (4), 983–1049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' 2016 [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Inés, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Domínguez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Heras, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Mata, and V.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Zhou, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Bousquet, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Lal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Weston and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Schölkopf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Learning with local and global consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 16, 321–328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' 2004 [5] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Zhu and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Ghahramani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Learning from Labeled and Unlabeled Data with Label Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' 2002 [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Zomorodian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Topological data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Advances in Applied and Computational Topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' 70, 1–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' 2012 Departamento de Matemáticas y Computación.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content=' Universidad de La Rioja Email address: {adrian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='ines, cesar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='dominguez, jonathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='heras}@unirioja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='es Email address: {gadea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='mata, julio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='rubio}@unirioja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} +page_content='es' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFJT4oBgHgl3EQf3C0d/content/2301.11658v1.pdf'} diff --git a/RdE2T4oBgHgl3EQfBwYp/content/2301.03605v1.pdf b/RdE2T4oBgHgl3EQfBwYp/content/2301.03605v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..670836508fc354d4592651367a59471ab0b0f00e --- /dev/null +++ b/RdE2T4oBgHgl3EQfBwYp/content/2301.03605v1.pdf @@ -0,0 +1,3 @@ +version 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PDEs require for stability a CFL-condition which implies that the time step size +depends on the size of the elements of the mesh. On cut-cell meshes, elements can become arbitrarily small and thus the time +step size cannot take the size of small cut-cells into account but has to be chosen based on the background mesh elements. +A remedy for this is the so called DoD (domain of dependence) stabilization for which several favorable theoretical +and numerical properties have been shown in one and two space dimensions [4, 9]. Up to now the method is restricted to +stabilization of cut-cells with exactly one inflow and one outflow face, i.e. triangular cut-cells with a no-flow face (see [4]). +We extend the DoD stabilization to cut-cells with multiple in- and outflow faces by properly considering the flow distribu- +tion inside the cut-cell. We further prove L2-stability for the semi-discrete formulation in space and present numerical results +to validate the proposed extension. +Copyright line will be provided by the publisher +1 +Introduction +To avoid the mesh generation process of complex geometries, cut-cell methods are an attractive alternative. The general idea +is to start with a simple, e.g. structured, background mesh and to cut out the desired geometry. This results in a mesh with +unstructured polyhedral cells, called cut-cells. Cut-cells can have an arbitrary shape and can become arbitrarily small, causing +the small cell problem. To use explicit time stepping schemes for solving hyperbolic conservation laws, the time step size +would need to be chosen based on the smallest cut-cell in the grid to ensure stability, which is in general not feasible. +Developing solution approaches to the small cell problem in the context of discontinuous Galerkin (DG) schemes is a very +recent research branch, including for example the work in [7,8,10]. In this contribution we focus on the domain of dependence +(DoD) stabilization, which was introduced in [4] for the linear transport equation in one and two space dimensions and was +extended to non-linear systems in one space dimension in [9]. It is based on a DG scheme in space to allow for higher- +order approximations and possesses several desirable theoretical properties. Numerical results show the expected higher-order +behavior in smooth flow and robustness around shocks. +Up to now, the DoD stabilization in two dimensions has only been used to stabilize small triangular cut-cells for linear +advection parallel to a ramp [4,11]. In this setup, the small stabilized cut-cells have exactly one inflow and one outflow face, +which was exploited in the design of the stabilization. When moving to non-linear or coupled linear problems, this does not +hold true anymore and one has to deal with multiple inflow and outflow faces. +In this work, we take the first step in that direction by considering the linear advection equation on a cut-cell mesh with +arbitrary flow directions, resulting in triangular cut-cells having 2 inflow and 1 outflow neighbor or reverse. As this causes +significant additional complications, we will only consider piecewise constant polynomials here. We will prove L2-stability +for the semi-discrete scheme and present numerical results to validate the new extension of the stabilization terms. +The outline of the paper is as follows: we will first describe the problem setup and then introduce the new extended +stabilization. Then we will show the L2-stability proof and conclude with numerical results. +2 +Problem setup +Let Ω ⊂ R2 be an open and connected domain. We consider linear hyperbolic systems of the form +ut + Aux + Buy = 0 +in Ω × (0, T), +(1a) +τu = g +on ∂Ω × (0, T), +(1b) +u = u0 +on Ω × {t = 0}, +(1c) +where u(t) ∈ Rm, and A, B ∈ Rm×m constant, and τ is an appropriate boundary operator such that we only impose inflow +boundary conditions on incoming waves (and not everywhere on ∂Ω). We require that for any unit vector n = (n1, n2)T ∈ S1 +the matrices C = (n1A + n2B) are symmetric and simultaneously diagonalizable over the reals, i.e. there is an orthogonal +matrix O ∈ Rm×m and diagonal matrices Λn ∈ Diag(Rm×m) such that n1A + n2B = OΛnOT ∀n ∈ S1. +∗ Corresponding author: e-mail g_birk01@wwu.de +Copyright line will be provided by the publisher +arXiv:2301.02715v1 [math.NA] 6 Jan 2023 + +2 +Gunnar Birke1,, Christian Engwer1,, Sandra May2,, and Florian Streitbürger3, +� +Mh +∩ +¯Ω +(x0, 0) +γ += +Mh +E ∈ � +Mh +E1 +E2 +Fig. 1: Construction of the mesh: Out of the structured grid � +Mh on the domain Ω the mesh Mh is constructed by introducing cut-cells +E1, E2 ⊂ E ∈ � +Mh along the cut such that ¯ +E1 ∪ ¯ +E2 = ¯E. +In our numerical tests we will choose Ω = [0, 1]2 and discretize it by a structured grid � +Mh. We then introduce an artificial +cut, a straight line going through the square, starting at (x0, 0) and having an angle γ relative to the x-axis. This creates an +internal boundary with two subdomains which we will resolve by a cut-cell mesh Mh. A sketch is contained in Fig. 1. So far +we have always tested with flow parallel to that cut. Here, we consider flow in various directions, keeping the cut fixed. +We define the sets of internal and external faces as +Fint +h = {F = ∂E1 ∩ ∂E2|E1, E2 ∈ Mh, E1 ̸= E2, |F| > 0}, +Fext +h += {F = ∂E ∩ ∂Ω|E ∈ Mh, |F| > 0}, +and the set of faces of an element E ∈ Mh by FE +h = {F ∈ Fint +h ∪ Fext +h |F ⊂ ∂E}. We choose a fixed local numbering on +each of these sets and denote the neighbor element of E corresponding to a face Fi ∈ FE +h by Ei. For the discretization in +space we choose the discrete function space +V0 +h(Mh) = {vh ∈ L2(Ω)m | (vh)i|E ∈ P0(E) ∀(E, i) ∈ Mh × {1, .., m}}. +For vh ∈ V0 +h(Mh) we denote by vE +h the value of vh on an element E ∈ Mh. +For interior faces F ∈ Fint +h we fix an orientation of the outer unit normal vector nF and denote the inner and outer element +of F by E1 and E2, respectively. We then define average and jump by +{{uh}} := 1 +2(u +E1 +h + u +E2 +h ), +�uh� := u +E1 +h − u +E2 +h . +For exterior faces F ∈ Fext +h we simply choose the unit outer normal and denote by uEF +h +the solution on the cell that lies in the +interior of the domain and contains face F. We define the flux matrix on a face F as +CF = (nF )1A + (nF )2B = OΛF OT , +(2) +where (nF )1,2 denote the first and second component of the unit normal vector nF on face F. Based on this, we define +matrices which encode the flux directions as +C+ +F = OΛ+ +F OT , +C− +F = OΛ− +F OT +with +(Λ+ +F )i,i = max(0, (ΛF )i,i) +and +(Λ− +F )i,i = min(0, (ΛF )i,i). +Note that CF = C+ +F +C− +F . We also introduce a generalization of the absolute value for such flux matrices by |CF | = C+ +F −C− +F . +The (unstabilized) upwind semi-discretization in space is then given as: Find uh(t) ∈ V0 +h(Mh) such that +(∂tuh(t), vh)L2(Ω) + aupw +h (uh(t), vh) + lh(vh) = 0 +∀ vh ∈ V0 +h(Mh) +(3) +with +aupw +h (uh, vh) = +� +F ∈Fext +h +� +F +⟨C+ +F uEF +h , vEF +h ⟩ds + +� +F ∈Fint +h +� +F +⟨CF {{uh}}, �vh�⟩ + 1 +2⟨|CF | �uh� , �vh�⟩ds, +lh(vh) = − +� +F ∈Fext +h +� +F +⟨C− +F g, vEF +h ⟩ds. +Here, ⟨·, ·⟩ denotes the scalar product in Rm. We obtain aupw +h +and lh by integration by parts, where the integral over internal +edges leads to jump terms (second sum) and the boundary integral is split into outgoing waves (first sum) and incoming waves +(right hand side). We then discretize in time using the explicit Euler scheme. +Copyright line will be provided by the publisher + +DoD Stabilization of linear hyperbolic PDEs on general cut-cell meshes +3 +F1 +E1 +F2 +E2 +F3 +E3 +F4 +E4 +Ecut +β +⟨β, nF2⟩+ � +F2 +� +ω1(uE1 +h − uEcut +h +) ++ ω4(uE4 +h − uEcut +h +) +� +�v�ds +Fig. 2: Domain of dependence extension illustrated on a four-sided cut-cell: We introduce a direct mass transport from cells E1 and E4 into +E2. The colored regions indicate the coupling between the faces, the corresponding parts of the stabilization term on F2 are highlighted +accordingly. From the graphic one can see that flow orientation and geometry play a central part in determining the right mass redistribution +and the ωi will have to be chosen accordingly. The face F3 needs to be stabilized as well. Note that there should not be any coupling +between between F2 and F3 as both are outflow faces. +3 +Stabilization +To deal with small cut-cells, an additional term J0 +h = � +Ecut∈I J0,Ecut +h +with I being the set of small cut-cells that require +stabilization is added to the space semi-discretization (we will comment on our choice in the numerical results below). This +results in the following scheme: Find uh(t) ∈ V0 +h(Mh) such that +(∂tuh(t), vh)L2(Ω) + aupw +h (uh, vh) + J0 +h(uh, vh) + lh(vh) = 0 +∀ vh ∈ V0 +h(Mh). +(4) +Let Ecut be a small cut-cell that requires stabilization. The idea behind J0,Ecut +h +is the following: When the time step size is not +chosen to respect the small size of Ecut, the domain of dependence of an outflow neighbor E of Ecut will extend beyond Ecut. +We therefore extend the numerical DoD of E such that it receives information directly from the inflow neighbors of Ecut. The +amount of mass passed directly between neighbors of Ecut is chosen such that the update on Ecut becomes stable. +To explain the main concept we consider the scalar linear transport equation ut + ∇ · (βu) = 0 with constant velocity +β = (β1, β2)T ∈ R2, i.e., we use A = β1, B = β2 in (1). Consider Fig. 2 where the domain of dependence of E2 potentially +reaches into E1 and E4. In that case, for choosing the time step based on the size of background cells, mass (physically) is +moved from E1 and from E4 to E2 in a single time step and this coupling must be mimicked by the stabilization. +We introduce an extension operator Lext +E′(uh)(x) = uE′ +h (x) for E′ ∈ Mh which acts on functions uh ∈ V0 +h(Mh) and +corresponds to evaluating the (constant) polynomial of cell E′ outside its original support. Using this operator, the following +stabilization term was introduced in [4] for triangular cut-cells with single outflow face F2 and single inflow face F1: +J0,Ecut +h +(uh, vh) = ηEcut +� +F2 +⟨β, nF2⟩(Lext +E1(uh) − uEcut +h +) �vh� ds. +(5) +This term introduces a direct coupling between E1 and E2. The parameter ηEcut ∈ [0, 1] controls how much mass is transported +via this coupling. In the case of exactly one inflow face F1 and one outflow face F2 (and one face with no-penetration b.c.), +as considered in [4], this term suffices to ensure stability. Here, however, the cut-cell is allowed to have two inflow faces as +illustrated in Fig. 2. Furthermore, these inflow faces will in general influence multiple neighbors of the cut-cell where the +degree of influence depends on the geometry and flow direction. To handle this we propose the following extension of (5) +J0,Ecut +h +(uh, vh) = ηEcut +� +Fj∈FEcut +h +� +Fj +� +Fi∈FEcut +h +ωi⟨β, nFj⟩+(Lext +Ei(uh) − uEcut +h +) �vh� ds. +(6) +Here the ωi ∈ R provide information about the flow distribution for incoming flow of the face Fi. Note that the extended +solutions of all neighbor elements are evaluated on all faces, and ωi = 0 if Fi is not an inflow face. We provide a specific +formula of how to choose these weights for triangular cut-cells below in section 3.2. +Going back to the system case we allow ωi ∈ Rm×m and arrive at our final formulation +J0,Ecut +h +(uh, vh) = ηEcut +� +Fj∈FEcut +h +� +Fj +� +Fi∈FEcut +h +⟨ωiC+ +F (Lext +Ei(uh) − uEcut +h +), �vh�⟩ds. +(7) +Copyright line will be provided by the publisher + +4 +Gunnar Birke1,, Christian Engwer1,, Sandra May2,, and Florian Streitbürger3, +Note that in defining J0,Ecut we assume that all normal vectors nj for Fj ∈ FEcut +h +correspond to outward normal vectors with +respect to Ecut. In order to ensure consistency and stability the weights ωi must fulfill +� +Fi∈FEcut +h +ωi = Idm×m, +(8) +� +Fj∈FEcut +h +� +Fj +ωiC+ +j ds = − +� +Fi +C− +i ds +∀ Fi ∈ FEcut +h +. +(9) +Additionally we require that ωiC+ +Fj is always symmetric and positive semi-definite. Equation (8) can be understood as an +assurance that the overall amount of mass moved over a face by our stabilization is correct. For the scalar case, we require the +ωi to build a convex combination. Equation (9) means that a portion of the inflow is exactly redistributed over all outflow face +candidates. This in particular prevents overshoots on small cut-cells for appropriate choices of ηEcut. +3.1 +L2-stability +Equipped with the aforementioned properties of our stabilization we can show L2-stability for the semi-discrete scheme in +space. For brevity we consider homogeneous inflow boundary conditions and assume that exact and discrete solution vanish +on the boundary, so that we can ignore any domain boundary terms and focus on the situation of cut-cells. For P 0 functions +this means that they are zero in all boundary cells. +Theorem 3.1 Consider (1) with homogeneous boundary conditions. Assume that the discrete solution uh(t) vanishes on +the boundary ∂Ω for all t ∈ (0, T). Let uh(t) ∈ V0 +h(Mh) be the solution to the semi-discrete problem (4). Then it holds +||uh(t)||L2(Ω) ≤ ||uh(0)||L2(Ω) +∀ t ∈ (0, T). +P r o o f. We choose vh = uh(t) in (4). Any boundary terms vanish as trace(uh(t)) = 0 on ∂Ω. This yields +(∂tuh(t), uh(t))L2(Ω) + aupw +h (uh(t), uh(t)) + J0 +h(uh(t), uh(t)) = 0. +By the fundamental theorem of calculus +� t +0 +(∂τuh(τ), uh(τ))L2(Ω)dτ = +� t +0 +d +dτ +1 +2||uh(τ)||2 +L2(Ω)dτ = 1 +2||uh(t)||2 +L2(Ω) − 1 +2||uh(0)||2 +L2(Ω). +To ease notation we will write u = uh(t) in the following. The goal now is to show that aupw +h (u, u) + J0 +h(u, u) ≥ 0. We first +consider aupw +h (u, u), which expands into +aupw +h (u, u) = +� +F ∈Fint +h +� +F +⟨CF {{u}}, �u�⟩ + ⟨ 1 +2|CF |�u�, �u�⟩ds = +� +F ∈Fint +h +� +F +⟨C+ +F uE1 + C− +F uE2, uE1 − uE2⟩ds += +� +F ∈Fint +h +� +F +⟨C+ +F uE1, uE1⟩ − ⟨C+ +F uE1, uE2⟩ + ⟨C− +F uE2, uE1⟩ − ⟨C− +F uE2, uE2⟩ds. +Now we add zeros (in form of ± 1 +2⟨C− +F uE1, uE1⟩ and ± 1 +2⟨C+ +F uE2, uE2⟩) to get += +� +F ∈Fint +h +� +F +1 +2⟨(C+ +F − C− +F )uE1, uE1⟩ + 1 +2⟨(C+ +F + C− +F )uE1, uE1⟩ − ⟨C+ +F uE1, uE2⟩ ++ ⟨C− +F uE2, uE1⟩ + 1 +2⟨(C+ +F − C− +F )uE2, uE2⟩ − 1 +2⟨(C+ +F + C− +F )uE2, uE2⟩ds += +� +F ∈Fint +h +� +F +1 +2⟨|CF |(uE1 − uE2), uE1 − uE2⟩ + 1 +2⟨CF uE1, uE1⟩ − 1 +2⟨CF uE2, uE2⟩ +− 1 +2⟨CF uE1, uE2⟩ + 1 +2⟨CF uE2, uE1⟩ds. +Due to the symmetry of CF , the terms in the last line cancel each other. For the last two terms in the second to last line we +use the divergence theorem. Since uh(t) is elementwise constant, on E ∈ Mh it holds that +0 = +� +E +∇ · (⟨AuE, uE⟩, ⟨BuE, uE⟩)dx = +� +F ∈Fint +h ∪Fext +h ,F ∩∂E̸=∅ +� +F +⟨CF uE, uE⟩ds, +Copyright line will be provided by the publisher + +DoD Stabilization of linear hyperbolic PDEs on general cut-cell meshes +5 +and therefore, these terms vanish as well. Finally, due to |CF | being positive semi-definite, we obtain positivity of aupw +h (u, u): +aupw +h (u, u) = +� +F ∈Fint +h +� +F +1 +2⟨|CF |(uE1 − uE2), uE1 − uE2⟩ds ≥ 0. +We now investigate J0 +h. For a small cut-cell Ecut ∈ I we have due to (8) +J0,Ecut +h +(u, u) =ηEcut +� +Fj∈FEcut +h +� +Fj +⟨( +� +Fi∈FEcut +h +ωiC+ +FjuEi) − C+ +FjuEcut, uEcut − uEj⟩ds +=ηEcut +� +Fj∈FEcut +h +� +Fj +⟨ +� +Fi∈FEcut +h +ωiC+ +FjuEi, uEcut⟩ − ⟨ +� +Fi∈FEcut +h +ωiC+ +FjuEi, uEj⟩ +− ⟨C+ +FjuEcut, uEcut⟩ + ⟨C+ +FjuEcut, uEj⟩ds. +Adding again zeros (in form of ±⟨� +Fi∈FEcut +h +ωiC+ +FjuEi, uEi⟩ and ±⟨C+ +FjuEj, uEj⟩) and reordering gives += − 1 +2ηEcut +� +Fj∈FEcut +h +� +Fj +⟨C+ +FjuEcut, uEcut⟩ − 2⟨C+ +FjuEcut, uEj⟩ + ⟨C+ +FjuEj, uEj⟩ ++ ⟨ +� +Fi∈FEcut +h +ωiC+ +FjuEi, uEi⟩ − 2⟨ +� +Fi∈FEcut +h +ωiC+ +FjuEi, uEcut⟩ + ⟨C+ +FjuEcut, uEcut⟩ +− ⟨ +� +Fi∈FEcut +h +ωiC+ +FjuEi, uEi⟩ + 2⟨ +� +Fi∈FEcut +h +ωiC+ +FjuEi, uEj⟩ − ⟨C+ +FjuEj, uEj⟩ds += − 1 +2ηEcut +� +Fj∈FEcut +h +� +Fj +⟨C+ +Fj(uEcut − uEj), uEcut − uEj⟩ ++ +� +Fi∈FEcut +h +⟨ωiC+ +Fj(uEi − uEcut), uEi − uEcut⟩ +(C+ +Fj, ωiC+ +Fj symm. and (8)) +− +� +Fi∈FEcut +h +⟨ωiC+ +Fj(uEi − uEj), uEi − uEj⟩ds +(9)= − 1 +2ηEcut +� +Fj∈FEcut +h +� +Fj +⟨|CFj|(uEcut − uEj), uEcut − uEj⟩ +− +� +Fi∈FEcut +h +⟨ωiC+ +Fj(uEi − uEj), uEi − uEj⟩ds. +Since ηEcut ∈ [0, 1] the first term inside the sum can be compensated with terms from aupw +h . The second term is always non- +negative since ωiC+ +Fj is always positive semi-definite. Note that the second term corresponds to dissipation introduced by an +extended jump. This concludes the proof. +3.2 +Choice of parameters +To perform actual computations we need to select concrete ωi in (7) that fulfill properties (8) and (9). For the situation of linear +simultaneously diagonalizable hyperbolic systems and triangular cut-cells we suggest ωi = |Fi|C− +Fi(� +Fk∈FEcut +h +|Fk|C− +Fk)−1 +for each Fi ∈ FEcut +h +. For linear advection, this would result in ωi = 0 for an outflow edge Fi and ωi corresponding to some +sort of weighted proportion of the total inflow for an inflow edge Fi. We also need to set ηEcut for Ecut ∈ I. A stable but not +necessarily optimal choice is ηEcut = || � +F ∈FEcut +h +� +F Λ− +F ds||∞. +4 +Numerical results +For the numerical tests we select angles γ, θ, ρ1, ρ2 ∈ [0, 2π) where γ is the angle of the cut, see Fig. 1, and set +Λ1 = +�cos(ρ1) +0 +0 +cos(ρ2) +� +, Λ2 = +�sin(ρ1) +0 +0 +sin(ρ2) +� +, O = +�cos(θ) +− sin(θ) +sin(θ) +cos(θ) +� +. +Copyright line will be provided by the publisher + +6 +References +102 +2 × 102 +3 × 102 +4 × 102 +N +10−2 +10−1 +error +Setup 1: ρ1 = 7π +4 , ρ2 = π, γ = 40° +u1 L1-error 0.98 +u2 L1-error 0.99 +u1 L∞-error 0.99 +u2 L∞-error 0.99 +order 1 +102 +2 × 102 +3 × 102 +4 × 102 +N +10−2 +10−1 +error +Setup 2: ρ1 = 7π +4 , ρ2 = 3π +2 , γ = 30° +u1 L1-error 0.98 +u2 L1-error 0.99 +u1 L∞-error 0.99 +u2 L∞-error 0.97 +order 1 +Fig. 3: Plots of the error, measured in the L1- (blue, red) and L∞-norm (orange, transparent purple). Left plot has values ρ1 = 7π +4 , ρ2 = π +and γ = 40°, right plot has ρ1 = 7π +4 , ρ2 = 3π +2 and γ = 30°. In all cases we have chosen θ = 4π +3 . The green line is for reference. +Then our system matrices will be given by A = OΛ1OT and B = OΛ2OT . +The cut starts at (x0, 0) = (0.2001, 0). As initial conditions we choose u0(x) = O +�sin(2π(x1 cos(ρ1) + x2 sin(ρ1))) +cos(2π(x1 cos(ρ2) + x2 sin(ρ2))) +� +. +Boundary conditions are given by the exact solution. Let N denote the number of background cells in either coordinate +direction. We compute the time step size via ∆t = 0.4 +h +maxn∈S1 ||n1Λ1+n2Λ2||∞ , where h = +1 +N . The factor of 0.4 allows that +bigger cut-cells do not need to be stabilized. We then choose I = {E ∈ Mh| |E| +h2 < 0.4} as the set of stabilized cut-cells. The +final time of our simulation is T = 0.5. +Our implementation is based on the DUNE framework (see [2], [1]), in particular the dune-udg (see [3], [5]) and dune- +pdelab modules. The local subtriangulations for the cut-cells are computed by the TPMC library (see [6]). +Fig. 3 shows convergence plots for two particular setups. Note that the flow directions have been chosen to not be parallel +to the ramp angle γ. We observe the expected order of convergence for a first order scheme in both the L1- and the L∞-norm. +In addition, all numerically computed solution values, including those on small cut-cells, stayed within the bounds of the +initial state during the simulation, confirming the added stability of the DoD stabilization. +5 +Discussion and Outlook +We have extended the DoD stabilization to cut-cells with multiple inflow/outflow faces for the case of component-wise and +piecewise constant trial and test functions and linear, simultaneously diagonalizable systems in two dimensions. We have +proven L2-stability for the semi-discrete setting. Numerically we observe full first-order convergence in different numerical +tests and no over/undershoot on cut-cells. In future work, we plan to extend our method to more general systems, e.g., the +acoustics and Euler equations. An extension of the presented formulation to higher-order approximations is ongoing research. +Acknowledgements +This work has been partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) +as project 439956613 under contract numbers EN 1042/5-1 and MA 7773/4-1/2. +References +[1] P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, R. Kornhuber, M. Ohlberger, and O. Sander, Computing, 82, 121-138 (2008) +[2] P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, M. Ohlberger, and O. Sander, Computing, 82, 103-119 (2008) +[3] P. Bastian and C. Engwer, Int. Jour. for Num. Meth. in Eng., 79, 12, 1557-1576 (2009) +[4] C. Engwer, S. May, A. Nüßing, and F. Streitbürger, SIAM J. Sci. Comput. 42, 6, A3677-A3703 (2020). +[5] C. Engwer and F. Heimann, Proceedings of the DUNE user meeting, Stuttgart, Germany, Advances in DUNE (Springer Berlin, Heidel- +berg, 2012) pp. 89-100 +[6] C. Engwer and A. Nüßing, ACM Trans. on Math. Soft., 44, 2, Art. No. 14 (2018) +[7] P. Fu and G. Kreiss, SIAM J. Sci. Comput., 43,4, A2404–A2424, 2021. +[8] A. Giuliani, SIAM J. Sci. Comput. 44, 1, A389-A415 (2022) +[9] S. May, F. Streitbürger, Appl. Math. Comput. 419, Art. 126854 (2022). +[10] S. Schoeder, S. Sticko, G. Kreiss and M. Kronbichler, Int. J. Numer. Meth. Engrg. 121, 13, 2979-3003 (2020) +[11] F. Streitbürger, G. Birke, C. Engwer, and S. May, Spectral and High Order Methods for Partial Differential Equations ICOSAHOM +2020+1., 137 (Springer International Publishing, 2023). +Copyright line will be provided by the publisher + diff --git a/U9E0T4oBgHgl3EQf2wIM/content/tmp_files/load_file.txt b/U9E0T4oBgHgl3EQf2wIM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..01cb2c02bd2d9651dff389e2f6d179999b70097c --- /dev/null +++ b/U9E0T4oBgHgl3EQf2wIM/content/tmp_files/load_file.txt @@ -0,0 +1,269 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf,len=268 +page_content='DoD Stabilization of linear hyperbolic PDEs on general cut-cell meshes Gunnar Birke1,∗, Christian Engwer1,, Sandra May2,, and Florian Streitbürger3, 1 Applied Mathematics Münster, Münster University 2 Department of Information Technology, Uppsala University 3 Department of Mathematics, Dortmund University Standard numerical methods for hyperbolic PDEs require for stability a CFL-condition which implies that the time step size depends on the size of the elements of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' On cut-cell meshes, elements can become arbitrarily small and thus the time step size cannot take the size of small cut-cells into account but has to be chosen based on the background mesh elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' A remedy for this is the so called DoD (domain of dependence) stabilization for which several favorable theoretical and numerical properties have been shown in one and two space dimensions [4, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Up to now the method is restricted to stabilization of cut-cells with exactly one inflow and one outflow face, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' triangular cut-cells with a no-flow face (see [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We extend the DoD stabilization to cut-cells with multiple in- and outflow faces by properly considering the flow distribu- tion inside the cut-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We further prove L2-stability for the semi-discrete formulation in space and present numerical results to validate the proposed extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Copyright line will be provided by the publisher 1 Introduction To avoid the mesh generation process of complex geometries, cut-cell methods are an attractive alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The general idea is to start with a simple, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' structured, background mesh and to cut out the desired geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' This results in a mesh with unstructured polyhedral cells, called cut-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Cut-cells can have an arbitrary shape and can become arbitrarily small, causing the small cell problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' To use explicit time stepping schemes for solving hyperbolic conservation laws, the time step size would need to be chosen based on the smallest cut-cell in the grid to ensure stability, which is in general not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Developing solution approaches to the small cell problem in the context of discontinuous Galerkin (DG) schemes is a very recent research branch, including for example the work in [7,8,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' In this contribution we focus on the domain of dependence (DoD) stabilization, which was introduced in [4] for the linear transport equation in one and two space dimensions and was extended to non-linear systems in one space dimension in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' It is based on a DG scheme in space to allow for higher- order approximations and possesses several desirable theoretical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Numerical results show the expected higher-order behavior in smooth flow and robustness around shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Up to now, the DoD stabilization in two dimensions has only been used to stabilize small triangular cut-cells for linear advection parallel to a ramp [4,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' In this setup, the small stabilized cut-cells have exactly one inflow and one outflow face, which was exploited in the design of the stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' When moving to non-linear or coupled linear problems, this does not hold true anymore and one has to deal with multiple inflow and outflow faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' In this work, we take the first step in that direction by considering the linear advection equation on a cut-cell mesh with arbitrary flow directions, resulting in triangular cut-cells having 2 inflow and 1 outflow neighbor or reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' As this causes significant additional complications, we will only consider piecewise constant polynomials here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We will prove L2-stability for the semi-discrete scheme and present numerical results to validate the new extension of the stabilization terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The outline of the paper is as follows: we will first describe the problem setup and then introduce the new extended stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Then we will show the L2-stability proof and conclude with numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 2 Problem setup Let Ω ⊂ R2 be an open and connected domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We consider linear hyperbolic systems of the form ut + Aux + Buy = 0 in Ω × (0, T), (1a) τu = g on ∂Ω × (0, T), (1b) u = u0 on Ω × {t = 0}, (1c) where u(t) ∈ Rm, and A, B ∈ Rm×m constant, and τ is an appropriate boundary operator such that we only impose inflow boundary conditions on incoming waves (and not everywhere on ∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We require that for any unit vector n = (n1, n2)T ∈ S1 the matrices C = (n1A + n2B) are symmetric and simultaneously diagonalizable over the reals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' there is an orthogonal matrix O ∈ Rm×m and diagonal matrices Λn ∈ Diag(Rm×m) such that n1A + n2B = OΛnOT ∀n ∈ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' ∗ Corresponding author: e-mail g_birk01@wwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='de Copyright line will be provided by the publisher arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='02715v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='NA] 6 Jan 2023 2 Gunnar Birke1,, Christian Engwer1,, Sandra May2,, and Florian Streitbürger3, � Mh ∩ ¯Ω (x0, 0) γ = Mh E ∈ � Mh E1 E2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 1: Construction of the mesh: Out of the structured grid � Mh on the domain Ω the mesh Mh is constructed by introducing cut-cells E1, E2 ⊂ E ∈ � Mh along the cut such that ¯ E1 ∪ ¯ E2 = ¯E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' In our numerical tests we will choose Ω = [0, 1]2 and discretize it by a structured grid � Mh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We then introduce an artificial cut, a straight line going through the square, starting at (x0, 0) and having an angle γ relative to the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' This creates an internal boundary with two subdomains which we will resolve by a cut-cell mesh Mh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' A sketch is contained in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' So far we have always tested with flow parallel to that cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Here, we consider flow in various directions, keeping the cut fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We define the sets of internal and external faces as Fint h = {F = ∂E1 ∩ ∂E2|E1, E2 ∈ Mh, E1 ̸= E2, |F| > 0}, Fext h = {F = ∂E ∩ ∂Ω|E ∈ Mh, |F| > 0}, and the set of faces of an element E ∈ Mh by FE h = {F ∈ Fint h ∪ Fext h |F ⊂ ∂E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We choose a fixed local numbering on each of these sets and denote the neighbor element of E corresponding to a face Fi ∈ FE h by Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' For the discretization in space we choose the discrete function space V0 h(Mh) = {vh ∈ L2(Ω)m | (vh)i|E ∈ P0(E) ∀(E, i) ∈ Mh × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='., m}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' For vh ∈ V0 h(Mh) we denote by vE h the value of vh on an element E ∈ Mh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' For interior faces F ∈ Fint h we fix an orientation of the outer unit normal vector nF and denote the inner and outer element of F by E1 and E2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We then define average and jump by {{uh}} := 1 2(u E1 h + u E2 h ), �uh� := u E1 h − u E2 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' For exterior faces F ∈ Fext h we simply choose the unit outer normal and denote by uEF h the solution on the cell that lies in the interior of the domain and contains face F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We define the flux matrix on a face F as CF = (nF )1A + (nF )2B = OΛF OT , (2) where (nF )1,2 denote the first and second component of the unit normal vector nF on face F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Based on this, we define matrices which encode the flux directions as C+ F = OΛ+ F OT , C− F = OΛ− F OT with (Λ+ F )i,i = max(0, (ΛF )i,i) and (Λ− F )i,i = min(0, (ΛF )i,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Note that CF = C+ F +C− F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We also introduce a generalization of the absolute value for such flux matrices by |CF | = C+ F −C− F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The (unstabilized) upwind semi-discretization in space is then given as: Find uh(t) ∈ V0 h(Mh) such that (∂tuh(t), vh)L2(Ω) + aupw h (uh(t), vh) + lh(vh) = 0 ∀ vh ∈ V0 h(Mh) (3) with aupw h (uh, vh) = � F ∈Fext h � F ⟨C+ F uEF h , vEF h ⟩ds + � F ∈Fint h � F ⟨CF {{uh}}, �vh�⟩ + 1 2⟨|CF | �uh� , �vh�⟩ds, lh(vh) = − � F ∈Fext h � F ⟨C− F g, vEF h ⟩ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Here, ⟨·, ·⟩ denotes the scalar product in Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We obtain aupw h and lh by integration by parts, where the integral over internal edges leads to jump terms (second sum) and the boundary integral is split into outgoing waves (first sum) and incoming waves (right hand side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We then discretize in time using the explicit Euler scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Copyright line will be provided by the publisher DoD Stabilization of linear hyperbolic PDEs on general cut-cell meshes 3 F1 E1 F2 E2 F3 E3 F4 E4 Ecut β ⟨β, nF2⟩+ � F2 � ω1(uE1 h − uEcut h ) + ω4(uE4 h − uEcut h ) � �v�ds Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 2: Domain of dependence extension illustrated on a four-sided cut-cell: We introduce a direct mass transport from cells E1 and E4 into E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The colored regions indicate the coupling between the faces, the corresponding parts of the stabilization term on F2 are highlighted accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' From the graphic one can see that flow orientation and geometry play a central part in determining the right mass redistribution and the ωi will have to be chosen accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The face F3 needs to be stabilized as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Note that there should not be any coupling between between F2 and F3 as both are outflow faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 3 Stabilization To deal with small cut-cells, an additional term J0 h = � Ecut∈I J0,Ecut h with I being the set of small cut-cells that require stabilization is added to the space semi-discretization (we will comment on our choice in the numerical results below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' This results in the following scheme: Find uh(t) ∈ V0 h(Mh) such that (∂tuh(t), vh)L2(Ω) + aupw h (uh, vh) + J0 h(uh, vh) + lh(vh) = 0 ∀ vh ∈ V0 h(Mh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' (4) Let Ecut be a small cut-cell that requires stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The idea behind J0,Ecut h is the following: When the time step size is not chosen to respect the small size of Ecut, the domain of dependence of an outflow neighbor E of Ecut will extend beyond Ecut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We therefore extend the numerical DoD of E such that it receives information directly from the inflow neighbors of Ecut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The amount of mass passed directly between neighbors of Ecut is chosen such that the update on Ecut becomes stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' To explain the main concept we consider the scalar linear transport equation ut + ∇ · (βu) = 0 with constant velocity β = (β1, β2)T ∈ R2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=', we use A = β1, B = β2 in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Consider Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 2 where the domain of dependence of E2 potentially reaches into E1 and E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' In that case, for choosing the time step based on the size of background cells, mass (physically) is moved from E1 and from E4 to E2 in a single time step and this coupling must be mimicked by the stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We introduce an extension operator Lext E′(uh)(x) = uE′ h (x) for E′ ∈ Mh which acts on functions uh ∈ V0 h(Mh) and corresponds to evaluating the (constant) polynomial of cell E′ outside its original support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Using this operator, the following stabilization term was introduced in [4] for triangular cut-cells with single outflow face F2 and single inflow face F1: J0,Ecut h (uh, vh) = ηEcut � F2 ⟨β, nF2⟩(Lext E1(uh) − uEcut h ) �vh� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' (5) This term introduces a direct coupling between E1 and E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The parameter ηEcut ∈ [0, 1] controls how much mass is transported via this coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' In the case of exactly one inflow face F1 and one outflow face F2 (and one face with no-penetration b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' ), as considered in [4], this term suffices to ensure stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Here, however, the cut-cell is allowed to have two inflow faces as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Furthermore, these inflow faces will in general influence multiple neighbors of the cut-cell where the degree of influence depends on the geometry and flow direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' To handle this we propose the following extension of (5) J0,Ecut h (uh, vh) = ηEcut � Fj∈FEcut h � Fj � Fi∈FEcut h ωi⟨β, nFj⟩+(Lext Ei(uh) − uEcut h ) �vh� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' (6) Here the ωi ∈ R provide information about the flow distribution for incoming flow of the face Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Note that the extended solutions of all neighbor elements are evaluated on all faces, and ωi = 0 if Fi is not an inflow face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We provide a specific formula of how to choose these weights for triangular cut-cells below in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Going back to the system case we allow ωi ∈ Rm×m and arrive at our final formulation J0,Ecut h (uh, vh) = ηEcut � Fj∈FEcut h � Fj � Fi∈FEcut h ⟨ωiC+ F (Lext Ei(uh) − uEcut h ), �vh�⟩ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' (7) Copyright line will be provided by the publisher 4 Gunnar Birke1,, Christian Engwer1,, Sandra May2,, and Florian Streitbürger3, Note that in defining J0,Ecut we assume that all normal vectors nj for Fj ∈ FEcut h correspond to outward normal vectors with respect to Ecut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' In order to ensure consistency and stability the weights ωi must fulfill � Fi∈FEcut h ωi = Idm×m, (8) � Fj∈FEcut h � Fj ωiC+ j ds = − � Fi C− i ds ∀ Fi ∈ FEcut h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' (9) Additionally we require that ωiC+ Fj is always symmetric and positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Equation (8) can be understood as an assurance that the overall amount of mass moved over a face by our stabilization is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' For the scalar case, we require the ωi to build a convex combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Equation (9) means that a portion of the inflow is exactly redistributed over all outflow face candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' This in particular prevents overshoots on small cut-cells for appropriate choices of ηEcut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='1 L2-stability Equipped with the aforementioned properties of our stabilization we can show L2-stability for the semi-discrete scheme in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' For brevity we consider homogeneous inflow boundary conditions and assume that exact and discrete solution vanish on the boundary, so that we can ignore any domain boundary terms and focus on the situation of cut-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' For P 0 functions this means that they are zero in all boundary cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='1 Consider (1) with homogeneous boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Assume that the discrete solution uh(t) vanishes on the boundary ∂Ω for all t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Let uh(t) ∈ V0 h(Mh) be the solution to the semi-discrete problem (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Then it holds ||uh(t)||L2(Ω) ≤ ||uh(0)||L2(Ω) ∀ t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' P r o o f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We choose vh = uh(t) in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Any boundary terms vanish as trace(uh(t)) = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' This yields (∂tuh(t), uh(t))L2(Ω) + aupw h (uh(t), uh(t)) + J0 h(uh(t), uh(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' By the fundamental theorem of calculus � t 0 (∂τuh(τ), uh(τ))L2(Ω)dτ = � t 0 d dτ 1 2||uh(τ)||2 L2(Ω)dτ = 1 2||uh(t)||2 L2(Ω) − 1 2||uh(0)||2 L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' To ease notation we will write u = uh(t) in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The goal now is to show that aupw h (u, u) + J0 h(u, u) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We first consider aupw h (u, u), which expands into aupw h (u, u) = � F ∈Fint h � F ⟨CF {{u}}, �u�⟩ + ⟨ 1 2|CF |�u�, �u�⟩ds = � F ∈Fint h � F ⟨C+ F uE1 + C− F uE2, uE1 − uE2⟩ds = � F ∈Fint h � F ⟨C+ F uE1, uE1⟩ − ⟨C+ F uE1, uE2⟩ + ⟨C− F uE2, uE1⟩ − ⟨C− F uE2, uE2⟩ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Now we add zeros (in form of ± 1 2⟨C− F uE1, uE1⟩ and ± 1 2⟨C+ F uE2, uE2⟩) to get = � F ∈Fint h � F 1 2⟨(C+ F − C− F )uE1, uE1⟩ + 1 2⟨(C+ F + C− F )uE1, uE1⟩ − ⟨C+ F uE1, uE2⟩ + ⟨C− F uE2, uE1⟩ + 1 2⟨(C+ F − C− F )uE2, uE2⟩ − 1 2⟨(C+ F + C− F )uE2, uE2⟩ds = � F ∈Fint h � F 1 2⟨|CF |(uE1 − uE2), uE1 − uE2⟩ + 1 2⟨CF uE1, uE1⟩ − 1 2⟨CF uE2, uE2⟩ − 1 2⟨CF uE1, uE2⟩ + 1 2⟨CF uE2, uE1⟩ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Due to the symmetry of CF , the terms in the last line cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' For the last two terms in the second to last line we use the divergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Since uh(t) is elementwise constant, on E ∈ Mh it holds that 0 = � E ∇ · (⟨AuE, uE⟩, ⟨BuE, uE⟩)dx = � F ∈Fint h ∪Fext h ,F ∩∂E̸=∅ � F ⟨CF uE, uE⟩ds, Copyright line will be provided by the publisher DoD Stabilization of linear hyperbolic PDEs on general cut-cell meshes 5 and therefore, these terms vanish as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Finally, due to |CF | being positive semi-definite, we obtain positivity of aupw h (u, u): aupw h (u, u) = � F ∈Fint h � F 1 2⟨|CF |(uE1 − uE2), uE1 − uE2⟩ds ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We now investigate J0 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' For a small cut-cell Ecut ∈ I we have due to (8) J0,Ecut h (u, u) =ηEcut � Fj∈FEcut h � Fj ⟨( � Fi∈FEcut h ωiC+ FjuEi) − C+ FjuEcut, uEcut − uEj⟩ds =ηEcut � Fj∈FEcut h � Fj ⟨ � Fi∈FEcut h ωiC+ FjuEi, uEcut⟩ − ⟨ � Fi∈FEcut h ωiC+ FjuEi, uEj⟩ − ⟨C+ FjuEcut, uEcut⟩ + ⟨C+ FjuEcut, uEj⟩ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Adding again zeros (in form of ±⟨� Fi∈FEcut h ωiC+ FjuEi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' uEi⟩ and ±⟨C+ FjuEj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' uEj⟩) and reordering gives = − 1 2ηEcut � Fj∈FEcut h � Fj ⟨C+ FjuEcut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' uEcut⟩ − 2⟨C+ FjuEcut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' uEj⟩ + ⟨C+ FjuEj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' uEj⟩ + ⟨ � Fi∈FEcut h ωiC+ FjuEi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' uEi⟩ − 2⟨ � Fi∈FEcut h ωiC+ FjuEi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' uEcut⟩ + ⟨C+ FjuEcut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' uEcut⟩ − ⟨ � Fi∈FEcut h ωiC+ FjuEi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' uEi⟩ + 2⟨ � Fi∈FEcut h ωiC+ FjuEi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' uEj⟩ − ⟨C+ FjuEj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' uEj⟩ds = − 1 2ηEcut � Fj∈FEcut h � Fj ⟨C+ Fj(uEcut − uEj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' uEcut − uEj⟩ + � Fi∈FEcut h ⟨ωiC+ Fj(uEi − uEcut),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' uEi − uEcut⟩ (C+ Fj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' ωiC+ Fj symm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' and (8)) − � Fi∈FEcut h ⟨ωiC+ Fj(uEi − uEj), uEi − uEj⟩ds (9)= − 1 2ηEcut � Fj∈FEcut h � Fj ⟨|CFj|(uEcut − uEj), uEcut − uEj⟩ − � Fi∈FEcut h ⟨ωiC+ Fj(uEi − uEj), uEi − uEj⟩ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Since ηEcut ∈ [0, 1] the first term inside the sum can be compensated with terms from aupw h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The second term is always non- negative since ωiC+ Fj is always positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Note that the second term corresponds to dissipation introduced by an extended jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='2 Choice of parameters To perform actual computations we need to select concrete ωi in (7) that fulfill properties (8) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' For the situation of linear simultaneously diagonalizable hyperbolic systems and triangular cut-cells we suggest ωi = |Fi|C− Fi(� Fk∈FEcut h |Fk|C− Fk)−1 for each Fi ∈ FEcut h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' For linear advection, this would result in ωi = 0 for an outflow edge Fi and ωi corresponding to some sort of weighted proportion of the total inflow for an inflow edge Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We also need to set ηEcut for Ecut ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' A stable but not necessarily optimal choice is ηEcut = || � F ∈FEcut h � F Λ− F ds||∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 4 Numerical results For the numerical tests we select angles γ, θ, ρ1, ρ2 ∈ [0, 2π) where γ is the angle of the cut, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 1, and set Λ1 = �cos(ρ1) 0 0 cos(ρ2) � , Λ2 = �sin(ρ1) 0 0 sin(ρ2) � , O = �cos(θ) − sin(θ) sin(θ) cos(θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Copyright line will be provided by the publisher 6 References 102 2 × 102 3 × 102 4 × 102 N 10−2 10−1 error Setup 1: ρ1 = 7π 4 , ρ2 = π, γ = 40° u1 L1-error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='98 u2 L1-error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='99 u1 L∞-error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='99 u2 L∞-error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='99 order 1 102 2 × 102 3 × 102 4 × 102 N 10−2 10−1 error Setup 2: ρ1 = 7π 4 , ρ2 = 3π 2 , γ = 30° u1 L1-error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='98 u2 L1-error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='99 u1 L∞-error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='99 u2 L∞-error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='97 order 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 3: Plots of the error, measured in the L1- (blue, red) and L∞-norm (orange, transparent purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Left plot has values ρ1 = 7π 4 , ρ2 = π and γ = 40°, right plot has ρ1 = 7π 4 , ρ2 = 3π 2 and γ = 30°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' In all cases we have chosen θ = 4π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The green line is for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Then our system matrices will be given by A = OΛ1OT and B = OΛ2OT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The cut starts at (x0, 0) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='2001, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' As initial conditions we choose u0(x) = O �sin(2π(x1 cos(ρ1) + x2 sin(ρ1))) cos(2π(x1 cos(ρ2) + x2 sin(ρ2))) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Boundary conditions are given by the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Let N denote the number of background cells in either coordinate direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We compute the time step size via ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='4 h maxn∈S1 ||n1Λ1+n2Λ2||∞ , where h = 1 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='4 allows that bigger cut-cells do not need to be stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We then choose I = {E ∈ Mh| |E| h2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='4} as the set of stabilized cut-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The final time of our simulation is T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Our implementation is based on the DUNE framework (see [2], [1]), in particular the dune-udg (see [3], [5]) and dune- pdelab modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' The local subtriangulations for the cut-cells are computed by the TPMC library (see [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 3 shows convergence plots for two particular setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Note that the flow directions have been chosen to not be parallel to the ramp angle γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We observe the expected order of convergence for a first order scheme in both the L1- and the L∞-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' In addition, all numerically computed solution values, including those on small cut-cells, stayed within the bounds of the initial state during the simulation, confirming the added stability of the DoD stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 5 Discussion and Outlook We have extended the DoD stabilization to cut-cells with multiple inflow/outflow faces for the case of component-wise and piecewise constant trial and test functions and linear, simultaneously diagonalizable systems in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' We have proven L2-stability for the semi-discrete setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Numerically we observe full first-order convergence in different numerical tests and no over/undershoot on cut-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' In future work, we plan to extend our method to more general systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=', the acoustics and Euler equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' An extension of the presented formulation to higher-order approximations is ongoing research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Acknowledgements This work has been partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) as project 439956613 under contract numbers EN 1042/5-1 and MA 7773/4-1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 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Engrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' 121, 13, 2979-3003 (2020) [11] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Streitbürger, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Birke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Engwer, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' May, Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=', 137 (Springer International Publishing, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} +page_content=' Copyright line will be provided by the publisher' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E0T4oBgHgl3EQf2wIM/content/2301.02715v1.pdf'} diff --git a/UtAyT4oBgHgl3EQfhfhI/vector_store/index.pkl b/UtAyT4oBgHgl3EQfhfhI/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..2499902644e74caf1152d0bc3d6c4ec4ced7f43a --- /dev/null +++ b/UtAyT4oBgHgl3EQfhfhI/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:37c6859aaa674b5f127a5847b34e672acdef3c7cfb1905d645bc970d0619a7dc +size 84376 diff --git a/UtE4T4oBgHgl3EQfMQxc/content/tmp_files/2301.04945v1.pdf.txt b/UtE4T4oBgHgl3EQfMQxc/content/tmp_files/2301.04945v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9be95435868d1fda8f28185f5379f31ac3d8d339 --- /dev/null +++ b/UtE4T4oBgHgl3EQfMQxc/content/tmp_files/2301.04945v1.pdf.txt @@ -0,0 +1,1353 @@ +Leveraging Rights of Data Subjects for Social Media Analysis: +Studying TikTok via Data Donations +Savvas Zannettou1, Olivia-Nemes Nemeth2, Oshrat Ayalon2, +Angelica Goetzen2, Krishna P. Gummadi2, Elissa M. Redmiles2, and Franziska Roesner3 +1TU Delft, 2Max Planck Institute for Software Systems, 3University of Washington +s.zannettou@tudelft.nl, {onemes, oayalon, agoetzen, gummadi, eredmiles}@mpi-sws.org, +franzi@cs.washington.edu +Abstract +TikTok is a relatively novel and widely popular media plat- +form. In response to its expanding user base and cultural im- +pact, researchers are turning to study the platform; however, +TikTok, like many social media platforms, restricts external +access to data. Prior works have acquired data from scraping +the platform, user self-reports, and from accounts created by +researchers for the study’s purpose. Existing techniques, while +yielding important insights, contain limitations for gathering +large-scale quantitative insights on how real TikTok users be- +have on the platform. We bridge this research gap by imple- +menting a data donation system to collect TikTok data. Our +system leverages users’ right to access their data enabled by +the EU’s GDPR regulation. We recruit 347 TikTok users, ask +them to request their data from TikTok, and then use our sys- +tem to customize, anonymize, and donate their data. We collect +4.9M videos viewed 9.2M times by our participants – and as- +sociated engagement metrics – to analyze how people consume +content on TikTok, how prevalent liking behavior is on TikTok, +and whether there are substantial differences across our partic- +ipants’ demographics. We conclude our work by discussing +the lessons learned and future avenues for implementing data +donation systems, which we believe offer a promising avenue +for collecting user behavioral traces to understand social phe- +nomena through the lens of the Web. +1 +Introduction +The Web is a constantly evolving ecosystem, with new media +platforms proliferating and changing the way people consume +information. One recent and notable addition to the ecosystem +is TikTok, a short-form video platform revolutionizing the way +people get entertained online by offering an endless stream of +video recommendations. TikTok is widely popular, currently +sitting at 1.3 billion users worldwide [16]. Academic research +has quickly turned to focus on the platform, seeking to under- +stand TikTok’s algorithm (e.g., [6]), content (e.g., [48, 19, 1]) +and users (e.g., [18, 22, 26]). +Due to TikTok’s lack of external access (e.g., API) through +which to conduct direct measurements, all of this research +must be conducted with external data. Prior work has gath- +ered data by scraping the platform (e.g., [2, 19, 20]), an ap- +proach that can only collect a few thousands of videos, re- +lies on publicly available information that are included on +the web page’s source, and is usually biased towards popular +videos; from self-reports (e.g., [18, 22, 26]), which suffer from +known biases in social media research [27, 42, 9, 12]; or from +researcher-created accounts [6, 35], which is a promising tech- +nique, but may yield data that ultimately lacks the authentic- +ity, diversity, and account history that real user accounts would +contain. +In this work, we offer a method for overcoming such data +collection limitations via a novel measurement system for di- +rectly measuring the TikTok platform. Using this system, we +conduct the first, to the best of our knowledge, measurement +of people’s consumption and engagement with TikTok con- +tent. Our measurement approach combines the power of col- +lective action with the protections offered by the EU’s General +Data Protection Regulation (GDPR) [11] to design a TikTok +data donation system. The GDPR describes the right of ac- +cess by the data subject (Article 15), which allows individuals +(i.e., users) to request and get access to all the data about them +that the controller (i.e., social media platform) collects and pro- +cesses. Should users have a means of donating their data for +research, this opens new possibilities for undertaking research +studies. Indeed, previous work leverage the rights of data sub- +jects for data donations to study advertisements on Twitter [43] +and the use of Instagram by adolescents [10]. These efforts, +however, do not design and implement a data donation system; +they rely simply on users uploading their files to a server. As +the data includes many personal identifiers, having a data dona- +tion system is essential to better balance user privacy with data +donation (i.e., by removing personal identifiers client-side), +which may increase the likelihood of user data donation. +Our data donation system facilitates donations from TikTok +users who request their downloadable data from the platform. +Our donation system accepts data generated by the TikTok +platform and allows users to anonymize and customize their +data (i.e., select which fields of the data they want to donate) +before donating their data to our infrastructure. This approach +allows us to collect data on a user’s activity from the beginning +1 +arXiv:2301.04945v1 [cs.SI] 12 Jan 2023 + +of their TikTok usage to the time they requested their data. The +data includes a wide variety of fields, including the entire video +viewing history and engagement activities (e.g., likes, shares, +and comments on videos). These datasets are rich and provide +us with an invaluable opportunity to study how people use Tik- +Tok, while maintaining user privacy. +In this work, we use our donation system to conduct a study +of TikTok user behavior that seeks to understand how people +behave on TikTok (e.g., how many videos people watch and +how much time they spent on the platform), how prevalent is +video engagement on TikTok, and whether there are signif- +icant differences across our participants’ demographics. We +recruited 347 TikTok users (our study participants) to donate +their data using our system, ultimately compensating our par- +ticipants with $6.9K USD. We find that participants watch a +relatively large number of videos per day (median of 90 videos +per day) and spend on average 27 minutes per day on Tik- +Tok. Compared to YouTube, this daily consumption is larger +(based on one report, users spent 19 average daily minutes on +YouTube in 2022 [24]), despite the fact that YouTube’s videos +have a longer duration. Also, we find that younger partici- +pants (aged between 18-24 years old) spent more time on Tik- +Tok compared to older participants (e.g., aged between 25-34). +This result prompts the need for further investigation into how +younger people use TikTok, particularly, to investigate poten- +tial social media addiction. With regards to user engagement, +we find that participants typically engaged with videos by lik- +ing them (vs. commenting or sharing) and that younger par- +ticipants (aged between 18-24 years old) tend to ‘like’ more +videos compared to older participants (aged between 25-34 +years old). +Contributions. The contributions of this work are two-fold. +First, to the best of our knowledge, we perform the first re- +search study on TikTok that is based on data donated by real +users. We shed light on how TikTok users behave on the plat- +form and how prevalent their engagement is via video liking. +As TikTok becomes increasingly popular, especially among +younger people, such studies allow us to understand better +how people are using this emerging platform. Second, we in- +vestigate the possibility of performing social media studies by +leveraging data donated by the users themselves. By design- +ing, implementing, and using a data donation system, we expe- +rience and surface challenges (e.g., malicious users) that exist +in obtaining and analyzing data donated by real users. We dis- +cuss these challenges and the lessons we learned, which we +believe will be of interest to the research community focusing +on understanding closed social media platforms. +Paper Organization. Section 2 reviews previous work, while +Section 3 describes our data donation system that facilitates +data donations from real users. In Section 4.1 we report our +recruitment procedures and in Section 5 we discuss our ethi- +cal considerations. Section 6 includes our analysis and results +aiming to provide insights into how people behave on TikTok. +Finally, we conclude and discuss our findings, potential impli- +cations, and lessons learned in Section 7. +2 +Background & Related Work +2.1 +TikTok +Known as Douyin in China, TikTok is a short-form video plat- +form owned by parent company ByteDance. ByteDance first +launched Douyin for China-based users in 2016, and launched +TikTok as its international counterpart shortly after [39, 37]. +The company acquired Musical.ly, a similar short-form video +platform, in 2018, and in doing so enabled TikTok to expand +its user base worldwide [39]. Since then, TikTok usage has +continued to grow, rising to be the most downloaded app of +2020 [4]. At the time of this writing, TikTok sits at 1.3 billion +monthly active users worldwide [16]. +As a platform, TikTok allows users to both watch and cre- +ate short-form videos up to 3 minutes in length (currently, the +company is rolling out a 10-minute length option for select +users [23]). +TikTok offers multiple editing capabilities for +creators, and allows users to connect with peers via follow- +ing, messaging, and sharing content. One of the app’s most +prominent characteristics is its ability to recommend relevant +video content to viewers [33, 18]; when using TikTok, a user +may scroll through two different content feeds, one contain- +ing videos posted by the people they follow (“Following”), +the other a curated feed of content from many different cre- +ators (“For You”). Contents on the For You feed are served to +users based on a user’s account information and behavior, the +specifics of which are largely unknown to the public. +Much of the prior work on TikTok has studied the recom- +mendation algorithm [40, 1, 6, 18, 35, 34], the content on the +platform [3, 13, 28, 46, 19, 48, 37, 21, 44, 45, 31], and per- +formed user studies on TikTok users [49, 32, 17, 18, 26, 8, +30, 22]. To the best of our knowledge, our work is the first +to investigate user behavior through the lens of data donated +directly from users using the rights of data subjects. These +datasets allow us to obtain a holistic and unique view into how +people behave on TikTok and how they engage with content on +TikTok. +2.2 +Data Donation +Per the EU’s GDPR [11], most major digital platforms now +provide their users with electronic access to the personal data +they have on each user, via downloadable data packages [7]. A +prominent movement in the medical field [5, 36], researchers +studying digital platforms are beginning to leverage the rich +information in these packages by requesting that users donate +them for study. Data donations offer unique insights into dig- +ital platforms [41]: for example, uncovering widely-used ad +targeting mechanisms on Twitter that were largely ignored by +prior work [43] and gaining new insights into how adolescents +use Instagram [10] have all been possible via user data dona- +tion. +Motivations for using user-donated data stem from limita- +tions of other methods. People’s perceptions of their own on- +line behavior, for instance, can be unreliable [27, 42, 9, 12]. +Additionally, researcher-created accounts on digital platforms +may lack the authenticity, diversity and history that real user +2 + +accounts have. Further, scraping TikTok data yields fruitful +data sets, yet has capacity limitations and totally relies on pub- +lic – and typically, popular – content available on the platform. +User-donated TikTok data can provide further insight into how +real TikTok users are consuming content on TikTok. In this +work, we design and implement a data donation system that +facilitates data donations from TikTok users directly. +3 +Data Donation System +We implement a data donation system, Social Media Donator +(SMD), where users can get information on how they can re- +quest their data from the TikTok mobile application. We pro- +vide details on how users can request their TikTok data and +what is included in the data in Appendix A. After users down- +load their data, they can use SMD to anonymize and customize +their data before transferring the data to our infrastructure. +Data Anonymization. The data collected from TikTok in- +cludes personal information and identifiers for each user, such +as phone numbers and email addresses. +Due to this, it is +essential to ensure proper anonymization of the data before +transferring it to our infrastructure. +To ensure the data is +properly anonymized before it is transferred to our infrastruc- +ture, our SMD system removes certain information by default. +This includes the user’s profile information, direct messages, +information about videos uploaded, IP addresses and device +information, purchase information, and account status. The +anonymization process is done on the client side, and we em- +phasize that we only transfer the anonymized dataset to our +backend. Additionally, we provide a Python script that allows +users to anonymize and customize their dataset offline with- +out using SMD. This script is identical to the one available in +SMD and is intended for use by participants who are concerned +about privacy. +Data Customization. TikTok users may have different lev- +els of comfort in sharing certain data fields. For example, a +user who frequently posts comments with personal informa- +tion may not feel comfortable sharing their comment-related +data. To address this, we have implemented a customization +feature in SMD, which allows users to choose which fields of +their data they are comfortable donating. The only manda- +tory field is the video viewing history, which includes only the +URLs of the videos watched and the timestamps. Additionally, +there are some fields that users are not able to donate, as they +contain personal information and identifiers (as outlined in the +data anonymization procedure). For the remaining fields that +a user can choose to donate, we provide clear explanations of +what data is included, with specific examples and a description +of how we plan to use each field in our analysis. Additionally, +for data fields that may contain sensitive information, we have +added warning labels to alert users that the field may poten- +tially include private information. For example, for the search +history, we added a warning label that says "This information +may be sensitive if you did uncommon searches for things re- +lated to your real identity, e.g., searching for videos of a family +member’s small sports team." Similar warnings were added for +the followers, following, and comments data since these fields +may reveal the user’s identity through their follower network +and comments made on public videos. +SMD calculates the compensation for the user based on +their selections of which fields of data they opt-in to donate. +The mandatory video viewing history is compensated with $5, +while all the optional fields such as Like History, Search His- +tory, Share History, Login Information, App Settings, Com- +ments, Favorites, Following, Followers, and Ads Information +are compensated with $1 each, except for comments. For com- +ments, users have the option to either donate their comment +timestamps and content for $2 or only the timestamps for $1. +The total compensation for each user ranges from $5 to $16, +depending on their selections. +Data Donation & Survey. Users can donate and transfer their +anonymized and customized data to our infrastructure with a +single click on the SMD interface. After the data donation, +we present all users with an optional survey that includes gen- +eral demographic questions and questions about their usage of +the TikTok platform and their perceptions of the TikTok algo- +rithm’s recommendations. This survey helps us to gain extra +context on the users such as their age, gender, and location. It is +important to note that all questions in the survey are optional, +and users can choose to not answer by selecting the "Prefer +not to say" option. All users who choose to fill out the survey +will receive an additional compensation of $4 regardless of the +questions they choose not to answer. +4 +Data Collection +We present our approach to collecting data from TikTok users. +We describe our recruitment process, metadata collection for +TikTok videos, and our efforts to assess the quality of donated +data. +4.1 +User Recruitment +We recruited participants for the study in two ways: 1) by shar- +ing the study through the authors’ networks on Twitter, and +2) by running Facebook Ads targeting people over 18 years +old living in the U.S. who Facebook had tagged with the "Tik- +Tok" interest category. For the former, we created a poster to +advertise the study and shared a single tweet that was amplified +by all the authors’ Twitter accounts. The tweet was shared in +January 2022 and received 64.5K impressions on Twitter. For +the latter, we ran Facebook Ads between January 21, 2022, +and February 13, 2022, with an average budget of $8.5 per +day. Using these two methods, we recruited 347 participants, +whom we compensated with an overall amount of $6.9K in the +form of Amazon gift cards sent via email. +Fig. 1 shows the percentage of participants that opted-in to +donate each potential field that exists in their TikTok data. As +we can observe, most of our participants chose to donate al- +most all the fields, as all fields appear in at least 95% of all +the donations. +Participants were less willing to share their +Search History (95%) and Followers (96%), Following (98%), +3 + +Ads +Settings +Login History +Like History +Share History +Favorites +Comments +Followings +Followers +Search History +0 +20 +40 +60 +80 +100 +%Donations +99% +99% +99% +99% +99% +98% +98% +98% +96% +95% +Figure 1: Percentage of donations that opt-in to donate to each field +included in the TikTok data. +Region +# (%) +Age +# (%) +Africa +174 (52.2%) +25-34 years old +162 (48.6%) +N/C America +108 (32.4%) +18-24 years old +145 (43.5%) +South America +22 (6.6%) +35-44 years old +13 (3.9%) +Prefer not to say +19 (5.7%) +Prefer not to say +12 (3.6%) +Europe +10 (3.0%) +45-64 years old +1 (0.3%) +Gender +# (%) +Education +# (%) +Men +183 (54.9%) +Bachelor’s or above +181 (54.3%) +Women +144 (43.2%) +Associate’s degree +58 (17.4%) +Non-binary +3 (0.9%) +Some college +51 (15.3%) +Prefer not to say +2 (0.6%) +HS or below +26 (7.8%) +Self-described +1 (0.3%) +Prefer not to say +15 (4.5%) +- +- +Trade school +2 (0.6%) +Table 1: Demographics of the participants that completed our survey +(96% of all participants). “Prefer not to say” refers to users that opted +out from answering that question in the survey, while “N/C America” +refers to “North/Central America.” +and Comments (98%). This is potentially due to the warning +labels associated with these fields in our SMD interface, ex- +plaining that some information included in these fields might +be sensitive (see Section 3). This result suggests that most par- +ticipants perceive the trade-off between the compensation and +the donation of additional fields as worthwhile. +Participants’ Demographics. Participants self-reported their +demographics via an optional survey (see Section 3). Note that +14 out of 347 participants (4%) opted out from completing the +survey, hence we do not have any demographic information +about them. Table 1 provides an overview of our participants’ +demographics. +About the reported region in which they reside, we observe +that just over half of our participants are from Africa (52%), +while the remainder is from North/Central America (32%), +South America (6.6%), and Europe (3%). Interestingly, many +of our participants were from Africa despite the fact that we +mainly targeted Facebook users living in the U.S. We hypoth- +esize that the monetary incentives may have been relatively +more substantial for users from Africa highlighting that people +from under-developed countries can be attracted by the mone- +tary incentives to participate in research by donating their data. +With regards to the age of our participants, we note that the +majority of our participants are 34 years or younger (91%), +#Participants +#Actions +Video Viewing History +347 +9,212,100 +Like History +328 +1,120,716 +Search History +332 +13,282 +Share History +253 +24,944 +Comments +227 +52,436 +Following +333 +84,654 +Followers +295 +43,642 +Table 2: Overview of our dataset. We report the number of partici- +pants with at least one action. +with 48.6% of the participants aged between 25-34 years old +and 43% of the participants aged between 18 and 24 years old. +Note that given the popularity of these age groups in our par- +ticipant set, we focus our demographic analysis in Section 6 on +the 18-24 years old and 25-34 years old age groups only. Our +participants set is somewhat gender-balanced with 55% of the +participants being men, while 43.2% of our participants are +women and 1.2% are non-binary or self-described their gen- +der. Note that we compare only men and women in the gen- +der analyses presented in Section 6, as we do not have suffi- +cient statistical power to make comparisons between people of +other genders. Finally, most of our participants are educated +with 54% having a bachelor’s degree or above (e.g., MSc or +PhD), 32% have completed some post-high school coursework +or completed an associate’s degree or country equivalent. +Dataset. Table 2 reports an overview of our dataset; it in- +cludes 9.2M video views from 347 participants, between July +26, 2020 and February 21, 2022. Our dataset also includes +1.1M like actions from 328 participants, 13K search actions +from 332 participants, 24.9K shares from 253 participants, +52.4K comments from 227 participants, 84.6K following ac- +tions (i.e., the participant started following a user) from 333 +users, and 43K follower actions (i.e., a TikTok user started fol- +lowing one of our participants) from 295 participants. Also, +we note that by looking into how these actions appear over +time in our dataset, we find that there is a 2-month gap for the +like actions data. That is, there is no likes information for any +of the 347 recruited participants, likely because of issues with +the logging infrastructure within TikTok. Also, we find that +information about sharing videos is only included in the data +after July 28, 2021. +4.2 +Video Metadata Collection +Each participant’s data includes information about their activ- +ity, including the videos they watched, liked, shared, etc. Note +that for each video, the data only includes a video URL. To ob- +tain more context and metadata about the TikTok videos that +are included in our donations, we scrape each video’s Tik- +Tok webpage. Specifically, we use an unofficial Python API +wrapper [38] that essentially uses Selenium to scrape the Tik- +Tok page for each video and provide the video’s metadata in +JSON format. Overall, in our 347 donations, we find 4,938,805 +unique TikTok videos viewed by our participants. +We at- +tempted to collect the metadata for all of the videos, man- +4 + +aging to obtain metadata for 4,122,038 videos (83.4% of all +videos). The rest of the videos were either deleted by the up- +loader or TikTok itself, or the account that posted the videos +set their account into private mode. Since participants’ data +donations include their entire activity, we collect videos from +2020, which are more likely to be deleted compared to newer +videos. We ran our metadata collection between January 17, +2022, and March 12, 2022. For each video, we collected the +following metadata: 1) Video creation timestamp; 2) Video +description and title (defined by the uploader); 3) Information +about the uploader such as the uploader’s username and unique +identifier; 4) Video metadata such as video duration, video for- +mat, definition (e.g., 720p), etc.; and 5) Video statistics such as +the number of shares, comments, and number of times that the +video was viewed (at the time of our crawl). +4.3 +Assessing the “Quality” of Donations +SMD requires participants to provide an email, for the pur- +poses of sending the compensation. We also use the MD5 +hash of each email as the unique identifier for each donation. +Note that we do not link the email address explicitly with the +donation for anonymity purposes. As expected with studies +that offer a monetary incentive, users may try to “trick” the +system to earn money easily. Indeed, during our recruitment, +we noticed that some TikTok users were trying to earn more +money by donating duplicate or near-duplicate data under dif- +ferent email addresses. To detect such malicious donations, +SMD calculates all the pairwise Jaccard similarities between +the video URLs and timestamps that are included in the video +viewing history. Then, SMD flags the donations that have a +Jaccard similarity of over 0.2 for either the video URLs or the +timestamps, and then we manually check the donations to ver- +ify that are indeed duplicate or near-duplicate donations. This +process is done before sending the compensation. Overall, we +received 31 duplicate or near-duplicate donations (all of them +having a Jaccard similarity of 0.9 or more) that would have +cost us $571. For these donations, we did not pay users, and +we informed them via email that they would not receive com- +pensation for all their subsequent duplicate donations (as they +already received compensation previously). +Another concern that we have is that malicious users can +generate fake video URLs and fake timestamps to try and do- +nate fake data for compensation. To verify if this is happening +in our recruitment, we check for what percentage of each do- +nation’s videos we were able to obtain video metadata. Here, +we hypothesize that if a malicious user generates fake URLs +then we will not be able to get any metadata for the videos in- +cluded in the donation (since the URLs will not exist). We did +not find any evidence of users submitting fabricated donations, +since for all the donations we were able to obtain metadata for +a large percentage of the videos — at least 70% for every do- +nation, with a median of 90%. As discussed above, the rest of +the videos were inaccessible because they were either deleted +or the account became private. +5 +Ethical Considerations +Before recruiting participants and collecting any data from real +TikTok users, we obtained approvals from two different Ethi- +cal Review Board committees (one from a university in Eu- +rope and one from a university in the US). We provided to +the Ethical Review Board committees a detailed document +explaining all the various data fields included in the TikTok +data, our anonymization/customization procedures, the con- +sent form that is presented to the users, as well as our adver- +tisements for recruiting users. +For each recruited user, we obtained explicit consent by pro- +viding them with a consent form (which can be downloaded +via our SMD donation system) and by requesting the users to +acknowledge that they fully understand the content of the con- +sent form and that they allow us to store their data. Addition- +ally, as noted in Section 3, for each data field we explained to +the participants how we are planning to use the data included +in their data and the potential privacy implications of donating +some fields of the data (e.g., comments, search history, and fol- +lower/following network). As mentioned in the previous sec- +tion, all users that donated their data got a compensation using +Amazon gift cards between $5-$20 depending on the fields of +their data they opted-in to donate and whether they filled in the +optional small survey. +We will permanently delete all user data within 36 months +after the end of our project and the collected data will not be +distributed to any third parties. Also, we emphasize that the +users have the power on their hands with regard to what spe- +cific fields of data they opt-in to donate. For our analysis, we +follow standard ethical guidelines [29] like reporting our re- +sults in aggregate and not attempting to track users across plat- +forms. Finally, our metadata collection focuses only on pub- +licly accessible videos that are available on the TikTok plat- +form during the data collection period (i.e., we do not collect +any information about private videos or deleted videos). +6 +Results +In this section, we present our analysis and results focusing +on understanding how users behave on TikTok, particularly, +assessing how many videos they watch, how much time they +spent on the platform, how many videos each participant liked, +and whether there are substantial differences on user behavior +across demographics. +Method. We compute two metrics to measure consumptive +behavior: Time Spent and Volume. To compute Time Spent on +the platform, we need to infer an approximation of the watch- +ing duration of each video view based on the timestamps since +the donated data only contains information on when each par- +ticipant started watching each video (see Appendix A on what +is included in the data). To do this, for each video, we calcu- +late the number of seconds between the time that the partici- +pant started watching a video and the time that the participant +started watching the next video. This allows us to obtain an ap- +proximation of the viewing duration of each video, except for +5 + +102 +103 +104 +Average daily time spent for each participant (seconds) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CDF +(a) All +102 +103 +104 +Average daily time spent for each participant (seconds) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CDF +18-24 years old +25-34 years old +(b) Age +102 +103 +104 +Average daily time spent for each participant (seconds) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CDF +Men +Women +(c) Gender +Figure 2: CDF of the average time (in seconds) that our participants spent watching TikTok videos per day. +100 +101 +102 +103 +Average number of daily video views per participant +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CDF +(a) All +100 +101 +102 +103 +Average number of daily video views per participant +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CDF +18-24 years old +25-34 years old +(b) Age +100 +101 +102 +103 +Average number of daily video views per participant +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CDF +Men +Women +(c) Gender +Figure 3: CDF of the average number of daily video views per participant. +the last video of each session: when the user takes a break, we +would infer the viewing time of the last video of the session +to be unrealistically long. Unfortunately, we cannot simply +use the duration of the video itself to identify session breaks, +because we do not have duration metadata for all videos, and +because on TikTok, videos automatically start again and loop +if users do not scroll to the next one. +Instead, we set the +threshold for identifying session breaks to 105 seconds (1.75 +minutes) following the methodology by Halfaker et al. [15]. +The main idea of the method is that one can perform user +session identification (i.e., when users take breaks) based on +inter-activity times by performing a two-component Gaussian +mixture model clustering using the expectation-maximization +algorithm [25]. +By obtaining the results from this clustering, we can iden- +tify the cutoff threshold between the two clusters, which in our +TikTok dataset is equal to 105 seconds. This means that when- +ever a participant appears to spend more than 105 seconds on +a video, then we infer that they took a break and this marks the +end of a session — in this case, we are unable to infer the time +that the participant devoted to that specific video. This thresh- +old is substantially smaller compared to the one found by Hal- +faker et al. [15] on Wikipedia (about 1 hour). This is mainly +due to the fundamental differences between the two platforms, +as people tend to spend more time on a Wikipedia article than +a TikTok video. Indeed, by looking at the inferred viewing du- +rations and all video durations (obtained from the video meta- +data), we find that 98.5% of all the inferred viewing durations +are 105 seconds or less. At the same time, only a small fraction +of the videos have a duration of over 105 seconds (1.8% of all +videos). Note that we experimented with larger thresholds (be- +tween 100 seconds and 600 seconds); the main insights from +our results were the same, hence we report the results with 105 +seconds as our threshold. Second, we compute a Volume met- +ric: the average number of videos each participant watched per +day. +Time Spent. We start our analysis by looking into how much +time our participants spend on TikTok. Fig. 2 shows the Cumu- +lative Distribution Function (CDF) of the average time spent +per day for each participant. For each participant, we calculate +the sum of the inferred durations for each day that they used +TikTok, and then we calculate the average time across all days. +For all participants (see Fig. 2(a)), we find a median value of +1,622 seconds per day (27 minutes), with 25% of our partici- +pants (Q1) spending on average less than 834 seconds per day +(13.9 minutes) on TikTok, while the 25% most active (Q3) par- +ticipants spent 2,891 seconds (48 minutes) or more (σ: 1,864 +seconds). The most active participant in our dataset spent, on +average, 12,186 seconds per day (3.3 hours). +We also investigate if there are differences across partici- +pants based on their demographic information (see Fig. 2(b) +for age and Fig. 2(c) for gender). We find that participants +aged between 18-24 years old spent more time on TikTok (me- +dian: 1,821 seconds, Q1: 948 seconds, Q3: 3,131 seconds, +σ: 2,008 seconds) compared to participants aged between 25- +34 years old (median: 1,423 seconds, Q1: 749 seconds, Q3: +2,667 seconds, σ: 1,771 seconds). To assess the statistical sig- +nificance of our results, we perform a Mann-Whitney U test +on the distributions finding that the differences between partic- +ipants aged between 18-24 years old and 25-34 years old are +statistically significant (p < 0.05). With regards to the gender- +related results (see Fig. 2(c)), we find that women (median: +6 + +0 +10 +20 +30 +40 +50 +60 +70 +%videos that users engaged with per donation +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CDF +Figure 4: Percentage of videos that the participants liked per dona- +tion. +1,917 seconds, Q1: 962 seconds, Q3: 3,243 seconds, σ: 2,109 +seconds) spent more time on the platform compared to men +(median: 1,360 seconds, Q1: 728 seconds, Q3: 2,432 seconds, +σ: 1,452 seconds). These results are statistically significant, as +confirmed by a Mann-Whitney U test (p < 0.01). +Volume. Next, we analyze the volume of videos watched by +participants. Fig. 3(a) shows the CDF of the average number of +video views per participant, for all participants, while Fig. 3(b) +and Fig. 3(c) show the same results for participants grouped +by the gender and age groups we compare, respectively. We +observe that, on average, our participants watch a substantial +number of videos per day, with a median of 89.9 videos per +day (Q1: 40.7 videos, Q3: 170.3 videos σ: 128.9 videos). This +is somewhat expected since TikTok videos are usually short in +length, hence users can watch a large number of videos with- +out spending much time (compared to other video platforms +like YouTube). We also investigate the differences across age +groups and gender (see Fig. 3(b) and Fig. 3(c)), finding that, +among our participants, women watched significantly (MWU +all tests p < 0.01) more videos on TikTok than men. +Liking behavior. From the statistics in Table 2, we can ex- +tract some valuable insights into user engagement on TikTok. +Our dataset’s most popular engagement action is liking videos, +with 1.1M actions. This is a substantially larger number than +other engagement actions, such as shares (24K) and comments +(52K). Given the popularity of these engagement actions and +their temporal coverage (see Dataset in Section 4), through- +out the rest of the paper, we focus on likes when we study our +participants’ engagement with videos. +We look into the prevalence of likes on a per-participant +level. Fig. 4 shows the Cumulative Distribution Function of +the percentage of videos that were liked per participant (for the +ones that opted-in for donating to each of these engagement- +related fields). We find a median of 4% of videos that are liked +per donation (Q1: 0.87%, Q3: 14.09%, σ : 15.03%). +We also investigate differences in liking prevalence on Tik- +Tok across our participants’ demographic attributes. Fig. 5 +shows the CDF of the percentage of videos that are liked by +each participant across the ages and genders we compare. We +observe similar results as with the consumptive behavior fo- +cusing on the number of videos or time spent on the platform; +10 +2 +10 +1 +100 +101 +102 +%videos that users liked per donation +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CDF +18-24 years old +25-34 years old +(a) Age +10 +2 +10 +1 +100 +101 +102 +%videos that users liked per donation +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CDF +Men +Women +(b) Gender +Figure 5: Percentage of videos that the participants liked across our +demographic attributes. +1) Participants aged between 18-24 years old tend to like more +videos (median: 5.8%, Q1: 1.2%, Q3: 23.6%, σ: 16.6%) +compared to the participants that are between 25-34 years old +(median: 3.6%, Q1: 0.9%, Q3: 10.5%, σ : 13.5%) or older +(median: 4%, Q1: 1.5%, Q3: 14.5%, σ: 14.8%). This dif- +ference in engagement between the age groups is statistically +significant with p < 0.05; 2) We find no statistically signifi- +cant differences (p = 0.15) in participants’ liking behavior for +men and women. +Take-aways: The main take-away points from our analysis +are: +• Our participants spent a considerable amount of time +watching videos on TikTok per day (median time spent: +27 minutes, median number of videos: 90 videos). This +consumption is larger compared to other video platforms +like YouTube, where people spend on average 19 min- +utes [24]. Also, we find that participants aged between +18-24 years old tend to spend more time on the platform +than participants aged between 25-34 years old. This dif- +ference in consumptive behavior highlights the need for +more research efforts on understanding user consump- +tive behavior in young people, especially in teens (an age +group that we do not analyze here and might be more +prone to addiction from TikTok). Finally, we find that +women tend to spend more time and watch more videos +on TikTok compared to men. +• By far the most popular engagement action is liking a +video with sharing and commenting on videos having a +7 + +substantially smaller prevalence in our dataset. Also, we +find statistically significant differences in the liking be- +havior of participants aged between 18-24 years old com- +pared to participants aged 25-34 years old (i.e., younger +participants tend to like more content on TikTok). +7 +Discussion & Future Work +Below, we discuss our TikTok user behavior measurements +and lessons for future data donation efforts. +7.1 +Observing TikTok Behavior +Our work is a stepping stone to understanding how people con- +sume and engage with content on TikTok, based on traces from +real users via data donations. Our analysis shows that our par- +ticipants spend a considerable amount of time per day on Tik- +Tok and watch many videos, more so than on other popular +video platforms like YouTube [24]. They engage with videos +to a relatively limited extent: a median of 4% of videos were +liked per donation, and likes were significantly more prevalent +than other engagement actions such as shares and comments. +Our quantitative findings and data collection methodology +can assist future investigations that analyze content and/or par- +ticipant experiences. For example: What is the impact of Tik- +Tok’s algorithm on users, and to what extent do the recom- +mendations lead to informational rabbit holes? What is the +content that people of different demographics engage with? +What actions do users take to influence TikTok’s recommen- +dations (e.g., intentionally skipping videos or following cer- +tain accounts), and how effective are those actions? What is +the role of ads on TikTok [14, 47]? Future research may also +wish to consider additional demographics than those explored +in our study. +7.2 +Data Donation System +Here, we summarize future directions for and lessons learned +from designing and deploying our data donation system. +User recruitment. In this work, our initial goal was to re- +cruit participants exclusively from the U.S., which is why we +targeted people from the U.S. on Facebook and explicitly men- +tioned that we were recruiting U.S. people in our Twitter post. +Despite our initial goal and our ads, our data donation sys- +tem was publicly available to anyone, and we did not limit +access to people coming from specific regions. Hence, we ob- +tained a substantial number of participants from other regions +as well (e.g., Africa). Future work that aims to recruit partic- +ipants from specific regions or with other restrictions should +implement measures that ensure participants meet the study’s +requirements. One way to do this is by limiting the access +to the system to specific IP addresses that come from the re- +gion/country of interest. +Malicious users and assessing the quality of donations. +During our recruitment and data donation procedures, we no- +ticed several instances of malicious users trying to “trick the +system” to get extra monetary incentives. We identified three +cases of malicious users: 1) users donating duplicate data (i.e., +donating the same data multiple times with different email ad- +dresses) to get extra monetary incentives; 2) users donating +their data, then using TikTok for a few more days, and then +trying to donate their data again (i.e., duplicate data for the en- +tire period except the few extra days at the end of the data); +and 3) users creating new TikTok accounts, watching only a +few videos on TikTok, and then trying to donate their data via +our donation system. Based on these observations, we make +the following recommendations. +Researchers implementing +data donation systems should design and implement specific +countermeasures to detect malicious users that try to donate +“useless” data. Concretely, developers of data donation sys- +tems should implement features to detect duplicate or near- +duplicate donations, similar to the ones we implemented. In +addition, to overcome the problem of people creating new ac- +counts for the sake of donating their data, data donation sys- +tems can be developed in such a way so that they only ac- +cept donations that meet a minimum number of days of activ- +ity. For instance, in our case, our donation system rejected all +donations where the donation lifespan (i.e., the difference be- +tween the last video view and first video view) was less than +three months. Finally, in our work, we did not notice any at- +tempts to donate completely fabricated data (e.g., randomly +computer-generated video URLs and timestamps). However, +we argue that this possibility exists, and tech-savvy malicious +users may use some techniques in the future depending on how +much money they can make from this activity. +Pricing mechanisms. Our experience recruiting people to do- +nate their data for research purposes with monetary incentives, +highlights that those from countries where the incentives offer +more purchasing power (e.g., African countries) may be more +willing to donate their data. More broadly, an essential aspect +of data donation infrastructures is the underlying pricing mech- +anism, since this may affect how willing potential participants +will be to donate their data. For our data donation infrastruc- +ture, we set some pre-defined compensation amounts for the +viewing video history ($5), as well as all optional fields ($1 for +each field). We thought that these prices are reasonable for our +purpose and what was asked from the participants. However, +future work is needed to empirically understand the interplay +between data donation and pricing. +Trustworthiness of the data donation system. The degree of +trust between the participants and the data donation system is +likely to affect participant recruitment. We made two design +choices in an effort to enhance participant trust. First, we pro- +vided offline tools that participants can run to anonymize and +customize their data before using our data donation system. +Second, we offer users control over which fields to donate. +Nevertheless, future work on data donation systems should in- +vestigate which, if any, of these measures enhance user trust +and explore additional measures that may appeal to users such +as stronger formal guarantees that the data donation code is +working as described and allowing only the access and queries +participants expect to be run on their data. +Missing data and the need for compliance audits. By col- +8 + +lecting and analyzing user-donated data from TikTok, we no- +ticed some missing data for all the participants (i.e., there were +no like data for two months). The missing data might be due +to failures in the logging infrastructure within TikTok or due +to lost data. Nevertheless, this prompts the need for systematic +audits of social media platforms to assess the platforms’ com- +pliance with the access rights of data subjects. Future work can +design controlled experiments to assess how accurate and com- +prehensive the data provided by platforms is; e.g., by using the +platform to perform some pre-defined actions, then accessing +the provided data and comparing it with the set of pre-defined +actions. +7.3 +Limitations +Naturally, our work has some limitations. First, our recruit- +ment procedure and research budget allowed us to obtain data +from a small number of participants (347). We did not cover a +full range of user demographics (e.g., age, gender, geography, +etc.). Due to a lack of public information about the demo- +graphics of TikTok’s user base, we cannot draw conclusions +about the representativeness of our sample. Despite this crit- +ical limitation, our results offer a first measurement-based in- +vestigation into how people consume and engage with content +on the TikTok platform. At the same time, the lessons learned +from designing and implementing a data donation system can +provide valuable insights to researchers that aim to collect data +using similar approaches. Second, our video metadata collec- +tion is (necessarily) done post-hoc, hence we are unable to ob- +tain a holistic view of all videos referenced in the data dona- +tions since approximately 17% of all videos were not accessi- +ble during our video metadata collection period. Third, even +though the data donations are quite detailed, some of our re- +sults are based on inferences, such as the inferred time that +each participant spent watching videos. Due to this, we are un- +able to accurately infer the viewing durations for video views +that are above our determined threshold following the method- +ology by [15]. Therefore, our results based on the time spent +on the platform should be considered lower-bound results, as +we exclude all the video views at the end of each inferred ses- +sion. +Many of our limitations reflect fundamental challenges with +data donation as a way to externally study and audit closed +platforms: our investigation was often limited by intentional or +unintentional (e.g., the gap in “like” data) omissions in the data +provided by TikTok’s data download feature. Hence, as the re- +search community and data protection regulation communities +consider the role of data donation in platform transparency and +auditing, we must also critically assess the quality and goals of +the data downloads that platforms provide to comply with data +protection laws. +References +[1] J. Bandy and N. Diakopoulos. # tulsaflop: A case study +of algorithmically-influenced collective action on tiktok. +arXiv preprint arXiv:2012.07716, 2020. +[2] C. H. Basch, Z. Meleo-Erwin, J. Fera, C. 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What twitter knows: Characterizing ad tar- +geting practices, user perceptions, and ad explanations +through users’ own twitter data. In 29th USENIX Secu- +rity Symposium (USENIX Security 20), pages 145–162, +2020. +[44] G. Weimann and N. Masri. +Research note: spreading +hate on tiktok. Studies in conflict & terrorism, pages 1– +14, 2020. +[45] G. Weimann and N. Masri. +Tiktok’s spiral of anti- +semitism. Journalism and Media, 2(4):697–708, 2021. +10 + +[46] A. +Yeung, +E. +Ng, +and +E. +Abi-Jaoude. +Tik- +tok +and +attention-deficit/hyperactivity +disorder: +A +cross-sectional study of social media content qual- +ity. +The Canadian Journal of Psychiatry, +page +07067437221082854, 2022. +[47] E. Zeng, T. Kohno, and F. Roesner. What makes a ‘bad’ +ad? user perceptions of problematic online advertising. +In CHI Conference on Human Factors in Computing Sys- +tems, 2021. +[48] J. Zeng and C. Abidin. +‘# okboomer, time to meet +the zoomers’: studying the memefication of intergener- +ational politics on tiktok. Information, Communication +& Society, 24(16):2459–2481, 2021. +[49] J. Zeng and D. B. V. Kaye. From content moderation +to visibility moderation: A case study of platform gover- +nance on tiktok. Policy & Internet, 14(1):79–95, 2022. +A +Requesting Data from TikTok +TikTok enables its users to request a comprehensive dataset of +their activity on the platform and other personal information +the platform has on them. The request can be made through +the TikTok mobile application’s settings menu, and the data +will be provided in either JSON or a human-readable format +based on the user’s preference. The process takes around 3-4 +days for the data to be ready for download. Here we outline the +various fields of information included in a user’s TikTok data +download. +• Video Viewing History: A list of videos and the times- +tamp that the user started watching each video. +• Like History: A list of videos that the user liked and the +timestamp of each like action. +• Search History: A list of search queries that the user per- +formed in TikTok, including a timestamp of each search +action. +• Share History: A list of videos that were shared by the +user, along with the timestamp of the action, and the +method for the share (e.g., shared via WhatsApp or Face- +book Messenger, etc.). +• Login Information: A list of login information for each +time the user started using the TikTok app. For each lo- +gin, the file includes the timestamp, the IP address from +where the user performed the login, information about the +device (device model and device system), network type +(e.g., WiFi, mobile data, etc.), and the carrier. +• App Settings: Information about the settings that the user +set on TikTok. For instance, the interests that the user +pre-defined in the TikTok application and privacy settings +(e.g., whether the account is private, who can comment +on their videos, who can message them, etc.) +• Comments: A list of comments made by the user along +with the timestamp of the action. Note that the file does +not include the specific video where the comment was +posted. +• Favorites: A list of videos, effects, hashtags, sounds, +and videos that the user favorited on TikTok including the +timestamp of each action. +• Following/Followers: A list of accounts that the user fol- +lows/follow the user along with the specific timestamp of +the follow action +• Ads Information: Information about the advertisers that +targeted the user. +• Profile Information: A set of profile attributes that the +user set on the TikTok application. It includes the user’s +bio, email address, telephone number, username, profile +photo, etc. +• Direct Messages: A list of messages shared via private +chats between the user and other TikTok users. +• Video Uploads: A list of the user’s uploaded videos. +• Purchase History: +Information about the purchases +made by the user within the TikTok platform. +• Account Status: Information about the status of the Tik- +Tok application on the user’s phone (e.g., application ver- +sion, screen resolution, etc. ) +11 + diff --git a/UtE4T4oBgHgl3EQfMQxc/content/tmp_files/load_file.txt b/UtE4T4oBgHgl3EQfMQxc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d4a5919cd48b56bffaef6700c50ddb7359e4996d --- /dev/null +++ b/UtE4T4oBgHgl3EQfMQxc/content/tmp_files/load_file.txt @@ -0,0 +1,928 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf,len=927 +page_content='Leveraging Rights of Data Subjects for Social Media Analysis: Studying TikTok via Data Donations Savvas Zannettou1, Olivia-Nemes Nemeth2, Oshrat Ayalon2, Angelica Goetzen2, Krishna P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Gummadi2, Elissa M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Redmiles2, and Franziska Roesner3 1TU Delft, 2Max Planck Institute for Software Systems, 3University of Washington s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='zannettou@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='nl, {onemes, oayalon, agoetzen, gummadi, eredmiles}@mpi-sws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='org, franzi@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='edu Abstract TikTok is a relatively novel and widely popular media plat- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' In response to its expanding user base and cultural im- pact, researchers are turning to study the platform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' however, TikTok, like many social media platforms, restricts external access to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Prior works have acquired data from scraping the platform, user self-reports, and from accounts created by researchers for the study’s purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Existing techniques, while yielding important insights, contain limitations for gathering large-scale quantitative insights on how real TikTok users be- have on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We bridge this research gap by imple- menting a data donation system to collect TikTok data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Our system leverages users’ right to access their data enabled by the EU’s GDPR regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We recruit 347 TikTok users, ask them to request their data from TikTok, and then use our sys- tem to customize, anonymize, and donate their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We collect 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='9M videos viewed 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2M times by our participants – and as- sociated engagement metrics – to analyze how people consume content on TikTok, how prevalent liking behavior is on TikTok, and whether there are substantial differences across our partic- ipants’ demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We conclude our work by discussing the lessons learned and future avenues for implementing data donation systems, which we believe offer a promising avenue for collecting user behavioral traces to understand social phe- nomena through the lens of the Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 1 Introduction The Web is a constantly evolving ecosystem, with new media platforms proliferating and changing the way people consume information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' One recent and notable addition to the ecosystem is TikTok, a short-form video platform revolutionizing the way people get entertained online by offering an endless stream of video recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' TikTok is widely popular, currently sitting at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='3 billion users worldwide [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Academic research has quickly turned to focus on the platform, seeking to under- stand TikTok’s algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', [6]), content (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', [48, 19, 1]) and users (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', [18, 22, 26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Due to TikTok’s lack of external access (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', API) through which to conduct direct measurements, all of this research must be conducted with external data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Prior work has gath- ered data by scraping the platform (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', [2, 19, 20]), an ap- proach that can only collect a few thousands of videos, re- lies on publicly available information that are included on the web page’s source, and is usually biased towards popular videos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' from self-reports (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', [18, 22, 26]), which suffer from known biases in social media research [27, 42, 9, 12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' or from researcher-created accounts [6, 35], which is a promising tech- nique, but may yield data that ultimately lacks the authentic- ity, diversity, and account history that real user accounts would contain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' In this work, we offer a method for overcoming such data collection limitations via a novel measurement system for di- rectly measuring the TikTok platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Using this system, we conduct the first, to the best of our knowledge, measurement of people’s consumption and engagement with TikTok con- tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Our measurement approach combines the power of col- lective action with the protections offered by the EU’s General Data Protection Regulation (GDPR) [11] to design a TikTok data donation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The GDPR describes the right of ac- cess by the data subject (Article 15), which allows individuals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', users) to request and get access to all the data about them that the controller (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', social media platform) collects and pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Should users have a means of donating their data for research, this opens new possibilities for undertaking research studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Indeed, previous work leverage the rights of data sub- jects for data donations to study advertisements on Twitter [43] and the use of Instagram by adolescents [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' These efforts, however, do not design and implement a data donation system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' they rely simply on users uploading their files to a server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' As the data includes many personal identifiers, having a data dona- tion system is essential to better balance user privacy with data donation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', by removing personal identifiers client-side), which may increase the likelihood of user data donation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Our data donation system facilitates donations from TikTok users who request their downloadable data from the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Our donation system accepts data generated by the TikTok platform and allows users to anonymize and customize their data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', select which fields of the data they want to donate) before donating their data to our infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This approach allows us to collect data on a user’s activity from the beginning 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='04945v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='SI] 12 Jan 2023 of their TikTok usage to the time they requested their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The data includes a wide variety of fields, including the entire video viewing history and engagement activities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', likes, shares, and comments on videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' These datasets are rich and provide us with an invaluable opportunity to study how people use Tik- Tok, while maintaining user privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' In this work, we use our donation system to conduct a study of TikTok user behavior that seeks to understand how people behave on TikTok (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', how many videos people watch and how much time they spent on the platform), how prevalent is video engagement on TikTok, and whether there are signif- icant differences across our participants’ demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We recruited 347 TikTok users (our study participants) to donate their data using our system, ultimately compensating our par- ticipants with $6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='9K USD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We find that participants watch a relatively large number of videos per day (median of 90 videos per day) and spend on average 27 minutes per day on Tik- Tok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Compared to YouTube, this daily consumption is larger (based on one report, users spent 19 average daily minutes on YouTube in 2022 [24]), despite the fact that YouTube’s videos have a longer duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Also, we find that younger partici- pants (aged between 18-24 years old) spent more time on Tik- Tok compared to older participants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', aged between 25-34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This result prompts the need for further investigation into how younger people use TikTok, particularly, to investigate poten- tial social media addiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' With regards to user engagement, we find that participants typically engaged with videos by lik- ing them (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' commenting or sharing) and that younger par- ticipants (aged between 18-24 years old) tend to ‘like’ more videos compared to older participants (aged between 25-34 years old).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The contributions of this work are two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' First, to the best of our knowledge, we perform the first re- search study on TikTok that is based on data donated by real users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We shed light on how TikTok users behave on the plat- form and how prevalent their engagement is via video liking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' As TikTok becomes increasingly popular, especially among younger people, such studies allow us to understand better how people are using this emerging platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Second, we in- vestigate the possibility of performing social media studies by leveraging data donated by the users themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' By design- ing, implementing, and using a data donation system, we expe- rience and surface challenges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', malicious users) that exist in obtaining and analyzing data donated by real users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We dis- cuss these challenges and the lessons we learned, which we believe will be of interest to the research community focusing on understanding closed social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Paper Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Section 2 reviews previous work, while Section 3 describes our data donation system that facilitates data donations from real users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='1 we report our recruitment procedures and in Section 5 we discuss our ethi- cal considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Section 6 includes our analysis and results aiming to provide insights into how people behave on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Finally, we conclude and discuss our findings, potential impli- cations, and lessons learned in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 2 Background & Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='1 TikTok Known as Douyin in China, TikTok is a short-form video plat- form owned by parent company ByteDance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' ByteDance first launched Douyin for China-based users in 2016, and launched TikTok as its international counterpart shortly after [39, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The company acquired Musical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='ly, a similar short-form video platform, in 2018, and in doing so enabled TikTok to expand its user base worldwide [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Since then, TikTok usage has continued to grow, rising to be the most downloaded app of 2020 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' At the time of this writing, TikTok sits at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='3 billion monthly active users worldwide [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' As a platform, TikTok allows users to both watch and cre- ate short-form videos up to 3 minutes in length (currently, the company is rolling out a 10-minute length option for select users [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' TikTok offers multiple editing capabilities for creators, and allows users to connect with peers via follow- ing, messaging, and sharing content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' One of the app’s most prominent characteristics is its ability to recommend relevant video content to viewers [33, 18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' when using TikTok, a user may scroll through two different content feeds, one contain- ing videos posted by the people they follow (“Following”), the other a curated feed of content from many different cre- ators (“For You”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Contents on the For You feed are served to users based on a user’s account information and behavior, the specifics of which are largely unknown to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Much of the prior work on TikTok has studied the recom- mendation algorithm [40, 1, 6, 18, 35, 34], the content on the platform [3, 13, 28, 46, 19, 48, 37, 21, 44, 45, 31], and per- formed user studies on TikTok users [49, 32, 17, 18, 26, 8, 30, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' To the best of our knowledge, our work is the first to investigate user behavior through the lens of data donated directly from users using the rights of data subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' These datasets allow us to obtain a holistic and unique view into how people behave on TikTok and how they engage with content on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2 Data Donation Per the EU’s GDPR [11], most major digital platforms now provide their users with electronic access to the personal data they have on each user, via downloadable data packages [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' A prominent movement in the medical field [5, 36], researchers studying digital platforms are beginning to leverage the rich information in these packages by requesting that users donate them for study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Data donations offer unique insights into dig- ital platforms [41]: for example, uncovering widely-used ad targeting mechanisms on Twitter that were largely ignored by prior work [43] and gaining new insights into how adolescents use Instagram [10] have all been possible via user data dona- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Motivations for using user-donated data stem from limita- tions of other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' People’s perceptions of their own on- line behavior, for instance, can be unreliable [27, 42, 9, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Additionally, researcher-created accounts on digital platforms may lack the authenticity, diversity and history that real user 2 accounts have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Further, scraping TikTok data yields fruitful data sets, yet has capacity limitations and totally relies on pub- lic – and typically, popular – content available on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' User-donated TikTok data can provide further insight into how real TikTok users are consuming content on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' In this work, we design and implement a data donation system that facilitates data donations from TikTok users directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 3 Data Donation System We implement a data donation system, Social Media Donator (SMD), where users can get information on how they can re- quest their data from the TikTok mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We pro- vide details on how users can request their TikTok data and what is included in the data in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' After users down- load their data, they can use SMD to anonymize and customize their data before transferring the data to our infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Data Anonymization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The data collected from TikTok in- cludes personal information and identifiers for each user, such as phone numbers and email addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Due to this, it is essential to ensure proper anonymization of the data before transferring it to our infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' To ensure the data is properly anonymized before it is transferred to our infrastruc- ture, our SMD system removes certain information by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This includes the user’s profile information, direct messages, information about videos uploaded, IP addresses and device information, purchase information, and account status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The anonymization process is done on the client side, and we em- phasize that we only transfer the anonymized dataset to our backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Additionally, we provide a Python script that allows users to anonymize and customize their dataset offline with- out using SMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This script is identical to the one available in SMD and is intended for use by participants who are concerned about privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Data Customization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' TikTok users may have different lev- els of comfort in sharing certain data fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For example, a user who frequently posts comments with personal informa- tion may not feel comfortable sharing their comment-related data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' To address this, we have implemented a customization feature in SMD, which allows users to choose which fields of their data they are comfortable donating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The only manda- tory field is the video viewing history, which includes only the URLs of the videos watched and the timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Additionally, there are some fields that users are not able to donate, as they contain personal information and identifiers (as outlined in the data anonymization procedure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For the remaining fields that a user can choose to donate, we provide clear explanations of what data is included, with specific examples and a description of how we plan to use each field in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Additionally, for data fields that may contain sensitive information, we have added warning labels to alert users that the field may poten- tially include private information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For example, for the search history, we added a warning label that says "This information may be sensitive if you did uncommon searches for things re- lated to your real identity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', searching for videos of a family member’s small sports team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='" Similar warnings were added for the followers, following, and comments data since these fields may reveal the user’s identity through their follower network and comments made on public videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' SMD calculates the compensation for the user based on their selections of which fields of data they opt-in to donate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The mandatory video viewing history is compensated with $5, while all the optional fields such as Like History, Search His- tory, Share History, Login Information, App Settings, Com- ments, Favorites, Following, Followers, and Ads Information are compensated with $1 each, except for comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For com- ments, users have the option to either donate their comment timestamps and content for $2 or only the timestamps for $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The total compensation for each user ranges from $5 to $16, depending on their selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Data Donation & Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Users can donate and transfer their anonymized and customized data to our infrastructure with a single click on the SMD interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' After the data donation, we present all users with an optional survey that includes gen- eral demographic questions and questions about their usage of the TikTok platform and their perceptions of the TikTok algo- rithm’s recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This survey helps us to gain extra context on the users such as their age, gender, and location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' It is important to note that all questions in the survey are optional, and users can choose to not answer by selecting the "Prefer not to say" option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' All users who choose to fill out the survey will receive an additional compensation of $4 regardless of the questions they choose not to answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 4 Data Collection We present our approach to collecting data from TikTok users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We describe our recruitment process, metadata collection for TikTok videos, and our efforts to assess the quality of donated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='1 User Recruitment We recruited participants for the study in two ways: 1) by shar- ing the study through the authors’ networks on Twitter, and 2) by running Facebook Ads targeting people over 18 years old living in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' who Facebook had tagged with the "Tik- Tok" interest category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For the former, we created a poster to advertise the study and shared a single tweet that was amplified by all the authors’ Twitter accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The tweet was shared in January 2022 and received 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='5K impressions on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For the latter, we ran Facebook Ads between January 21, 2022, and February 13, 2022, with an average budget of $8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='5 per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Using these two methods, we recruited 347 participants, whom we compensated with an overall amount of $6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='9K in the form of Amazon gift cards sent via email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 1 shows the percentage of participants that opted-in to donate each potential field that exists in their TikTok data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' As we can observe, most of our participants chose to donate al- most all the fields, as all fields appear in at least 95% of all the donations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Participants were less willing to share their Search History (95%) and Followers (96%), Following (98%), 3 Ads Settings Login History Like History Share History Favorites Comments Followings Followers Search History 0 20 40 60 80 100 %Donations 99% 99% 99% 99% 99% 98% 98% 98% 96% 95% Figure 1: Percentage of donations that opt-in to donate to each field included in the TikTok data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Region # (%) Age # (%) Africa 174 (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2%) 25-34 years old 162 (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6%) N/C America 108 (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='4%) 18-24 years old 145 (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='5%) South America 22 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6%) 35-44 years old 13 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='9%) Prefer not to say 19 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='7%) Prefer not to say 12 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6%) Europe 10 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0%) 45-64 years old 1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='3%) Gender # (%) Education # (%) Men 183 (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='9%) Bachelor’s or above 181 (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='3%) Women 144 (43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2%) Associate’s degree 58 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='4%) Non-binary 3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='9%) Some college 51 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='3%) Prefer not to say 2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6%) HS or below 26 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='8%) Self-described 1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='3%) Prefer not to say 15 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='5%) Trade school 2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6%) Table 1: Demographics of the participants that completed our survey (96% of all participants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' “Prefer not to say” refers to users that opted out from answering that question in the survey, while “N/C America” refers to “North/Central America.” and Comments (98%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This is potentially due to the warning labels associated with these fields in our SMD interface, ex- plaining that some information included in these fields might be sensitive (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This result suggests that most par- ticipants perceive the trade-off between the compensation and the donation of additional fields as worthwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Participants’ Demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Participants self-reported their demographics via an optional survey (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Note that 14 out of 347 participants (4%) opted out from completing the survey, hence we do not have any demographic information about them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Table 1 provides an overview of our participants’ demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' About the reported region in which they reside, we observe that just over half of our participants are from Africa (52%), while the remainder is from North/Central America (32%), South America (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6%), and Europe (3%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Interestingly, many of our participants were from Africa despite the fact that we mainly targeted Facebook users living in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We hypoth- esize that the monetary incentives may have been relatively more substantial for users from Africa highlighting that people from under-developed countries can be attracted by the mone- tary incentives to participate in research by donating their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' With regards to the age of our participants, we note that the majority of our participants are 34 years or younger (91%), #Participants #Actions Video Viewing History 347 9,212,100 Like History 328 1,120,716 Search History 332 13,282 Share History 253 24,944 Comments 227 52,436 Following 333 84,654 Followers 295 43,642 Table 2: Overview of our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We report the number of partici- pants with at least one action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' with 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6% of the participants aged between 25-34 years old and 43% of the participants aged between 18 and 24 years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Note that given the popularity of these age groups in our par- ticipant set, we focus our demographic analysis in Section 6 on the 18-24 years old and 25-34 years old age groups only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Our participants set is somewhat gender-balanced with 55% of the participants being men, while 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2% of our participants are women and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2% are non-binary or self-described their gen- der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Note that we compare only men and women in the gen- der analyses presented in Section 6, as we do not have suffi- cient statistical power to make comparisons between people of other genders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Finally, most of our participants are educated with 54% having a bachelor’s degree or above (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', MSc or PhD), 32% have completed some post-high school coursework or completed an associate’s degree or country equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Table 2 reports an overview of our dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' it in- cludes 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2M video views from 347 participants, between July 26, 2020 and February 21, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Our dataset also includes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='1M like actions from 328 participants, 13K search actions from 332 participants, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='9K shares from 253 participants, 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='4K comments from 227 participants, 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6K following ac- tions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', the participant started following a user) from 333 users, and 43K follower actions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', a TikTok user started fol- lowing one of our participants) from 295 participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Also, we note that by looking into how these actions appear over time in our dataset, we find that there is a 2-month gap for the like actions data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' That is, there is no likes information for any of the 347 recruited participants, likely because of issues with the logging infrastructure within TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Also, we find that information about sharing videos is only included in the data after July 28, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2 Video Metadata Collection Each participant’s data includes information about their activ- ity, including the videos they watched, liked, shared, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Note that for each video, the data only includes a video URL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' To ob- tain more context and metadata about the TikTok videos that are included in our donations, we scrape each video’s Tik- Tok webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Specifically, we use an unofficial Python API wrapper [38] that essentially uses Selenium to scrape the Tik- Tok page for each video and provide the video’s metadata in JSON format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Overall, in our 347 donations, we find 4,938,805 unique TikTok videos viewed by our participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We at- tempted to collect the metadata for all of the videos, man- 4 aging to obtain metadata for 4,122,038 videos (83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='4% of all videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The rest of the videos were either deleted by the up- loader or TikTok itself, or the account that posted the videos set their account into private mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Since participants’ data donations include their entire activity, we collect videos from 2020, which are more likely to be deleted compared to newer videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We ran our metadata collection between January 17, 2022, and March 12, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For each video, we collected the following metadata: 1) Video creation timestamp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 2) Video description and title (defined by the uploader);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 3) Information about the uploader such as the uploader’s username and unique identifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 4) Video metadata such as video duration, video for- mat, definition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', 720p), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' and 5) Video statistics such as the number of shares, comments, and number of times that the video was viewed (at the time of our crawl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='3 Assessing the “Quality” of Donations SMD requires participants to provide an email, for the pur- poses of sending the compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We also use the MD5 hash of each email as the unique identifier for each donation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Note that we do not link the email address explicitly with the donation for anonymity purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' As expected with studies that offer a monetary incentive, users may try to “trick” the system to earn money easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Indeed, during our recruitment, we noticed that some TikTok users were trying to earn more money by donating duplicate or near-duplicate data under dif- ferent email addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' To detect such malicious donations, SMD calculates all the pairwise Jaccard similarities between the video URLs and timestamps that are included in the video viewing history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Then, SMD flags the donations that have a Jaccard similarity of over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2 for either the video URLs or the timestamps, and then we manually check the donations to ver- ify that are indeed duplicate or near-duplicate donations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This process is done before sending the compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Overall, we received 31 duplicate or near-duplicate donations (all of them having a Jaccard similarity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='9 or more) that would have cost us $571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For these donations, we did not pay users, and we informed them via email that they would not receive com- pensation for all their subsequent duplicate donations (as they already received compensation previously).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Another concern that we have is that malicious users can generate fake video URLs and fake timestamps to try and do- nate fake data for compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' To verify if this is happening in our recruitment, we check for what percentage of each do- nation’s videos we were able to obtain video metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Here, we hypothesize that if a malicious user generates fake URLs then we will not be able to get any metadata for the videos in- cluded in the donation (since the URLs will not exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We did not find any evidence of users submitting fabricated donations, since for all the donations we were able to obtain metadata for a large percentage of the videos — at least 70% for every do- nation, with a median of 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' As discussed above, the rest of the videos were inaccessible because they were either deleted or the account became private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 5 Ethical Considerations Before recruiting participants and collecting any data from real TikTok users, we obtained approvals from two different Ethi- cal Review Board committees (one from a university in Eu- rope and one from a university in the US).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We provided to the Ethical Review Board committees a detailed document explaining all the various data fields included in the TikTok data, our anonymization/customization procedures, the con- sent form that is presented to the users, as well as our adver- tisements for recruiting users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For each recruited user, we obtained explicit consent by pro- viding them with a consent form (which can be downloaded via our SMD donation system) and by requesting the users to acknowledge that they fully understand the content of the con- sent form and that they allow us to store their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Addition- ally, as noted in Section 3, for each data field we explained to the participants how we are planning to use the data included in their data and the potential privacy implications of donating some fields of the data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', comments, search history, and fol- lower/following network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' As mentioned in the previous sec- tion, all users that donated their data got a compensation using Amazon gift cards between $5-$20 depending on the fields of their data they opted-in to donate and whether they filled in the optional small survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We will permanently delete all user data within 36 months after the end of our project and the collected data will not be distributed to any third parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Also, we emphasize that the users have the power on their hands with regard to what spe- cific fields of data they opt-in to donate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For our analysis, we follow standard ethical guidelines [29] like reporting our re- sults in aggregate and not attempting to track users across plat- forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Finally, our metadata collection focuses only on pub- licly accessible videos that are available on the TikTok plat- form during the data collection period (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', we do not collect any information about private videos or deleted videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 6 Results In this section, we present our analysis and results focusing on understanding how users behave on TikTok, particularly, assessing how many videos they watch, how much time they spent on the platform, how many videos each participant liked, and whether there are substantial differences on user behavior across demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We compute two metrics to measure consumptive behavior: Time Spent and Volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' To compute Time Spent on the platform, we need to infer an approximation of the watch- ing duration of each video view based on the timestamps since the donated data only contains information on when each par- ticipant started watching each video (see Appendix A on what is included in the data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' To do this, for each video, we calcu- late the number of seconds between the time that the partici- pant started watching a video and the time that the participant started watching the next video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This allows us to obtain an ap- proximation of the viewing duration of each video, except for 5 102 103 104 Average daily time spent for each participant (seconds) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 CDF (a) All 102 103 104 Average daily time spent for each participant (seconds) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 CDF 18-24 years old 25-34 years old (b) Age 102 103 104 Average daily time spent for each participant (seconds) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 CDF Men Women (c) Gender Figure 2: CDF of the average time (in seconds) that our participants spent watching TikTok videos per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 100 101 102 103 Average number of daily video views per participant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 CDF (a) All 100 101 102 103 Average number of daily video views per participant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 CDF 18-24 years old 25-34 years old (b) Age 100 101 102 103 Average number of daily video views per participant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 CDF Men Women (c) Gender Figure 3: CDF of the average number of daily video views per participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' the last video of each session: when the user takes a break, we would infer the viewing time of the last video of the session to be unrealistically long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Unfortunately, we cannot simply use the duration of the video itself to identify session breaks, because we do not have duration metadata for all videos, and because on TikTok, videos automatically start again and loop if users do not scroll to the next one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Instead, we set the threshold for identifying session breaks to 105 seconds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='75 minutes) following the methodology by Halfaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The main idea of the method is that one can perform user session identification (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', when users take breaks) based on inter-activity times by performing a two-component Gaussian mixture model clustering using the expectation-maximization algorithm [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' By obtaining the results from this clustering, we can iden- tify the cutoff threshold between the two clusters, which in our TikTok dataset is equal to 105 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This means that when- ever a participant appears to spend more than 105 seconds on a video, then we infer that they took a break and this marks the end of a session — in this case, we are unable to infer the time that the participant devoted to that specific video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This thresh- old is substantially smaller compared to the one found by Hal- faker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' [15] on Wikipedia (about 1 hour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This is mainly due to the fundamental differences between the two platforms, as people tend to spend more time on a Wikipedia article than a TikTok video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Indeed, by looking at the inferred viewing du- rations and all video durations (obtained from the video meta- data), we find that 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='5% of all the inferred viewing durations are 105 seconds or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' At the same time, only a small fraction of the videos have a duration of over 105 seconds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='8% of all videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Note that we experimented with larger thresholds (be- tween 100 seconds and 600 seconds);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' the main insights from our results were the same, hence we report the results with 105 seconds as our threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Second, we compute a Volume met- ric: the average number of videos each participant watched per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Time Spent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We start our analysis by looking into how much time our participants spend on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 2 shows the Cumu- lative Distribution Function (CDF) of the average time spent per day for each participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For each participant, we calculate the sum of the inferred durations for each day that they used TikTok, and then we calculate the average time across all days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For all participants (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 2(a)), we find a median value of 1,622 seconds per day (27 minutes), with 25% of our partici- pants (Q1) spending on average less than 834 seconds per day (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='9 minutes) on TikTok, while the 25% most active (Q3) par- ticipants spent 2,891 seconds (48 minutes) or more (σ: 1,864 seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The most active participant in our dataset spent, on average, 12,186 seconds per day (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='3 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We also investigate if there are differences across partici- pants based on their demographic information (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 2(b) for age and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 2(c) for gender).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We find that participants aged between 18-24 years old spent more time on TikTok (me- dian: 1,821 seconds, Q1: 948 seconds, Q3: 3,131 seconds, σ: 2,008 seconds) compared to participants aged between 25- 34 years old (median: 1,423 seconds, Q1: 749 seconds, Q3: 2,667 seconds, σ: 1,771 seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' To assess the statistical sig- nificance of our results, we perform a Mann-Whitney U test on the distributions finding that the differences between partic- ipants aged between 18-24 years old and 25-34 years old are statistically significant (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' With regards to the gender- related results (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 2(c)), we find that women (median: 6 0 10 20 30 40 50 60 70 %videos that users engaged with per donation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 CDF Figure 4: Percentage of videos that the participants liked per dona- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 1,917 seconds, Q1: 962 seconds, Q3: 3,243 seconds, σ: 2,109 seconds) spent more time on the platform compared to men (median: 1,360 seconds, Q1: 728 seconds, Q3: 2,432 seconds, σ: 1,452 seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' These results are statistically significant, as confirmed by a Mann-Whitney U test (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Next, we analyze the volume of videos watched by participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 3(a) shows the CDF of the average number of video views per participant, for all participants, while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 3(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 3(c) show the same results for participants grouped by the gender and age groups we compare, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We observe that, on average, our participants watch a substantial number of videos per day, with a median of 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='9 videos per day (Q1: 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='7 videos, Q3: 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='3 videos σ: 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='9 videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This is somewhat expected since TikTok videos are usually short in length, hence users can watch a large number of videos with- out spending much time (compared to other video platforms like YouTube).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We also investigate the differences across age groups and gender (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 3(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 3(c)), finding that, among our participants, women watched significantly (MWU all tests p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='01) more videos on TikTok than men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Liking behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' From the statistics in Table 2, we can ex- tract some valuable insights into user engagement on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Our dataset’s most popular engagement action is liking videos, with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='1M actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This is a substantially larger number than other engagement actions, such as shares (24K) and comments (52K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Given the popularity of these engagement actions and their temporal coverage (see Dataset in Section 4), through- out the rest of the paper, we focus on likes when we study our participants’ engagement with videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We look into the prevalence of likes on a per-participant level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 4 shows the Cumulative Distribution Function of the percentage of videos that were liked per participant (for the ones that opted-in for donating to each of these engagement- related fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We find a median of 4% of videos that are liked per donation (Q1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='87%, Q3: 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='09%, σ : 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='03%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We also investigate differences in liking prevalence on Tik- Tok across our participants’ demographic attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 5 shows the CDF of the percentage of videos that are liked by each participant across the ages and genders we compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We observe similar results as with the consumptive behavior fo- cusing on the number of videos or time spent on the platform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 10 2 10 1 100 101 102 %videos that users liked per donation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 CDF 18-24 years old 25-34 years old (a) Age 10 2 10 1 100 101 102 %videos that users liked per donation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='0 CDF Men Women (b) Gender Figure 5: Percentage of videos that the participants liked across our demographic attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 1) Participants aged between 18-24 years old tend to like more videos (median: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='8%, Q1: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2%, Q3: 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6%, σ: 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6%) compared to the participants that are between 25-34 years old (median: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='6%, Q1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='9%, Q3: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='5%, σ : 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='5%) or older (median: 4%, Q1: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='5%, Q3: 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='5%, σ: 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='8%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This dif- ference in engagement between the age groups is statistically significant with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 2) We find no statistically signifi- cant differences (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='15) in participants’ liking behavior for men and women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Take-aways: The main take-away points from our analysis are: Our participants spent a considerable amount of time watching videos on TikTok per day (median time spent: 27 minutes, median number of videos: 90 videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This consumption is larger compared to other video platforms like YouTube, where people spend on average 19 min- utes [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Also, we find that participants aged between 18-24 years old tend to spend more time on the platform than participants aged between 25-34 years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' This dif- ference in consumptive behavior highlights the need for more research efforts on understanding user consump- tive behavior in young people, especially in teens (an age group that we do not analyze here and might be more prone to addiction from TikTok).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Finally, we find that women tend to spend more time and watch more videos on TikTok compared to men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' By far the most popular engagement action is liking a video with sharing and commenting on videos having a 7 substantially smaller prevalence in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Also, we find statistically significant differences in the liking be- havior of participants aged between 18-24 years old com- pared to participants aged 25-34 years old (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', younger participants tend to like more content on TikTok).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 7 Discussion & Future Work Below, we discuss our TikTok user behavior measurements and lessons for future data donation efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='1 Observing TikTok Behavior Our work is a stepping stone to understanding how people con- sume and engage with content on TikTok, based on traces from real users via data donations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Our analysis shows that our par- ticipants spend a considerable amount of time per day on Tik- Tok and watch many videos, more so than on other popular video platforms like YouTube [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' They engage with videos to a relatively limited extent: a median of 4% of videos were liked per donation, and likes were significantly more prevalent than other engagement actions such as shares and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Our quantitative findings and data collection methodology can assist future investigations that analyze content and/or par- ticipant experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For example: What is the impact of Tik- Tok’s algorithm on users, and to what extent do the recom- mendations lead to informational rabbit holes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' What is the content that people of different demographics engage with?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' What actions do users take to influence TikTok’s recommen- dations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', intentionally skipping videos or following cer- tain accounts), and how effective are those actions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' What is the role of ads on TikTok [14, 47]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Future research may also wish to consider additional demographics than those explored in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='2 Data Donation System Here, we summarize future directions for and lessons learned from designing and deploying our data donation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' User recruitment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' In this work, our initial goal was to re- cruit participants exclusively from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', which is why we targeted people from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' on Facebook and explicitly men- tioned that we were recruiting U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' people in our Twitter post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Despite our initial goal and our ads, our data donation sys- tem was publicly available to anyone, and we did not limit access to people coming from specific regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Hence, we ob- tained a substantial number of participants from other regions as well (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', Africa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Future work that aims to recruit partic- ipants from specific regions or with other restrictions should implement measures that ensure participants meet the study’s requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' One way to do this is by limiting the access to the system to specific IP addresses that come from the re- gion/country of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Malicious users and assessing the quality of donations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' During our recruitment and data donation procedures, we no- ticed several instances of malicious users trying to “trick the system” to get extra monetary incentives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We identified three cases of malicious users: 1) users donating duplicate data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', donating the same data multiple times with different email ad- dresses) to get extra monetary incentives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 2) users donating their data, then using TikTok for a few more days, and then trying to donate their data again (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', duplicate data for the en- tire period except the few extra days at the end of the data);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' and 3) users creating new TikTok accounts, watching only a few videos on TikTok, and then trying to donate their data via our donation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Based on these observations, we make the following recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Researchers implementing data donation systems should design and implement specific countermeasures to detect malicious users that try to donate “useless” data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Concretely, developers of data donation sys- tems should implement features to detect duplicate or near- duplicate donations, similar to the ones we implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' In addition, to overcome the problem of people creating new ac- counts for the sake of donating their data, data donation sys- tems can be developed in such a way so that they only ac- cept donations that meet a minimum number of days of activ- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For instance, in our case, our donation system rejected all donations where the donation lifespan (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', the difference be- tween the last video view and first video view) was less than three months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Finally, in our work, we did not notice any at- tempts to donate completely fabricated data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', randomly computer-generated video URLs and timestamps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' However, we argue that this possibility exists, and tech-savvy malicious users may use some techniques in the future depending on how much money they can make from this activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Pricing mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Our experience recruiting people to do- nate their data for research purposes with monetary incentives, highlights that those from countries where the incentives offer more purchasing power (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', African countries) may be more willing to donate their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' More broadly, an essential aspect of data donation infrastructures is the underlying pricing mech- anism, since this may affect how willing potential participants will be to donate their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For our data donation infrastruc- ture, we set some pre-defined compensation amounts for the viewing video history ($5), as well as all optional fields ($1 for each field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We thought that these prices are reasonable for our purpose and what was asked from the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' However, future work is needed to empirically understand the interplay between data donation and pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Trustworthiness of the data donation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The degree of trust between the participants and the data donation system is likely to affect participant recruitment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We made two design choices in an effort to enhance participant trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' First, we pro- vided offline tools that participants can run to anonymize and customize their data before using our data donation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Second, we offer users control over which fields to donate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Nevertheless, future work on data donation systems should in- vestigate which, if any, of these measures enhance user trust and explore additional measures that may appeal to users such as stronger formal guarantees that the data donation code is working as described and allowing only the access and queries participants expect to be run on their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Missing data and the need for compliance audits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' By col- 8 lecting and analyzing user-donated data from TikTok, we no- ticed some missing data for all the participants (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', there were no like data for two months).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The missing data might be due to failures in the logging infrastructure within TikTok or due to lost data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Nevertheless, this prompts the need for systematic audits of social media platforms to assess the platforms’ com- pliance with the access rights of data subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Future work can design controlled experiments to assess how accurate and com- prehensive the data provided by platforms is;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', by using the platform to perform some pre-defined actions, then accessing the provided data and comparing it with the set of pre-defined actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='3 Limitations Naturally, our work has some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' First, our recruit- ment procedure and research budget allowed us to obtain data from a small number of participants (347).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' We did not cover a full range of user demographics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', age, gender, geography, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Due to a lack of public information about the demo- graphics of TikTok’s user base, we cannot draw conclusions about the representativeness of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Despite this crit- ical limitation, our results offer a first measurement-based in- vestigation into how people consume and engage with content on the TikTok platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' At the same time, the lessons learned from designing and implementing a data donation system can provide valuable insights to researchers that aim to collect data using similar approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Second, our video metadata collec- tion is (necessarily) done post-hoc, hence we are unable to ob- tain a holistic view of all videos referenced in the data dona- tions since approximately 17% of all videos were not accessi- ble during our video metadata collection period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Third, even though the data donations are quite detailed, some of our re- sults are based on inferences, such as the inferred time that each participant spent watching videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Due to this, we are un- able to accurately infer the viewing durations for video views that are above our determined threshold following the method- ology by [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Therefore, our results based on the time spent on the platform should be considered lower-bound results, as we exclude all the video views at the end of each inferred ses- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Many of our limitations reflect fundamental challenges with data donation as a way to externally study and audit closed platforms: our investigation was often limited by intentional or unintentional (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', the gap in “like” data) omissions in the data provided by TikTok’s data download feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Hence, as the re- search community and data protection 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Computers in Human Behavior Reports, 3:100090, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' [43] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Wei, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Stamos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Veys, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Reitinger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Goodman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Herman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Filipczuk, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Weinshel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Mazurek, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' What twitter knows: Characterizing ad tar- geting practices, user perceptions, and ad explanations through users’ own twitter data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' In 29th USENIX Secu- rity Symposium (USENIX Security 20), pages 145–162, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' [44] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Weimann and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Masri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Research note: spreading hate on tiktok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Studies in conflict & terrorism, pages 1– 14, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' [45] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Weimann and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Masri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Tiktok’s spiral of anti- semitism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Journalism and Media, 2(4):697–708, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' 10 [46] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Yeung, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Ng, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Abi-Jaoude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Tik- tok and attention-deficit/hyperactivity disorder: A cross-sectional study of social media content qual- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The Canadian Journal of Psychiatry, page 07067437221082854, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' [47] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Zeng, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Kohno, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Roesner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' What makes a ‘bad’ ad?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' user perceptions of problematic online advertising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' In CHI Conference on Human Factors in Computing Sys- tems, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' [48] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Zeng and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Abidin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' ‘# okboomer, time to meet the zoomers’: studying the memefication of intergener- ational politics on tiktok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Information, Communication & Society, 24(16):2459–2481, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' [49] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Zeng and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Kaye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' From content moderation to visibility moderation: A case study of platform gover- nance on tiktok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Policy & Internet, 14(1):79–95, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' A Requesting Data from TikTok TikTok enables its users to request a comprehensive dataset of their activity on the platform and other personal information the platform has on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The request can be made through the TikTok mobile application’s settings menu, and the data will be provided in either JSON or a human-readable format based on the user’s preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' The process takes around 3-4 days for the data to be ready for download.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Here we outline the various fields of information included in a user’s TikTok data download.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Video Viewing History: A list of videos and the times- tamp that the user started watching each video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Like History: A list of videos that the user liked and the timestamp of each like action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Search History: A list of search queries that the user per- formed in TikTok, including a timestamp of each search action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Share History: A list of videos that were shared by the user, along with the timestamp of the action, and the method for the share (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', shared via WhatsApp or Face- book Messenger, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Login Information: A list of login information for each time the user started using the TikTok app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For each lo- gin, the file includes the timestamp, the IP address from where the user performed the login, information about the device (device model and device system), network type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', WiFi, mobile data, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' ), and the carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' App Settings: Information about the settings that the user set on TikTok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' For instance, the interests that the user pre-defined in the TikTok application and privacy settings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', whether the account is private, who can comment on their videos, who can message them, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=') Comments: A list of comments made by the user along with the timestamp of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Note that the file does not include the specific video where the comment was posted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Favorites: A list of videos, effects, hashtags, sounds, and videos that the user favorited on TikTok including the timestamp of each action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Following/Followers: A list of accounts that the user fol- lows/follow the user along with the specific timestamp of the follow action Ads Information: Information about the advertisers that targeted the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Profile Information: A set of profile attributes that the user set on the TikTok application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' It includes the user’s bio, email address, telephone number, username, profile photo, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Direct Messages: A list of messages shared via private chats between the user and other TikTok users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Video Uploads: A list of the user’s uploaded videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Purchase History: Information about the purchases made by the user within the TikTok platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' Account Status: Information about the status of the Tik- Tok application on the user’s phone (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=', application ver- sion, screen resolution, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} +page_content=' ) 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE4T4oBgHgl3EQfMQxc/content/2301.04945v1.pdf'} diff --git a/VdE3T4oBgHgl3EQf0Qv6/content/2301.04737v1.pdf b/VdE3T4oBgHgl3EQf0Qv6/content/2301.04737v1.pdf new file mode 100644 index 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Pointe Dr. +Laguna Hills, CA 92653 +wangjuan313@gmail.com +Bin Xia +Shenzhen SiBright Co., Ltd. +Tinwe Industrial Park, No. 6 Liufang Rd. +Shenzhen, Guangdong 518052 +b.xia@sibionics.com +ABSTRACT +Weakly supervised image segmentation approaches in the literature usually achieve high segmenta- +tion performance using tight bounding box supervision and decrease the performance greatly when +supervised by loose bounding boxes. However, compared with loose bounding box, it is much +more difficult to acquire tight bounding box due to its strict requirements on the precise locations +of the four sides of the box. To resolve this issue, this study investigates whether it is possible +to maintain good segmentation performance when loose bounding boxes are used as supervision. +For this purpose, this work extends our previous parallel transformation based multiple instance +learning (MIL) for tight bounding box supervision by integrating an MIL strategy based on polar +transformation to assist image segmentation. The proposed polar transformation based MIL formu- +lation works for both tight and loose bounding boxes, in which a positive bag is defined as pixels +in a polar line of a bounding box with one endpoint located inside the object enclosed by the box +and the other endpoint located at one of the four sides of the box. Moreover, a weighted smooth +maximum approximation is introduced to incorporate the observation that pixels closer to the ori- +gin of the polar transformation are more likely to belong to the object in the box. The proposed +approach was evaluated on two public datasets using dice coefficient when bounding boxes at differ- +ent precision levels were considered in the experiments. The results demonstrate that the proposed +approach achieves state-of-the-art performance for bounding boxes at all precision levels and is ro- +bust to mild and moderate errors in the loose bounding box annotations. The codes are available at +https://github.com/wangjuan313/wsis-beyond-tightBB. +Keywords Multiple instance learning, Weakly supervised image segmentation, Polar transformation, Bounding box, +Deep neural networks +1 +Introduction +Image segmentation is the process of partitioning a digital image into multiple image segments such that pixels in an +image segment share certain characteristics and are assigned the same category label. It has been widely studied in +all kinds of applications [1] since 1965. In recent years, with the development of the deep learning in medical image +analysis [2,3,4,5], deep neural networks (DNNs) have been used to successfully tackle a variety of image segmentation +problems in a fully-supervised manner [6, 7, 8]. However, it is labor-intensive and expensive to collect large-scale +dataset with precise pixel-wise annotations for DNN training, thus limiting the value of the image segmentation in real +applications. This is especially true in medical image analysis due to the difficulty of the segmentation problems and +the difficulty in recruiting qualified annotators. +To resolve the problem mentioned above, great interests have been made in the literature to develop weakly supervised +image segmentation (WSIS), the purpose of which is to substitute costly pixel-wise annotations into cost-effective +annotations as supervision for image segmentation. Several types of annotations have been investigated, including +image labels [9, 10], points [11], scribbles [12], and bounding boxes [13, 14]. This work considers bounding boxes +as supervision for image segmentation. In the literature, some efforts have been made to develop WSIS adopting +arXiv:2301.12053v1 [cs.CV] 28 Jan 2023 + +WSIS beyond tight BB +Figure 1: Demonstration of tight (red color) and loose (green color) bounding boxes for the object sheep. +bounding box supervision. For example, Rajchl et al. [13] designed an iterative optimization approach for image +segmentation, in which a neural network classifier was trained using bounding box supervision. Hsu et al. [15] +exploited mask R-CNN to simultaneously conduct object detection and image segmentation, in which the bounding +box supervision was formulated as multiple instance learning (MIL) for image segmentation. Kervadec et al. [14] +imposed a set of constraints on the network output by leveraging the tightness prior of bounding boxes as supervision +for image segmentation. +Depending on the relationship between an object and its bounding box annotation, as shown in Figure 1, bounding +box annotations can be divided into two categories: tight bounding box and loose bounding box. The tight bounding +box has its four sides touching the object, thus the size of the tight bounding box is same as the size of the object; +in contrast, the size of the loose bounding box is larger than the size of the object, thus at least one side of the loose +bounding box does not touch the object. Compared with the loose bounding boxes, the tight bounding boxes have +strict requirements on the precise locations of the four sides of the bounding box annotations, thus it is much more +difficult and time-consuming to annotate tight bounding boxes. However, most methods in the literature, if not all, +achieve high segmentation performance using tight bounding box supervision, and decrease the performance greatly +when supervised by loose bounding boxes [14]. Such inconsistency in the difficulty of annotation acquisition and +the segmentation performance for tight and loose bounding boxes poses a problem in bounding box supervision. To +conquer this issue, in this paper we investigate whether it is possible to maintain good segmentation performance when +loose bounding boxes are used as supervision. +Recently, in our previous study we developed a WSIS approach using tight bounding box supervision by exploiting +the properties of tight bounding boxes for MIL formulation [16] and achieved state-of-the-art segmentation perfor- +mance [16,17]. Building on our previous success in [16], this study extends tight bounding box supervision to loose +bounding box supervision for image segmentation. For this purpose, we propose an MIL formulation based on polar +transformation of the image region in the bounding box to assist the approach in [16] for image segmentation, and +develop a weighted smooth maximum approximation to incorporate the observation that pixels closer to the origin of +the polar transformation are more likely to belong to the object in the bounding box. In the end, the segmentation +assistance is conducted by combining the loss derived from the polar transformation based MIL into the loss in [16]. +The polar transformation based MIL strategy developed in this study is valid for both tight and loose bounding boxes, +contributing on the good segmentation performance of the proposed approach for both tight and loose bounding box +supervision. In the experiments, the proposed approach is evaluated by two public datasets when both tight bound- +ing boxes and loose bounding boxes at different precision levels are used as supervision. The results demonstrate +that the proposed approach outperforms several existing methods in all precision levels of bounding boxes; and more +importantly, it is robust to mild and moderate errors in the loose bounding box annotations. +In summary, the contributions of this study are as follows: +1. First, we develop a WSIS approach beyond tight bounding box supervision. It achieves state-of-the-art per- +formance for bounding boxes at all precision levels and is robust to mild and moderate errors in the loose +bounding box annotations. +2. Second, we propose an MIL strategy based on polar transformation of the image regions of the bounding +boxes to incorporate the bounding box supervision into the network output to assist the image segmenta- +tion. The proposed polar transformation based MIL formulation works for both tight and loose bounding +boxes, contributing on the good segmentation performance of the proposed approach for both tight and loose +bounding box supervision. +2 + +WSIS beyond tight BB +(a) +(b) +Figure 2: Demonstration of crossing and polar lines for the object “sheep”. In these plots, crossing and polar lines are +marked by blue dashed lines and bounding boxes are denoted as red rectangles, in which the tight and loose bounding +boxes are shown in the upper and lower rows, respectively. The left column shows examples of crossing lines and the +right column shows examples of polar lines, in which points O are indicated by green dots. +3. Third, a weighted smooth maximum approximation is introduced for the bag prediction in the proposed +polar transformation based MIL to incorporate the observation that pixels closer to the origin of the polar +transformation are more likely to belong to the object in the box. +4. Finally, the proposed approach is evaluated on two public datasets when bounding boxes at different precision +levels are used as supervision. The results demonstrate the effectiveness of the proposed approach for image +segmentation in all precision levels of bounding boxes. +2 +Preliminaries +2.1 +Problem descriptions +This study investigates the use of bounding boxes as supervision for weakly supervised image segmentation (WSIS). +That is, for each object in the training set, a bounding box annotation is provided to supervise the model training. +2.1.1 +Bounding box and object +To avoid any ambiguity, the definitions of bounding box and object are first introduced as follows: +Bounding box is an imaginary rectangle enclosed a thing of interest in an image, which has been widely used in object +detection. For an object, its bounding box annotation encloses the whole object in the box such that it does not overlap +with the region outside its bounding box. Depending on the relationship between an object and its bounding box +annotation, bounding box annotations can be divided into two categories: one is tight bounding box, and the other is +loose bounding box. Tight bounding box is the smallest rectangle enclosing the whole object, thus the object must +touch the four sides of its bounding box. In the end, the size of the object is same as the size of its tight bounding box. +In contrary, loose bounding box is outside of the object, thus at least one side of the loose bounding box does not touch +the object. In the end, the size of the object is smaller than the size of its loose bounding box. +In a bounding box, two types of lines are considered in this study. For convenience, they are named as crossing line +and polar line. Crossing line of a bounding box is defined as a line with its two endpoints located on the opposite sides +of the box. Polar line of a bounding box is defined as a line with one endpoint (denoted as point O) located on a pixel +belonging to the object in the bounding box and the other endpoint located at one of the four sides of the bounding +box. As examples, Figure 2 demonstrates examples of crossing lines (left column) and polar lines (right column) for +both tight (upper row) and loose (lower row) bounding boxes. +Without loss of generality, object considered in this study is defined as a thing which covers a connected region in an +image. It is important to notice that an object does not include any disjointed parts of a thing. If there are multiple +disjointed parts in a thing, each part is treated as an independent object, thus a bounding box is annotated for each part. +3 + +WSIS beyond tight BB +2.1.2 +WSIS using bounding box supervision +For ease of development, we first introduce the fully supervised image segmentation (FSIS). Suppose I denotes an in- +put image and Y ∈ {1, 2, · · · , C} is its corresponding pixel-level category label for C categories under consideration, +the image segmentation task is to obtain the prediction of Y, denoted as P, for the input image I. For the problem of +FSIS, the pixel-wise category label Y is available for each image I in the training set for model optimization. +However, in the problem of WSIS using bounding box supervision, the pixel-level category label Y is unavailable; +instead, it provides the bounding box label B as supervision for model training. In this study, the bounding box label +B is denoted as B = {bm, ym}, m = 1, 2, · · · , M, in which M is the number of bounding box annotations, bm is a +4-dimensional vector denoting the top left and bottom right points of the mth bounding box, and ym ∈ {1, 2, · · · , C} +is the category label of the object in the mth bounding box. +This study considers a specific type of deep neural networks (DNNs) for image segmentation, such as UNet [6] and +FCN [7]. This type of DNNs is able to output pixel-wise prediction for the input image. Due to the possible overlaps +of objects of different categories in an image, which is especially true in medical images, this study formulates the +image segmentation problem as a multi-label classification problem. That is, for a location k in the input image, it +outputs a vector pk = [pk1, pk2, · · · , pkC], one element for a category; each element is converted to the range of [0, 1] +using the sigmoid function. +2.1.3 +Multiple Instance Learning +Multiple instance learning (MIL) is a form of weakly supervised learning in which training samples are arranged in +sets, called bags, and a category label is provided for the entire bag [18]. In MIL, supervision is only provided for +bags, and the labels of individual samples in the bags are not provided. For image segmentation, training samples are +individual pixels of images in the training set, thus a bag consists of a set of different individual pixels of an image. +In MIL, a bag is positive if it has at least one positive sample, and a bag is negative if all of its individual samples are +negative. Therefore, for a category c, the pixel with highest prediction in a positive bag tends to belong to category +c, while even the pixel with highest prediction in a negative bag does not in category c. Based on this observation, +suppose pkc is the network output of the kth pixel in bag b for category c, the bag prediction Pc(b) of bag b for category +c can be defined as +Pc(b) = +n−1 +max +k=0 pkc +(1) +where n is the number of pixels in the bag b. +2.2 +MIL baseline +This study considers the MIL baseline approach which employs tight bounding boxes for supervision. +2.2.1 +Positive and negative bags +In an input image I, for an object of category c and its tight bounding box, it can be easily noted that any vertical and +horizontal crossing line in the tight bounding box has at least one pixel belonging to the object in the box, hence pixels +on a vertical or horizontal crossing line of the tight bounding box compose a positive bag for category c. Furthermore, +pixels on a vertical or horizontal line of the image that do not overlap with any bounding boxes of category c in the +image do not belong to category c, hence pixels on a vertical or horizontal line of the image that does not overlap with +any bounding boxes of category c constitute a negative bag for category c. Based on these observations, for a category +c, the MIL baseline approach considers all of the horizontal and vertical crossing lines of the tight bounding boxes of +category c as positive bags, and all of the horizontal and vertical lines of the image that do not overlap any bounding +boxes of category c in the image as negative bags [15]. As examples, the positive and negative bags for MIL baseline +approach are demonstrated in Figure 3. +2.2.2 +MIL baseline loss +To optimize the various parameters associated with network, MIL baseline loss with two terms is employed [15]. For +a category c, suppose its positive and negative bags in the training set are denoted as B+ +c and B− +c , respectively, then +MIL baseline loss Lc is +Lc = φc(P; B+ +c , B− +c ) + λϕc(P) +(2) +where φc is the unary loss, ϕc is the pairwise loss, and λ is a constant value controlling the trade off between the unary +loss and the pairwise loss. +4 + +WSIS beyond tight BB +Figure 3: Demonstration of positive and negative bags for MIL baseline with the tight bounding box annotation. The +tight bounding box is indicated by the red rectangle for the object “sheep”. The examples of positive and negative +bags are marked by blue and green colors, respectively. +The unary loss φc is defined as: +φc = − +1 +|B+ +c | + |B− +c | +� +� � +b∈B+ +c +log Pc(b) + +� +b∈B− +c +log(1 − Pc(b)) +� +� +(3) +where |B| is the cardinality of B. Mathematically, the unary loss is a binary cross entropy loss for the bag prediction +Pc(b). It gets minimum when the the bag prediction Pc(b) is 1 for positive bags and 0 for negative bags. More +importantly, the unary loss adaptively selects one pixel per bag based on the network prediction for optimization, +yielding an adaptive sampling effect on the training samples during training. +However, using the unary loss alone is prone to segment merely the discriminative parts of an object rather than the +whole object. To resolve this problem, the pairwise loss is introduced as follows: +ϕc = 1 +|ε| +� +(k,k′)∈ε +(pkc − pk′c)2 +(4) +where ε is the set containing all neighboring pixel pairs. Complementary to the unary loss, the pairwise loss enforces +the piece-wise smoothness in the network prediction. +In the end, for all C categories, the MIL baseline loss L is +L = +C +� +c=1 +Lc +(5) +3 +Methods +As noted in the introduction, tight bounding boxes are difficult to acquire due to the strong constraint posed on the +precise locations of the four sides of the bounding box annotations. To deal with this issue, this study extends our +previous approach on WSIS using tight bounding box supervision [16], named as parallel transformation based MIL +in this study, by incorporating a polar transformation based MIL, which works for both tight and loose bounding boxes, +to assist the image segmentation. In the end, the total loss L for network optimization is as follows: +L = +C +� +c=1 +φpa +c (P; Bpa+ +c +, Bpa− +c +) + φpo +c (P; Bpo+ +c +, Bpo− +c +) + λϕc(P) +(6) +where φpa +c is the unary loss derived from the parallel transformation based MIL for its positive bags Bpa+ +c +and negative +bags Bpa− +c +(will be described in Section 3.1), φpo +c is the unary loss obtained from the polar transformation based MIL +for its positive bags Bpo+ +c +and negative bags Bpo− +c +(will be introduced in Section 3.2), and ϕc is the pairwise loss +defined in equation (4). +3.1 +Parallel transformation based MIL using tight bounding box supervision +The method described in this section was first introduced in [16]. More details are provided in this study regarding to +efficient calculation of the positive bag prediction. +5 + +WSIS beyond tight BB +Figure 4: Demonstration of positive bags in parallel transformation based MIL with tight bounding box annotation. +In this plot, the tight bounding box of the object “sheep” is indicated by the red rectangle; examples of positive bags +from two different angles are provided, in which those with θ′ = 25◦ are marked by purple dashed lines and those +with θ′ = 0◦ are given by green dashed lines. +3.1.1 +Positive bags Bpa+ +c +One issue associated with the positive bag definition in MIL baseline is that for an object of height H pixels and width +W pixels, it yields only H+W positive bags, the value of which is much smaller than the size of the object, hence +limiting the selected positive samples during training and resulting in a bottleneck for the segmentation performance. +Noticed that in an input image I, for an object of category c and its tight bounding box, any parallel crossing line of +the tight bounding box also has at least one pixel belonging to the object in the box, this study generalizes the positive +bag definition by considering a parallel crossing line of the tight bounding box as a positive bag for category c. A +parallel crossing line of a bounding box can be parameterized by an angle θ′ ∈ (−90◦, 90◦) with respect to the edges +of the box where its two endpoints located. For an angle θ′, two sets of parallel crossing lines can be obtained from the +bounding box, one crosses up and bottom edges of the box, and the other crosses left and right edges. In the end, the +positive bags Bpa+ +c +for category c are all parallel crossing lines of the tight bounding boxes of the objects of category c +on a set of different angles. As demonstration, Figure 4 shows examples of positive bags obtained from two different +angles, where those indicated by purple dashed lines are with θ′ = 25◦, and those marked by green dashed lines have +θ′ = 0◦. More importantly, by comparing Figures 3 and 4, the positive bags in MIL baseline are a subset and special +cases of the positive bags Bpa+ +c +with θ′ = 0◦. In the experiments, this study presets the set of angles as θ′ ∈ (a, b, s), +denoting evenly spaced angle values within interval (a, b) with step s. +In implementation, it is inefficient to directly calculate the bag prediction Pc(b) of positive bags Bpa+ +c +from parallel +crossing lines in the input image. To facilitate it, we propose to transform the parallel crossing lines with angle θ′ in +the input image into vertical or horizontal lines by rotating the input image by angle θ′. In this study, this process of +obtaining parallel crossing lines is named as parallel transformation to emphasize that the transformation is targeted +at crossing lines of bounding boxes. As examples, Figure 5(a) shows the process of parallel transformation, in which +examples of parallel crossing lines with angle θ′ = 25◦ in the upper image are transformed into horizontal and vertical +lines in the lower image. +However, for efficient calculation of the bag prediction, the parallel transformation of the input image alone is not +enough. As shown in Figure 5(a), the vertical lines in the rotated image are no longer aligned at the same starting and +ending points along the horizontal direction; and the same problem also exists for the horizontal lines in the rotated +images. Therefore, an indicator has to be provided for each pixel in the rotated image to determine whether it is in a +positive bag or not. For this purpose, we construct a box-mask image for each tight bounding box in the input image. +The box-mask image has same size as the input image, its value is set to be 1 for the pixels in the box and 0 for those +outside the box. Afterwards, the same parallel transformation is applied to the box-mask image to determine whether +each pixel is in a positive bag. In the rotated box-mask image, the pixels with value 1 in a vertical or horizontal line +corresponds to a parallel crossing line, thus consisting of a positive bag. As examples, in Figures 5(b), we also show +the parallel transformation of the box-mask image of Figures 5(a). From Figure 5(b), the white pixels along each +horizontal or vertical line (denoted by dashed blue line) consist of a positive bag in the rotated box-mask image. +Finally, to further speed up the calculation, for a tight bounding box of an object, the parallel transformation is only +applied to the cropped region of the input image around the box and its corresponding box-mask image, in which a +small margin is added to four sides of the box to avoid information loss during rotation. +6 + +WSIS beyond tight BB +(a) +(b) +Figure 5: Demonstration of parallel transformation, in which the images and their results of parallel transformation +are provided in the upper and lower rows, respectively. In these plots, examples of paralleled crossing lines with angle +θ′ = 25◦ are marked by purple dashed lines. (a) input image, in which the tight bounding box of “sheep” is marked +by red rectangle, (b) the box-mask image of the tight bounding box in (a). +3.1.2 +Negative bags Bpa− +c +Similar as the positive bag definition, the negative bag definition in MIL baseline also has the problem of limited +samples. Notice in an input image I, for a category c, any individual pixels outside of any bounding boxes of category +c in the image do not belong to category c, we define a negative bag for category c as an individual pixel outside any +bounding boxes of category c. In the end, negative bags Bpa− +c +for category c consist of all of individual pixels outside +all of the bounding boxes of category c. This definition greatly increases the number of negative bags for training, and +forces the network to learn every pixel outside bounding boxes. +3.1.3 +Unary loss φpa +c +The parallel transformed based MIL formulation above will inevitably lead to imbalance between positive and negative +bags. To eliminate this issue, we borrow the concept of focal loss [19] and define the unary loss as follows: +φpa +c += − 1 +N + +� +� � +b∈Bpa+ +c +β (1 − Pc(b))γ log Pc(b)+ +� +b∈Bpa− +c +(1 − β)Pc(b)γ log(1 − Pc(b)) +� +� +(7) +where N + = max(1, |Bpa+ +c +|), β ∈ [0, 1] is the weighting factor, and γ ≥ 0 is the focusing parameter. Mathematically, +the unary loss φpa +c +is focal loss for bag prediction Pc(b), it gets minimum when Pc(b) is 1 for positive bags and 0 for +negative bags. +3.2 +Polar transformation based MIL using tight or loose bounding box supervision +This study proposes polar transformation based MIL to assist the parallel transformation based MIL for image seg- +mentation. The proposed polar transformation based MIL works for both tight and loose bounding boxes, contributing +on the good segmentation performance of the proposed approach for both tight and loose bounding box supervision. +Its details are provided in this section. +3.2.1 +Positive bags Bpo+ +c +To extend the positive bag definition beyond the tight bounding box supervision, we consider the polar line of the +bounding box. For an object of category c, any polar line of its bounding box has at least one pixel belonging to +7 + +WSIS beyond tight BB +(a) +(b) +Figure 6: Demonstration of positive bags in polar transformation based MIL for (a) tight and (b) loose bounding boxes +of the object “sheep”. In each plot, the bounding box is marked by the red rectangle, the point O is denoted by the +green dot, and examples of positive bags are indicated by blue dashed lines. +category c, thus this study considers pixels in a polar line of the bounding box as a positive bag for category c. This +definition does not employ any prior information of the bounding box, thus is valid for both tight and loose bounding +boxes of the object. In the end, the positive bags Bpo+ +c +for category c are defined as all of the polar lines of the +bounding boxes of the objects with category c. As examples, Figure 6 shows examples of positive bags for both tight +and loose bounding boxes. +Mathematically, the bag prediction Pc(b) of positive bags of an object from the corresponding polar lines can be +efficiently obtained by applying polar transformation to the image. The polar transformation of an image transfers the +image from the Cartesian coordinate system to the polar coordinate system, providing a pixel-wise representation in +the polar coordinate system. In this study, it transfers a polar line of the bounding box of an object in an image into a +horizontal line in the polar image. Suppose (u, v) is the Cartesian coordinate of a pixel in the the Cartesian coordinate +domain with respect to an origin Op and a radius Rp, and the polar coordinate is (r, θ), where r > 0 and θ ∈ [0, 2π) +are the radial and angular coordinates, respectively. The polar transformation maps the pixel (u, v) in the Cartesian +coordinate plane to the corresponding pixel (r, θ) in the polar coordinate plane as follows: +r = +√ +u2 + v2 +θ = tan−1(v/u) +(8) +In polar transformation, the size of the polar image in the polar coordinate plane is preset by user during experiments. +Suppose it is Nr × Nθ, where Nr is the dimension of the polar axis and Nθ is the dimension of the angle axis. +In this study, the polar transformation is applied to the cropped region of the input image around the bounding box. A +small margin is added to four sides of the bounding box for region cropping to avoid possible information loss during +transformation. For the cropped region, during polar transformation, the origin Op is set as the point O defined in +Section 2.1.1 and the radius Rp is set as half length of the diagonal line of the bounding box R, which is the radius of +the minimum circle enclosing the bounding box. With such settings, in polar transformation of the cropped region, the +radial coordinate r is evenly distributed in [0, R] with step R/Nr, and the angular coordinate θ is evenly distributed in +[0, 2π) with step 2π/Nθ. As demonstration, Figures 7(a) and (c) show polar transformation for examples of positive +bags from tight and loose bounding boxes, respectively. +Similarly, as shown in Figures 7(a) and (c), the horizontal lines in a polar image are no longer aligned at the ending +points along the vertical direction. To determine whether each pixel in the polar image is in a positive bag or not, we +also introduce the box-mask image of the cropped region and applied the same polar transformation to the box-mask +image as indicator of pixels in positive bags. As examples, Figures 7(b) and (d) show the box-mask images and its +corresponding polar images for the cropped regions in Figures 7(a) and (c), respectively. In the lower row of Figures +7(b) and (d), the white pixels along a horizontal line (denoted by dashed blue lines) consist of a positive bag. +Finally, for bounding box annotation, the origin O is unknown and should be determined during experiments. Based +on the fact that the origin O is inside the object in the bounding box, it is selected as the pixel with maximum network +output among all of the pixels in the bounding box during training in the experiments. Such design is intuitive since +the pixel with highest prediction are more likely belong to the object. +3.2.2 +Negative bags Bpo− +c +This study employs the same negative bag definition as in that in Section 3.1.2 for the polar transformation based MIL, +therefore, we have Bpo− +c += Bpa− +c +. +8 + +WSIS beyond tight BB +(a) +(b) +(c) +(d) +Figure 7: Demonstration of polar transformation for examples of positive bags, in which the images and their polar +images are provided in the upper and lower rows, respectively. (a) The cropped region of the object “sheep” from tight +bounding box, (b) the box-mask image of (a), (c) the cropped region of the object “sheep” from a loose bounding box, +(d) the box-mask image of (c). +3.2.3 +Unary loss φpo +c +Similar as unary loss φpa +c defined in Section 3.1.3, the unary loss φpo +c for polar transformation based MIL is +φpo +c = − 1 +N + +� +� � +b∈Bpo+ +c +β (1 − Pc(b))γ log Pc(b)+ +� +b∈Bpo− +c +(1 − β)Pc(b)γ log(1 − Pc(b)) +� +� +(9) +where N + = max(1, |Bpo+ +c +|). +3.3 +Smooth maximum approximation +In both unary losses φpa +c +and φpo +c , the bag prediction Pc(b) = +n−1 +max +k=0 pkc is the maximum prediction value of pixels in +a bag. However, the derivative ∂Pc/∂pkc is discontinuous, leading to numerical instability. To conquer this issue, +we introduce a technique called smooth maximum approximation to replace the maximum function by its smooth +maximum approximation [20]. In this study, we consider two variants of smooth maximum approximation for Pc(b) +as follows: +(1) α-softmax function: +Sα(b) = +�n−1 +k=0 pkceαpkc +�n−1 +k=0 eαpkc +(10) +where α > 0 is a constant. The higher α value denotes the closer approximation of Sα(b) to Pc(b). +(2) α-quasimax function: +Qα(b) = 1 +α log +�n−1 +� +k=0 +eαpkc +� +− log n +α +(11) +where α > 0 is a constant. The higher α value also denotes closer the approximation of Qα(b) to Pc(b). It can be +easily proved that Qα(b) ≤ Pc(b) always holds. +For image segmentation problem, the smooth maximum approximation has an extra advantage as follows: different +from the maximum function Pc with ∂Pc/∂pkc > 0 at only one pixel, the smooth maximum approximation has +∂Sα/∂pkc > 0 and ∂Qα/∂pkc > 0 for all pkc, thus it is able to learn all pixels together in a bag for model optimization. +9 + +50 +100 +150 +200 +250 +300 +350 +0 +50 +100 +150 +r0 +50 +100 +150 +200 +250 +300 +350 +0 +50 +100 +150 +r0 +50 +100 +150 +200 +250 +300 +350 +0 +50 +100 +150 +200 +r0 +50 +100 +150 +200 +250 +300 +350 +0 +50 +100 +150 +200 +rWSIS beyond tight BB +Moreover, a positive bag usually has more than one positive pixel in real segmentation problem, thus this property is +beneficial as well considering this fact. Therefore, besides the advantage in numerical stability, the smooth maximum +approximation is also helpful for performance improvement. +3.3.1 +Weighted smooth maximum approximation for Bpo+ +c +For the pixels in a polar line, the origin O is inside the object and the pixels closer to the origin O are more likely +belonging to the object. To incorporate this fact in optimization, a weight is introduced to the smooth maximum +approximation for positive bags Bpo+ +c +. In particular, a weight wk is assigned to prediction pkc of each pixel in the +positive bag, yielding weighted smooth maximum approximation. The weight wk is defined as follows: +wk = e−k2/(2σ2) +(12) +where σ = (Nr − 1)/√−2 log wmin and wmin is a preset parameter for the minimum weight of the pixel in positive +bags of an object. +4 +Experiments +4.1 +Datasets +This study made use of two public medical datasets as follows for performance evaluation: one is PROMISE12 dataset +[21] for prostate segmentation, and the other is ATLAS dataset [22] for brain lesion segmentation. +PROMISE12: The PROMISE12 dataset was released in MICCAI 2012 grand challenge [21]. It consists of transversal +T2-weighted MR images and their pixel-wise annotations from 50 patients, including both benign and prostate cancer +cases. The MR images were acquired at different centers with multiple MRI vendors and different scanning protocols. +The dataset was divided into two non-overlapping subsets, one subset with 40 patients for training and the other with +10 patients for validation. +ATLAS: The ATLAS dataset was developed by University of Southern California [22]. It consists of 229 T1-weighted +MR images and their pixel-wise annotations from 220 patients, acquired from different cohorts and different scanners. +The dataset was divided into two non-overlapping subsets, one subset with 203 images from 195 patients for training +and the other with 26 images from 25 patients for validation. +For fairness of comparison, for both datasets, the images in the training and validation subsets are exactly same as those +in studies [14] and [16]. Moreover, same as studies [14] and [16], this study reports the segmentation performance for +the validation subset in the results. +4.2 +Performance evaluation +This study employs dice coefficient to evaluate the performance of the proposed approach for image segmentation. +The dice coefficient has been widely used as a standard performance metric in medical image segmentation. It is in +the range of [0, 1]. The higher dice coefficient represents better segmentation performance. In this study, the dice +coefficient is calculated based on 3D MR images by stacking predictions of the corresponding 2D slices together. +4.3 +Bounding box settings +To evaluate the performance of the proposed approach supervised by bounding boxes at different precision levels, +this study considers both tight and loose bounding boxes in the experiments. The loose bounding box of an object is +obtained by adding a margin (denoted as m) on each side of its corresponding tight bounding box. Specifically, the +bounding boxes at the following four different precision levels are investigated: 1) tight bounding boxes (denoted as +m=0), 2) loose bounding boxes obtained by adding 5 pixels to each side of the corresponding tight bounding boxes +(denoted as m=5), 3) loose bounding boxes which add 10 pixels to each side of the corresponding tight bonding +boxes (denoted as m=10), and 4) loose bounding boxes acquired by adding random number of pixels, generated +from uniform distribution in the range of [0, 10], to each side of the corresponding tight bounding boxes (denoted as +m∼U(0, 10)). In the experiments, m=0, m=5, and m=10 are used to quantitatively investigate the effect of precision +levels of the bounding boxes on the segmentation performance, and m∼U(0, 10) stimulates bounding boxes acquired +in real annotation task, in which the margins provided by annotators are usually different and random among different +objects. +To measure the precision of a bounding box annotation, mean absolute relative difference (MARD) is introduced. This +study defines MARD as the average of the absolute errors in height and width between the object and its bounding +10 + +WSIS beyond tight BB +Table 1: Mean and standard deviation (in bracket) of the MARD values for the bounding boxes at four different +precision levels for PROMISE12 and ATLAS datasets. +Bounding box settings +PROMISE12 +ATLAS +m=0 +0.00% (0.00%) +0.00% (0.00%) +m=5 +22.28% (10.62%) +122.22% (96.68%) +m=10 +44.56% (21.23%) +244.44% (193.36%) +m∼U(0, 10) +22.23% (12.90%) +122.67% (109.41%) +box as follows: +MARD = 1 +2 × +�mx1 + mx2 +w ++ my1 + my2 +h +� +(13) +where w and h are width and height of the object, respectively; mx1, mx2, my1, and my2 are the margin added to the +left, right, up, and down sides of the tight bounding box. +In Table 1, the mean and standard deviation of the MARD values for the bounding boxes at four different precision +levels are provided for PROMISE12 dataset, in which the results are calculated based on all of the bounding boxes +in the training subset. As can be seen, mean MARD values are close to 22% for m=5 and m∼U(0, 10), indicating +that the bounding boxes are accurate and there is only mild error in the bounding box annotations. For m=10, mean +MARD value increases into 44.56%, indicating the sizes of the bounding boxes are almost 1.5 times of the sizes of the +objects on average, a moderate error in the bounding box annotations. +Table 1 also lists the mean and standard deviation of the MARD values for ATLAS dataset. It can be seen, for m=5 +and m∼U(0, 10), the mean MARD values are close to 122.5%, suggesting a severe error in bounding box annotations. +Lastly, for m=10, the mean MARD value is 244.44%. It represents that the sizes of the bounding boxes are almost +2.5 times larger than the sizes of the objects on average, indicating a very severe error in bounding box annotations. +4.4 +Methods for comparison +To demonstrate the overall performance of the proposed approach for image segmentation, this study considers the +following methods for comparison: +1) Fully supervised image segmentation (FSIS): FSIS employs pixel-wise annotations as supervision for image seg- +mentation. It can be treated as the upper bound of the segmentation performance for WSIS due to the use of fully +supervised learning based on costly pixel-wise annotations. +2) MIL baseline: It is a WSIS approach supervised by tight bounding boxes. It is described in Section 2.2. +3) Deep cut [13]: It is a WSIS approach using bounding box supervision for image segmentation. It trains neural +network classifier in an iterative optimization way. +4) Global constraint [14]: It is a WSIS approach adopting tight bounding boxes as supervision. It imposes a set of +constraints on the network outputs based on the tightness prior of bounding boxes for image segmentation. +5) Parallel transformation based MIL (denoted as PA): It is a WSIS approach with tight bounding box supervision, +which was developed in [16]. This study also describes it in Section 3.1 for completeness. The loss of this approach +is L = �C +c=1 φpa +c (P; Bpa+ +c +, Bpa− +c +) + λϕc(P). Besides being an existing method for comparison, this approach also +serves as an ablation study of the proposed approach which removes the component of polar transformation based +MIL. +6) Polar transformation based MIL (denoted as PO): It is a WSIS approach supervised by bounding boxes. It is +described in Section 3.2, optimized by the loss L = �C +c=1 φpo +c (P; Bpo+ +c +, Bpo− +c +) + λϕc(P) during training. This +method is an ablation study of the proposed approach after removing the component of parallel transformation based +MIL. +Overall, the summary of the methods for comparison is listed in Table 2. For fairness of comparison, the network +structures used in a comparison study are same for all methods. +11 + +WSIS beyond tight BB +Table 2: Summary of the methods for comparison. +Methods +Supervision +Properties of supervision +FSIS +masks +pixel-wise +Deep cut [13] +bounding boxes +tight/loose +Global constraint [14] +bounding boxes +tight +MIL baseline +bounding boxes +tight +PA [16] +bounding boxes +tight +PO +bounding boxes +tight/loose +Proposed approach +bounding boxes +tight/loose +4.5 +Implementation details +4.5.1 +Experimental setups +In this study, all experiments were implemented using PyTorch and the experimental codes are available at https: +//github.com/wangjuan313/wsis-beyond-tightBB. Image segmentation was conducted on the 2D slices of +MR images. As indicated below, most experimental setups were set to be same as those in [14] and [16] for fairness +of comparison. +For the PROMISE12 dataset, all images were resized to 256×256 pixels. A residual version of UNet [6] was employed +for segmentation. The models were trained by Adam optimizer [23] with parameters as follows: batch size = 16, initial +learning rate = 10−4, β1 = 0.9, and β2 = 0.99. An off-line data augmentation procedure was performed to the images +in the training set, and the following operations were considered: 1) mirroring, 2) flipping, and 3) rotation. +For the ATLAS dataset, all images were resized to 208 × 256 pixels. ENet [24] was used as backbone for image +segmentation. The models were trained by Adam optimizer with following parameters: batch size = 80, initial learning +rate = 5 × 10−4, β1 = 0.9, and β2 = 0.99. No augmentation was conducted during training. +4.5.2 +Hyperparameters +The weight λ of the pairwise loss ϕc(P) was set as λ = 10 based on experience in all experiments. +The +parameters in the unary losses φpa +c +and φpo +c +were set as β = 0.25 and γ = 2 according to the focal loss +[19]. +For parallel transformation based MIL, the parameters θ′ for parallel crossing lines and α for smooth +maximum approximation were obtained by grid search with the following values: +α ∈ {4, 6, 8} and θ′ +∈ +{(−40◦, 40◦, 10◦), (−40◦, 40◦, 20◦), (−60◦, 60◦, 30◦)}. For polar transformation based MIL, the parameters Nr and +Nθ in polar transformation and wmin and α in weighted smooth maximum approximation were obtained by grid search +as follows: Nr ∈ {10, 20, 30, 40}, Nθ ∈ {60, 90, 120}, wmin ∈ {0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8}, and α ∈ {0.5, 1, 2}. +5 +Results +5.1 +Performance comparison for PROMISE12 dataset +Table 3 gives dice coefficients of the proposed approach supervised by bounding boxes at different precision levels +for PROMISE12 dataset. Two models are considered for each level, one employing α-softmax function and the other +adopting α-quasimax function. As can be seen, the proposed approach has only minor decrease in dice coefficients for +m=5, m=10, and m∼U(0, 10) when compared with m=0, indicating that the proposed approach is robust to minor +and moderate errors in bounding box annotations. +For comparison, we also report results of the MIL baseline in Table 3. It gets dice coefficient of 0.859 for m=0, 0.840 +for m=5, 0.795 for m=10, and 0.832 for m∼U(0, 10). These values are much lower than their counterparts of the +proposed approach for all precision levels. +Moreover, the results of PA and PO approaches are listed in Table 3 as well. For both approaches, two models are +considered for each precision level, one using α-softmax function and the other adopting α-quasimax function. As +can be seen, both approaches have lower dice coefficients when compared with the proposed approach at different +precision levels. More importantly, comparing with both of these two approaches, the proposed approach has greater +performance improvements when m increases from m=0 to m=5 and m=10. These results suggest that both parallel +12 + +WSIS beyond tight BB +Table 3: Comparison of dice coefficients among different methods for the PROMISE12 dataset when bounding boxes +at different precision levels are considered, in which the standard deviation of dice coefficients among different MR +images is reported in the bracket. NA denotes that the result is not applicable and the symbol “-” indicates unavailable +result. +Methods +m = 0 +m = 5 +m = 10 +m ∼ U(0, 10) +FSIS +0.894 (0.021) +NA +NA +NA +Deep cut [13] +0.827 (0.085) +- +0.684 (0.069) +- +Global constraint [14] +0.835 (0.032) +- +0.778 (0.047) +- +MIL baseline +0.859 (0.038) +0.840 (0.046) +0.795 (0.023) +0.832 (0.047) +PA (α-softmax) [16] +0.878 (0.031) +0.868 (0.033) +0.862 (0.041) +0.869 (0.043) +PA (α-quasimax) [16] +0.880 (0.024) +0.871 (0.030) +0.856 (0.039) +0.875 (0.031) +PO (α-softmax) +0.867 (0.022) +0.858 (0.034) +0.843 (0.035) +0.858 (0.032) +PO (α-quasimax) +0.871 (0.019) +0.859 (0.030) +0.841 (0.038) +0.860 (0.021) +Proposed approach (α-softmax) +0.887 (0.027) +0.882 (0.023) +0.875 (0.034) +0.874 (0.026) +Proposed approach (α-quasimax) +0.887 (0.017) +0.880 (0.029) +0.869 (0.026) +0.876 (0.033) +Table 4: Comparison of dice coefficients among different methods for the ATLAS dataset when bounding boxes at +different precision levels are considered. +Methods +m = 0 +m = 5 +m = 10 +m ∼ U(0, 10) +FSIS +0.512 (0.292) +NA +NA +NA +Deep cut [13] +0.375 (0.246) +- +- +- +Global constraint [14] +0.474 (0.245) +- +- +- +MIL baseline +0.408 (0.249) +0.395 (0.240) +0.357 (0.223) +0.379 (0.237) +PA (α-softmax) [16] +0.494 (0.236) +0.451 (0.248) +0.400 (0.254) +0.456 (0.269) +PA (α-quasimax) [16] +0.488 (0.240) +0.448 (0.250) +0.412 (0.241) +0.437 (0.267) +PO (α-softmax) +0.463 (0.241) +0.417 (0.225) +0.373 (0.220) +0.432 (0.215) +PO (α-quasimax) +0.470 (0.221) +0.407 (0.241) +0.381 (0.230) +0.437 (0.218) +Proposed approach (α-softmax) +0.491 (0.233) +0.464 (0.251) +0.418 (0.246) +0.487 (0.246) +Proposed approach (α-quasimax) +0.503 (0.245) +0.462 (0.265) +0.409 (0.236) +0.464 (0.267) +transformation based MIL and polar transformation based MIL are effective in the proposed approach, and they are +especially helpful when the error in the bounding box annotations is larger. +Furthermore, Table 3 also provides the results of Deep cut and Global constraint approaches, which are cited from +study [14]. The results demonstrate that the proposed approach outperforms these two methods at a large margin for +both m=0 and m=10. +Lastly, FSIS gets dice coefficient of 0.894, the upper bound of performance for WSIS. As can be seen, the proposed +approach achieves performance close to FSIS for m=0 and m=5, and slightly lower performance for m=10 and +m∼U(0, 10). +5.2 +Performance comparison for ATLAS dataset +Table 4 reports dice coefficients of the proposed approach supervised by bounding boxes at different precision levels +for the ATLAS dataset. As can be noted, the proposed approach gets decreased performance for m=5, m=10, and +m∼U(0, 10) when compared with m=0. In particular, for m=10, the proposed approach has dice coefficient of 0.418 +(a 14.87% reduction in performance) when using α-softmax function and 0.409 (a 18.69% reduction) when employing +α-quasimax function. These results show that severe or very severe errors in bounding boxes could decrease the +segmentation performance greatly for the proposed approach. +13 + +WSIS beyond tight BB +(a) PROMISE12, m = 0 +(b) PROMISE12, m ∼ U(0, 10) +(c) ATLAS, m = 0 +(d) ATLAS, m ∼ U(0, 10) +Figure 8: Selected origins in the polar transformation, where the selected origins are denoted by red plus signs and the +pixel-wise ground truths of segmentation are marked by blue color. +For comparison, Table 4 also shows results of the MIL baseline. It gets much lower dice coefficients when compared +with the proposed approach at all precision levels. +In Table 4, we also report the results of PA and PA approaches. Both approaches get much lower dice coefficients +when compared with the proposed approach at different precision levels. These results certify the effectiveness of +both parallel transformation based MIL and polar transformation based MIL in the proposed approach. +Moreover, Table 4 also lists the results of Deep cut and Global constraint approaches reported in study [14], where +only the results for m=0 are available. The dice coefficients of these two methods are much lower than those of the +proposed approach. +Finally, FSIS achieves dice coefficient of 0.512, which is close to the results of the proposed approach for m=0, and +much higher for m=5, m=10, and m∼U(0, 10). +5.3 +Visualization of the origin in the polar transformation +In the proposed approach, the origin O in the polar transformation was determined during training. To verify that +the such automatic selection process indeed yields valid origin O (i.e. it is located inside the object in the bounding +box), we show the selected origins of several example images in Figure 8. In these plots, the models of the proposed +approach with α-softmax function obtained at the end of each epoch are considered for origin selection. The selected +origins are marked by red plus signs and the pixel-wise ground truths of segmentation are indicated by blue color. In +Figure 8, bounding boxes at two different precision levels are considered: m=0 denoting the use of accurate bounding +14 + +***++WSIS beyond tight BB +boxes and m∼U(0, 10) indicating the use of simulated real bounding box annotations. From Figure 8, all selected +origins are located in the object, verifying that the proposed approach is able to select origins correctly during training. +6 +Conclusion +This study investigates whether it is possible to maintain good segmentation performance for loose bounding box +supervision. Extending the previous parallel transformation based MIL, it developed an MIL strategy based on polar +transformation to assist image segmentation. 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Enet: A deep neural network archi- +tecture for real-time semantic segmentation. arXiv preprint arXiv:1606.02147, 2016. +16 + diff --git a/Y9FLT4oBgHgl3EQfVi9P/content/tmp_files/load_file.txt b/Y9FLT4oBgHgl3EQfVi9P/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..23bd512ccfb8e149f3573d3f1e895a41ac9cde41 --- /dev/null +++ b/Y9FLT4oBgHgl3EQfVi9P/content/tmp_files/load_file.txt @@ -0,0 +1,631 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf,len=630 +page_content='WEAKLY SUPERVISED IMAGE SEGMENTATION BEYOND TIGHT BOUNDING BOX ANNOTATIONS Juan Wang Horizon Med Innovation, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 23421 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Pointe Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Laguna Hills, CA 92653 wangjuan313@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='com Bin Xia Shenzhen SiBright Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=', Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Tinwe Industrial Park, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 6 Liufang Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Shenzhen, Guangdong 518052 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='xia@sibionics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='com ABSTRACT Weakly supervised image segmentation approaches in the literature usually achieve high segmenta- tion performance using tight bounding box supervision and decrease the performance greatly when supervised by loose bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' However, compared with loose bounding box, it is much more difficult to acquire tight bounding box due to its strict requirements on the precise locations of the four sides of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' To resolve this issue, this study investigates whether it is possible to maintain good segmentation performance when loose bounding boxes are used as supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For this purpose, this work extends our previous parallel transformation based multiple instance learning (MIL) for tight bounding box supervision by integrating an MIL strategy based on polar transformation to assist image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The proposed polar transformation based MIL formu- lation works for both tight and loose bounding boxes, in which a positive bag is defined as pixels in a polar line of a bounding box with one endpoint located inside the object enclosed by the box and the other endpoint located at one of the four sides of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Moreover, a weighted smooth maximum approximation is introduced to incorporate the observation that pixels closer to the ori- gin of the polar transformation are more likely to belong to the object in the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The proposed approach was evaluated on two public datasets using dice coefficient when bounding boxes at differ- ent precision levels were considered in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The results demonstrate that the proposed approach achieves state-of-the-art performance for bounding boxes at all precision levels and is ro- bust to mild and moderate errors in the loose bounding box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The codes are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='com/wangjuan313/wsis-beyond-tightBB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Keywords Multiple instance learning, Weakly supervised image segmentation, Polar transformation, Bounding box, Deep neural networks 1 Introduction Image segmentation is the process of partitioning a digital image into multiple image segments such that pixels in an image segment share certain characteristics and are assigned the same category label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It has been widely studied in all kinds of applications [1] since 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In recent years, with the development of the deep learning in medical image analysis [2,3,4,5], deep neural networks (DNNs) have been used to successfully tackle a variety of image segmentation problems in a fully-supervised manner [6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' However, it is labor-intensive and expensive to collect large-scale dataset with precise pixel-wise annotations for DNN training, thus limiting the value of the image segmentation in real applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' This is especially true in medical image analysis due to the difficulty of the segmentation problems and the difficulty in recruiting qualified annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' To resolve the problem mentioned above, great interests have been made in the literature to develop weakly supervised image segmentation (WSIS), the purpose of which is to substitute costly pixel-wise annotations into cost-effective annotations as supervision for image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Several types of annotations have been investigated, including image labels [9, 10], points [11], scribbles [12], and bounding boxes [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' This work considers bounding boxes as supervision for image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the literature, some efforts have been made to develop WSIS adopting arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='12053v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='CV] 28 Jan 2023 WSIS beyond tight BB Figure 1: Demonstration of tight (red color) and loose (green color) bounding boxes for the object sheep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' bounding box supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For example, Rajchl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' [13] designed an iterative optimization approach for image segmentation, in which a neural network classifier was trained using bounding box supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' [15] exploited mask R-CNN to simultaneously conduct object detection and image segmentation, in which the bounding box supervision was formulated as multiple instance learning (MIL) for image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Kervadec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' [14] imposed a set of constraints on the network output by leveraging the tightness prior of bounding boxes as supervision for image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Depending on the relationship between an object and its bounding box annotation, as shown in Figure 1, bounding box annotations can be divided into two categories: tight bounding box and loose bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The tight bounding box has its four sides touching the object, thus the size of the tight bounding box is same as the size of the object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' in contrast, the size of the loose bounding box is larger than the size of the object, thus at least one side of the loose bounding box does not touch the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Compared with the loose bounding boxes, the tight bounding boxes have strict requirements on the precise locations of the four sides of the bounding box annotations, thus it is much more difficult and time-consuming to annotate tight bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' However, most methods in the literature, if not all, achieve high segmentation performance using tight bounding box supervision, and decrease the performance greatly when supervised by loose bounding boxes [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Such inconsistency in the difficulty of annotation acquisition and the segmentation performance for tight and loose bounding boxes poses a problem in bounding box supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' To conquer this issue, in this paper we investigate whether it is possible to maintain good segmentation performance when loose bounding boxes are used as supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Recently, in our previous study we developed a WSIS approach using tight bounding box supervision by exploiting the properties of tight bounding boxes for MIL formulation [16] and achieved state-of-the-art segmentation perfor- mance [16,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Building on our previous success in [16], this study extends tight bounding box supervision to loose bounding box supervision for image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For this purpose, we propose an MIL formulation based on polar transformation of the image region in the bounding box to assist the approach in [16] for image segmentation, and develop a weighted smooth maximum approximation to incorporate the observation that pixels closer to the origin of the polar transformation are more likely to belong to the object in the bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the end, the segmentation assistance is conducted by combining the loss derived from the polar transformation based MIL into the loss in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The polar transformation based MIL strategy developed in this study is valid for both tight and loose bounding boxes, contributing on the good segmentation performance of the proposed approach for both tight and loose bounding box supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the experiments, the proposed approach is evaluated by two public datasets when both tight bound- ing boxes and loose bounding boxes at different precision levels are used as supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The results demonstrate that the proposed approach outperforms several existing methods in all precision levels of bounding boxes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' and more importantly, it is robust to mild and moderate errors in the loose bounding box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In summary, the contributions of this study are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' First, we develop a WSIS approach beyond tight bounding box supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It achieves state-of-the-art per- formance for bounding boxes at all precision levels and is robust to mild and moderate errors in the loose bounding box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Second, we propose an MIL strategy based on polar transformation of the image regions of the bounding boxes to incorporate the bounding box supervision into the network output to assist the image segmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The proposed polar transformation based MIL formulation works for both tight and loose bounding boxes, contributing on the good segmentation performance of the proposed approach for both tight and loose bounding box supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 2 WSIS beyond tight BB (a) (b) Figure 2: Demonstration of crossing and polar lines for the object “sheep”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In these plots, crossing and polar lines are marked by blue dashed lines and bounding boxes are denoted as red rectangles, in which the tight and loose bounding boxes are shown in the upper and lower rows, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The left column shows examples of crossing lines and the right column shows examples of polar lines, in which points O are indicated by green dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Third, a weighted smooth maximum approximation is introduced for the bag prediction in the proposed polar transformation based MIL to incorporate the observation that pixels closer to the origin of the polar transformation are more likely to belong to the object in the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Finally, the proposed approach is evaluated on two public datasets when bounding boxes at different precision levels are used as supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The results demonstrate the effectiveness of the proposed approach for image segmentation in all precision levels of bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1 Problem descriptions This study investigates the use of bounding boxes as supervision for weakly supervised image segmentation (WSIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' That is, for each object in the training set, a bounding box annotation is provided to supervise the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1 Bounding box and object To avoid any ambiguity, the definitions of bounding box and object are first introduced as follows: Bounding box is an imaginary rectangle enclosed a thing of interest in an image, which has been widely used in object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For an object, its bounding box annotation encloses the whole object in the box such that it does not overlap with the region outside its bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Depending on the relationship between an object and its bounding box annotation, bounding box annotations can be divided into two categories: one is tight bounding box, and the other is loose bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Tight bounding box is the smallest rectangle enclosing the whole object, thus the object must touch the four sides of its bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the end, the size of the object is same as the size of its tight bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In contrary, loose bounding box is outside of the object, thus at least one side of the loose bounding box does not touch the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the end, the size of the object is smaller than the size of its loose bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In a bounding box, two types of lines are considered in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For convenience, they are named as crossing line and polar line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Crossing line of a bounding box is defined as a line with its two endpoints located on the opposite sides of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Polar line of a bounding box is defined as a line with one endpoint (denoted as point O) located on a pixel belonging to the object in the bounding box and the other endpoint located at one of the four sides of the bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As examples, Figure 2 demonstrates examples of crossing lines (left column) and polar lines (right column) for both tight (upper row) and loose (lower row) bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Without loss of generality, object considered in this study is defined as a thing which covers a connected region in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It is important to notice that an object does not include any disjointed parts of a thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' If there are multiple disjointed parts in a thing, each part is treated as an independent object, thus a bounding box is annotated for each part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 3 WSIS beyond tight BB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2 WSIS using bounding box supervision For ease of development, we first introduce the fully supervised image segmentation (FSIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Suppose I denotes an in- put image and Y ∈ {1, 2, · · · , C} is its corresponding pixel-level category label for C categories under consideration, the image segmentation task is to obtain the prediction of Y, denoted as P, for the input image I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For the problem of FSIS, the pixel-wise category label Y is available for each image I in the training set for model optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' However, in the problem of WSIS using bounding box supervision, the pixel-level category label Y is unavailable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' instead, it provides the bounding box label B as supervision for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In this study, the bounding box label B is denoted as B = {bm, ym}, m = 1, 2, · · · , M, in which M is the number of bounding box annotations, bm is a 4-dimensional vector denoting the top left and bottom right points of the mth bounding box, and ym ∈ {1, 2, · · · , C} is the category label of the object in the mth bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' This study considers a specific type of deep neural networks (DNNs) for image segmentation, such as UNet [6] and FCN [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' This type of DNNs is able to output pixel-wise prediction for the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Due to the possible overlaps of objects of different categories in an image, which is especially true in medical images, this study formulates the image segmentation problem as a multi-label classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' That is, for a location k in the input image, it outputs a vector pk = [pk1, pk2, · · · , pkC], one element for a category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' each element is converted to the range of [0, 1] using the sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='3 Multiple Instance Learning Multiple instance learning (MIL) is a form of weakly supervised learning in which training samples are arranged in sets, called bags, and a category label is provided for the entire bag [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In MIL, supervision is only provided for bags, and the labels of individual samples in the bags are not provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For image segmentation, training samples are individual pixels of images in the training set, thus a bag consists of a set of different individual pixels of an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In MIL, a bag is positive if it has at least one positive sample, and a bag is negative if all of its individual samples are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Therefore, for a category c, the pixel with highest prediction in a positive bag tends to belong to category c, while even the pixel with highest prediction in a negative bag does not in category c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Based on this observation, suppose pkc is the network output of the kth pixel in bag b for category c, the bag prediction Pc(b) of bag b for category c can be defined as Pc(b) = n−1 max k=0 pkc (1) where n is the number of pixels in the bag b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2 MIL baseline This study considers the MIL baseline approach which employs tight bounding boxes for supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1 Positive and negative bags In an input image I, for an object of category c and its tight bounding box, it can be easily noted that any vertical and horizontal crossing line in the tight bounding box has at least one pixel belonging to the object in the box, hence pixels on a vertical or horizontal crossing line of the tight bounding box compose a positive bag for category c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Furthermore, pixels on a vertical or horizontal line of the image that do not overlap with any bounding boxes of category c in the image do not belong to category c, hence pixels on a vertical or horizontal line of the image that does not overlap with any bounding boxes of category c constitute a negative bag for category c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Based on these observations, for a category c, the MIL baseline approach considers all of the horizontal and vertical crossing lines of the tight bounding boxes of category c as positive bags, and all of the horizontal and vertical lines of the image that do not overlap any bounding boxes of category c in the image as negative bags [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As examples, the positive and negative bags for MIL baseline approach are demonstrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2 MIL baseline loss To optimize the various parameters associated with network, MIL baseline loss with two terms is employed [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For a category c, suppose its positive and negative bags in the training set are denoted as B+ c and B− c , respectively, then MIL baseline loss Lc is Lc = φc(P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' B+ c , B− c ) + λϕc(P) (2) where φc is the unary loss, ϕc is the pairwise loss, and λ is a constant value controlling the trade off between the unary loss and the pairwise loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 4 WSIS beyond tight BB Figure 3: Demonstration of positive and negative bags for MIL baseline with the tight bounding box annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The tight bounding box is indicated by the red rectangle for the object “sheep”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The examples of positive and negative bags are marked by blue and green colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The unary loss φc is defined as: φc = − 1 |B+ c | + |B− c | � � � b∈B+ c log Pc(b) + � b∈B− c log(1 − Pc(b)) � � (3) where |B| is the cardinality of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Mathematically, the unary loss is a binary cross entropy loss for the bag prediction Pc(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It gets minimum when the the bag prediction Pc(b) is 1 for positive bags and 0 for negative bags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' More importantly, the unary loss adaptively selects one pixel per bag based on the network prediction for optimization, yielding an adaptive sampling effect on the training samples during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' However, using the unary loss alone is prone to segment merely the discriminative parts of an object rather than the whole object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' To resolve this problem, the pairwise loss is introduced as follows: ϕc = 1 |ε| � (k,k′)∈ε (pkc − pk′c)2 (4) where ε is the set containing all neighboring pixel pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Complementary to the unary loss, the pairwise loss enforces the piece-wise smoothness in the network prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the end, for all C categories, the MIL baseline loss L is L = C � c=1 Lc (5) 3 Methods As noted in the introduction, tight bounding boxes are difficult to acquire due to the strong constraint posed on the precise locations of the four sides of the bounding box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' To deal with this issue, this study extends our previous approach on WSIS using tight bounding box supervision [16], named as parallel transformation based MIL in this study, by incorporating a polar transformation based MIL, which works for both tight and loose bounding boxes, to assist the image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the end, the total loss L for network optimization is as follows: L = C � c=1 φpa c (P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Bpa+ c , Bpa− c ) + φpo c (P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Bpo+ c , Bpo− c ) + λϕc(P) (6) where φpa c is the unary loss derived from the parallel transformation based MIL for its positive bags Bpa+ c and negative bags Bpa− c (will be described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1), φpo c is the unary loss obtained from the polar transformation based MIL for its positive bags Bpo+ c and negative bags Bpo− c (will be introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2), and ϕc is the pairwise loss defined in equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1 Parallel transformation based MIL using tight bounding box supervision The method described in this section was first introduced in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' More details are provided in this study regarding to efficient calculation of the positive bag prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 5 WSIS beyond tight BB Figure 4: Demonstration of positive bags in parallel transformation based MIL with tight bounding box annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In this plot, the tight bounding box of the object “sheep” is indicated by the red rectangle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' examples of positive bags from two different angles are provided, in which those with θ′ = 25◦ are marked by purple dashed lines and those with θ′ = 0◦ are given by green dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1 Positive bags Bpa+ c One issue associated with the positive bag definition in MIL baseline is that for an object of height H pixels and width W pixels, it yields only H+W positive bags, the value of which is much smaller than the size of the object, hence limiting the selected positive samples during training and resulting in a bottleneck for the segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Noticed that in an input image I, for an object of category c and its tight bounding box, any parallel crossing line of the tight bounding box also has at least one pixel belonging to the object in the box, this study generalizes the positive bag definition by considering a parallel crossing line of the tight bounding box as a positive bag for category c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' A parallel crossing line of a bounding box can be parameterized by an angle θ′ ∈ (−90◦, 90◦) with respect to the edges of the box where its two endpoints located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For an angle θ′, two sets of parallel crossing lines can be obtained from the bounding box, one crosses up and bottom edges of the box, and the other crosses left and right edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the end, the positive bags Bpa+ c for category c are all parallel crossing lines of the tight bounding boxes of the objects of category c on a set of different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As demonstration, Figure 4 shows examples of positive bags obtained from two different angles, where those indicated by purple dashed lines are with θ′ = 25◦, and those marked by green dashed lines have θ′ = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' More importantly, by comparing Figures 3 and 4, the positive bags in MIL baseline are a subset and special cases of the positive bags Bpa+ c with θ′ = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the experiments, this study presets the set of angles as θ′ ∈ (a, b, s), denoting evenly spaced angle values within interval (a, b) with step s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In implementation, it is inefficient to directly calculate the bag prediction Pc(b) of positive bags Bpa+ c from parallel crossing lines in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' To facilitate it, we propose to transform the parallel crossing lines with angle θ′ in the input image into vertical or horizontal lines by rotating the input image by angle θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In this study, this process of obtaining parallel crossing lines is named as parallel transformation to emphasize that the transformation is targeted at crossing lines of bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As examples, Figure 5(a) shows the process of parallel transformation, in which examples of parallel crossing lines with angle θ′ = 25◦ in the upper image are transformed into horizontal and vertical lines in the lower image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' However, for efficient calculation of the bag prediction, the parallel transformation of the input image alone is not enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As shown in Figure 5(a), the vertical lines in the rotated image are no longer aligned at the same starting and ending points along the horizontal direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' and the same problem also exists for the horizontal lines in the rotated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Therefore, an indicator has to be provided for each pixel in the rotated image to determine whether it is in a positive bag or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For this purpose, we construct a box-mask image for each tight bounding box in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The box-mask image has same size as the input image, its value is set to be 1 for the pixels in the box and 0 for those outside the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Afterwards, the same parallel transformation is applied to the box-mask image to determine whether each pixel is in a positive bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the rotated box-mask image, the pixels with value 1 in a vertical or horizontal line corresponds to a parallel crossing line, thus consisting of a positive bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As examples, in Figures 5(b), we also show the parallel transformation of the box-mask image of Figures 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' From Figure 5(b), the white pixels along each horizontal or vertical line (denoted by dashed blue line) consist of a positive bag in the rotated box-mask image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Finally, to further speed up the calculation, for a tight bounding box of an object, the parallel transformation is only applied to the cropped region of the input image around the box and its corresponding box-mask image, in which a small margin is added to four sides of the box to avoid information loss during rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 6 WSIS beyond tight BB (a) (b) Figure 5: Demonstration of parallel transformation, in which the images and their results of parallel transformation are provided in the upper and lower rows, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In these plots, examples of paralleled crossing lines with angle θ′ = 25◦ are marked by purple dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' (a) input image, in which the tight bounding box of “sheep” is marked by red rectangle, (b) the box-mask image of the tight bounding box in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2 Negative bags Bpa− c Similar as the positive bag definition, the negative bag definition in MIL baseline also has the problem of limited samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Notice in an input image I, for a category c, any individual pixels outside of any bounding boxes of category c in the image do not belong to category c, we define a negative bag for category c as an individual pixel outside any bounding boxes of category c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the end, negative bags Bpa− c for category c consist of all of individual pixels outside all of the bounding boxes of category c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' This definition greatly increases the number of negative bags for training, and forces the network to learn every pixel outside bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='3 Unary loss φpa c The parallel transformed based MIL formulation above will inevitably lead to imbalance between positive and negative bags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' To eliminate this issue, we borrow the concept of focal loss [19] and define the unary loss as follows: φpa c = − 1 N + � � � b∈Bpa+ c β (1 − Pc(b))γ log Pc(b)+ � b∈Bpa− c (1 − β)Pc(b)γ log(1 − Pc(b)) � � (7) where N + = max(1, |Bpa+ c |), β ∈ [0, 1] is the weighting factor, and γ ≥ 0 is the focusing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Mathematically, the unary loss φpa c is focal loss for bag prediction Pc(b), it gets minimum when Pc(b) is 1 for positive bags and 0 for negative bags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2 Polar transformation based MIL using tight or loose bounding box supervision This study proposes polar transformation based MIL to assist the parallel transformation based MIL for image seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The proposed polar transformation based MIL works for both tight and loose bounding boxes, contributing on the good segmentation performance of the proposed approach for both tight and loose bounding box supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Its details are provided in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1 Positive bags Bpo+ c To extend the positive bag definition beyond the tight bounding box supervision, we consider the polar line of the bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For an object of category c, any polar line of its bounding box has at least one pixel belonging to 7 WSIS beyond tight BB (a) (b) Figure 6: Demonstration of positive bags in polar transformation based MIL for (a) tight and (b) loose bounding boxes of the object “sheep”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In each plot, the bounding box is marked by the red rectangle, the point O is denoted by the green dot, and examples of positive bags are indicated by blue dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' category c, thus this study considers pixels in a polar line of the bounding box as a positive bag for category c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' This definition does not employ any prior information of the bounding box, thus is valid for both tight and loose bounding boxes of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the end, the positive bags Bpo+ c for category c are defined as all of the polar lines of the bounding boxes of the objects with category c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As examples, Figure 6 shows examples of positive bags for both tight and loose bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Mathematically, the bag prediction Pc(b) of positive bags of an object from the corresponding polar lines can be efficiently obtained by applying polar transformation to the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The polar transformation of an image transfers the image from the Cartesian coordinate system to the polar coordinate system, providing a pixel-wise representation in the polar coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In this study, it transfers a polar line of the bounding box of an object in an image into a horizontal line in the polar image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Suppose (u, v) is the Cartesian coordinate of a pixel in the the Cartesian coordinate domain with respect to an origin Op and a radius Rp, and the polar coordinate is (r, θ), where r > 0 and θ ∈ [0, 2π) are the radial and angular coordinates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The polar transformation maps the pixel (u, v) in the Cartesian coordinate plane to the corresponding pixel (r, θ) in the polar coordinate plane as follows: r = √ u2 + v2 θ = tan−1(v/u) (8) In polar transformation, the size of the polar image in the polar coordinate plane is preset by user during experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Suppose it is Nr × Nθ, where Nr is the dimension of the polar axis and Nθ is the dimension of the angle axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In this study, the polar transformation is applied to the cropped region of the input image around the bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' A small margin is added to four sides of the bounding box for region cropping to avoid possible information loss during transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For the cropped region, during polar transformation, the origin Op is set as the point O defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1 and the radius Rp is set as half length of the diagonal line of the bounding box R, which is the radius of the minimum circle enclosing the bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' With such settings, in polar transformation of the cropped region, the radial coordinate r is evenly distributed in [0, R] with step R/Nr, and the angular coordinate θ is evenly distributed in [0, 2π) with step 2π/Nθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As demonstration, Figures 7(a) and (c) show polar transformation for examples of positive bags from tight and loose bounding boxes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Similarly, as shown in Figures 7(a) and (c), the horizontal lines in a polar image are no longer aligned at the ending points along the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' To determine whether each pixel in the polar image is in a positive bag or not, we also introduce the box-mask image of the cropped region and applied the same polar transformation to the box-mask image as indicator of pixels in positive bags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As examples, Figures 7(b) and (d) show the box-mask images and its corresponding polar images for the cropped regions in Figures 7(a) and (c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the lower row of Figures 7(b) and (d), the white pixels along a horizontal line (denoted by dashed blue lines) consist of a positive bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Finally, for bounding box annotation, the origin O is unknown and should be determined during experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Based on the fact that the origin O is inside the object in the bounding box, it is selected as the pixel with maximum network output among all of the pixels in the bounding box during training in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Such design is intuitive since the pixel with highest prediction are more likely belong to the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2 Negative bags Bpo− c This study employs the same negative bag definition as in that in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2 for the polar transformation based MIL, therefore, we have Bpo− c = Bpa− c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 8 WSIS beyond tight BB (a) (b) (c) (d) Figure 7: Demonstration of polar transformation for examples of positive bags, in which the images and their polar images are provided in the upper and lower rows, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' (a) The cropped region of the object “sheep” from tight bounding box, (b) the box-mask image of (a), (c) the cropped region of the object “sheep” from a loose bounding box, (d) the box-mask image of (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='3 Unary loss φpo c Similar as unary loss φpa c defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='3, the unary loss φpo c for polar transformation based MIL is φpo c = − 1 N + � � � b∈Bpo+ c β (1 − Pc(b))γ log Pc(b)+ � b∈Bpo− c (1 − β)Pc(b)γ log(1 − Pc(b)) � � (9) where N + = max(1, |Bpo+ c |).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='3 Smooth maximum approximation In both unary losses φpa c and φpo c , the bag prediction Pc(b) = n−1 max k=0 pkc is the maximum prediction value of pixels in a bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' However, the derivative ∂Pc/∂pkc is discontinuous, leading to numerical instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' To conquer this issue, we introduce a technique called smooth maximum approximation to replace the maximum function by its smooth maximum approximation [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In this study, we consider two variants of smooth maximum approximation for Pc(b) as follows: (1) α-softmax function: Sα(b) = �n−1 k=0 pkceαpkc �n−1 k=0 eαpkc (10) where α > 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The higher α value denotes the closer approximation of Sα(b) to Pc(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' (2) α-quasimax function: Qα(b) = 1 α log �n−1 � k=0 eαpkc � − log n α (11) where α > 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The higher α value also denotes closer the approximation of Qα(b) to Pc(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It can be easily proved that Qα(b) ≤ Pc(b) always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For image segmentation problem, the smooth maximum approximation has an extra advantage as follows: different from the maximum function Pc with ∂Pc/∂pkc > 0 at only one pixel, the smooth maximum approximation has ∂Sα/∂pkc > 0 and ∂Qα/∂pkc > 0 for all pkc, thus it is able to learn all pixels together in a bag for model optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 9 50 100 150 200 250 300 350 0 50 100 150 r0 50 100 150 200 250 300 350 0 50 100 150 r0 50 100 150 200 250 300 350 0 50 100 150 200 r0 50 100 150 200 250 300 350 0 50 100 150 200 rWSIS beyond tight BB Moreover, a positive bag usually has more than one positive pixel in real segmentation problem, thus this property is beneficial as well considering this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Therefore, besides the advantage in numerical stability, the smooth maximum approximation is also helpful for performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1 Weighted smooth maximum approximation for Bpo+ c For the pixels in a polar line, the origin O is inside the object and the pixels closer to the origin O are more likely belonging to the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' To incorporate this fact in optimization, a weight is introduced to the smooth maximum approximation for positive bags Bpo+ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In particular, a weight wk is assigned to prediction pkc of each pixel in the positive bag, yielding weighted smooth maximum approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The weight wk is defined as follows: wk = e−k2/(2σ2) (12) where σ = (Nr − 1)/√−2 log wmin and wmin is a preset parameter for the minimum weight of the pixel in positive bags of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1 Datasets This study made use of two public medical datasets as follows for performance evaluation: one is PROMISE12 dataset [21] for prostate segmentation, and the other is ATLAS dataset [22] for brain lesion segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' PROMISE12: The PROMISE12 dataset was released in MICCAI 2012 grand challenge [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It consists of transversal T2-weighted MR images and their pixel-wise annotations from 50 patients, including both benign and prostate cancer cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The MR images were acquired at different centers with multiple MRI vendors and different scanning protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The dataset was divided into two non-overlapping subsets, one subset with 40 patients for training and the other with 10 patients for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' ATLAS: The ATLAS dataset was developed by University of Southern California [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It consists of 229 T1-weighted MR images and their pixel-wise annotations from 220 patients, acquired from different cohorts and different scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The dataset was divided into two non-overlapping subsets, one subset with 203 images from 195 patients for training and the other with 26 images from 25 patients for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For fairness of comparison, for both datasets, the images in the training and validation subsets are exactly same as those in studies [14] and [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Moreover, same as studies [14] and [16], this study reports the segmentation performance for the validation subset in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2 Performance evaluation This study employs dice coefficient to evaluate the performance of the proposed approach for image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The dice coefficient has been widely used as a standard performance metric in medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It is in the range of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The higher dice coefficient represents better segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In this study, the dice coefficient is calculated based on 3D MR images by stacking predictions of the corresponding 2D slices together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='3 Bounding box settings To evaluate the performance of the proposed approach supervised by bounding boxes at different precision levels, this study considers both tight and loose bounding boxes in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The loose bounding box of an object is obtained by adding a margin (denoted as m) on each side of its corresponding tight bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' the bounding boxes at the following four different precision levels are investigated: 1) tight bounding boxes (denoted as m=0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 2) loose bounding boxes obtained by adding 5 pixels to each side of the corresponding tight bounding boxes (denoted as m=5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 3) loose bounding boxes which add 10 pixels to each side of the corresponding tight bonding boxes (denoted as m=10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' and 4) loose bounding boxes acquired by adding random number of pixels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' generated from uniform distribution in the range of [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 10],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' to each side of the corresponding tight bounding boxes (denoted as m∼U(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the experiments, m=0, m=5, and m=10 are used to quantitatively investigate the effect of precision levels of the bounding boxes on the segmentation performance, and m∼U(0, 10) stimulates bounding boxes acquired in real annotation task, in which the margins provided by annotators are usually different and random among different objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' To measure the precision of a bounding box annotation, mean absolute relative difference (MARD) is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' This study defines MARD as the average of the absolute errors in height and width between the object and its bounding 10 WSIS beyond tight BB Table 1: Mean and standard deviation (in bracket) of the MARD values for the bounding boxes at four different precision levels for PROMISE12 and ATLAS datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Bounding box settings PROMISE12 ATLAS m=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='00% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='00%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='00% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='00%) m=5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='28% (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='62%) 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='22% (96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='68%) m=10 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='56% (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='23%) 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='44% (193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='36%) m∼U(0, 10) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='23% (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='90%) 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='67% (109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='41%) box as follows: MARD = 1 2 × �mx1 + mx2 w + my1 + my2 h � (13) where w and h are width and height of the object, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' mx1, mx2, my1, and my2 are the margin added to the left, right, up, and down sides of the tight bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In Table 1, the mean and standard deviation of the MARD values for the bounding boxes at four different precision levels are provided for PROMISE12 dataset, in which the results are calculated based on all of the bounding boxes in the training subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As can be seen, mean MARD values are close to 22% for m=5 and m∼U(0, 10), indicating that the bounding boxes are accurate and there is only mild error in the bounding box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For m=10, mean MARD value increases into 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='56%, indicating the sizes of the bounding boxes are almost 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='5 times of the sizes of the objects on average, a moderate error in the bounding box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Table 1 also lists the mean and standard deviation of the MARD values for ATLAS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It can be seen, for m=5 and m∼U(0, 10), the mean MARD values are close to 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='5%, suggesting a severe error in bounding box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Lastly, for m=10, the mean MARD value is 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='44%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It represents that the sizes of the bounding boxes are almost 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='5 times larger than the sizes of the objects on average, indicating a very severe error in bounding box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='4 Methods for comparison To demonstrate the overall performance of the proposed approach for image segmentation, this study considers the following methods for comparison: 1) Fully supervised image segmentation (FSIS): FSIS employs pixel-wise annotations as supervision for image seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It can be treated as the upper bound of the segmentation performance for WSIS due to the use of fully supervised learning based on costly pixel-wise annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 2) MIL baseline: It is a WSIS approach supervised by tight bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It is described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 3) Deep cut [13]: It is a WSIS approach using bounding box supervision for image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It trains neural network classifier in an iterative optimization way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 4) Global constraint [14]: It is a WSIS approach adopting tight bounding boxes as supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It imposes a set of constraints on the network outputs based on the tightness prior of bounding boxes for image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 5) Parallel transformation based MIL (denoted as PA): It is a WSIS approach with tight bounding box supervision, which was developed in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' This study also describes it in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1 for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The loss of this approach is L = �C c=1 φpa c (P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Bpa+ c , Bpa− c ) + λϕc(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Besides being an existing method for comparison, this approach also serves as an ablation study of the proposed approach which removes the component of polar transformation based MIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 6) Polar transformation based MIL (denoted as PO): It is a WSIS approach supervised by bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It is described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2, optimized by the loss L = �C c=1 φpo c (P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Bpo+ c , Bpo− c ) + λϕc(P) during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' This method is an ablation study of the proposed approach after removing the component of parallel transformation based MIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Overall, the summary of the methods for comparison is listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For fairness of comparison, the network structures used in a comparison study are same for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 11 WSIS beyond tight BB Table 2: Summary of the methods for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Methods Supervision Properties of supervision FSIS masks pixel-wise Deep cut [13] bounding boxes tight/loose Global constraint [14] bounding boxes tight MIL baseline bounding boxes tight PA [16] bounding boxes tight PO bounding boxes tight/loose Proposed approach bounding boxes tight/loose 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='5 Implementation details 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1 Experimental setups In this study, all experiments were implemented using PyTorch and the experimental codes are available at https: //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='com/wangjuan313/wsis-beyond-tightBB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Image segmentation was conducted on the 2D slices of MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As indicated below, most experimental setups were set to be same as those in [14] and [16] for fairness of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For the PROMISE12 dataset, all images were resized to 256×256 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' A residual version of UNet [6] was employed for segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The models were trained by Adam optimizer [23] with parameters as follows: batch size = 16, initial learning rate = 10−4, β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='9, and β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' An off-line data augmentation procedure was performed to the images in the training set, and the following operations were considered: 1) mirroring, 2) flipping, and 3) rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For the ATLAS dataset, all images were resized to 208 × 256 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' ENet [24] was used as backbone for image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The models were trained by Adam optimizer with following parameters: batch size = 80, initial learning rate = 5 × 10−4, β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='9, and β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' No augmentation was conducted during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2 Hyperparameters The weight λ of the pairwise loss ϕc(P) was set as λ = 10 based on experience in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The parameters in the unary losses φpa c and φpo c were set as β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='25 and γ = 2 according to the focal loss [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For parallel transformation based MIL, the parameters θ′ for parallel crossing lines and α for smooth maximum approximation were obtained by grid search with the following values: α ∈ {4, 6, 8} and θ′ ∈ {(−40◦, 40◦, 10◦), (−40◦, 40◦, 20◦), (−60◦, 60◦, 30◦)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For polar transformation based MIL, the parameters Nr and Nθ in polar transformation and wmin and α in weighted smooth maximum approximation were obtained by grid search as follows: Nr ∈ {10, 20, 30, 40}, Nθ ∈ {60, 90, 120}, wmin ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='8}, and α ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='5, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 5 Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='1 Performance comparison for PROMISE12 dataset Table 3 gives dice coefficients of the proposed approach supervised by bounding boxes at different precision levels for PROMISE12 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Two models are considered for each level, one employing α-softmax function and the other adopting α-quasimax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As can be seen, the proposed approach has only minor decrease in dice coefficients for m=5, m=10, and m∼U(0, 10) when compared with m=0, indicating that the proposed approach is robust to minor and moderate errors in bounding box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For comparison, we also report results of the MIL baseline in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It gets dice coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='859 for m=0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='840 for m=5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='795 for m=10, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='832 for m∼U(0, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' These values are much lower than their counterparts of the proposed approach for all precision levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Moreover, the results of PA and PO approaches are listed in Table 3 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For both approaches, two models are considered for each precision level, one using α-softmax function and the other adopting α-quasimax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As can be seen, both approaches have lower dice coefficients when compared with the proposed approach at different precision levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' More importantly, comparing with both of these two approaches, the proposed approach has greater performance improvements when m increases from m=0 to m=5 and m=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' These results suggest that both parallel 12 WSIS beyond tight BB Table 3: Comparison of dice coefficients among different methods for the PROMISE12 dataset when bounding boxes at different precision levels are considered, in which the standard deviation of dice coefficients among different MR images is reported in the bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' NA denotes that the result is not applicable and the symbol “-” indicates unavailable result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Methods m = 0 m = 5 m = 10 m ∼ U(0, 10) FSIS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='894 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='021) NA NA NA Deep cut [13] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='827 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='085) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='684 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='069) Global constraint [14] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='835 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='032) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='778 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='047) MIL baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='859 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='038) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='840 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='046) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='795 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='023) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='832 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='047) PA (α-softmax) [16] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='878 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='031) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='868 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='033) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='862 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='041) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='869 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='043) PA (α-quasimax) [16] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='880 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='024) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='871 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='030) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='856 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='039) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='875 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='031) PO (α-softmax) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='867 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='858 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='034) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='843 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='035) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='858 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='032) PO (α-quasimax) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='871 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='019) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='859 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='030) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='841 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='038) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='860 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='021) Proposed approach (α-softmax) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='887 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='027) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='882 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='023) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='875 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='034) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='874 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='026) Proposed approach (α-quasimax) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='887 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='017) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='880 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='029) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='869 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='026) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='876 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='033) Table 4: Comparison of dice coefficients among different methods for the ATLAS dataset when bounding boxes at different precision levels are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Methods m = 0 m = 5 m = 10 m ∼ U(0, 10) FSIS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='512 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='292) NA NA NA Deep cut [13] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='375 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='246) Global constraint [14] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='474 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='245) MIL baseline 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='237) PA (α-softmax) [16] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='494 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='236) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='451 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='248) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='400 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='254) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='456 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='269) PA (α-quasimax) [16] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='488 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='240) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='448 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='250) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='412 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='241) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='437 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='267) PO (α-softmax) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='463 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='241) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='417 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='225) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='373 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='220) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='432 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='215) PO (α-quasimax) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='470 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='221) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='407 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='241) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='381 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='230) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='437 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='218) Proposed approach (α-softmax) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='491 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='233) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='464 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='251) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='418 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='246) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='487 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='246) Proposed approach (α-quasimax) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='503 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='245) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='462 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='265) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='409 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='236) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='464 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='267) transformation based MIL and polar transformation based MIL are effective in the proposed approach, and they are especially helpful when the error in the bounding box annotations is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Furthermore, Table 3 also provides the results of Deep cut and Global constraint approaches, which are cited from study [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The results demonstrate that the proposed approach outperforms these two methods at a large margin for both m=0 and m=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Lastly, FSIS gets dice coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='894, the upper bound of performance for WSIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As can be seen, the proposed approach achieves performance close to FSIS for m=0 and m=5, and slightly lower performance for m=10 and m∼U(0, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='2 Performance comparison for ATLAS dataset Table 4 reports dice coefficients of the proposed approach supervised by bounding boxes at different precision levels for the ATLAS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' As can be noted, the proposed approach gets decreased performance for m=5, m=10, and m∼U(0, 10) when compared with m=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In particular, for m=10, the proposed approach has dice coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='418 (a 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='87% reduction in performance) when using α-softmax function and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='409 (a 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='69% reduction) when employing α-quasimax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' These results show that severe or very severe errors in bounding boxes could decrease the segmentation performance greatly for the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 13 WSIS beyond tight BB (a) PROMISE12, m = 0 (b) PROMISE12, m ∼ U(0, 10) (c) ATLAS, m = 0 (d) ATLAS, m ∼ U(0, 10) Figure 8: Selected origins in the polar transformation, where the selected origins are denoted by red plus signs and the pixel-wise ground truths of segmentation are marked by blue color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' For comparison, Table 4 also shows results of the MIL baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' It gets much lower dice coefficients when compared with the proposed approach at all precision levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In Table 4, we also report the results of PA and PA approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Both approaches get much lower dice coefficients when compared with the proposed approach at different precision levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' These results certify the effectiveness of both parallel transformation based MIL and polar transformation based MIL in the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Moreover, Table 4 also lists the results of Deep cut and Global constraint approaches reported in study [14], where only the results for m=0 are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The dice coefficients of these two methods are much lower than those of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Finally, FSIS achieves dice coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='512, which is close to the results of the proposed approach for m=0, and much higher for m=5, m=10, and m∼U(0, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='3 Visualization of the origin in the polar transformation In the proposed approach, the origin O in the polar transformation was determined during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' To verify that the such automatic selection process indeed yields valid origin O (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' it is located inside the object in the bounding box), we show the selected origins of several example images in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In these plots, the models of the proposed approach with α-softmax function obtained at the end of each epoch are considered for origin selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The selected origins are marked by red plus signs and the pixel-wise ground truths of segmentation are indicated by blue color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In Figure 8, bounding boxes at two different precision levels are considered: m=0 denoting the use of accurate bounding 14 ***++WSIS beyond tight BB boxes and m∼U(0, 10) indicating the use of simulated real bounding box annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' From Figure 8, all selected origins are located in the object, verifying that the proposed approach is able to select origins correctly during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' 6 Conclusion This study investigates whether it is possible to maintain good segmentation performance for loose bounding box supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Extending the previous parallel transformation based MIL, it developed an MIL strategy based on polar transformation to assist image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' Moreover, a weighted smooth maximum approximation was introduced to incorporate the observation that pixels closer to the origin of the polar transformation are more likely to belong to the object in the bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' In the experiments, the proposed approach was evaluated on two public datasets using dice coefficient for bounding boxes at different precision levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FLT4oBgHgl3EQfVi9P/content/2301.12053v1.pdf'} +page_content=' The results demonstrate the superior performance of the proposed approach for bounding boxes at different precision levels and the robustness of the proposed approach for bounding boxes with mild and moderate errors.' metadata={'source': 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sampling, +which hinder trackers to predict accurate bounding boxes +(BBoxes). Exploring an approach that seeks to maximize +the preservation of object points and their object-aware +features is of particular significance. +Motivated by this, +we propose an Object Preserving Siamese Network (OP- +SNet), which can significantly maintain object integrity +and boost tracking performance. Firstly, the object high- +lighting module enhances the object-aware features and +extracts discriminative features from template and search +area. Then, the object-preserved sampling selects object +candidates to obtain object-preserved search area seeds +and drop the background points that contribute less to +tracking. Finally, the object localization network precisely +locates 3D BBoxes based on the object-preserved search +area seeds. Extensive experiments demonstrate our method +outperforms the state-of-the-art performance (∼9.4% and +∼2.5% success gain on KITTI and Waymo Open Dataset +respectively). +1. Introduction +Single object tracking (SOT) has become a significant +issue of computer vision, which contributes widely to var- +ious fields, such as autonomous driving [15, 18], security +surveillance [26,30] and robotics [2,11], etc. With the fast +development of 3D sensors, such as LiDAR, 3D data reflect- +ing real-world coordinates and object sizes can be captured. +Therefore, 3D computer vision tasks attract more and more +attention in recent years [10, 16]. 3D single object track- +ing is also an essential part of 3D computer vision, which +aims to improve autonomous cars’ safety via predicting ac- +curate object trajectories. At present, many 2D single object +tracking algorithms have been proposed and proved their ef- +ficiency, but they cannot directly handle 3D data since 3D +point clouds are of sparsity and non-uniformity [23]. +Figure 1. Exemplified illustration to show how our OPSNet works, +from the object highlighting module to the object-preserved sam- +pling and localization module. +Mainstream 3D SOT approaches are based on the +Siamese network [7–9, 23, 32, 34] that consists of two +weight-shared backbone branches to respectively handle +template and search area. Most of the Siamese-based ap- +proaches use PointNet++ [22] as the backbone which hier- +archically samples points and aggregates local features from +raw point clouds. STNet [9] firstly adopts Transformer as +the backbone for encoding all the points of template and +search area. Considering that the tracking tasks require to +be conducted in real-time and tracking speed is also a sig- +nificant evaluation indicator, the PointNet++ is of higher ef- +ficiency and less computational burden [22]. +Since PointNet++ applies furthest points sampling (FPS) +or random sampling, all the approaches based on Point- +Net++ suffer from the negative effects brought by randomly +dropped object points. To overcome this issue, PTTR [34] +samples the search area points that are most similar to tem- +plate points in the embedding space for object preservation. +However, once PTTR regresses an inaccurate 3D BBox as +a template during inference, search area sampling will be +1 +arXiv:2301.12057v1 [cs.CV] 28 Jan 2023 + +Points +Highlighting score +RawTemplate +Object Highlighting +Module +Highlighted +3DBBox +Search Area Seeds ++ +Object-preserved +Object Localization +Sampling +Network +RawSearchArea +Object Candidates +Object-Preserved +Search Area Seedssimultaneously misled. Exploring a robust approach that +seeks to maximize the preservation of object points and +their object-aware features for tracking is of particular sig- +nificance. We propose our novel OPSNet for object high- +lighting, object preservation, and accurate object localiza- +tion. +We exemplify how our OPSNet works in Fig. 1. The +object highlighting module consists of three main parts: 1) +The backbone first obtains template and search area seeds +(each seed is represented as [x; f] ∈ R3+d1, where x de- +notes the 3D coordinates of the seed and f is the seed fea- +tures vector), then feature-targeted transformation converts +the search area seeds into two feature-targeted seed subsets, +2) adaptive cross-correlation utilizes two branches to re- +spectively handle two feature-targeted subsets, one branch +highlights object-aware features from the consistency be- +tween the subset and the template; the other branch extracts +discriminative features from the discrepancy between the +subset and the template, 3) object augmentation concate- +nates the two subsets and further obtains highlighted search +area seeds with the cross attention mechanism. Each seed is +appended with a highlighting score predicted by MLP, the +score is supervised by smooth point-wise one-hot label and +Focal Loss [14]. +Obtained the highlighted search area seeds and the high- +lighting scores, the object-preserved sampling selects the +seeds with top k highlighting scores as object candidates, +Then, each object candidate clusters the neighbor seeds +to aggregate local features, and we can obtain the object- +preserved search area seeds. Object integrity and object- +aware features can be effectively preserved and the back- +ground points that cause redundancy have been dropped +simultaneously. +Finally, the object localization network +performs effective BBoxes prediction based on the object- +preserved search area seeds without generating multiple +proposals. +Experimental results (average success gain of ∼9.4% +and ∼2.5% over the SOTA method on KITTI [6] and +Waymo Open Dataset [25]) demonstrate that OPSNet +achieves high-performance tracking, handles sparse point +clouds effectively, and performs robust tracking on long se- +quences. +The contributions of this paper are as follows: +• Object highlighting module adaptively extracts not +only highlighted object-aware features but also dis- +criminative features from template and search area. +• Object-preserved sampling significantly maintains the +object integrity and reduces the misleading back- +ground points and redundancy. +• Object localization network performs more effective +BBoxes prediction than other voting-based regression +approaches without generating multiple proposals. +2. Related Work +Deep learning on point clouds. +Since Qi et al. [21] +proposed their inspirational paradigm PointNet, 3D deep +learning on point clouds has stimulated the interest of re- +searchers. +3D deep learning on point clouds methods +mainly consist of point-based [4, 22, 23, 31], voxel-based +[13, 35, 36], graph-based [20, 27]. We choose the point- +based method since the point-based method learns distinct +object-aware features, handles sparse scenes well and re- +duces computational burden [22]. +3D Single object tracking. Recently, with the rapid devel- +opment of sensors, 3D sensors such as LiDAR can obtain +3D point cloud data for 3D single object tracking. Gian- +cola et al. [7] first proposed a shape-completion-based 3D +Siamese tracker (SC3D), which originally encodes shape +information into the template for matching template and +search area. However, SC3D cannot be end-to-end trained. +Qi et al. [23] proposed a point-to-box (P2B) Siamese track- +ing scheme, which adopted the target-specific feature aug- +mentation module to learn template and search area cross- +correlation and utilized Hough Voting modules to regress +potential target centers in the search area. Zhou et al. [34] +proposed PTTR that firstly adopted relation-aware sampling +(RAS) in the Siamese network, search area branch obtains a +feature map shared by the template for sampling reference, +which effectively preserves object points in the search area. +Furthermore, Hui et al. [8] proposed a novel BBox regres- +sion module, which compressed voxelized sparse 3D fea- +tures for 2D detection and performs effective object track- +ing. +Object segmentation. +Object-preserved sampling em- +ployed in OPSNet is similar to object segmentation, but they +differ in task-driven purposes. Recently proposed object +segmentation approaches [1, 17, 19, 24] are based on deep +learning. Wu et al. [28] formulated object segmentation +as a point-wise multi-class classification problem, and they +propose an end-to-end pipeline called SqueezeSeg based on +convolutional neural networks. Our object-preserved sam- +pling also aims at point-wise classification, but the task- +driven purpose is selecting object candidates by referring +to the point-wise highlighting scores. +3. Method +3.1. Overview +In Siamese-based tracking methods, the backbone down- +samples template and search area to obtain their seeds by +set abstraction layers that consist of sampling and group- +ing [22], each seed is represented as [x; f] ∈ R3+d1, where +x denotes the 3D coordinates of the seed and f is the seed +features vector. We aim to retain object points by object- +preserved sampling and highlight object-aware features in +the search area for accurate object localization. OPSNet +2 + +Figure 2. The main pipeline of OPSNet. OPSNet consists of two main parts: 1) object highlighting module, and 2) object-preserved +sampling and localization. Firstly, the set abstraction layers output template and search area seeds, respectively. Then, with the help of +template seeds object highlighting module enhances the object-aware features and extracts the discriminative features from search area +seeds. Finally, the object-preserved sampling selects object candidates from the highlighted search area seeds for object preservation and +accurate 3D BBox localization. +has two main parts (Fig. 2): 1) object highlighting mod- +ule, and 2) object-preserved sampling and localization. The +object highlighting module aims to enhance object-aware +features and extract discriminative features from template +and search area, which can assist the object-preserved sam- +pling to achieve object-background recognition. +Object- +preserved sampling and localization maintain object in- +tegrity and reduce redundancy for effectively predicting ac- +curate BBoxes. +3.2. Object Highlighting Module +Raw points in template Ptmp (of size N) and search +area Psea (of size M) are fed into a feature backbone to +obtain N1 template seeds Q = {qi}N1 +i=1 and search area +seeds R = {rj}M1 +j=1, the backbone consists of two set ab- +straction layers. Then feature-targeted transformation mod- +ule converts the search area seeds into two feature-targeted +seed subsets. +After that, the adaptive cross-correlation +utilizes two branches to respectively handle two subsets, +one branch highlights object-aware features with consis- +tent cross-correlation; the other branch extracts discrimina- +tive features with discrepant cross-correlation. Finally, the +object augmentation module further enhances the object- +aware features. +Feature-targeted transformation. Feature-targeted trans- +formation initializes two independent MLPs with input and +output dimension {M1, M1 +2 }, then the transposed search +area seeds RT are fed into them to generate two feature- +targeted seed subsets RT +1 and RT +2 , respectively. For dimen- +sion alignment, we transpose RT +1 and RT +2 to obtain feature- +targeted subsets R1 (of size M1 +2 ) and R2 (of size M1 +2 ). +Adaptive Cross-correlation. +Adaptive cross-correlation +consists of consistent and discrepant cross-correlation. +Consistent cross-correlation. Consistent cross-correlation +computes the cosine similarity Fcos between a seed subset +R1 and the template seeds Q, which focuses more on con- +sistent semantic correlation. +sij = Fcos(qi, rj) = +qiT · rj +∥qi∥2 · ∥rj∥2 +(1) +where qi, rj respectively denotes certain point in template +seeds Q and feature-targeted subset R1. sij denotes the co- +sine similarity between points qi, rj. Then we gather the +most similar point in template seeds with each point in the +subset R1, as Fig. 3 shows, which can be written as : +rj +′ = MLP[rj, skj, qk, xk] +k = argmax +i=1,2,...,N1 +{Fcos(qi, rj)} , ∀rj ∈ R1 +(2) +where k is the index of the maximum similarity +Fcos(qi, rj) value, and skj, xk are the corresponding max- +imum value and coordinate (x, y, z), respectively. [·, ·, ·, ·] +denotes concatenation. +Here, we obtain a consistent cross-correlated feature vec- +tor rj′ of j-th point in the subset R1 after MLP and in- +tegrate them to generate consistent cross-correlated seeds +subset R1 +′. +Discrepant cross-correlation. Discrepant cross-correlation +extracts discrepant semantics between Q and feature- +targeted subset R2 into R2. We firstly learn a discrepancy +map by computing subtracted difference and using an MLP, +which is formulated as: +dij = Fsub(qi, rj) = MLP(qi − rj), ∀qi ∈ Q, rj ∈ R2 +(3) +3 + +Mix(3 +di) +Template Seeds +Template +N×3 +Set Abstraction +Ni×(3 +di) +Layers + Augmentation +M2×(3 + d2) +Object +Object-Preserved +Adaptive Cross-correlation +Highlighted +Sampling +Search Area Seeds +M1 +×(3 + dl) +M1 +-×(3 + d) +2 +Object Localization +Top M2 +Feature-targeted +1×1 +Network +Search Area +Transformation +M +M×3 +Set Abstraction +M×(3 + d) +Final 3D BBox +Layers +Search Area Seeds +八 +Highlighting Scores +Object Highlighting Module +Object-Preserved Sampling & LocalizationFigure 3. Illustration of the object highlighting module. Object highlighting module can be divided into three main parts: 1) feature-targeted +transformation converts search area seeds into two feature-targeted seed subsets, 2) adaptive cross-correlation extracts the consistent and +discrepant features into two subsets respectively with the help of the template seeds, 3) object augmentation further enhances the object +features by employing the cross attention mechanism. +where qi, rj respectively denote certain point in the tem- +plate seeds Q and subset R2. dij denotes the discrepant +weight between points qi, rj, as Fig. 3 shows. +The dis- +crepant cross-correlated targeted point rj′′ is given by: +rj +′′ = MLP( +MAX +i=1,2,...,N1 {qi · dij}) +(4) +here MAX is the max pooling operator. We obtain a dis- +crepant cross-correlated feature vector rj′′ of j-th point in +the subset R2 and integrate them to generate discrepant +cross-correlated seeds subset R2 +′. +Concatenating R1 +′ and R2 +′, the cross-correlated search +area seeds R′ are obtained. +Object augmentation. +We first subtract the cross- +correlated seeds R′ and search area seeds R, then put the +subtracted tensor into MLP for augmenting differentiation +features R∆. For further augmenting the object-aware fea- +tures in the cross-correlated seeds R′, cross attention mech- +anism is adopted, which can be formulated as: +�R = CrossAttention(R′ + Xs, R∆ + Xs, R∆ + Xs) +(5) +where the three inputs from left to right are used as query, +key, and value in Eq. (5), respectively. Xs denotes the po- +sition embedding. +Here the object-highlighted search area seeds �R += +{fj}M1 +j=1 are obtained. +3.3. Object Preserved Sampling +Obtained object-highlighted search area seeds �R += +{fj}M1 +j=1 and given original BBox annotations, point-wise +one-hot labels Y = {yi}M1 +i=1 can be generated. We pre- +dict a highlighting score for each seed: ˆyi = MLP(fi), i = +1, 2, ..., M1, and select seeds with highest M2 scores as can- +didates. In addition, smooth one-hot labels is formulated as: +yi +′ = +� yi − +ε +M1 , +yi = 1. +yi + +ε +M1 , +yi = 0. +(6) +Obtained the smooth one-hot labels, we compute the +Focal loss [14] for supervision. +Considering the search +area generally contains unbalanced objects and background +points number, the focal loss can be regarded as an inge- +nious solution, which is defined by: +Lcls = − +� +yi +′[yi = 1](1 − ˆyi)αlog( ˆyi)+ +yi +′[yi = 0] ˆyi +αlog(1 − ˆyi) +(7) +where yi′[cond.] is the indicator function, follows Eq. +(6), ˆyi is the predicted highlighting score. Besides, α = 2 +is empirically set in all experiments. +We select object candidates with top M2 highlighting +scores as ball query centers for grouping and aggregating +local features to obtain object-preserved search area seeds +Rop = {rop +j }M2 +j=1, rop +j += [xop +j ; f op +j ] ∈ R3+d2, where xop de- +notes the object-preserved seed coordinates and f op is the +feature vector. +3.4. Object Localization Network +Inspired by [5,8], we develop an object localization net- +work utilizing bird’s eyes view (BEV), and the object detec- +tion paradigm has been customized for tracking adaption. +Firstly, +sparse +search +area +seeds +Rop +re- +quires +to +be +segmented +within +the +BEV +range +4 + +Feature-targeted Transformation +Adaptive Cross-correlation +Object Augmentation +Cross-correlated +Feature +xyz +Template Seeds +MLP +Mix(3 + di) +G +Similarity map +专 +×(3 +d1) +Q +Cross Attention +2 +M1 +M1 +×(3 +d1) +Kt +(3 + d1)x +Consistent cross-correlation module +2 +V +Highlighted +(3 + d)xM1 +I Template Seeds + Maxpooling +Seeds +Search Area +MLP +T +M1×(3 + d2) +Seeds +M1 +Mi×(3 + d1) +Search Area +M +M1 +(3 + d1)× +×(3 + di) +Seeds +. +Discrepant cross-correlation module +Mi×(3 + d1) +Transpose[(xmin, xmax), (ymin, ymax)]. +We voxelize segmented +Rop with voxel side length as r, then we can obtain dense +BEV feature map E +∈ RH×W ×c in x-y plane, where +H = ⌊ xmax−xmin +r +⌋, W += ⌊ ymax−ymin +r +⌋, and c is the +channel of features that obtained by average pooling. Then +we locate the original 3D point coordinates (x, y, z) in +2D x-y plane presented by (u, v), where u = +x−xmin +r +and v = +y−ymin +r +. +Since the dense map uses discrete +coordinates, we need discrete (¯u, ¯v) = (⌊u⌋, ⌊v⌋) as well. +Here ⌊·⌋ denotes floor operation. +Therefore, we can localize 2D-center based on the dense +BEV feature map, we discretize ground truth bounding box +(GTBB) center �pc (x, y, z) into pc (¯u, ¯v), then we expand +pc into a center map M ∈ RH×W ×1 that defined by: +� pij = +1 +d+1, +pij ∈ B. +pij = 0, +pij /∈ B. +(8) +where d presents the Euclidean distance between pij and +GTBB center pc; pij ∈ B and pij /∈ B mean the pixel pij +inside or outside the 2D GTBB B, respectively. +Pervasive distractors can easily mislead object center lo- +calization, and we need to localize the only object required +to track with, thus we conduct train & test syncretic loss +punishment on wrongly located centers. We firstly predict +n stacked center maps +ˆ +Mn ∈ RH×W ×n from BEV fea- +ture map E but we only enforce the first center map +ˆ +M0 ∈ +RH×W ×1 to approach the ground truth M by using Focal +loss [14], which denoted by Lcenter. As for other n − 1 +center maps, we compute Euclidean distances summation +between the predicted centers ˆpc +i in +ˆ +Mi, (i = 1, 2, ..., n) +and GT, which is defined by: +Lloc = 1 +n +k +� +i=1 +∥pc − ˆpc +i∥2 +(9) +where pc +i denotes i-th predicted center in ˆ +Mi. +Since the 2D BEV-based feature map generates a dis- +crete 2D-center, the offset of the continuous ground truth +center requires to be regressed. We surround the predicted +object center with a square of radius l. Similar to 2D-center +prediction, predicted offset map ˆO ∈ RH×W ×2 is gener- +ated from BEV feature map E, which is supervised by: +Loff = +l +� +δx=−l +l +� +δy=−l +| ˆOpc+(δx,δy) − [ �pc − pc + (δx, δy)]| +(10) +where �pc and pc present the ground truth center with con- +tinuous and discrete coordinates respectively. +Two individual CNNs are applied to directly regress the +z-axis coordinate and the rotation angle θ. Given two pre- +dicted z-axis map ˆZ ∈ RH×W ×1 and the rotation map +ˆΘ ∈ RH×W ×1, we use L1 loss to compute error: +Lz = | ˆ +Zc − z| +(11) +Lθ = |ˆΘ − θ| +(12) +We sum the all above losses as our final network loss: +L = λ1(Lcenter + Lloc + Loff + Lθ) + λ2Lz + λ3Lcls. +λ1, λ2, and λ3 are the hyper-parameters for loss regulation. +4. Experiments +Extensive experiments have been conducted to validate +the effectiveness of our OPSNet and its components, we +mainly focus on car and pedestrian tracking since cars and +pedestrians appear in large quantity and diversity. +4.1. Experimental Settings +Datasets. KITTI [6] and Waymo Open Dataset (WOD) [25] +are used for training and validation. Since the labels of the +KITTI test set are not provided, following previous works +[7, 23, 32], we use the training set to train and evaluate. +The training set is spilt as follows: scenes 0-16 for train- +ing, scenes 17-18 for validation, and scenes 19-20 for test- +ing. For WOD, we follow [34] to transform the WOD into +class-balanced tracklets that tracking paradigms can handle. +Evaluation Metrics. Our evaluation metrics use the One +Pass Evaluation (OPE) [29] from single object tracking. +The Success metrics are defined as the Area Under Curve +(AUC) of the IoU between the predicted box and the GT. +The Precision metrics are defined as the AUC of the dis- +tance between the centers for errors from 0 to 2m. +Implementation details. Following [23], template number +of points sets as N = 512 and search area sets as M = 1024 +by randomly points discarding and duplicating. We obtain +template and search area seeds by three set abstraction lay- +ers with query radii of 0.3, 0.5, and 0.7, yielding M1 = 256, +N1 = 128 and we set the seed feature dimension d1 = 128. +Object-preserved sampling selects highlighted search area +seeds with top 128 scores, yielding M2 = 128, and local +feature aggregation generates object-preserved seeds with +feature dimension d2 = 128. +Targeted-feature transformation module uses two 3-layer +MLPs with batch normalization (BN) and a ReLU activa- +tion layer. For highlighting score prediction, object pre- +served sampling uses 3-layer MLP and Sigmoid function +for object score prediction. In the object localization net- +work, we set the voxel side length r = 0.3m, the channel +of BEV feature map c = 128, and we generate 4 stacked +center maps (n = 4). For loss regulation, we set λ1 = 1.0, +λ2 = 2.0, and λ3 = 0.5. +Training and testing. +We aggregate the first GTBB of +the current tracklet and the previous GTBB as the tem- +plate, which is applied with a random offset on the previous +GTBB to augment data. The search areas are augmented +5 + +Table 1. Comparison with other state-of-the-art approaches on the +KITTI dataset. The instance frame number of each category is +shown under category names, and the mean denotes the average +results of all categories. Bold presents the best performance. +Category +Car +Ped +Van +Cyclist +Mean +Frame num. +6424 +6088 +1248 +308 +14068 +Success +P2B [23] +56.2 +40.8 +40.8 +32.1 +39.5 +BAT [32] +60.5 +42.1 +52.4 +33.7 +47.2 +LTTR [3] +65.0 +33.2 +35.8 +66.2 +50.0 +V2B [8] +70.5 +48.3 +50.1 +40.8 +58.4 +PTTR [34] +65.2 +50.9 +52.5 +65.1 +58.4 +STNet [9] +72.1 +49.9 +58.0 +73.5 +61.3 +M2-Track [33] +65.5 +61.5 +53.8 +73.2 +62.9 +OPSNet (ours) +77.0 +65.6 +77.7 +85.5 +72.3 +Precision +P2B [23] +72.8 +49.6 +48.4 +44.7 +53.9 +BAT [32] +77.7 +70.1 +67.0 +45.4 +65.1 +LTTR [3] +77.1 +56.8 +45.6 +89.9 +67.4 +V2B [8] +81.3 +73.5 +58.0 +49.7 +75.2 +PTTR [34] +77.4 +81.6 +61.8 +90.5 +77.8 +STNet [9] +84.0 +77.2 +70.6 +93.7 +80.1 +M2-Track [33] +80.8 +88.2 +70.7 +93.5 +83.4 +OPSNet (ours) +86.1 +90.3 +85.8 +93.7 +88.0 +similarly by enlarging the current GT with an offset, for all +experiments, this offset sets as 2m. Additionally, we ap- +ply the Adam optimizer [12] with an initial learning rate of +0.001 and decreased by 5 times after 10 epochs. We train +the OPSNet on a single NVIDIA RTX2080 Ti GPU with a +batchsize of 32. OPSNet can obtain satisfying results after +about 40 epochs. +4.2. Results +Evaluation on KITTI. As shown in Tab. 1, our OPSNet +shows a large improvement over SOTA methods, All the +methods besides M2-track [33] apply the Siamese-based +network. +Our OPSNet outperforms the SOTA tracking +paradigm M2-track with a large margin of ∼9.4% / ∼4.6% +on average success / precision and the SOTA PTTR [34] +based on PointNet++ of ∼13.9% / ∼10.2%. +Our OP- +SNet achieves the highest tracking performance in all four +categories. +As shown in Fig. 4, we compare the pro- +posed OPSNet against PTTR over the car and pedestrian +sequences since PTTR adopts relation-aware sampling for +object preservation. +The error gradually accumulates in +PTTR tracking since the bad template misleads the search +area sampling, but the object-preserved search area seeds +generated by object-preserved sampling assist the OPSNet +to perform robust tracking. +Evaluation on WOD. To validate the generalization ability +of our OPSNet, we use our OPSNet that pre-trained over the +KITTI dataset for testing tracking on Waymo Open Dataset +(WOD) following [8]. According to the point number of the +first frame, WOD can be divided into three subsets of dif- +ferent tracking levels of difficulty, including easy, medium, +Table 2. Comparison with other state-of-the-art approaches on the +WOD dataset. +Category +Vehicle +Level +Frame num. +Easy +67832 +Medium +61252 +Hard +56647 +Mean +185371 +Success +BAT [32] +61.0 +53.3 +48.9 +54.7 +V2B [8] +64.5 +55.1 +52.0 +57.6 +STNet [9] +65.9 +57.5 +54.6 +59.7 +OPSNet (ours) +68.8 +58.7 +55.2 +61.3 +Precision +BAT [32] +68.3 +60.9 +57.8 +62.7 +V2B [8] +71.5 +63.2 +62.0 +65.9 +STNet [9] +72.7 +66.0 +64.7 +68.0 +OPSNet (ours) +74.4 +66.9 +65.1 +69.1 +Category +Pedestrian +Level +Frame num. +Easy +85280 +Medium +82253 +Hard +74219 +Mean +241752 +Success +BAT [32] +19.3 +17.8 +17.2 +18.2 +V2B [8] +27.9 +22.5 +20.1 +23.7 +STNet [9] +29.2 +24.7 +22.2 +25.5 +OPSNet (ours) +32.2 +28.4 +25.1 +28.7 +Precision +BAT [32] +32.6 +29.8 +28.3 +30.3 +V2B [8] +43.9 +36.2 +33.1 +37.9 +STNet [9] +45.3 +38.2 +35.8 +39.9 +OPSNet (ours) +47.4 +41.9 +37.6 +42.5 +and hard. Even though tracking frames are far more than +KITTI has, our OPSNet still shows the generalization abil- +ity of the tracking performance on WOD, which outper- +forms the baseline BAT [32] with a gain of ∼6.6% / ∼6.4% +on the category Vehicle, results as Tab. 2 shows. +Tracking on sparse point clouds. Handling extreme sparse +scenes is a great challenge in 3D SOT tasks. We test our +OPSNet on the object of extreme sparse points, which is +divided into three object point number intervals including +[0, 50), [50, 100), [100, 150). We compare the tracking re- +sult of our OPSNet, PTTR [34], V2B [8], and STNet [9]. +As Fig. 5 shows, our OPSNet achieves the highest track- +ing performance on three intervals with ∼2.8% / ∼2.6%, +∼4.7% / ∼3.7%, ∼6.7% / ∼4.7% success / precision gain +over the V2B. Experiment results demonstrate that our OP- +SNet effectively preserves sparse objects and handles ex- +treme sparse point clouds well. +Tracking on long sequences. +Siamese-based methods +probably lose the object when tracking on long sequences. +Therefore the performance on long sequences evaluates +the robustness of SOT methods. +PTTR adopts their +relation-aware sampling for object preservation, therefore +we test our OPSNet and PTTR on sequence length inter- +vals [0, 50), [50, 200), [200, +∞) and compute average re- +call rate after sampling as the indicator of object preserva- +tion. As Fig. 6 shows, our OPSNet outperforms the PTTR +on three intervals with ∼11.8% / ∼7.7%, ∼10.1% / ∼6.4%, +6 + +Figure 4. Result comparison visualization with PTTR [34]. We can observe that the predicted BBoxes by OPSNet hold tight to the ground +truths. +Figure 5. Evaluated mean success / precision of the three extreme +sparse point number intervals on the KITTI Car category. +Figure 6. Evaluated mean success / precision and recall rate of the +three sequence length intervals on the KITTI Car category. +∼14.0% / ∼12.1% success / precision gain and ∼22.0%, +∼20.6%, ∼29.9% recall rate gain. +Running speed. We evaluate the inference speed of our +model on a single RTX2080 Ti GPU. Our method achieves +23 FPS, including 21 ms for pre-processing point clouds, +21 ms for network forward propagation, and 2 ms for post- +processing. We also test V2B, and STNet in the default +settings run with 20 FPS and 19 FPS, respectively. +4.3. Ablation Studies +We conduct ablation studies to evaluate the modules pro- +posed in OPSNet, the results are shown with OPE evalua- +tion metrics and based on the KITTI dataset. +Comparison on sampling strategies. We compare our pro- +posed object preserved sampling (OPS) with existing unsu- +pervised sampling strategies including relation-aware sam- +pling (RAS) [34], random sampling (RS) [23], and furthest +point sampling (FPS) [22] on sampling 128 candidates from +the 256 search area seeds, as Fig. 7 shows. We compute the +recall rates for object points after sampling which demon- +strate the effectiveness of our OPS. We also replace our OPS +with RAS, RS, and FPS in OPSNet, default OPSNet outper- +forms clear success / precision gain of ∼4.7% / ∼3.6% on +the car category and ∼9.0% / ∼8.8% on the pedestrian over +the FPS baseline, comparison shown in Tab. 3. +Ablation studies on object highlighting module. We con- +duct ablation experiments to validate the effectiveness of +the object highlighting components, respectively, as Tab. 4 +shows. We report the results of cars and pedestrians track- +ing on the KITTI dataset. Only a single cross-correlation +approach cannot perform the best tracking (∼6.9% / ∼3.5% +decrease with only consistent cross-correlation and ∼6.4% +/ ∼3.0% decrease with only discrepant cross-correlation on +Car). +For analyzing feature-targeted transformation, we +simply divide the search area into two subsets without uti- +lizing MLPs, so that the cross-correlation cannot be adap- +7 + +Timeline (Car) +Timeline (Pedestrian) +Ground Truth +OPSNet (ours) +PTTRSuccess +Precision +85 +95 +82.5 +91.5 +80 +90 +75.8 +75 +86.8 +74.3 +85.6 +85.5 +73.7 +85 +t.t8 +72 +83.2 +70 +70.6 +68.8 +% 80 +% +65 +75 +73 +60 +60.4 +57.6 +30.2 +57.4 +70 +55 +53.2 +67.2 +50 +65 +Interval +[0,50] +[50,100] +[100,150] +Interval +[0,50] +[50,100] +[100,150] +Frame +Frame +1938 +861 +494 +1938 +861 +494 +Number +Number +PTTR +_V2B +STNet +OPSNet (ours)PTTR +OPSNet +95 +95 +90 +90 +89.2 +86.6 +85 +81.5 +85 +81.6 +81.2 +82.8 +80.2 +80 +80 +79.5。 +78.4 +76.8 +75.3 +75 +75 +70.7 +% 70 +67.7 +66.7 +%70 +65 +61.3 +65 +60 +57.5 +56.2 +60 +55 +55 +50 +485 +50 +45 +45 +[0,50] +[50,200] +[200,+] +[0,50] +[50,200] +[200,+∞0] +Success +Precision- +-Recall +Success +Precision--RecallFigure 7. Comparison on sampling strategies: our object-preserved sampling (OPS), relation-aware sampling (RAS) proposed in PTTR +[34], random sampling (RS) employed in [23], and furthest points sampling (FPS) proposed in PointNet++ [22]. Note that red points +denote the object and black ones denote the background. +Table 3. Tracking performance comparison on different sampling +approaches. Shown by the evaluation metrics success / precision +on the category car and pedestrian. +Method +Car +Pedestrian +RS [23] +72.9 / 83.7 +58.2 / 82.1 +FPS [22] +71.2 / 81.8 +52.4 / 80.5 +RAS [34] +73.3 / 83.0 +61.0 / 84.5 +OPS (ours) +77.0 / 86.1 +65.6 / 90.3 +Table 4. Ablation studies results on object highlighting module. +Car +Pedestrian +w/o object augmentation +74.5 / 84.3 +62.2 / 86.9 +w/o feature-targeted transformation +69.3 / 82.5 +59.1 / 79.8 +only consistent cross-correlation +70.1 / 82.6 +59.9 / 80.1 +only discrepant cross-correlation +70.6 / 83.1 +60.3 / 80.5 +default setting +77.0 / 86.1 +65.6 / 90.3 +tively conducted (∼7.7% / ∼3.6% decrease without the +feature-targeted transformation on Car). +Results demon- +strate that object augmentation further improves tracking +performance (∼2.5% / ∼1.8% gain on the category Car). +Ablation studies on object localization network. +We +replace our object localization network (OLN) with the +VoteNet-based approaches including Region Proposal Net- +work (RPN) applied in [23,32] and coarse-to-refine regres- +sion (C2R) proposed in [34] to validate the tracking perfor- +mance. VoteNet regresses the offset from point to center +with a score as proposals. However, when facing sparse +scenes VoteNet is hard to generate high-quality proposals. +Table 5. Tracking performance comparison on common object +locating approaches. +Car +Pedestrian +RPN [23,32] +68.0 / 79.6 +48.8 / 72.5 +C2R [34] +71.5 / 83.2 +57.1 / 80.2 +OLN (ours) +77.0 / 86.1 +65.6 / 90.3 +Our object localization network does not require generating +numerous 3D proposals thus it can handle the above issue +and outperforms baseline RPN with ∼9.0% / ∼6.5% and +∼16.8% / ∼17.8% increase on the category Car and Pedes- +trian. +5. Conclusion +In this paper, we propose a novel Object Preserving +Siamese Network (OPSNet) for single object tracking on +point clouds. Our OPSNet aims to enhance the object-aware +features and extract discriminative features from template +and search area with the object highlighting module, main- +tain object integrity and drop redundant background points +with the object-preserved sampling, and predict accurate +BBoxes with the object localization network. +The experimental results (∼9.4% and ∼2.5% success +gain on KITTI and WOD respectively) demonstrate that +our OPSNet achieves high-performance object tracking and +possesses generalization ability, indicate that object high- +lighting module and object-preserved sampling effectively +handle sparse point clouds and assist the object localization +network to perform robust tracking on long sequences. +8 + +Search Area seeds +Object-preserved Sampling +Relation-aware Sampling +Random Sampling +Furthest Point Sampling +Mi = 256 +M2 = 128 +M2 = 128 +M2=128 +M2 = 128 +Points on object:113 +Points on object: 99 +Points on object: 71 +Points on object: 55 +Points on object: 47 +recall: 87.6% +recall: 62.8% +recall: 48.7% +recall: 41.7% +Car +Points on object :54 +Points on object: 54 +Points on object: 45 +Points on object: 42 +Points on object: 30 +** +recall: 100% +recall: 84.9% +recall: 77.8% +recall: 55.6% +Pedestrian +.s...... +".References +[1] Tristan Brodeur, Hadi AliAkbarpour, and Steve Suddarth. +Point cloud object segmentation using multi elevation-layer +2d bounding-boxes. 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International +Journal of Machine Learning and Cybernetics, pages 1–13, +2022. 2 +10 + diff --git a/Z9FLT4oBgHgl3EQfWi9I/content/tmp_files/load_file.txt b/Z9FLT4oBgHgl3EQfWi9I/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fa1324914543f0dead539b7d78e1c048216c041d --- /dev/null +++ b/Z9FLT4oBgHgl3EQfWi9I/content/tmp_files/load_file.txt @@ -0,0 +1,783 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf,len=782 +page_content='Object Preserving Siamese Network for Single Object Tracking on Point Clouds Kaijie Zhao, Haitao Zhao, Zhongze Wang, Jingchao Peng, Zhengwei Hu East China University of Science and Technology No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='130, Meilong Rd, Shanghai Abstract Obviously, the object is the key factor of the 3D single object tracking (SOT) task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' However, previous Siamese- based trackers overlook the negative effects brought by ran- domly dropped object points during backbone sampling, which hinder trackers to predict accurate bounding boxes (BBoxes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Exploring an approach that seeks to maximize the preservation of object points and their object-aware features is of particular significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Motivated by this, we propose an Object Preserving Siamese Network (OP- SNet), which can significantly maintain object integrity and boost tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Firstly, the object high- lighting module enhances the object-aware features and extracts discriminative features from template and search area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Then, the object-preserved sampling selects object candidates to obtain object-preserved search area seeds and drop the background points that contribute less to tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Finally, the object localization network precisely locates 3D BBoxes based on the object-preserved search area seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Extensive experiments demonstrate our method outperforms the state-of-the-art performance (∼9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4% and ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5% success gain on KITTI and Waymo Open Dataset respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Introduction Single object tracking (SOT) has become a significant issue of computer vision, which contributes widely to var- ious fields, such as autonomous driving [15, 18], security surveillance [26,30] and robotics [2,11], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' With the fast development of 3D sensors, such as LiDAR, 3D data reflect- ing real-world coordinates and object sizes can be captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Therefore, 3D computer vision tasks attract more and more attention in recent years [10, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 3D single object track- ing is also an essential part of 3D computer vision, which aims to improve autonomous cars’ safety via predicting ac- curate object trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' At present, many 2D single object tracking algorithms have been proposed and proved their ef- ficiency, but they cannot directly handle 3D data since 3D point clouds are of sparsity and non-uniformity [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Exemplified illustration to show how our OPSNet works, from the object highlighting module to the object-preserved sam- pling and localization module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Mainstream 3D SOT approaches are based on the Siamese network [7–9, 23, 32, 34] that consists of two weight-shared backbone branches to respectively handle template and search area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Most of the Siamese-based ap- proaches use PointNet++ [22] as the backbone which hier- archically samples points and aggregates local features from raw point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' STNet [9] firstly adopts Transformer as the backbone for encoding all the points of template and search area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Considering that the tracking tasks require to be conducted in real-time and tracking speed is also a sig- nificant evaluation indicator, the PointNet++ is of higher ef- ficiency and less computational burden [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Since PointNet++ applies furthest points sampling (FPS) or random sampling, all the approaches based on Point- Net++ suffer from the negative effects brought by randomly dropped object points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' To overcome this issue, PTTR [34] samples the search area points that are most similar to tem- plate points in the embedding space for object preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' However, once PTTR regresses an inaccurate 3D BBox as a template during inference, search area sampling will be 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='12057v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='CV] 28 Jan 2023 Points Highlighting score RawTemplate Object Highlighting Module Highlighted 3DBBox Search Area Seeds + Object-preserved Object Localization Sampling Network RawSearchArea Object Candidates Object-Preserved Search Area Seedssimultaneously misled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Exploring a robust approach that seeks to maximize the preservation of object points and their object-aware features for tracking is of particular sig- nificance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We propose our novel OPSNet for object high- lighting, object preservation, and accurate object localiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We exemplify how our OPSNet works in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' The object highlighting module consists of three main parts: 1) The backbone first obtains template and search area seeds (each seed is represented as [x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' f] ∈ R3+d1, where x de- notes the 3D coordinates of the seed and f is the seed fea- tures vector), then feature-targeted transformation converts the search area seeds into two feature-targeted seed subsets, 2) adaptive cross-correlation utilizes two branches to re- spectively handle two feature-targeted subsets, one branch highlights object-aware features from the consistency be- tween the subset and the template;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' the other branch extracts discriminative features from the discrepancy between the subset and the template, 3) object augmentation concate- nates the two subsets and further obtains highlighted search area seeds with the cross attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Each seed is appended with a highlighting score predicted by MLP, the score is supervised by smooth point-wise one-hot label and Focal Loss [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Obtained the highlighted search area seeds and the high- lighting scores, the object-preserved sampling selects the seeds with top k highlighting scores as object candidates, Then, each object candidate clusters the neighbor seeds to aggregate local features, and we can obtain the object- preserved search area seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Object integrity and object- aware features can be effectively preserved and the back- ground points that cause redundancy have been dropped simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Finally, the object localization network performs effective BBoxes prediction based on the object- preserved search area seeds without generating multiple proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Experimental results (average success gain of ∼9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4% and ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5% over the SOTA method on KITTI [6] and Waymo Open Dataset [25]) demonstrate that OPSNet achieves high-performance tracking, handles sparse point clouds effectively, and performs robust tracking on long se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' The contributions of this paper are as follows: Object highlighting module adaptively extracts not only highlighted object-aware features but also dis- criminative features from template and search area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Object-preserved sampling significantly maintains the object integrity and reduces the misleading back- ground points and redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Object localization network performs more effective BBoxes prediction than other voting-based regression approaches without generating multiple proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Related Work Deep learning on point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Since Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' [21] proposed their inspirational paradigm PointNet, 3D deep learning on point clouds has stimulated the interest of re- searchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 3D deep learning on point clouds methods mainly consist of point-based [4, 22, 23, 31], voxel-based [13, 35, 36], graph-based [20, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We choose the point- based method since the point-based method learns distinct object-aware features, handles sparse scenes well and re- duces computational burden [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 3D Single object tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Recently, with the rapid devel- opment of sensors, 3D sensors such as LiDAR can obtain 3D point cloud data for 3D single object tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Gian- cola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' [7] first proposed a shape-completion-based 3D Siamese tracker (SC3D), which originally encodes shape information into the template for matching template and search area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' However, SC3D cannot be end-to-end trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' [23] proposed a point-to-box (P2B) Siamese track- ing scheme, which adopted the target-specific feature aug- mentation module to learn template and search area cross- correlation and utilized Hough Voting modules to regress potential target centers in the search area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' [34] proposed PTTR that firstly adopted relation-aware sampling (RAS) in the Siamese network, search area branch obtains a feature map shared by the template for sampling reference, which effectively preserves object points in the search area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Furthermore, Hui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' [8] proposed a novel BBox regres- sion module, which compressed voxelized sparse 3D fea- tures for 2D detection and performs effective object track- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Object segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Object-preserved sampling em- ployed in OPSNet is similar to object segmentation, but they differ in task-driven purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Recently proposed object segmentation approaches [1, 17, 19, 24] are based on deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' [28] formulated object segmentation as a point-wise multi-class classification problem, and they propose an end-to-end pipeline called SqueezeSeg based on convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Our object-preserved sam- pling also aims at point-wise classification, but the task- driven purpose is selecting object candidates by referring to the point-wise highlighting scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Overview In Siamese-based tracking methods, the backbone down- samples template and search area to obtain their seeds by set abstraction layers that consist of sampling and group- ing [22], each seed is represented as [x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' f] ∈ R3+d1, where x denotes the 3D coordinates of the seed and f is the seed features vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We aim to retain object points by object- preserved sampling and highlight object-aware features in the search area for accurate object localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' OPSNet 2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' The main pipeline of OPSNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' OPSNet consists of two main parts: 1) object highlighting module, and 2) object-preserved sampling and localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Firstly, the set abstraction layers output template and search area seeds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Then, with the help of template seeds object highlighting module enhances the object-aware features and extracts the discriminative features from search area seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Finally, the object-preserved sampling selects object candidates from the highlighted search area seeds for object preservation and accurate 3D BBox localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' has two main parts (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 2): 1) object highlighting mod- ule, and 2) object-preserved sampling and localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' The object highlighting module aims to enhance object-aware features and extract discriminative features from template and search area, which can assist the object-preserved sam- pling to achieve object-background recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Object- preserved sampling and localization maintain object in- tegrity and reduce redundancy for effectively predicting ac- curate BBoxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Object Highlighting Module Raw points in template Ptmp (of size N) and search area Psea (of size M) are fed into a feature backbone to obtain N1 template seeds Q = {qi}N1 i=1 and search area seeds R = {rj}M1 j=1, the backbone consists of two set ab- straction layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Then feature-targeted transformation mod- ule converts the search area seeds into two feature-targeted seed subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' After that, the adaptive cross-correlation utilizes two branches to respectively handle two subsets, one branch highlights object-aware features with consis- tent cross-correlation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' the other branch extracts discrimina- tive features with discrepant cross-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Finally, the object augmentation module further enhances the object- aware features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Feature-targeted transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Feature-targeted trans- formation initializes two independent MLPs with input and output dimension {M1, M1 2 }, then the transposed search area seeds RT are fed into them to generate two feature- targeted seed subsets RT 1 and RT 2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' For dimen- sion alignment, we transpose RT 1 and RT 2 to obtain feature- targeted subsets R1 (of size M1 2 ) and R2 (of size M1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Adaptive Cross-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Adaptive cross-correlation consists of consistent and discrepant cross-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Consistent cross-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Consistent cross-correlation computes the cosine similarity Fcos between a seed subset R1 and the template seeds Q, which focuses more on con- sistent semantic correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' sij = Fcos(qi, rj) = qiT · rj ∥qi∥2 · ∥rj∥2 (1) where qi, rj respectively denotes certain point in template seeds Q and feature-targeted subset R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' sij denotes the co- sine similarity between points qi, rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Then we gather the most similar point in template seeds with each point in the subset R1, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 3 shows, which can be written as : rj ′ = MLP[rj, skj, qk, xk] k = argmax i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=',N1 {Fcos(qi, rj)} , ∀rj ∈ R1 (2) where k is the index of the maximum similarity Fcos(qi, rj) value, and skj, xk are the corresponding max- imum value and coordinate (x, y, z), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' [·, ·, ·, ·] denotes concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Here, we obtain a consistent cross-correlated feature vec- tor rj′ of j-th point in the subset R1 after MLP and in- tegrate them to generate consistent cross-correlated seeds subset R1 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Discrepant cross-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Discrepant cross-correlation extracts discrepant semantics between Q and feature- targeted subset R2 into R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We firstly learn a discrepancy map by computing subtracted difference and using an MLP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' which is formulated as: dij = Fsub(qi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' rj) = MLP(qi − rj),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' ∀qi ∈ Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' rj ∈ R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Mix(3 +di) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Template Seeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Template ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='N×3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Set Abstraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Ni×(3 +di) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Augmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='M2×(3 + d2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Object ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Object-Preserved ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Adaptive Cross-correlation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Highlighted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Sampling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Search Area Seeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='M1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='×(3 + dl) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='M1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='×(3 + d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Object Localization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Top M2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Feature-targeted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1×1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Search Area ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Transformation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='M×3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Set Abstraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='M×(3 + d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Final 3D BBox ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Search Area Seeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='八 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Highlighting Scores ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Object Highlighting Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Object-Preserved Sampling & LocalizationFigure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Illustration of the object highlighting module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Object highlighting module can be divided into three main parts: 1) feature-targeted transformation converts search area seeds into two feature-targeted seed subsets, 2) adaptive cross-correlation extracts the consistent and discrepant features into two subsets respectively with the help of the template seeds, 3) object augmentation further enhances the object features by employing the cross attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' where qi, rj respectively denote certain point in the tem- plate seeds Q and subset R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' dij denotes the discrepant weight between points qi, rj, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 3 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' The dis- crepant cross-correlated targeted point rj′′ is given by: rj ′′ = MLP( MAX i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=',N1 {qi · dij}) (4) here MAX is the max pooling operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We obtain a dis- crepant cross-correlated feature vector rj′′ of j-th point in the subset R2 and integrate them to generate discrepant cross-correlated seeds subset R2 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Concatenating R1 ′ and R2 ′, the cross-correlated search area seeds R′ are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Object augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We first subtract the cross- correlated seeds R′ and search area seeds R, then put the subtracted tensor into MLP for augmenting differentiation features R∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' For further augmenting the object-aware fea- tures in the cross-correlated seeds R′, cross attention mech- anism is adopted, which can be formulated as: �R = CrossAttention(R′ + Xs, R∆ + Xs, R∆ + Xs) (5) where the three inputs from left to right are used as query, key, and value in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' (5), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Xs denotes the po- sition embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Here the object-highlighted search area seeds �R = {fj}M1 j=1 are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Object Preserved Sampling Obtained object-highlighted search area seeds �R = {fj}M1 j=1 and given original BBox annotations, point-wise one-hot labels Y = {yi}M1 i=1 can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We pre- dict a highlighting score for each seed: ˆyi = MLP(fi), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=', M1, and select seeds with highest M2 scores as can- didates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' In addition, smooth one-hot labels is formulated as: yi ′ = � yi − ε M1 , yi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' yi + ε M1 , yi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' (6) Obtained the smooth one-hot labels, we compute the Focal loss [14] for supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Considering the search area generally contains unbalanced objects and background points number, the focal loss can be regarded as an inge- nious solution, which is defined by: Lcls = − � yi ′[yi = 1](1 − ˆyi)αlog( ˆyi)+ yi ′[yi = 0] ˆyi αlog(1 − ˆyi) (7) where yi′[cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='] is the indicator function, follows Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' (6), ˆyi is the predicted highlighting score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Besides, α = 2 is empirically set in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We select object candidates with top M2 highlighting scores as ball query centers for grouping and aggregating local features to obtain object-preserved search area seeds Rop = {rop j }M2 j=1, rop j = [xop j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' f op j ] ∈ R3+d2, where xop de- notes the object-preserved seed coordinates and f op is the feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Object Localization Network Inspired by [5,8], we develop an object localization net- work utilizing bird’s eyes view (BEV), and the object detec- tion paradigm has been customized for tracking adaption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Firstly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='sparse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='search ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='area ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='seeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Rop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='re- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='quires ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='be ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='segmented ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='within ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='BEV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='range ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Feature-targeted Transformation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Adaptive Cross-correlation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Object Augmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Cross-correlated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='xyz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Template Seeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='MLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Mix(3 + di) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Similarity map ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='专 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='×(3 +d1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Cross Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='M1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='M1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='×(3 +d1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Kt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='(3 + d1)x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Consistent cross-correlation module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Highlighted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='(3 + d)xM1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='I Template Seeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Maxpooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Seeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Search Area ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='MLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='M1×(3 + d2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Seeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='M1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Mi×(3 + d1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Search Area ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='M1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='(3 + d1)× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='×(3 + di) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='Seeds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Discrepant cross-correlation module Mi×(3 + d1) Transpose[(xmin, xmax), (ymin, ymax)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We voxelize segmented Rop with voxel side length as r, then we can obtain dense BEV feature map E ∈ RH×W ×c in x-y plane, where H = ⌊ xmax−xmin r ⌋, W = ⌊ ymax−ymin r ⌋, and c is the channel of features that obtained by average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Then we locate the original 3D point coordinates (x, y, z) in 2D x-y plane presented by (u, v), where u = x−xmin r and v = y−ymin r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Since the dense map uses discrete coordinates, we need discrete (¯u, ¯v) = (⌊u⌋, ⌊v⌋) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Here ⌊·⌋ denotes floor operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Therefore, we can localize 2D-center based on the dense BEV feature map, we discretize ground truth bounding box (GTBB) center �pc (x, y, z) into pc (¯u, ¯v), then we expand pc into a center map M ∈ RH×W ×1 that defined by: � pij = 1 d+1, pij ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' pij = 0, pij /∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' (8) where d presents the Euclidean distance between pij and GTBB center pc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' pij ∈ B and pij /∈ B mean the pixel pij inside or outside the 2D GTBB B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Pervasive distractors can easily mislead object center lo- calization, and we need to localize the only object required to track with, thus we conduct train & test syncretic loss punishment on wrongly located centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We firstly predict n stacked center maps ˆ Mn ∈ RH×W ×n from BEV fea- ture map E but we only enforce the first center map ˆ M0 ∈ RH×W ×1 to approach the ground truth M by using Focal loss [14], which denoted by Lcenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' As for other n − 1 center maps, we compute Euclidean distances summation between the predicted centers ˆpc i in ˆ Mi, (i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=', n) and GT, which is defined by: Lloc = 1 n k � i=1 ∥pc − ˆpc i∥2 (9) where pc i denotes i-th predicted center in ˆ Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Since the 2D BEV-based feature map generates a dis- crete 2D-center, the offset of the continuous ground truth center requires to be regressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We surround the predicted object center with a square of radius l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Similar to 2D-center prediction, predicted offset map ˆO ∈ RH×W ×2 is gener- ated from BEV feature map E, which is supervised by: Loff = l � δx=−l l � δy=−l | ˆOpc+(δx,δy) − [ �pc − pc + (δx, δy)]| (10) where �pc and pc present the ground truth center with con- tinuous and discrete coordinates respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Two individual CNNs are applied to directly regress the z-axis coordinate and the rotation angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Given two pre- dicted z-axis map ˆZ ∈ RH×W ×1 and the rotation map ˆΘ ∈ RH×W ×1, we use L1 loss to compute error: Lz = | ˆ Zc − z| (11) Lθ = |ˆΘ − θ| (12) We sum the all above losses as our final network loss: L = λ1(Lcenter + Lloc + Loff + Lθ) + λ2Lz + λ3Lcls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' λ1, λ2, and λ3 are the hyper-parameters for loss regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Experiments Extensive experiments have been conducted to validate the effectiveness of our OPSNet and its components, we mainly focus on car and pedestrian tracking since cars and pedestrians appear in large quantity and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Experimental Settings Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' KITTI [6] and Waymo Open Dataset (WOD) [25] are used for training and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Since the labels of the KITTI test set are not provided, following previous works [7, 23, 32], we use the training set to train and evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' The training set is spilt as follows: scenes 0-16 for train- ing, scenes 17-18 for validation, and scenes 19-20 for test- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' For WOD, we follow [34] to transform the WOD into class-balanced tracklets that tracking paradigms can handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Our evaluation metrics use the One Pass Evaluation (OPE) [29] from single object tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' The Success metrics are defined as the Area Under Curve (AUC) of the IoU between the predicted box and the GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' The Precision metrics are defined as the AUC of the dis- tance between the centers for errors from 0 to 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Following [23], template number of points sets as N = 512 and search area sets as M = 1024 by randomly points discarding and duplicating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We obtain template and search area seeds by three set abstraction lay- ers with query radii of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7, yielding M1 = 256, N1 = 128 and we set the seed feature dimension d1 = 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Object-preserved sampling selects highlighted search area seeds with top 128 scores, yielding M2 = 128, and local feature aggregation generates object-preserved seeds with feature dimension d2 = 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Targeted-feature transformation module uses two 3-layer MLPs with batch normalization (BN) and a ReLU activa- tion layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' For highlighting score prediction, object pre- served sampling uses 3-layer MLP and Sigmoid function for object score prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' In the object localization net- work, we set the voxel side length r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3m, the channel of BEV feature map c = 128, and we generate 4 stacked center maps (n = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' For loss regulation, we set λ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0, λ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0, and λ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Training and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We aggregate the first GTBB of the current tracklet and the previous GTBB as the tem- plate, which is applied with a random offset on the previous GTBB to augment data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' The search areas are augmented 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Comparison with other state-of-the-art approaches on the KITTI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' The instance frame number of each category is shown under category names, and the mean denotes the average results of all categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Bold presents the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Category Car Ped Van Cyclist Mean Frame num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 6424 6088 1248 308 14068 Success P2B [23] 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 BAT [32] 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 LTTR [3] 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 V2B [8] 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 PTTR [34] 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 STNet [9] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 M2-Track [33] 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 OPSNet (ours) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 Precision P2B [23] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 BAT [32] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 LTTR [3] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 V2B [8] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 PTTR [34] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 STNet [9] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 M2-Track [33] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 OPSNet (ours) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 similarly by enlarging the current GT with an offset, for all experiments, this offset sets as 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Additionally, we ap- ply the Adam optimizer [12] with an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='001 and decreased by 5 times after 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We train the OPSNet on a single NVIDIA RTX2080 Ti GPU with a batchsize of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' OPSNet can obtain satisfying results after about 40 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Results Evaluation on KITTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 1, our OPSNet shows a large improvement over SOTA methods, All the methods besides M2-track [33] apply the Siamese-based network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Our OPSNet outperforms the SOTA tracking paradigm M2-track with a large margin of ∼9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4% / ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6% on average success / precision and the SOTA PTTR [34] based on PointNet++ of ∼13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9% / ∼10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Our OP- SNet achieves the highest tracking performance in all four categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 4, we compare the pro- posed OPSNet against PTTR over the car and pedestrian sequences since PTTR adopts relation-aware sampling for object preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' The error gradually accumulates in PTTR tracking since the bad template misleads the search area sampling, but the object-preserved search area seeds generated by object-preserved sampling assist the OPSNet to perform robust tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Evaluation on WOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' To validate the generalization ability of our OPSNet, we use our OPSNet that pre-trained over the KITTI dataset for testing tracking on Waymo Open Dataset (WOD) following [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' According to the point number of the first frame, WOD can be divided into three subsets of dif- ferent tracking levels of difficulty, including easy, medium, Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Comparison with other state-of-the-art approaches on the WOD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Category Vehicle Level Frame num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Easy 67832 Medium 61252 Hard 56647 Mean 185371 Success BAT [32] 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 V2B [8] 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 STNet [9] 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 OPSNet (ours) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 Precision BAT [32] 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 V2B [8] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 STNet [9] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 OPSNet (ours) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 Category Pedestrian Level Frame num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Easy 85280 Medium 82253 Hard 74219 Mean 241752 Success BAT [32] 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 V2B [8] 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 STNet [9] 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 OPSNet (ours) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 Precision BAT [32] 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 V2B [8] 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 STNet [9] 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 OPSNet (ours) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 and hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Even though tracking frames are far more than KITTI has, our OPSNet still shows the generalization abil- ity of the tracking performance on WOD, which outper- forms the baseline BAT [32] with a gain of ∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6% / ∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4% on the category Vehicle, results as Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 2 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Tracking on sparse point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Handling extreme sparse scenes is a great challenge in 3D SOT tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We test our OPSNet on the object of extreme sparse points, which is divided into three object point number intervals including [0, 50), [50, 100), [100, 150).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We compare the tracking re- sult of our OPSNet, PTTR [34], V2B [8], and STNet [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 5 shows, our OPSNet achieves the highest track- ing performance on three intervals with ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8% / ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6%, ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7% / ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7%, ∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7% / ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7% success / precision gain over the V2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Experiment results demonstrate that our OP- SNet effectively preserves sparse objects and handles ex- treme sparse point clouds well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Tracking on long sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Siamese-based methods probably lose the object when tracking on long sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Therefore the performance on long sequences evaluates the robustness of SOT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' PTTR adopts their relation-aware sampling for object preservation, therefore we test our OPSNet and PTTR on sequence length inter- vals [0, 50), [50, 200), [200, +∞) and compute average re- call rate after sampling as the indicator of object preserva- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 6 shows, our OPSNet outperforms the PTTR on three intervals with ∼11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8% / ∼7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7%, ∼10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1% / ∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4%, 6 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Result comparison visualization with PTTR [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We can observe that the predicted BBoxes by OPSNet hold tight to the ground truths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Evaluated mean success / precision of the three extreme sparse point number intervals on the KITTI Car category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Evaluated mean success / precision and recall rate of the three sequence length intervals on the KITTI Car category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' ∼14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0% / ∼12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1% success / precision gain and ∼22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0%, ∼20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6%, ∼29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9% recall rate gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Running speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We evaluate the inference speed of our model on a single RTX2080 Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Our method achieves 23 FPS, including 21 ms for pre-processing point clouds, 21 ms for network forward propagation, and 2 ms for post- processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We also test V2B, and STNet in the default settings run with 20 FPS and 19 FPS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Ablation Studies We conduct ablation studies to evaluate the modules pro- posed in OPSNet, the results are shown with OPE evalua- tion metrics and based on the KITTI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Comparison on sampling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We compare our pro- posed object preserved sampling (OPS) with existing unsu- pervised sampling strategies including relation-aware sam- pling (RAS) [34], random sampling (RS) [23], and furthest point sampling (FPS) [22] on sampling 128 candidates from the 256 search area seeds, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 7 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We compute the recall rates for object points after sampling which demon- strate the effectiveness of our OPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We also replace our OPS with RAS, RS, and FPS in OPSNet, default OPSNet outper- forms clear success / precision gain of ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7% / ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6% on the car category and ∼9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0% / ∼8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8% on the pedestrian over the FPS baseline, comparison shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Ablation studies on object highlighting module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We con- duct ablation experiments to validate the effectiveness of the object highlighting components, respectively, as Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 4 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We report the results of cars and pedestrians track- ing on the KITTI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Only a single cross-correlation approach cannot perform the best tracking (∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9% / ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5% decrease with only consistent cross-correlation and ∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4% / ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0% decrease with only discrepant cross-correlation on Car).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' For analyzing feature-targeted transformation, we simply divide the search area into two subsets without uti- lizing MLPs, so that the cross-correlation cannot be adap- 7 Timeline (Car) Timeline (Pedestrian) Ground Truth OPSNet (ours) PTTRSuccess Precision 85 95 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 80 90 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 75 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 85 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='t8 72 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 70 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 % 80 % 65 75 73 60 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 70 55 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 50 65 Interval [0,50] [50,100] [100,150] Interval [0,50] [50,100] [100,150] Frame Frame 1938 861 494 1938 861 494 Number Number PTTR _V2B STNet OPSNet (ours)PTTR OPSNet 95 95 90 90 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 85 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 85 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 80 80 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 75 75 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 % 70 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 %70 65 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 65 60 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 60 55 55 50 485 50 45 45 [0,50] [50,200] [200,+] [0,50] [50,200] [200,+∞0] Success Precision- Recall Success Precision--RecallFigure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Comparison on sampling strategies: our object-preserved sampling (OPS), relation-aware sampling (RAS) proposed in PTTR [34], random sampling (RS) employed in [23], and furthest points sampling (FPS) proposed in PointNet++ [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Note that red points denote the object and black ones denote the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Tracking performance comparison on different sampling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Shown by the evaluation metrics success / precision on the category car and pedestrian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Method Car Pedestrian RS [23] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 / 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 / 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 FPS [22] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 / 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4 / 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 RAS [34] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 / 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 / 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 OPS (ours) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 / 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 / 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Ablation studies results on object highlighting module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Car Pedestrian w/o object augmentation 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 / 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 / 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 w/o feature-targeted transformation 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 / 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 / 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 only consistent cross-correlation 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 / 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9 / 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 only discrepant cross-correlation 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 / 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 / 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 default setting 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 / 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 / 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 tively conducted (∼7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7% / ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6% decrease without the feature-targeted transformation on Car).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Results demon- strate that object augmentation further improves tracking performance (∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5% / ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8% gain on the category Car).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Ablation studies on object localization network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' We replace our object localization network (OLN) with the VoteNet-based approaches including Region Proposal Net- work (RPN) applied in [23,32] and coarse-to-refine regres- sion (C2R) proposed in [34] to validate the tracking perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' VoteNet regresses the offset from point to center with a score as proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' However, when facing sparse scenes VoteNet is hard to generate high-quality proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Tracking performance comparison on common object locating approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Car Pedestrian RPN [23,32] 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 / 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8 / 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 C2R [34] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5 / 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 / 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='2 OLN (ours) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0 / 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6 / 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='3 Our object localization network does not require generating numerous 3D proposals thus it can handle the above issue and outperforms baseline RPN with ∼9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='0% / ∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5% and ∼16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8% / ∼17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8% increase on the category Car and Pedes- trian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Conclusion In this paper, we propose a novel Object Preserving Siamese Network (OPSNet) for single object tracking on point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' Our OPSNet aims to enhance the object-aware features and extract discriminative features from template and search area with the object highlighting module, main- tain object integrity and drop redundant background points with the object-preserved sampling, and predict accurate BBoxes with the object localization network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' The experimental results (∼9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='4% and ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='5% success gain on KITTI and WOD respectively) demonstrate that our OPSNet achieves high-performance object tracking and possesses generalization ability, indicate that object high- lighting module and object-preserved sampling effectively handle sparse point clouds and assist the object localization network to perform robust tracking on long sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content=' 8 Search Area seeds Object-preserved Sampling Relation-aware Sampling Random Sampling Furthest Point Sampling Mi = 256 M2 = 128 M2 = 128 M2=128 M2 = 128 Points on object:113 Points on object: 99 Points on object: 71 Points on object: 55 Points on object: 47 recall: 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6% recall: 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8% recall: 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7% recall: 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='7% Car Points on object :54 Points on object: 54 Points on object: 45 Points on object: 42 Points on object: 30 ** recall: 100% recall: 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='9% recall: 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='8% recall: 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='6% Pedestrian .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FLT4oBgHgl3EQfWi9I/content/2301.12057v1.pdf'} +page_content='. 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Hu1,2, Y. Luan1, M. E. Scott3, J. Yan4,5, D. G. Mandrus4,5, X. Xu3,6, Z. Fei1,2 + +1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA +2Division of Materials Sciences and Engineering, Ames Laboratory, U.S. DOE, Iowa State +University, Ames, Iowa 50011, USA +3Department of Physics, University of Washington, Seattle, Washington 98195, USA +4Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, +Tennessee 37831, USA +5Department of Materials Science and Engineering, University of Tennessee, Knoxville, +Tennessee 37996, USA +6Department of Materials Science and Engineering, University of Washington, Seattle, +Washington 98195, USA + + +The exciton polariton (EP), a half-light and half-matter quasiparticle, is potentially an +important element for future photonic and quantum technologies1-4. It provides both strong +light-matter interactions and long-distance propagation that is necessary for applications +associated with energy or information transfer. Recently, strongly-coupled cavity EPs at +room temperature have been demonstrated in van der Waals (vdW) materials due to their +strongly-bound excitons5-9. Here we report a nano-optical imaging study of waveguide EPs +in MoSe2, a prototypical vdW semiconductor. The measured propagation length of the EPs +is sensitive to the excitation photon energy and reaches over 12 μm. The polariton wavelength +can be conveniently altered from 600 nm down to 300 nm by controlling the waveguide +thickness. Furthermore, we found an intriguing mode back-bending dispersion close to the +exciton resonance. The observed EPs in vdW semiconductors could be useful in future +nanophotonic circuits operating in the near-infrared to visible spectral regions. + + +In recent years, van der Waals (vdW) materials have emerged as a new material system +supporting various types of polaritons with unique properties10,11. For example, graphene was +discovered to support surface plasmon polaritons with high confinement, long lifetime and gate +tunability10-15. Thin flakes of hexagonal boron nitride were proven to support hyperbolic phonon +polaritons with wavelengths down to a few hundred nanometres10,11,16-18. These unique polaritons +make both materials promising for nanophotonic applications in the terahertz to mid-infrared +frequency regime. Group IVB transition-metal dichalcogenides (TMDs) with chemical formula +MX2 (M = Mo, W; X = S, Se) are vdW semiconductors with sizable bandgaps and strongly +bounded excitons19-22. These excitons can couple with photons to form half-light and half-matter +quasiparticles, namely exciton polaritons (EPs)1-4. Due to the large exciton binding energy, +polaritons in TMDs are expected to be stable and robust at ambient conditions, thus suitable for +technological applications. Indeed, far-field optical studies of TMDs embedded in micro-cavities +captured the spectroscopic signatures of strongly-coupled cavity EPs5-8. Real-space characteristics +(e.g. propagation, confinement, and interference) of the TMD polaritons, on the other hand, have +not been addressed. Very recently, an imaging study of WSe2 with the aperture-type scanning +near-field optical microscope was reported23, where interactions between waveguide photons and +excitons were observed. Nevertheless, the characteristic dispersion relation of the waveguide EPs +was not observed. + + +In this work, we performed nano-optical imaging studies of TMD planar waveguides, +where EPs were formed due to the strong coupling between excitons and waveguide photons24-28. +In order to probe these waveguide EPs, we used a scattering-type scanning near-field optical +microscope (s-SNOM) that is built based on a tapping-mode atomic force microscope (AFM) with +a sharp metallized tip (Fig. 1a). The spatial resolution of the s-SNOM defined by the radius of +curvature of the tip apex is about 25 nm. In addition, the AFM tip is illuminated by a p-polarized +laser beam. The high spatial resolution and p-polarized excitation of the s-SNOM enables effective +dispersion mapping of individual TM waveguide modes. For signal detection, we use a concave +mirror to collect photons back-scattered off the coupled tip-sample system (Fig. 1b) and these +photons are counted by an amplified silicon photodetector. The samples studied here are exfoliated +MoSe2 thin flakes on standard SiO2/Si wafers. In order to cover EPs due to the A excitons (~1.55 +eV) of MoSe2 (Supplementary Fig. S4), we used a continuous-wave Ti:Al2O3 laser that can be +tunable from 1.3 to 1.8 eV. + +In Fig. 1c-g, we show a selected dataset of s-SNOM images taken on a 156-nm-thick +MoSe2 planar waveguide, where we plot the normalized near-field amplitude (s) at various +excitation laser energies (). Within these images, we see clear interference fringes on MoSe2 +parallel to its edge (dashed lines), and these fringes demonstrate a clear energy dependence. First, +we find that the fringe period increases systematically with decreasing E. In addition, the fringe +intensity shows a significant enhancement at lower E. Moreover, we notice that the fringes extend +further into the sample interior as E decreases. For example, at E = 1.35 eV (Fig. 1g and +Supplementary Fig. S8), fringes can be seen 30 m away from the sample edge. + +Based on the above observations, we hypothesize that these fringes are generated due to +the interference between photons collected by the detector from different paths. As illustrated in +Fig. 1b, the collected photons come from two major paths. In the first path (marked with ‘P1’), +incident photons are scattered back directly by the s-SNOM tip. In the second path (marked with +‘P2’), the laser-illuminated tip launches in-plane propagative modes inside the sample. As +discussed in detail below, these in-plane modes correspond to the TM0 waveguide modes. The +waveguide modes propagate radially away from the tip and then get scattered into photons by the +sample edge. Photons collected from paths ‘P1’ and ‘P2’ have a phase delay that scales with the +distance between the tip and the sample edge. Therefore as the tip scans towards the edge of MoSe2, +one expects to see oscillations of photon intensity due to the interference of photons from the two +photon paths. Other possible photon paths play less important roles as discussed in the +Supplementary Information. + The above hypothesis implies a sensitive dependence of the fringes pattern on the +orientation of the sample edge relative to the incident beam. In the configuration described in Fig. +1b and Fig. 2a, the laser beam is in the x-z plane and the sample edge is along the y direction, so +the laser beam is perpendicular to the sample edge (referred to as ‘perpendicular configuration’). +In this configuration, photons collected through path ‘P2’ (Fig. 2a) are mainly from waveguide +modes (marked with ‘w.m.’) propagating along the -x direction (Supplementary Information). +Therefore, the fringe period in the perpendicular configuration (⊥) is expected to be +1 +0 +1 ( +/ +)cos +p +p + + + + + +− +⊥ + + + +− + + , (1) +where p is the wavelength of the waveguide mode, 0 is the excitation laser wavelength, and  ≈ +30 is the incident angle of the laser beam relative to the x-y plane. In another configuration (Fig. +2b), the sample edge is along the x direction and thus the in-plane projection of the incident beam +is parallel to the sample edge (referred to as ‘parallel configuration’). Here, photons in path ‘P2’ + +are mainly those scattered from waveguide modes traveling in an angle of  relative to the y +direction (Fig. 2b), where +1 +0 +sin [( +/ +)cos ] +p + + + + +− += + obtained by matching the boundary condition +(momentum conservation along the edge direction) (Supplementary Information). Therefore, the +fringe period in the parallel configuration (//) is expected to be +1 +// +0 +1/ cos +( +/ +)tan cos +p +p + + + + + + + +− + + + +− + + . (2) +Based on Eqs. 1 and 2, we know that +// + is smaller than ⊥ . Therefore, edge orientation +dependence study provides a convenient way to test our hypothesis about fringe formation. + +Figures 2c,d show near-field amplitude images taken at E = 1.38 eV (corresponding to 0 += 900 nm) in the perpendicular and parallel configurations, respectively. Apparently, the fringes +obtained in the parallel configuration (Fig. 2d) are denser than those in the perpendicular +configuration (Fig. 2c). For the purpose of quantitative comparison, we extracted line profiles +perpendicular to the fringes directly from Fig. 2c,d, and then performed Fourier Transform (FT) +analysis on these fringe profiles to accurately determine the fringe periods. Thus-obtained fringe +profiles are plotted in Fig. 2e,f and the corresponding FT profiles are given in Fig. 2g,h. For +convenience, we set the horizontal axis of the FT profiles to be +// +1/  and 1/ ⊥ for parallel and +perpendicular configurations, respectively. Considering that both +// + and ⊥ are clearly smaller +than 1 m, we only pay attention to the FT peaks above 1 m-1. In this regime, we can locate a +dominant FT peak at 1.64 m-1 for 1/ ⊥ and 2.51 m-1 for +// +1/  . Therefore we have +610 nm +⊥ = + +and +// +398 nm + = +, based on which we can calculate p to be 383 nm and 377 nm for perpendicular +and parallel configurations, respectively (Eqs. 1 and 2). The values of p acquired from the two +configurations are highly consistent with a deviation less than 2%, which validates our hypothesis +and analysis. +Following the above methodology, we can now analyze the s-SNOM imaging data taken +at all other laser energies. Figures 3a plots the fringe profiles taken at excitation energies from 1.35 +to 1.77 eV. Their corresponding FT profiles are shown in Fig. 3b, where we can locate the +dominant peaks (marked with arrows) due to the waveguide mode. By accurately measuring the +FT peak positions, we can extract ⊥ and then calculate p with Eq. 1. In addition, the propagation +length (Lp) of the waveguide mode can be estimated by measuring the linewidths of the FT peaks +(Methods). Thus-obtained Lp, plotted in Fig. 3c as squares, is at least over 12 m at low energies, +currently limited by our device size (Supplementary Information). At higher energies close to or +above the A exciton energy, Lp drops rapidly to 2 m or less. The general trend of the experimental +Lp is consistent with the theoretical estimation (solid curve in Fig. 3c) (Methods). +Based on the extracted p through fringe analyses, we construct the energy (E) - momentum +(qp = 2/p) dispersion relation of the waveguide mode in Fig. 1. The obtained experimental (qp, +E) data points (blue squares) are overlaid on top the calculated dispersion color map (Fig. 3d). As +introduced in the Methods, the bright regions in the color maps represent various +photonic/polaritonic modes existing in the sample/substrate system. For convenience, we use the +free-space photon wavevector k0 = 2/0 as the momentum unit, which leads to vertical dispersions +of photons in air (q = k0) and SiO2 (q = 1.46k0) marked by the green and blue dashed lines in Fig. +3d. Here in the dispersion map, we find a good agreement between experimental data points +(squares) with a confined mode close to q  2.5k0 in the color map. According to mode analysis +(Supplementary Fig. S5), this mode corresponds to the TM0 waveguide mode inside MoSe2. + +To reveal the detailed features of the TM0 mode, we show a zoomed-in view (2.2k0 < q < +2.8k0) of Fig. 3d in Fig. 3e, where a back-bending dispersion of the waveguide mode is clearly +visualized near the A exciton energy (see also the dispersion data of the 110-nm-thick MoSe2 +sample in Supplementary Fig. S6). Such an ‘anomalous’ dispersion is in fact the characteristic +behaviour of the EPs under measurements with fixed excitation energies (imaging experiments +with a continuous-wave laser in our case). The commonly-accepted anti-crossing dispersion of the +EPs, on the other hand, can be obtained by measurements at fixed momenta (e.g. spectroscopic +studies of cavity polaritons at fixed incident angles)1-8. The fixed-energy imaging measurements +intend to determine the polariton momenta (qp) by searching horizontally the dispersion map (e.g. +along horizontal dashed lines in Fig. 3f), while the fixed-momentum spectroscopic experiments +are to locate the polariton energy (Ep) by sweeping vertically the dispersion map (e.g. along the +vertical dashed lines in Fig. 3g). With both methods, one can obtain a series of (qp, Ep) data points +(blue crosses in Fig. 3f,g). The polariton dispersion reflected by these data points demonstrates +either back-bending (Fig. 3f) or anti-crossing (Fig. 3g) features. Note that the back-bending +dispersion also suggests that polaritons are subject to broadening, which introduces finite photonic +spectral weight at the gap between the top and bottom polaritonic branches. The broadening is +mainly due to scatterings of EPs with longitudinal optical phonons that can be strongly suppressed +at cryogenic temperature (Supplementary Fig. S7). Back-bending dispersion has been observed +previously in both plasmon and phonon polaritons29,30. Our experiment proves that the EPs also +share this phenomenon. Based on the back-bended dispersion data points, we estimate a Rabi +splitting energy (ERabi) of ~100 meV (yellow arrow in Fig. 3e), indicating a strong coupling +between excitons and waveguide photons. The ERabi value measured here in bulk MoSe2 appear to +be larger than those of atomic layers of TMDs5-8 and smaller than that of bulk WS29. + +Finally, we performed s-SNOM imaging of MoSe2 waveguides with different thicknesses. +As shown in Fig. 4a,b, we plot the near-field images of two additional MoSe2 waveguides with +thicknesses of 77 and 110 nm, respectively. They are both taken at E = 1.48 eV in the perpendicular +configuration. As described above, we determined the fringe period ( ⊥ ) by extracting fringe +profiles (Fig. 4c) by FT analysis (Fig. 4d). Employing Eq. 1, we obtained p versus waveguide +thickness (Fig. 4e), which shows good consistency with theory (black curve) (Supplementary +Material). From Fig. 4e, we found that p can be altered from 600 to 300 nm by controlling the +waveguide thickness. Note that the observed TM0 polariton mode is cut off in MoSe2 waveguides +with a thickness less than ~70 nm (Fig. 4e). In order to explore polaritons in thinner flakes or even +atomic layers of TMDs, other type of waveguide modes (e.g. TE0 mode23) or other coupling +methods (e.g. micro-cavity coupling1-8) have to be adopted. + +By combining the s-SNOM technique with rigorous theoretical analyses, we uncovered the +real-space characteristics of EPs in MoSe2 waveguides. The observed polaritons have shown a +small wavelength (down to 300 nm) and a long propagation length (up to 12 m or above) under +ambient conditions. These characteristics observed in our first generation devices are comparable +to or even better than surface plasmon polaritons in graphene12-15 and hyperbolic phonon polaritons +in hexagonal boron nitride16-18. Through careful design and engineering, the TMD waveguides +with tailored polaritonic modes could potentially be applied in miniaturized nanophotonic circuits +for information or energy transfer in the near-infrared to visible regions. In addition, it will be +interesting to perform polariton nano-imaging at cryogenic temperatures, where one could possibly +visualize EPs with stronger coupling strength and longer propagation length (Supplementary Fig. +S7). Future studies are also promising to explore new polaritonic characteristics and functionalities +by patterning TMD flakes into nano-resonators or other types of photonic structures (e.g. photonic + +crystals). Our work opens up new avenues for studies of EPs and paves the way for future +applications of TMDs in optoelectronics and nanophotonics. + +References +1. Weisbuch, C. et al. Observation of the coupled exciton–photon mode splitting in a +semiconductor quantum microcavity. Phys. Rev. Lett. 69, 3314–3317 (1992). +2. Gibbs, H.M., Khitrova, G. & Koch, S.W. Exciton-polariton light-semiconductor coupling +effects. Nature Photon. 5, 275-282 (2011). +3. Tassone, F., Bassani, F. & Andreani, L.C. Quantum-well reflectivity and exciton-polariton +dispersion. Phys. Rev. B 45, 6023-6030 (1992). +4. Deng, H., Haug, H. & Yamamoto, Y. Exciton-polariton Bose-Einstein condensation. Rev. Mod. +Phys. 82, 1489-1537 (2010). +5. Liu, X. et al. Strong light-matter coupling in two-dimensional atomic crystals. Nature Photon. +9, 30-34 (2015). +6. Dufferwiel, S. et al. Exciton–polaritons in van der Waals heterostructures embedded in tunable +microcavities. Nature Commun. 6, 8579 (2015). +7. Lundt, N. et al. 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Strong coupling between excitons in J- +aggregates and waveguide modes in thin polymer films. Appl. Phys. Lett. 98, 261103 (2011). +29. Arakawa, E.T., Williams, M.W., Hamm, R.N. & Ritchie, R.H. Effect of damping on surface +plasmon dispersion. Phys. Rev. Lett. 31, 1127-1129 (1973). +30. Schuller, E., Falge, H.J. & Borstel, G. Dispersion curves of surface phonon-polaritons with +backbending. Phys. Lett. A 54, 317-318 (1975). + +Acknowledgements +F.H., Y.L. and Z.F. acknowledge startup support from Iowa State University and Ames Laboratory. +The nano-optical imaging setup was partially supported by the W. M. Keck foundation. The work +at UW was supported by the U.S. DOE Basic Energy Sciences, Materials Sciences and +Engineering Division (DE-SC0008145 and SC0012509). The work at ORNL (JQY and DGM) +was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, +Materials Sciences and Engineering Division. + +Author contributions +Z.F. conceived the ideas and designed the experiments. F.H. carried out the s-SNOM experiments +and collected the data. Z.F., F.H. and Y.L. performed theoretical analyses and modeling of the data. +X.X., M.E.S., J.Y. and D.G.M. synthesized the MoSe2 crystals and fabricated the waveguide +devices. Z.F., X.X., F.H. and Y.L. wrote the paper. + +Additional information +The authors declare no competing financial interests. Reprints and permission information is +available online at http://npg.nature.com/reprintsandpermissions. Correspondence and requests +for materials should be addressed to Z.F. (zfei@iastate.edu). + +Figure Legends + + +Figure 1. Nano-optical imaging of a MoSe2 planar waveguide. a, Schematics of concentric +waveguide modes in MoSe2 launched by the laser-illuminated s-SNOM tip. b, Illustration of the +experimental setup where the incident beam is aligned perpendicular to the sample edge that is +along the y direction. We also sketch here the two major paths (‘P1’ and ‘P2’) where photons can +be collected by the concave mirror. c-g, Selected s-SNOM imaging data of a 156-nm-thick MoSe2 +planar waveguide taken at various laser energies (E). Here we plot the near-field amplitude (s) +normalized to that of the SiO2/Si substrate. The dashed lines mark the sample edge. The scale bars +represent 1 m. + + +a +tip +C +E = 1.77 eV +d +E=1.63eV +MoSe2 +MoSe2 +e +E +=1.48eV +E += +1.41eV +MoSe, +20 +S +concave mirror +b +laserbeam +MoSe2 +1 +MoSe2 +% +tip +Z += 1.35 eV +MoSe2 +SiO2 +Silicon +MoSe2 +Figure 2. Edge-orientation dependence study. a, Illustration of the perpendicular configuration, +where the incident beam (black arrow) is perpendicular to the sample edge. b, Illustration of the +parallel configuration, where the x-y plane projection of the incident beam is parallel to the sample +edge. The labeling ‘w.m.’ in a and b represents waveguide modes. c,d, Near-field amplitude +images of MoSe2 taken at E = 1.38 eV in the perpendicular and parallel configurations, respectively. +The dashed lines mark the sample edge. The scale bars represent 1 m. e,f, Real-space line profiles +extracted perpendicular to the fringes in c and d, respectively. Here d is the distance between the +tip and the sample edge. g,h, Fourier transform (FT) analysis of the real-space profiles in e and f, +respectively. + + +a +Perpendicular +b +Parallel +Tip +dil +MoSe2 +X +w.m. +X +edge +SiO2 +MoSe2 +SiO2 +c +d +MoSe2 +20 +MoSe +e +edge +edge +16 +(ou) +6 +8 +S +S +0 +0 +0 +2 +4 +6 +0 +2 +4 +6 +d (μm) +d (μm) +9 +3 +h +(a.u.) +2 +(a.u.) +FT +0 +0 +1 +2 +3 +0 +1 +2 +3 +1/p (μml) +1/pu (μm-1) +Figure 3. Dispersion analysis. a,b, Real-space fringe profiles and the corresponding FT profiles +of the 156-nm-thick MoSe2 waveguide taken at various excitation energies (1.35 - 1.77 eV) in the +perpendicular configuration. All the profiles are displaced vertically for clarity. The arrows in b +mark the FT peak associated with the measured waveguide mode in MoSe2. c, Propagation length +(Lp) of the measured waveguide mode from both experiment (squares) and theory (curve). d, +Experimental dispersion data points (blue squares) overlaid on the calculated dispersion color map. +e, A zoomed-in version of panel d at the q range from 2.2k0 to 2.8k0. The yellow arrow marks the +Rabi splitting energy. f, Illustration of the fixed-E experiments with horizontal line cuts across the +dispersion map. g, Illustration of the fixed-q experiments with vertical line cuts across the +dispersion map. The blue crosses in f and g mark the positions with maximum photonic spectral +weight along the line cuts. The color maps in d-g plot the imaginary part the p-polarized reflection +coefficient Im(rp) (Methods). + + +a +Fringe profiles +b +FT profiles +C 100 +IA +1.77 eV +1.75 eV +1.72 eV +口 +1.70 eV +1.68 eV +1.65 eV +1 +1.63 eV +1 +1.3 +1.4 +1.5 +1.6 +1.7 +1.8 +(a.u.) +1.61 eV +(n'e). +E (eV) +1.59 eV +d +:1.46ko +1.57 eV +1.8 +1.55 eV +1.7 +1.51 eV +1.48 eV +1.44 eV +E +1.5 +1.41eV +0 +1.38 eV +1.4 +1.35eV +1.3 +- +TMo +0 +2 +4 +6 +1 +2 +3 +1 +1.5 +2 +2.5 +3 +d (μm) +1/p, (μm-1) +q (ko) +e +f +g +1.8 +1.8 +1.8 +1.7 +1.7 +1.7 +> +1.6 +≤1.6 +≤ 1.6 +E +E +E +1.5 +1.5 +1.5 +1.4 +1.4 +1.4 +1,3 +TMo +1.3 +1.3 +2.22 +2.32.42.52.6 +2.72.8 +2.2 +2.3 +2.4 +2.5 +2.6 +2.7 +2.8 +2.22.32 +2.42.52.62.72.8 +q (ko) +q (ko) +q (ko) +Figure 4. Waveguide-thickness dependence study. a,b, Near-field imaging data of MoSe2 planar +waveguides with thicknesses of 77 and 110 nm, respectively. These images were taken at E = 1.48 +eV in the perpendicular configuration. The scale bars represent 1 m. c,d, Real-space fringe +profiles and the corresponding FT profiles of MoSe2 with various thicknesses. e, Thickness +dependence of polariton wavelengths (p) from both experiment (data points) and theory (black +curve) at E = 1.48 eV. + +Methods +Experimental setup. For nano-optical imaging experiments, we used a scattering-type scanning +near-field optical microscope (s-SNOM, Neaspec) that is built based on a tapping-mode atomic +force microscope (AFM) operating with a tapping frequency of about 270 kHz and a tapping +amplitude of about 50 nm. A pseudo-heterodyne interferometric detection module is implemented +in our s-SNOM to extract both the scattering amplitude (s) and phase () of the near-field signal. +In this work, we discuss mainly the s signal that is sufficient for describing all the characteristics +of the EPs. In order to subtract the background signal, we demodulated the near-field signal at the +3rd harmonics of the tapping frequency of the AFM tip. In all the displayed near-field images, we +plotted s normalized to that of the SiO2/Si substrate. For optical excitation, we used a Ti:Al2O3 +laser operating in the continuous-wave mode. The photon energy of the Ti:Al2O3 laser can be +conveniently tunable from 1.3 to 1.8 eV. The samples studied here are MoSe2 planar waveguides +fabricated using the mechanical exfoliation method. The substrates used for these samples are +standard Si wafers with a 300-nm-thick thermal oxide layer on the top. Our s-SNOM experiments +were all performed at ambient conditions. + +Dispersion calculation and analysis. The dispersion color maps shown in Fig. 3d-g were obtained +by evaluating numerically the imaginary part of the reflection coefficients Im(rp) of the multilayer +sample/substrate system. The bright curves shown in the color map correspond to various photonic +and polaritonic modes in the entire system. Considering that the electric field right underneath the +s-SNOM probe is perpendicular to the sample surface, only transverse magnetic (TM) waveguide +modes are excited. Therefore, we only consider p-polarized reflection coefficient rp(q, E) in the +dispersion calculation. By using the transfer matrix method, we numerically calculate Im(rp) of the + +a +b +Thickness=77nm +Thickness=110nm +20 +S +MoSe2 +MoSe2 +C +d +e +600 +3 +E = 1.48 eV +30 +156nm +156nm +500 +(cn: +2 +20 +(nm +a +110 nm +110 nm +400 +F +10 +77 nm +77 nm +0 +300 +0 +1 +0 +2 +4 +6 +0 +1 +2 +3 +0 +100 +200 +300 +d (μm) +1/p(μm1) +Thickness (nm)entire air/MoSe2/SiO2/Si system. The photonic/polaritonic modes appear at the (q, E) positions +where Im(rp) diverges or maximizes. Therefore, we can conveniently estimate numerically the +mode wavelength (p = 2/qp) through the calculated dispersion color maps. In order to fit the +experimental dispersion data points (Fig. 3d), we adopted the experimental ab-plane dielectric +constants of MoSe2 from previous optical measurement (Supplementary S4). The c-axis dielectric +constant (c) of MoSe2 is a fitting parameter, which was set to be 8.3 throughout the entire spectral +range of our experiment (1.3 – 1.8 eV). The good agreement between experimental data points and +calculated dispersion plot validates such an assumption. + +Determining the propagation length of the EPs. In order to determine quantitatively the +propagation length (Lp) of the measured waveguide EPs, we first extract the linewidths (plotted in +Supplementary Fig. S9a) of the FT peaks of the waveguide mode (Fig. 3b) and then determine Lp +using Eq. S11 in the Supplementary Information. Thus-obtained Lp data points are plotted in Fig. +3c. Note that Lp at the lowest energy (E = 1.35 eV) is most likely underestimated due to limited +resolution of the FT profiles originated from the finite size (~ 30 m) of our device (Supplementary +Information). In order to estimate theoretically Lp, we approximate the tip-launched waveguide +mode as cylinder waves with a wave function of +1/2 +( +) +i qx +t +Ax +e + +− +− +. Here, q = q1 + iq2 is the complex +in-plane momentum of the waveguide mode. Therefore, the amplitude of the wave decays as: +p +/(2 +) +1/2 + +x +L +Ax +e +− +− +, where the propagation length Lp equals to (2q2)-1. For an anisotropic material +like MoSe2 with an ab-plane dielectric function of ab = 1 + i2 (Supplementary Fig. S4) and an +c-axis dielectric constant of c  8.3, we have +2 +2 +0 +( +/ +) +c +c +ab +z +q +k +k + + + += +− +. Based on this equation, we +have an analytic formula of + + +1 +2 +2 +2 +2 +2 +1 1 +2 +1 +0 +1 +/ (2 +) 1 +1 ( +/ +)( +/ +1) +p +c +L +q +k +q + + + + + +− + + += +− +− +− + + +. By adopting the +q1=2/p data of the TM0 waveguide mode (Fig. 3), we can calculate Lp using the above formula. +The calculated result is plotted in Fig. 3c as the solid curve. The general trend of the theoretical +curve is consistent with the experimental data (squares in Fig. 3c). + +Data availability. The data that support the plots within this paper and other findings of this study +are available from the corresponding author upon reasonable request. + +Supplementary Information for +“Imaging exciton-polariton transport in MoSe2 waveguides” + +F. Hu1,2, Y. Luan1, M. E. Scott3, J. Yan4,5, D. G. Mandrus4,5, X. Xu3,6, Z. Fei1,2* + +1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA +2Division of Materials Sciences and Engineering, Ames Laboratory, U.S. DOE, Iowa State +University, Ames, Iowa 50011, USA +3Department of Physics, University of Washington, Seattle, Washington 98195, USA +4Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, +Tennessee 37831, USA +5Department of Materials Science and Engineering, University of Tennessee, Knoxville, +Tennessee 37996, USA +6Department of Materials Science and Engineering, University of Washington, Seattle, +Washington 98195, USA + + +*Correspondence to: zfei@iastate.edu. + + +List of contents + + +1. Discussion and analysis about fringe formation + + +2. Field-distribution calculation confirming TM0 waveguide modes + + +3. Polariton dispersion of the 110-nm-thick MoSe2 sample + + +4. Effects of exciton linewidth & temperature on polariton dispersion + + +5. Propagation length of the waveguide exciton polaritons + + + + + + + + + + + +1. Discussion and analysis about fringe formation +1.1 Photon path ‘P3’ +In the main text, we discuss two major photon paths ('P1’ and ‘P2’) for both perpendicular +and parallel configurations (Fig. 1, Fig. 2 and Supplementary Fig. S1). In path ‘P1’, photons are +scattered directly by the tip back to the detector. In path ‘P2’, photons first transfer into propagative +waveguide exciton polaritons (EPs) and then scatter to photons when reaching the edge of the +sample. In addition to the two photon paths, there is another possible photon path ‘P3’ that could +contribute to the signal collection process. In photon path ‘P3’, the photons first transfer into EPs +and then reflected backward to the tip after reaching the sample edge. The reflected EP modes can +be scattered back to detector by the AFM tip. This path has been extensively discussed in previous +nano-infrared imaging studies of graphene plasmons1 and hexagonal boron nitride (hBN) +polaritons2. In the current work, ‘P3’ plays a less significant role compared to ‘P2’ for the +following two reasons. First, the momenta (q) of the modes involved in our experiments is much +closer to the free-space photon wavevector (k0). As a result, only a small portion of the EP modes +are reflected back by the sample edge. In addition, the round-trip propagation of the EP modes in + +‘P3’ also suffers more damping compared to ‘P2’ due to the longer traveling distance. As a result, +we didn’t see clear experimental evidence of fringes due to the interference between photons from +paths ‘P1’ and ‘P3’. + +1.2 Edge excitation +The photon paths ‘P2’ and ‘P3’ discussed above all assume the SNOM tip as the only +launcher of the EPs. In fact, the sample edge can also launch polaritons when illuminated by the +laser beam. Nevertheless, due to the small size of the focused laser beam (~ 2 m), edge launching +is only possible when the tip-edge distance (d) is very small (d < 1 m). Therefore, the fringes that +are over 1 m away from the edges are solely due to the effects of tip scattering or launching (‘P1’ +and ‘P2’). When performing FT analysis (Fig. 3b) on the fringe profiles (Fig. 3a), we cut off a +short part of the fringe profiles (~ 1 m close to the edge) in order to avoid the complications +involving edge launching processes. + + +Supplementary Figure S1. Illustration of various paths from which the photons can be collected +by the detectors. + +1.3 Perpendicular configuration +In Fig. 2a and related text in the manuscript, we discuss the perpendicular configuration of +the experimental setup, where the laser beam is perpendicular to the edge of sample. Based on this +configuration, we obtain an equation (Eq. 1 in the main text) that describes the relation between +the fringe period ( ⊥ ) and the wavelength of the EPs (p). Now we introduce in detail how this +equation is derived. As discussed above, the fringes observed in our experiments are mainly due +to the interference between collected photons from paths ‘P1’ and ‘P2’. In the latter path (‘P2’), +the tip-launched EPs propagate radially away from the tip, so they could in principle be scattered +into free-space photons from any positions along the sample edge. Nevertheless, the concave +mirror with a relatively small collection angle of less than 14 (beam size < 1 inch, focusing length +≈ 2 inches) collects mainly the scattered photons from waveguide EPs propagating along the +direction perpendicular to the edge. +In order to elucidate that, we illustrate in Supplementary Fig. S2 the process of edge +scattering where the incident laser beam is in the x-z plane and the sample edge is along the y +direction (perpendicular configuration). Here we assume, tip-launched EPs with a momentum of +qp propagate to the edge with a random angle () with respect to the x-axis. At the edge, the EPs +are scattered into free-space photons with a wavevector of k0. The angle between the scattered + +P1 +Laser beam +P3 +P2 +Tip +waveguide mode +Sample +SiO2 +Substratephotons and the optical axis of the concave mirror is . As discussed above,  should be within +the collection angle of the concave mirror:  < 14. Therefore, photon wavevector along the y +direction (ky) should satisfy ky < k0sin() < k0sin(14) = 0.242k0. In the edge scattering process, +momentum is conserved along the edge direction (y direction), so we have ky = qpsin() < 0.242k0. +Considering that the EPs of MoSe2 studied in the current work are more confined in space than +photon modes in SiO2: qp > 1.46k0, we have  < 9.8o. Since the above upper bound limit analysis +is far from stringent, the actual average  will be much closer to zero with a more careful analysis. +Furthermore, EPs with finite  will be subjected to higher propagation loss and larger reflection +coefficient than those with normal incidence towards the edge ( = 0). Therefore, we assume  = + = 0o in the following analysis. The error bar of the extracted p based on such an assumption is +less than 1 - cos(9.8o)  1.4%. +Under the assumption of  =  = 0o, we estimate the phase difference () between photons +collected from paths ‘P1’ and ‘P2’ to be: +( +) +0 +0 +cos +2 +1/ +cos +/ +, +p +p +q d +k d +d + + + +  + = +− += +− + (S1) +where d is the distance between the tip and the sample edge, p and 0 are the wavelengths of +waveguide EPs and free-space photons, respectively. If d changes by a distance that equals to the +fringe period ( ⊥ ),  will change by 2. Therefore we have +( +) +1 +0 +1 +/ +cos +p +p + + + + + +− +⊥ + + += +− + + . (S2) + + +Supplementary Figure S2. Illustration of the perpendicular configuration of the nano-optical +experimental setup. + +1.4 Parallel configuration +Figure S3 illustrates the parallel configuration of the experimental setup, where the incident +laser beam is kept in the x-z plane and the sample edge is along the x direction. Similar to the +perpendicular configuration, the fringes are formed due to the phase difference between photons +collected by the detector from paths ‘P1’ and ‘P2’. In path ‘P2’, tip-launched EPs with an in-plane +momentum of qp propagate toward the edge with an angle of  with respect to the y axis and get +scattered into free-space photons (k0) when reaching the sample edge. Again, the angle  between +scattered photons and the optical axis should be less than the collection angle of the concave mirror: + < 14. From Supplementary Fig. S3, we find that the x component of the wavevector of the +scattered photons (kx) should satisfy: k0cos( +14) < kx < k0cos( −14), where  =  is the +angle between the optical axis of the concave mirror and the x-y plane. Therefore, we have 0.72k0 + +Laserbeam +tip +opticalaxis +d +α +ko +P2 +0 +qp +SiO2 +edge +Z. +MoSe2 +x< kx < 0.96k0. Considering the momentum conservation along the direction of the scattering edge +(x direction): kx = qpsin(), so we have 0.72k0/qp < sin() < 0.96k0/qp. Based on this inequality, we +have 16.8<    in the case of qp ~ 2.5k0 (e.g. TM0 mode in the 156-nm-thick MoSe2 sample). +The angle deviation is quite small, so we assume  = 0 for convenience. In this case, kx = qpsin() += k0cos()  = 20.3 when qp ~ 2.5k0. The uncertainty of the estimated p due to such an +assumption is estimated to be less than cos(16.8) - cos(20.3)  1.9%. +Under the assumption of  = 0, we estimate the phase difference () between photons +collected from paths ‘P1’ and ‘P2’ to be: +( +) +0 +0 +/ cos +tan cos +2 +1/ cos / +tan cos +/ +p +p +q d +k d +d + + + + +  + +  + = +− += +− + (S3) +where +1 +0 +sin [( +/ +)cos ] +p + + + + +− += + obtained from kx = qpsin(). If d changes by a distance that equals +to the fringe period ( +// + ),  will change by 2. Therefore we have +1 +// +0 +1/ cos +( +/ +)tan +cos +p +p + + + + + + + +− + + + +− + + + (S4) + + +Supplementary Figure S3. Illustration of the parallel configuration of the nano-optical +experimental setup. + + + +Supplementary Figure S4. In-plane (ab-plane) dielectric function of MoSe2 for calculations of +the EPs. The black and red curves are real and imaginary parts of the dielectric function, +respectively. The solid lines are experimental data adopted from Ref. 3, which were measured in +bulk MoSe2 at room temperature. The dashed lines are modeled results that we constructed by + +Laserbeam +P1 +tip +optical axis +α +P2 +ko +MoSe2 +A +edge +Z +y +X +SiO2A +30 +B +20 +10 +1.3 +1.4 +1.5 +1.6 +1.7 +1.8 +Energy (eV)using single Lorentzian oscillator for both A and B excitons (see detailed discussions in Section 4 +of the Supplementary Information). The c-axis permittivity is set to be 8.3 throughout our energy +region that produces a good fit to experimental dispersion data points (Fig. 3d in the main text). + +2. Field-distribution calculation confirming TM0 waveguide modes +In order to confirm the nature of the measured waveguide EPs, we performed field +distribution calculations by matching boundary conditions in Maxwell’s wave equations. The +waveguide structure is plotted in Supplementary Fig. S5a. The calculations were performed +considering 156-nm-thick MoSe2 excited by a laser beam with a photon energy of 1.38 eV, but the +general results apply also to other thicknesses (110 nm and 77 nm) or other laser energies. The ab- +plane optical constants for MoSe2 used in the calculation are adopted from Ref. 3 (Supplementary +Fig. S4) and the c-axis permittivity is set to be 8.3 that is determined through dispersion fitting +(Fig. 3d in the main text). The results are shown in Supplementary Fig. S5b and S5c, where we +plot the y-component magnetic field (Hy) and z-component electric field (Ez) at various z positions, +respectively. Note that the waveguide EPs is set to be propagating along the -x direction. From the +field distribution, we know that the measured waveguide EPs correspond to TM0 mode. + + +Supplementary Figure S5. Field-distribution calculation confirming TM0 waveguide modes. +a, Illustration of the MoSe2 waveguide sandwiched by air and SiO2. The waveguide EPs are +propagating along the -x direction. b,c, Calculated Hy(z) and Ez(z) of the waveguide EPs in the +156-nm-thick MoSe2 sample (Figs. 1 and 2 in the main text). The blue dashed lines here mark the +top and bottom surfaces of the MoSe2 layer. + +3. Polariton dispersion of the 110-nm-thick MoSe2 sample + +In Fig. 3d,e of the main text, the dispersion data points (blue squares) were obtained from +near-field amplitude images of the 156-nm-thick MoSe2 waveguide. Here in Supplementary Fig. +S6, we show additional dispersion data (blue squares) measured from the 110-nm-thick MoSe2 +waveguide. Again, these data points are in good agreement with the theoretical dispersion color +map and they also show the back-bending feature close to the A exciton energy. We also notice +that the polariton mode of the 110-nm-thick MoSe2 sample shift to the low momentum region (~ +2k0) compared to that of the 156-nm-thick one (~ 2.5k0), indicating reduced mode confinement or +increased mode wavelength (Fig. 4e in the main text). + + +a +b +c +> +air +x +air +air +propagation direction +MoSe2 +MoSe2 +N +MoSe2 +N +SiO2 +SiO2 +SiO2 +0 +1 +2 +-1 +0 +1 +Hy (a.u.) +Ez (a.u.) +Supplementary Figure S6. Experimental dispersion data of waveguide EPs overlaid on the +calculated dispersion color map of a 110-nm-thick MoSe2 waveguide. + +4. Effects of exciton linewidth & temperature on polariton dispersion +As discussed in the main text, the back-bending dispersion is a characteristic behavior of +polaritons subject to broadening when measured by experiments under continuous-wave excitation +(fixed excitation energies). Here we wish to explore the effects of the exciton broadening or +linewidth on back-bending. For that purpose, we calculated the dispersion diagrams of 156-nm- +thick MoSe2 waveguide using a modeled ab-plane dielectric function with Lorentzian oscillators: +A +B +1 +2 +2 +2 +2 +2 +A +A +B +B +( ) +( ) +( ) +E +E +i +E +E +E +i +E +E +E +i +E + + + + + + += ++ += ++ ++ +− +−  +− +−  +. (S5) +To match the experimental optical constants from literature3 (Supplementary Fig. S4), the +oscillator parameters are set to be +21 +  = +, +2 +A +2 eV + = +, +A +1.55 eV +E = + (A exciton energy), +A +0.1 eV + = + (A exciton linewidth), +2 +B +1.53 eV + = + , +B +1.85 eV +E = + (B exciton energy), and +B +0.12 eV + = + (B exciton linewidth). + +According to Ref. 4, the exciton linewidth (broadening) in MoSe2 comes from both non- +thermal and thermal processes. The non-thermal processes include defect and impurity scattering +or electron-electron scattering. The thermal process is mainly due to interactions with longitudinal +optical phonons. The thermal contribution can be strongly suppressed at low temperature (LT). As +a result, exciton linewidth at temperature less than 100 K could be less than half of that at room +temperature (RT). Therefore, we adopted +A +0.05 eV + + + as a rough estimation of the linewidths of +A excitons of bulk MoSe2 at low temperature, which is consistent with the experimental results +given in Ref. 4. + + +ko +1.46ko +1.8 +1.7 +(eV) +1.6 +E +1.5 +0 +1.4 +TMo +1.3 +1 +1.5 +2 +2.5 +3 +q (ko) +Supplementary Figure S7. a,b, Calculated dispersion color maps of the 156-nm-thick MoSe2 +waveguide with modeled ab-plane optical constants derived from a Lorentzian oscillator with +exciton linewidths of  =0.1 eV and 0.05 eV, corresponding to room temperature (RT) and low +temperature (LT), respectively. c,d, Zoomed-in version of panels a and b respectively at the q +range from 2.2k0 to 2.8k0. The blue crosses in c and d mark the positions with maximum photonic +spectral weight along the horizontal line cuts. + + +The calculated dispersion diagrams with the modeled optical constants (Eq. S5) are given +in Supplementary Fig. S7, where effects of exciton linewidth on polariton dispersion are clearly +demonstrated. Here we compare the case of +A +0.1 eV +  + (corresponding to RT, Supplementary +Fig. S7a and S7c) with that of +A +0.05 eV + + +(corresponding to LT, Supplementary Fig. S7b and +S7d). In both cases, back-bending dispersion close to the A exciton energy is clearly seen, but the +polariton mode with +A +0.05 eV + + +is back-bended more dramatically, indicating stronger light- +exciton coupling. In addition, the momentum broadening (q) of the EP mode away from the +exciton energy becomes narrower at +A + = 0.05 eV suggesting that polaritons have smaller damping +or longer propagation length. + + +5. Propagation length of the waveguide exciton polaritons +In Supplementary Fig. S8a and S8b, we plot the full-scale image and fringe profile of the +156-nm-thick MoSe2 waveguide taken at E = 1.35 eV. Both the images (Fig. 1) and profiles (Fig. +3a) shown in the main text are truncated intentionally to fit the diagram. Here in Supplementary +Fig. S8a and S8b, fringes or oscillations are seen 30 m away from the sample edge, indicating a +long propagation length (Lp) of these modes. In order to determine quantitatively the propagation + +TA = 0.1 eV (RT) +TA = 0.05 eV (LT) +a +b +1.8 +1.8 +1.7 +1.7 +(eV) +1.6 +(eV) +1.6 +E +E +1.5 +1.5 +1.4 +1.4 +1.3 +1.3 +0.5 +1.5 +2.5 +3 +3.5 +0.5 +1 +1.5 +2.5 +3 +3.5 +q (ko) +q (ko) +TA = 0.1eV (RT) +TA = 0.05 eV (LT) +C +d +0 +1.8 +1.8 +1.7 +1.7 +1.6 +(eV) +1.6 +E +E +1.5 +1.5 +1.4 +1.4 +1.3 +1.3 +2.2 +2.3 +2.4 +2.5 +2.6 +2.7 +2.8 +2.2 +2.3 +2.4 +2.5 +2.6 +2.7 +2.8 +q (ko) +q (ko)length (Lp) of the measured waveguide EPs, we first extract the linewidths of the FT peaks shown +in Fig. 3b of the main text. The FT linewidth data, plotted in Supplementary Fig. S9a, shows an +increase with energy in the spectral range. Based on the linewidth data, we can then determine Lp +using Eq. S11 (see discussions below). Thus-obtained Lp data points are plotted in Fig. 3c of the +main text. Note that the resolution (~1/30 m-1) of the FT profiles (Fig. 3b in the main text) is +limited by the sample size (~30 m), so the linewidth data point in Supplementary Fig. S9a at the +lowest photon energy (E = 1.35 eV) is most likely overestimated due to the resolution limit. +Therefore, the extracted Lp data point at this energy is possibly underestimated compared to the +realistic values. In Supplementary Fig. S9b, we also plot the number of propagation cycles, namely +propagate length (Lp) versus polariton wavelength (p), at various photon energies. Here one can +see that the waveguide EPs can propagate over 30 cycles before losing ~63% (1-1/e) of the +polariton energy. + + +Supplementary Figure S8. a, Near-field amplitude image (s) of the 156-nm-thick MoSe2 planar +waveguide taken at E = 1.35 eV in the perpendicular configuration. The near-field amplitude is +normalized to that of the SiO2/Si substrate. The white dashed line marks the edge of the sample. +The scale bar represents 1 m. b, Line profile taken from a perpendicular to fringes along the +green dashed line. + + +Supplementary Figure S9. a, Linewidths of the FT peaks measured from the FT profiles in Fig. +3b of the main text. b, Estimated number of propagation cycles (Lp/p) of polaritons at various +photon energies. Vertical dashed lines mark the A exciton energy. + +a +20 +SiO +MoSe +b +15 +10 +S +5 +0 +5 +10 +15 +20 +25 +30 +d (um)a +b +40 +1 +A +- +TAI +1 +0.6 + Propagation cycle +口 +口 +口 +30 +.wm) +0.4 +吕 +口口 +Linewidth +口 +口 +20 +口 +口 +口 +口 +0.2 +口 +10 +口 +口 +口 +口 +口 +0.0 +1 +. +0 +1 +1 +1.3 +1.4 +1.5 +1.6 +1.7 +1.8 +1.3 +1.4 +1.5 +1.6 +1.7 +1.8 +E (eV) +E (eV) +Now we discuss how we convert the linewidths (W) into propagation lengths (Lp) in the +case of perpendicular configuration. As discussed in the main text and also in Section 1 of the +Supplementary Information, the total optical signal (Etot) responsible for the observed interference +fringes comes from photon paths ‘P1’ (E1) and ‘P2’ (E2): Etot = E1 + E2. In path ‘P1’, photons are +directly back-scattered by the tip to the detector. In path ‘P2’, photons first transfer into the +waveguide EPs and then scattered back to photons by the sample edge. The momentum of the EP +mode can be written as: qp = q1 + iq2, where q1 = 2/p and q2 = 1/(2Lp). Throughout the +measurement, tip is fixed and the sample is scanning, so both the amplitude and phase of E2 is +dependent on the tip-edge distance (d). Therefore, we can write the amplitude of the total optical +signal as: + +( ) +tot +1 +1 +2 +2 +| +( )| | +( )| | +( ) +| +i +d +d +d +A +A d e  += ++ += ++ +E +E +E +(x > 0) . (S6) +Here A1 is the amplitude of E1, +2 +1/2 +2 +2 +( ) +d +q +A d +B d +e +− +− += + is the amplitude of E2, + + +1 +0 +0 +( ) +cos( ) +d +q +k +d + + += +− ++ is the relative phase difference between E2 and E1 in the +perpendicular configuration (Eq. S1). For simplicity, we define +1 +0 cos( ) +q +k + + = +− + below. The +constant A1 describes the efficiency of photon back-scattering in ‘P1’. The constant B2 describes +the conversion efficiency from photons to EPs (via tip) times the conversion efficiency from EPs +to photons (via edge) in ‘P2’. By substituting the formula of +2( ) +A d and +( ) +d + + above into Eq. S6, +we have + +0 +2 +2 +2 +0 +( +) +( +) +2 +2 +1 +1/2 +1/2 +1 +2 +1 +2 +1 +2 +to +2 +t( ) +q +q +i +d +i +d +d +d +d +q +d +A +B d e +A B d +e +e +A B d +e +e +− +− +− + + +− + + +− +− +− += ++ ++ ++ +E +. (S7) +Considering the low conversion rates between photons and EPs, and also the exponential decaying +of the EPs with d, we are safe to assume +2 +1/2 +2 +2 +1 +q d +A +B d +e +A +− +− += + +, so Eq. S7 can be written +approximately as: + +0 +0 +2 +( +) +( +) +tot +1/2 +1 +2 +1 +( ) +2 +i +d +i +d +q d +d +A +B d +e +e +e + + +− + + +− +− + + + ++ ++ + + +E + . (S8) + +By performing Fourier transformation (FT) on Eq. S8, we have + + + + + +0 +1/2 +1/2 +t +2 +2 +2 +ot( ) +( +) +( +) +k +q +i +q +k +i +e +k +− +− +−  + + +− ++ +− ++ ++  +E +. (S9) +Equation S9 indicates that there should be two peaks at k =  when performing FT analysis of +the fringe profiles. They are identical in shape and well apart from each other (q2 <<  ), so we +only need to consider one peak: + +1/4 +2 +2 +2 +tot( ) +( +) +k +q +k +− + + + + + +− + ++ +E + . (S10) +The full width at half maximum of the peak given by Eq. S10 is +2 +2 15 +k +q + += +. Note that the unit +of the FT profiles in Figs. 2 and 3 of the main text is 1/⊥ instead of k = 2/ ⊥, so the measured +linewidth (W) should be scaled by a factor of 2, namely +2 +15 +/ +W +q + += +. Therefore, we have: + +1 +2 +(2 +) +15 / (2 +) +P +L +q +W + +− += += + . (S11) +Based on Eq. S11, we converted the measured FT linewidths (Supplementary Fig. S9a) into the +propagation lengths at various excitation energies (squares in Fig. 3c). + +Supplementary References +31. Fei, Z. et al. Gate-tuning of graphene plasmons revealed by infrared nano-imaging. Nature 487, + +82-85 (2012). +32. Dai, S. et al. Tunable phonon polaritons in atomically thin van der Waals crystals of boron +nitride. Science 343, 1125-1129 (2014). +33. Beal, A.R. & Hughes, H.P. Kramers-Kronig analysis of the reflectivity spectra of 2H-MoS2, +2H-MoSe2 and 2H-MoTe2, J. Phys. C: Solid State Phys. 12, 881-890 (1979). +34. Arora, A., Nogajewski, K., Molas, M., Koperski, M. & Potemski, M. Exciton band structure +in layered MoSe2: from a monolayer to the bulk limit. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Scott3, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Yan4,5, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Mandrus4,5, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Xu3,6, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Fei1,2 1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA 2Division of Materials Sciences and Engineering, Ames Laboratory, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' DOE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Iowa State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Ames,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Iowa 50011,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' USA 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Seattle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Washington 98195,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' USA 4Materials Science and Technology Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Oak Ridge National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Oak Ridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Tennessee 37831,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' USA 5Department of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' University of Tennessee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Knoxville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Tennessee 37996,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' USA 6Department of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Seattle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Washington 98195,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' USA The exciton polariton (EP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' a half-light and half-matter quasiparticle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' is potentially an important element for future photonic and quantum technologies1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' It provides both strong light-matter interactions and long-distance propagation that is necessary for applications associated with energy or information transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Recently, strongly-coupled cavity EPs at room temperature have been demonstrated in van der Waals (vdW) materials due to their strongly-bound excitons5-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Here we report a nano-optical imaging study of waveguide EPs in MoSe2, a prototypical vdW semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The measured propagation length of the EPs is sensitive to the excitation photon energy and reaches over 12 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The polariton wavelength can be conveniently altered from 600 nm down to 300 nm by controlling the waveguide thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Furthermore, we found an intriguing mode back-bending dispersion close to the exciton resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The observed EPs in vdW semiconductors could be useful in future nanophotonic circuits operating in the near-infrared to visible spectral regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In recent years, van der Waals (vdW) materials have emerged as a new material system supporting various types of polaritons with unique properties10,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' For example, graphene was discovered to support surface plasmon polaritons with high confinement, long lifetime and gate tunability10-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Thin flakes of hexagonal boron nitride were proven to support hyperbolic phonon polaritons with wavelengths down to a few hundred nanometres10,11,16-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' These unique polaritons make both materials promising for nanophotonic applications in the terahertz to mid-infrared frequency regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Group IVB transition-metal dichalcogenides (TMDs) with chemical formula MX2 (M = Mo, W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' X = S, Se) are vdW semiconductors with sizable bandgaps and strongly bounded excitons19-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' These excitons can couple with photons to form half-light and half-matter quasiparticles, namely exciton polaritons (EPs)1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Due to the large exciton binding energy, polaritons in TMDs are expected to be stable and robust at ambient conditions, thus suitable for technological applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Indeed, far-field optical studies of TMDs embedded in micro-cavities captured the spectroscopic signatures of strongly-coupled cavity EPs5-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Real-space characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' propagation, confinement, and interference) of the TMD polaritons, on the other hand, have not been addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Very recently, an imaging study of WSe2 with the aperture-type scanning near-field optical microscope was reported23, where interactions between waveguide photons and excitons were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Nevertheless, the characteristic dispersion relation of the waveguide EPs was not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In this work, we performed nano-optical imaging studies of TMD planar waveguides, where EPs were formed due to the strong coupling between excitons and waveguide photons24-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In order to probe these waveguide EPs, we used a scattering-type scanning near-field optical microscope (s-SNOM) that is built based on a tapping-mode atomic force microscope (AFM) with a sharp metallized tip (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The spatial resolution of the s-SNOM defined by the radius of curvature of the tip apex is about 25 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In addition, the AFM tip is illuminated by a p-polarized laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The high spatial resolution and p-polarized excitation of the s-SNOM enables effective dispersion mapping of individual TM waveguide modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' For signal detection, we use a concave mirror to collect photons back-scattered off the coupled tip-sample system (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1b) and these photons are counted by an amplified silicon photodetector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The samples studied here are exfoliated MoSe2 thin flakes on standard SiO2/Si wafers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In order to cover EPs due to the A excitons (~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='55 eV) of MoSe2 (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S4), we used a continuous-wave Ti:Al2O3 laser that can be tunable from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1c-g, we show a selected dataset of s-SNOM images taken on a 156-nm-thick MoSe2 planar waveguide, where we plot the normalized near-field amplitude (s) at various excitation laser energies (\uf045).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Within these images, we see clear interference fringes on MoSe2 parallel to its edge (dashed lines), and these fringes demonstrate a clear energy dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' First, we find that the fringe period increases systematically with decreasing E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In addition, the fringe intensity shows a significant enhancement at lower E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Moreover, we notice that the fringes extend further into the sample interior as E decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' For example, at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='35 eV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1g and Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S8), fringes can be seen 30 \uf06dm away from the sample edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Based on the above observations, we hypothesize that these fringes are generated due to the interference between photons collected by the detector from different paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1b, the collected photons come from two major paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In the first path (marked with ‘P1’), incident photons are scattered back directly by the s-SNOM tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In the second path (marked with ‘P2’), the laser-illuminated tip launches in-plane propagative modes inside the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' As discussed in detail below, these in-plane modes correspond to the TM0 waveguide modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The waveguide modes propagate radially away from the tip and then get scattered into photons by the sample edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Photons collected from paths ‘P1’ and ‘P2’ have a phase delay that scales with the distance between the tip and the sample edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore as the tip scans towards the edge of MoSe2, one expects to see oscillations of photon intensity due to the interference of photons from the two photon paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Other possible photon paths play less important roles as discussed in the Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The above hypothesis implies a sensitive dependence of the fringes pattern on the orientation of the sample edge relative to the incident beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In the configuration described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 2a, the laser beam is in the x-z plane and the sample edge is along the y direction, so the laser beam is perpendicular to the sample edge (referred to as ‘perpendicular configuration’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In this configuration, photons collected through path ‘P2’ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 2a) are mainly from waveguide modes (marked with ‘w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='m.’) propagating along the -x direction (Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore, the fringe period in the perpendicular configuration (\uf072⊥) is expected to be 1 0 1 ( / )cos p p \uf072 \uf06c \uf06c \uf06c \uf061 − ⊥ \uf0e9 \uf0f9 \uf0bb − \uf0eb \uf0fb , (1) where \uf06cp is the wavelength of the waveguide mode, \uf06c0 is the excitation laser wavelength, and \uf061 ≈ 30\uf0b0 is the incident angle of the laser beam relative to the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In another configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 2b), the sample edge is along the x direction and thus the in-plane projection of the incident beam is parallel to the sample edge (referred to as ‘parallel configuration’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Here, photons in path ‘P2’ are mainly those scattered from waveguide modes traveling in an angle of \uf066 relative to the y direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 2b), where 1 0 sin [( / )cos ] p \uf066 \uf06c \uf06c \uf061 − = obtained by matching the boundary condition (momentum conservation along the edge direction) (Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore, the fringe period in the parallel configuration (\uf072//) is expected to be 1 // 0 1/ cos ( / )tan cos p p \uf072 \uf06c \uf066 \uf06c \uf06c \uf066 \uf061 − \uf0e9 \uf0f9 \uf0bb − \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' (2) Based on Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1 and 2, we know that // \uf072 is smaller than \uf072⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore, edge orientation dependence study provides a convenient way to test our hypothesis about fringe formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Figures 2c,d show near-field amplitude images taken at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='38 eV (corresponding to \uf06c0 = 900 nm) in the perpendicular and parallel configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Apparently, the fringes obtained in the parallel configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 2d) are denser than those in the perpendicular configuration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' For the purpose of quantitative comparison, we extracted line profiles perpendicular to the fringes directly from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 2c,d, and then performed Fourier Transform (FT) analysis on these fringe profiles to accurately determine the fringe periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Thus-obtained fringe profiles are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 2e,f and the corresponding FT profiles are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 2g,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' For convenience, we set the horizontal axis of the FT profiles to be // 1/ \uf072 and 1/ \uf072⊥ for parallel and perpendicular configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Considering that both // \uf072 and \uf072⊥ are clearly smaller than 1 \uf06dm, we only pay attention to the FT peaks above 1 \uf06dm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In this regime, we can locate a dominant FT peak at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='64 \uf06dm-1 for 1/ \uf072⊥ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='51 \uf06dm-1 for // 1/ \uf072 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore we have 610 nm \uf072⊥ = and // 398 nm \uf072 = , based on which we can calculate \uf06cp to be 383 nm and 377 nm for perpendicular and parallel configurations, respectively (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The values of \uf06cp acquired from the two configurations are highly consistent with a deviation less than 2%, which validates our hypothesis and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Following the above methodology, we can now analyze the s-SNOM imaging data taken at all other laser energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Figures 3a plots the fringe profiles taken at excitation energies from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='35 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='77 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Their corresponding FT profiles are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3b, where we can locate the dominant peaks (marked with arrows) due to the waveguide mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' By accurately measuring the FT peak positions, we can extract \uf072⊥ and then calculate \uf06cp with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In addition, the propagation length (Lp) of the waveguide mode can be estimated by measuring the linewidths of the FT peaks (Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Thus-obtained Lp, plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3c as squares, is at least over 12 \uf06dm at low energies, currently limited by our device size (Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' At higher energies close to or above the A exciton energy, Lp drops rapidly to 2 \uf06dm or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The general trend of the experimental Lp is consistent with the theoretical estimation (solid curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3c) (Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Based on the extracted \uf06cp through fringe analyses, we construct the energy (E) - momentum (qp = 2\uf070/\uf06cp) dispersion relation of the waveguide mode in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The obtained experimental (qp, E) data points (blue squares) are overlaid on top the calculated dispersion color map (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' As introduced in the Methods, the bright regions in the color maps represent various photonic/polaritonic modes existing in the sample/substrate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' For convenience, we use the free-space photon wavevector k0 = 2\uf070/\uf06c0 as the momentum unit, which leads to vertical dispersions of photons in air (q = k0) and SiO2 (q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='46k0) marked by the green and blue dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Here in the dispersion map, we find a good agreement between experimental data points (squares) with a confined mode close to q \uf0bb 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5k0 in the color map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' According to mode analysis (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S5), this mode corresponds to the TM0 waveguide mode inside MoSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' To reveal the detailed features of the TM0 mode, we show a zoomed-in view (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='2k0 < q < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8k0) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3d in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3e, where a back-bending dispersion of the waveguide mode is clearly visualized near the A exciton energy (see also the dispersion data of the 110-nm-thick MoSe2 sample in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Such an ‘anomalous’ dispersion is in fact the characteristic behaviour of the EPs under measurements with fixed excitation energies (imaging experiments with a continuous-wave laser in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The commonly-accepted anti-crossing dispersion of the EPs, on the other hand, can be obtained by measurements at fixed momenta (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' spectroscopic studies of cavity polaritons at fixed incident angles)1-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The fixed-energy imaging measurements intend to determine the polariton momenta (qp) by searching horizontally the dispersion map (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' along horizontal dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3f), while the fixed-momentum spectroscopic experiments are to locate the polariton energy (Ep) by sweeping vertically the dispersion map (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' along the vertical dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' With both methods, one can obtain a series of (qp, Ep) data points (blue crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3f,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The polariton dispersion reflected by these data points demonstrates either back-bending (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3f) or anti-crossing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3g) features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Note that the back-bending dispersion also suggests that polaritons are subject to broadening, which introduces finite photonic spectral weight at the gap between the top and bottom polaritonic branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The broadening is mainly due to scatterings of EPs with longitudinal optical phonons that can be strongly suppressed at cryogenic temperature (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Back-bending dispersion has been observed previously in both plasmon and phonon polaritons29,30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Our experiment proves that the EPs also share this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Based on the back-bended dispersion data points, we estimate a Rabi splitting energy (ERabi) of ~100 meV (yellow arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3e), indicating a strong coupling between excitons and waveguide photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The ERabi value measured here in bulk MoSe2 appear to be larger than those of atomic layers of TMDs5-8 and smaller than that of bulk WS29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Finally, we performed s-SNOM imaging of MoSe2 waveguides with different thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 4a,b, we plot the near-field images of two additional MoSe2 waveguides with thicknesses of 77 and 110 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' They are both taken at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='48 eV in the perpendicular configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' As described above, we determined the fringe period ( \uf072⊥ ) by extracting fringe profiles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 4c) by FT analysis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Employing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1, we obtained \uf06cp versus waveguide thickness (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 4e), which shows good consistency with theory (black curve) (Supplementary Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 4e, we found that \uf06cp can be altered from 600 to 300 nm by controlling the waveguide thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Note that the observed TM0 polariton mode is cut off in MoSe2 waveguides with a thickness less than ~70 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 4e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In order to explore polaritons in thinner flakes or even atomic layers of TMDs, other type of waveguide modes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' TE0 mode23) or other coupling methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' micro-cavity coupling1-8) have to be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' By combining the s-SNOM technique with rigorous theoretical analyses, we uncovered the real-space characteristics of EPs in MoSe2 waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The observed polaritons have shown a small wavelength (down to 300 nm) and a long propagation length (up to 12 \uf06dm or above) under ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' These characteristics observed in our first generation devices are comparable to or even better than surface plasmon polaritons in graphene12-15 and hyperbolic phonon polaritons in hexagonal boron nitride16-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Through careful design and engineering, the TMD waveguides with tailored polaritonic modes could potentially be applied in miniaturized nanophotonic circuits for information or energy transfer in the near-infrared to visible regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In addition, it will be interesting to perform polariton nano-imaging at cryogenic temperatures, where one could possibly visualize EPs with stronger coupling strength and longer propagation length (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Future studies are also 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+page_content=' Dispersion curves of surface phonon-polaritons with backbending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' A 54, 317-318 (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Acknowledgements F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' acknowledge startup support from Iowa State University and Ames Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The nano-optical imaging setup was partially supported by the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Keck foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The work at UW was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' DOE Basic Energy Sciences, Materials Sciences and Engineering Division (DE-SC0008145 and SC0012509).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The work at ORNL (JQY and DGM) was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Author contributions Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' conceived the ideas and designed the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' carried out the s-SNOM experiments and collected the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' performed theoretical analyses and modeling of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' synthesized the MoSe2 crystals and fabricated the waveguide devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=', X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' wrote the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Additional information The authors declare no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Reprints and permission information is available online at http://npg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='com/reprintsandpermissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Correspondence and requests for materials should be addressed to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' (zfei@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Figure Legends Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Nano-optical imaging of a MoSe2 planar waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' a, Schematics of concentric waveguide modes in MoSe2 launched by the laser-illuminated s-SNOM tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' b, Illustration of the experimental setup where the incident beam is aligned perpendicular to the sample edge that is along the y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' We also sketch here the two major paths (‘P1’ and ‘P2’) where photons can be collected by the concave mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' c-g, Selected s-SNOM imaging data of a 156-nm-thick MoSe2 planar waveguide taken at various laser energies (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Here we plot the near-field amplitude (s) normalized to that of the SiO2/Si substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The dashed lines mark the sample edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The scale bars represent 1 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' a tip C E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='77 eV d E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='63eV MoSe2 MoSe2 e E =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='48eV E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='41eV MoSe, 20 S concave mirror b laserbeam MoSe2 1 MoSe2 % tip Z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='35 eV MoSe2 SiO2 Silicon MoSe2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Edge-orientation dependence study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' a, Illustration of the perpendicular configuration, where the incident beam (black arrow) is perpendicular to the sample edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' b, Illustration of the parallel configuration, where the x-y plane projection of the incident beam is parallel to the sample edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The labeling ‘w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='m.’ in a and b represents waveguide modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' c,d, Near-field amplitude images of MoSe2 taken at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='38 eV in the perpendicular and parallel configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The dashed lines mark the sample edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The scale bars represent 1 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' e,f, Real-space line profiles extracted perpendicular to the fringes in c and d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Here d is the distance between the tip and the sample edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' g,h, Fourier transform (FT) analysis of the real-space profiles in e and f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' a Perpendicular b Parallel Tip dil MoSe2 X w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' X edge SiO2 MoSe2 SiO2 c d MoSe2 20 MoSe e edge edge 16 (ou) 6 8 S S 0 0 0 2 4 6 0 2 4 6 d (μm) d (μm) 9 3 h (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=') 2 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=') FT 0 0 1 2 3 0 1 2 3 1/p (μml) 1/pu (μm-1) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Dispersion analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' a,b, Real-space fringe profiles and the corresponding FT profiles of the 156-nm-thick MoSe2 waveguide taken at various excitation energies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='35 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='77 eV) in the perpendicular configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' All the profiles are displaced vertically for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The arrows in b mark the FT peak associated with the measured waveguide mode in MoSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' c, Propagation length (Lp) of the measured waveguide mode from both experiment (squares) and theory (curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' d, Experimental dispersion data points (blue squares) overlaid on the calculated dispersion color map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' e, A zoomed-in version of panel d at the q range from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='2k0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The yellow arrow marks the Rabi splitting energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' f, Illustration of the fixed-E experiments with horizontal line cuts across the dispersion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' g, Illustration of the fixed-q experiments with vertical line cuts across the dispersion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The blue crosses in f and g mark the positions with maximum photonic spectral weight along the line cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The color maps in d-g plot the imaginary part the p-polarized reflection coefficient Im(rp) (Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' a Fringe profiles b FT profiles C 100 IA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='77 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='75 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='72 eV 口 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='70 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='68 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='65 eV 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='63 eV 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content="61 eV (n'e)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' E (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='59 eV d :1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='46ko 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='57 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='55 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='51 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='48 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='44 eV E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='41eV 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='38 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='35eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 TMo 0 2 4 6 1 2 3 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 3 d (μm) 1/p, (μm-1) q (ko) e f g 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 ≤1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 E E E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 1,3 TMo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 q (ko) q (ko) q (ko) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Waveguide-thickness dependence study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' a,b, Near-field imaging data of MoSe2 planar waveguides with thicknesses of 77 and 110 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' These images were taken at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='48 eV in the perpendicular configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The scale bars represent 1 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' c,d, Real-space fringe profiles and the corresponding FT profiles of MoSe2 with various thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' e, Thickness dependence of polariton wavelengths (\uf06cp) from both experiment (data points) and theory (black curve) at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='48 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Methods Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' For nano-optical imaging experiments, we used a scattering-type scanning near-field optical microscope (s-SNOM, Neaspec) that is built based on a tapping-mode atomic force microscope (AFM) operating with a tapping frequency of about 270 kHz and a tapping amplitude of about 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' A pseudo-heterodyne interferometric detection module is implemented in our s-SNOM to extract both the scattering amplitude (s) and phase (\uf079) of the near-field signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In this work, we discuss mainly the s signal that is sufficient for describing all the characteristics of the EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In order to subtract the background signal, we demodulated the near-field signal at the 3rd harmonics of the tapping frequency of the AFM tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In all the displayed near-field images, we plotted s normalized to that of the SiO2/Si substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' For optical excitation, we used a Ti:Al2O3 laser operating in the continuous-wave mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The photon energy of the Ti:Al2O3 laser can be conveniently tunable from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The samples studied here are MoSe2 planar waveguides fabricated using the mechanical exfoliation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The substrates used for these samples are standard Si wafers with a 300-nm-thick thermal oxide layer on the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Our s-SNOM experiments were all performed at ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Dispersion calculation and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The dispersion color maps shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3d-g were obtained by evaluating numerically the imaginary part of the reflection coefficients Im(rp) of the multilayer sample/substrate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The bright curves shown in the color map correspond to various photonic and polaritonic modes in the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Considering that the electric field right underneath the s-SNOM probe is perpendicular to the sample surface, only transverse magnetic (TM) waveguide modes are excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore, we only consider p-polarized reflection coefficient rp(q, E) in the dispersion calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' By using the transfer matrix method, we numerically calculate Im(rp) of the a b Thickness=77nm Thickness=110nm 20 S MoSe2 MoSe2 C d e 600 3 E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='48 eV 30 156nm 156nm 500 (cn: 2 20 (nm a 110 nm 110 nm 400 F 10 77 nm 77 nm 0 300 0 1 0 2 4 6 0 1 2 3 0 100 200 300 d (μm) 1/p(μm1) Thickness (nm)entire air/MoSe2/SiO2/Si system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The photonic/polaritonic modes appear at the (q, E) positions where Im(rp) diverges or maximizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore, we can conveniently estimate numerically the mode wavelength (\uf06cp = 2\uf070/qp) through the calculated dispersion color maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In order to fit the experimental dispersion data points (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3d), we adopted the experimental ab-plane dielectric constants of MoSe2 from previous optical measurement (Supplementary S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The c-axis dielectric constant (\uf065c) of MoSe2 is a fitting parameter, which was set to be 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 throughout the entire spectral range of our experiment (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The good agreement between experimental data points and calculated dispersion plot validates such an assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Determining the propagation length of the EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In order to determine quantitatively the propagation length (Lp) of the measured waveguide EPs, we first extract the linewidths (plotted in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S9a) of the FT peaks of the waveguide mode (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3b) and then determine Lp using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S11 in the Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Thus-obtained Lp data points are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Note that Lp at the lowest energy (E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='35 eV) is most likely underestimated due to limited resolution of the FT profiles originated from the finite size (~ 30 \uf06dm) of our device (Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In order to estimate theoretically Lp, we approximate the tip-launched waveguide mode as cylinder waves with a wave function of 1/2 ( ) i qx t Ax e \uf077 − − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Here, q = q1 + iq2 is the complex in-plane momentum of the waveguide mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore, the amplitude of the wave decays as: p /(2 ) 1/2 x L Ax e − − , where the propagation length Lp equals to (2q2)-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' For an anisotropic material like MoSe2 with an ab-plane dielectric function of \uf065ab = \uf0651 + i\uf0652 (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S4) and an c-axis dielectric constant of \uf065c \uf0bb 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3, we have 2 2 0 ( / ) c c ab z q k k \uf065 \uf065 \uf065 = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Based on this equation, we have an analytic formula of \uf05b \uf05d 1 2 2 2 2 2 1 1 2 1 0 1 / (2 ) 1 1 ( / )( / 1) p c L q k q \uf065 \uf065 \uf065 \uf065 \uf065 − \uf0e9 \uf0f9 = − − − \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' By adopting the q1=2\uf070/\uf06cp data of the TM0 waveguide mode (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3), we can calculate Lp using the above formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The calculated result is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3c as the solid curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The general trend of the theoretical curve is consistent with the experimental data (squares in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Supplementary Information for “Imaging exciton-polariton transport in MoSe2 waveguides” F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Hu1,2, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Luan1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Scott3, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Yan4,5, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Mandrus4,5, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Xu3,6, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Fei1,2* 1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA 2Division of Materials Sciences and Engineering, Ames Laboratory, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' DOE, Iowa State University, Ames, Iowa 50011, USA 3Department of Physics, University of Washington, Seattle, Washington 98195, USA 4Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 5Department of Materials Science and Engineering, University of Tennessee, Knoxville, Tennessee 37996, USA 6Department of Materials Science and Engineering, University of Washington, Seattle, Washington 98195, USA Correspondence to: zfei@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' List of contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Discussion and analysis about fringe formation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Field-distribution calculation confirming TM0 waveguide modes 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Polariton dispersion of the 110-nm-thick MoSe2 sample 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Effects of exciton linewidth & temperature on polariton dispersion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Propagation length of the waveguide exciton polaritons 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Discussion and analysis about fringe formation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content="1 Photon path ‘P3’ In the main text, we discuss two major photon paths ('P1’ and ‘P2’) for both perpendicular and parallel configurations (Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 2 and Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In path ‘P1’, photons are scattered directly by the tip back to the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In path ‘P2’, photons first transfer into propagative waveguide exciton polaritons (EPs) and then scatter to photons when reaching the edge of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In addition to the two photon paths, there is another possible photon path ‘P3’ that could contribute to the signal collection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In photon path ‘P3’, the photons first transfer into EPs and then reflected backward to the tip after reaching the sample edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The reflected EP modes can be scattered back to detector by the AFM tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' This path has been extensively discussed in previous nano-infrared imaging studies of graphene plasmons1 and hexagonal boron nitride (hBN) polaritons2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In the current work, ‘P3’ plays a less significant role compared to ‘P2’ for the following two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' First, the momenta (q) of the modes involved in our experiments is much closer to the free-space photon wavevector (k0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' As a result, only a small portion of the EP modes are reflected back by the sample edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In addition, the round-trip propagation of the EP modes in ‘P3’ also suffers more damping compared to ‘P2’ due to the longer traveling distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' As a result, we didn’t see clear experimental evidence of fringes due to the interference between photons from paths ‘P1’ and ‘P3’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='2 Edge excitation The photon paths ‘P2’ and ‘P3’ discussed above all assume the SNOM tip as the only launcher of the EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In fact, the sample edge can also launch polaritons when illuminated by the laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Nevertheless, due to the small size of the focused laser beam (~ 2 \uf06dm), edge launching is only possible when the tip-edge distance (d) is very small (d < 1 \uf06dm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore, the fringes that are over 1 \uf06dm away from the edges are solely due to the effects of tip scattering or launching (‘P1’ and ‘P2’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' When performing FT analysis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3b) on the fringe profiles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3a), we cut off a short part of the fringe profiles (~ 1 \uf06dm close to the edge) in order to avoid the complications involving edge launching processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Supplementary Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Illustration of various paths from which the photons can be collected by the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 Perpendicular configuration In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 2a and related text in the manuscript, we discuss the perpendicular configuration of the experimental setup, where the laser beam is perpendicular to the edge of sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Based on this configuration, we obtain an equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1 in the main text) that describes the relation between the fringe period ( \uf072⊥ ) and the wavelength of the EPs (\uf06cp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Now we introduce in detail how this equation is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' As discussed above, the fringes observed in our experiments are mainly due to the interference between collected photons from paths ‘P1’ and ‘P2’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In the latter path (‘P2’), the tip-launched EPs propagate radially away from the tip, so they could in principle be scattered into free-space photons from any positions along the sample edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Nevertheless, the concave mirror with a relatively small collection angle of less than 14\uf0b0 (beam size < 1 inch, focusing length ≈ 2 inches) collects mainly the scattered photons from waveguide EPs propagating along the direction perpendicular to the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In order to elucidate that, we illustrate in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S2 the process of edge scattering where the incident laser beam is in the x-z plane and the sample edge is along the y direction (perpendicular configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Here we assume, tip-launched EPs with a momentum of qp propagate to the edge with a random angle (\uf066) with respect to the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' At the edge, the EPs are scattered into free-space photons with a wavevector of k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The angle between the scattered P1 Laser beam P3 P2 Tip waveguide mode Sample SiO2 Substratephotons and the optical axis of the concave mirror is \uf071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' As discussed above, \uf071 should be within the collection angle of the concave mirror: \uf071 < 14\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore, photon wavevector along the y direction (ky) should satisfy ky < k0sin(\uf071) < k0sin(14\uf0b0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='242k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In the edge scattering process, momentum is conserved along the edge direction (y direction), so we have ky = qpsin(\uf066) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='242k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Considering that the EPs of MoSe2 studied in the current work are more confined in space than photon modes in SiO2: qp > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='46k0, we have \uf066 < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Since the above upper bound limit analysis is far from stringent, the actual average \uf066 will be much closer to zero with a more careful analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Furthermore, EPs with finite \uf066 will be subjected to higher propagation loss and larger reflection coefficient than those with normal incidence towards the edge (\uf066 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore, we assume \uf071 = \uf066 = 0o in the following analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The error bar of the extracted \uf06cp based on such an assumption is less than 1 - cos(9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8o) \uf0bb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Under the assumption of \uf071 = \uf066 = 0o, we estimate the phase difference (\uf046) between photons collected from paths ‘P1’ and ‘P2’ to be: ( ) 0 0 cos 2 1/ cos / , p p q d k d d \uf061 \uf070 \uf06c \uf061 \uf06c \uf046 = − = − (S1) where d is the distance between the tip and the sample edge, \uf06cp and \uf06c0 are the wavelengths of waveguide EPs and free-space photons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' If d changes by a distance that equals to the fringe period ( \uf072⊥ ), \uf046 will change by 2\uf070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore we have ( ) 1 0 1 / cos p p \uf072 \uf06c \uf06c \uf06c \uf061 − ⊥ \uf0e9 \uf0f9 = − \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' (S2) Supplementary Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Illustration of the perpendicular configuration of the nano-optical experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 Parallel configuration Figure S3 illustrates the parallel configuration of the experimental setup, where the incident laser beam is kept in the x-z plane and the sample edge is along the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Similar to the perpendicular configuration, the fringes are formed due to the phase difference between photons collected by the detector from paths ‘P1’ and ‘P2’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In path ‘P2’, tip-launched EPs with an in-plane momentum of qp propagate toward the edge with an angle of \uf066 with respect to the y axis and get scattered into free-space photons (k0) when reaching the sample edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Again, the angle \uf071 between scattered photons and the optical axis should be less than the collection angle of the concave mirror: \uf071 < 14\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' From Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S3, we find that the x component of the wavevector of the scattered photons (kx) should satisfy: k0cos(\uf061 +14\uf0b0) < kx < k0cos(\uf061 −14\uf0b0), where \uf061 = \uf033\uf030\uf0b0 is the angle between the optical axis of the concave mirror and the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore, we have 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='72k0 Laserbeam tip opticalaxis d α ko P2 0 qp SiO2 edge Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' MoSe2 x< kx < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='96k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Considering the momentum conservation along the direction of the scattering edge (x direction): kx = qpsin(\uf066), so we have 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='72k0/qp < sin(\uf066) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='96k0/qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Based on this inequality, we have 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8\uf0b0< \uf066 \uf03c \uf032\uf032\uf02e\uf036\uf0b0 in the case of qp ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5k0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' TM0 mode in the 156-nm-thick MoSe2 sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The angle deviation is quite small, so we assume \uf071 = 0 for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In this case, kx = qpsin(\uf066) = k0cos(\uf061)\uf02c \uf066 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3\uf0b0 when qp ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The uncertainty of the estimated \uf06cp due to such an assumption is estimated to be less than cos(16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8\uf0b0) - cos(20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3\uf0b0) \uf0bb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Under the assumption of \uf071 = 0, we estimate the phase difference (\uf046) between photons collected from paths ‘P1’ and ‘P2’ to be: ( ) 0 0 / cos tan cos 2 1/ cos / tan cos / p p q d k d d \uf066 \uf066 \uf061 \uf070 \uf066 \uf06c \uf066 \uf061 \uf06c \uf046 = − = − (S3) where 1 0 sin [( / )cos ] p \uf066 \uf06c \uf06c \uf061 − = obtained from kx = qpsin(\uf066).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' If d changes by a distance that equals to the fringe period ( // \uf072 ), \uf046 will change by 2\uf070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore we have 1 // 0 1/ cos ( / )tan cos p p \uf072 \uf06c \uf066 \uf06c \uf06c \uf066 \uf061 − \uf0e9 \uf0f9 \uf0bb − \uf0eb \uf0fb (S4) Supplementary Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Illustration of the parallel configuration of the nano-optical experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Supplementary Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In-plane (ab-plane) dielectric function of MoSe2 for calculations of the EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The black and red curves are real and imaginary parts of the dielectric function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The solid lines are experimental data adopted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3, which were measured in bulk MoSe2 at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The dashed lines are modeled results that we constructed by Laserbeam P1 tip optical axis α P2 ko MoSe2 A edge Z y X SiO2A 30 B 20 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 Energy (eV)using single Lorentzian oscillator for both A and B excitons (see detailed discussions in Section 4 of the Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The c-axis permittivity is set to be 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 throughout our energy region that produces a good fit to experimental dispersion data points (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3d in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Field-distribution calculation confirming TM0 waveguide modes In order to confirm the nature of the measured waveguide EPs, we performed field distribution calculations by matching boundary conditions in Maxwell’s wave equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The waveguide structure is plotted in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The calculations were performed considering 156-nm-thick MoSe2 excited by a laser beam with a photon energy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='38 eV, but the general results apply also to other thicknesses (110 nm and 77 nm) or other laser energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The ab- plane optical constants for MoSe2 used in the calculation are adopted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3 (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S4) and the c-axis permittivity is set to be 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 that is determined through dispersion fitting (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3d in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The results are shown in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S5b and S5c, where we plot the y-component magnetic field (Hy) and z-component electric field (Ez) at various z positions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Note that the waveguide EPs is set to be propagating along the -x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' From the field distribution, we know that the measured waveguide EPs correspond to TM0 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Supplementary Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Field-distribution calculation confirming TM0 waveguide modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' a, Illustration of the MoSe2 waveguide sandwiched by air and SiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The waveguide EPs are propagating along the -x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' b,c, Calculated Hy(z) and Ez(z) of the waveguide EPs in the 156-nm-thick MoSe2 sample (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1 and 2 in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The blue dashed lines here mark the top and bottom surfaces of the MoSe2 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Polariton dispersion of the 110-nm-thick MoSe2 sample In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3d,e of the main text, the dispersion data points (blue squares) were obtained from near-field amplitude images of the 156-nm-thick MoSe2 waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Here in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S6, we show additional dispersion data (blue squares) measured from the 110-nm-thick MoSe2 waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Again, these data points are in good agreement with the theoretical dispersion color map and they also show the back-bending feature close to the A exciton energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' We also notice that the polariton mode of the 110-nm-thick MoSe2 sample shift to the low momentum region (~ 2k0) compared to that of the 156-nm-thick one (~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5k0), indicating reduced mode confinement or increased mode wavelength (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 4e in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' a b c > air x air air propagation direction MoSe2 MoSe2 N MoSe2 N SiO2 SiO2 SiO2 0 1 2 1 0 1 Hy (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=') Ez (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=') Supplementary Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Experimental dispersion data of waveguide EPs overlaid on the calculated dispersion color map of a 110-nm-thick MoSe2 waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Effects of exciton linewidth & temperature on polariton dispersion As discussed in the main text, the back-bending dispersion is a characteristic behavior of polaritons subject to broadening when measured by experiments under continuous-wave excitation (fixed excitation energies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Here we wish to explore the effects of the exciton broadening or linewidth on back-bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' For that purpose, we calculated the dispersion diagrams of 156-nm- thick MoSe2 waveguide using a modeled ab-plane dielectric function with Lorentzian oscillators: A B 1 2 2 2 2 2 A A B B ( ) ( ) ( ) E E i E E E i E E E i E \uf065 \uf065 \uf065 \uf065\uf0a5 \uf057 \uf057 = + = + + − − \uf047 − − \uf047 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' (S5) To match the experimental optical constants from literature3 (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S4), the oscillator parameters are set to be 21 \uf065 \uf0a5 = , 2 A 2 eV \uf057 = , A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='55 eV E = (A exciton energy), A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='1 eV \uf047 = (A exciton linewidth), 2 B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='53 eV \uf057 = , B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='85 eV E = (B exciton energy), and B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='12 eV \uf047 = (B exciton linewidth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' According to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 4, the exciton linewidth (broadening) in MoSe2 comes from both non- thermal and thermal processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The non-thermal processes include defect and impurity scattering or electron-electron scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The thermal process is mainly due to interactions with longitudinal optical phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The thermal contribution can be strongly suppressed at low temperature (LT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' As a result, exciton linewidth at temperature less than 100 K could be less than half of that at room temperature (RT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore, we adopted A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='05 eV \uf047 \uf0bb as a rough estimation of the linewidths of A excitons of bulk MoSe2 at low temperature, which is consistent with the experimental results given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' ko 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='46ko 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 TMo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 3 q (ko) Supplementary Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' a,b, Calculated dispersion color maps of the 156-nm-thick MoSe2 waveguide with modeled ab-plane optical constants derived from a Lorentzian oscillator with exciton linewidths of \uf047\uf041 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='1 eV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='05 eV, corresponding to room temperature (RT) and low temperature (LT), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' c,d, Zoomed-in version of panels a and b respectively at the q range from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='2k0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The blue crosses in c and d mark the positions with maximum photonic spectral weight along the horizontal line cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The calculated dispersion diagrams with the modeled optical constants (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S5) are given in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S7, where effects of exciton linewidth on polariton dispersion are clearly demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Here we compare the case of A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='1 eV \uf047 \uf0bb (corresponding to RT, Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S7a and S7c) with that of A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='05 eV \uf047 \uf0bb (corresponding to LT, Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S7b and S7d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In both cases, back-bending dispersion close to the A exciton energy is clearly seen, but the polariton mode with A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='05 eV \uf047 \uf0bb is back-bended more dramatically, indicating stronger light- exciton coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In addition, the momentum broadening (\uf044q) of the EP mode away from the exciton energy becomes narrower at A \uf047 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='05 eV suggesting that polaritons have smaller damping or longer propagation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Propagation length of the waveguide exciton polaritons In Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S8a and S8b, we plot the full-scale image and fringe profile of the 156-nm-thick MoSe2 waveguide taken at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='35 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Both the images (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 1) and profiles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3a) shown in the main text are truncated intentionally to fit the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Here in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S8a and S8b, fringes or oscillations are seen 30 \uf06dm away from the sample edge, indicating a long propagation length (Lp) of these modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In order to determine quantitatively the propagation TA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='1 eV (RT) TA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='05 eV (LT) a b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 E E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 q (ko) q (ko) TA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='1eV (RT) TA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='05 eV (LT) C d 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 E E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 q (ko) q (ko)length (Lp) of the measured waveguide EPs, we first extract the linewidths of the FT peaks shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3b of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The FT linewidth data, plotted in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S9a, shows an increase with energy in the spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Based on the linewidth data, we can then determine Lp using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S11 (see discussions below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Thus-obtained Lp data points are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3c of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Note that the resolution (~1/30 \uf06dm-1) of the FT profiles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3b in the main text) is limited by the sample size (~30 \uf06dm), so the linewidth data point in Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S9a at the lowest photon energy (E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='35 eV) is most likely overestimated due to the resolution limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore, the extracted Lp data point at this energy is possibly underestimated compared to the realistic values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S9b, we also plot the number of propagation cycles, namely propagate length (Lp) versus polariton wavelength (\uf06cp), at various photon energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Here one can see that the waveguide EPs can propagate over 30 cycles before losing ~63% (1-1/e) of the polariton energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Supplementary Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' a, Near-field amplitude image (s) of the 156-nm-thick MoSe2 planar waveguide taken at E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='35 eV in the perpendicular configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The near-field amplitude is normalized to that of the SiO2/Si substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The white dashed line marks the edge of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The scale bar represents 1 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' b, Line profile taken from a perpendicular to fringes along the green dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Supplementary Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' a, Linewidths of the FT peaks measured from the FT profiles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3b of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' b, Estimated number of propagation cycles (Lp/\uf06cp) of polaritons at various photon energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Vertical dashed lines mark the A exciton energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' a 20 SiO MoSe b 15 10 S 5 0 5 10 15 20 25 30 d (um)a b 40 1 A TAI 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 Propagation cycle 口 口 口 30 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='wm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 吕 口口 Linewidth 口 口 20 口 口 口 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='2 口 10 口 口 口 口 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 0 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='8 E (eV) E (eV) Now we discuss how we convert the linewidths (W) into propagation lengths (Lp) in the case of perpendicular configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' As discussed in the main text and also in Section 1 of the Supplementary Information, the total optical signal (Etot) responsible for the observed interference fringes comes from photon paths ‘P1’ (E1) and ‘P2’ (E2): Etot = E1 + E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In path ‘P1’, photons are directly back-scattered by the tip to the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' In path ‘P2’, photons first transfer into the waveguide EPs and then scattered back to photons by the sample edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The momentum of the EP mode can be written as: qp = q1 + iq2, where q1 = 2\uf070/\uf06cp and q2 = 1/(2Lp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Throughout the measurement, tip is fixed and the sample is scanning, so both the amplitude and phase of E2 is dependent on the tip-edge distance (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore, we can write the amplitude of the total optical signal as: ( ) tot 1 1 2 2 | ( )| | ( )| | ( ) | i d d d A A d e \uf046 = + = + E E E (x > 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' (S6) Here A1 is the amplitude of E1, 2 1/2 2 2 ( ) d q A d B d e − − = is the amplitude of E2, \uf05b \uf05d 1 0 0 ( ) cos( ) d q k d \uf061 \uf046 = − +\uf046 is the relative phase difference between E2 and E1 in the perpendicular configuration (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' For simplicity, we define 1 0 cos( ) q k \uf061 \uf04c = − below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The constant A1 describes the efficiency of photon back-scattering in ‘P1’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' The constant B2 describes the conversion efficiency from photons to EPs (via tip) times the conversion efficiency from EPs to photons (via edge) in ‘P2’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' By substituting the formula of 2( ) A d and ( ) d \uf046 above into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S6, we have 0 2 2 2 0 ( ) ( ) 2 2 1 1/2 1/2 1 2 1 2 1 2 to 2 t( ) q q i d i d d d d q d A B d e A B d e e A B d e e − − − \uf04c +\uf046 − \uf04c +\uf046 − − − = + + + E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' (S7) Considering the low conversion rates between photons and EPs, and also the exponential decaying of the EPs with d, we are safe to assume 2 1/2 2 2 1 q d A B d e A − − = \uf03c\uf03c , so Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S7 can be written approximately as: 0 0 2 ( ) ( ) tot 1/2 1 2 1 ( ) 2 i d i d q d d A B d e e e \uf04c +\uf046 − \uf04c +\uf046 − − \uf0e9 \uf0f9 \uf0bb + + \uf0eb \uf0fb E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' (S8) By performing Fourier transformation (FT) on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S8, we have \uf05b \uf05d \uf05b \uf05d 0 1/2 1/2 t 2 2 2 ot( ) ( ) ( ) k q i q k i e k − − − \uf046 \uf04c \uf0b5 − + − + + \uf04c E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' (S9) Equation S9 indicates that there should be two peaks at k = \uf0b1\uf04c when performing FT analysis of the fringe profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' They are identical in shape and well apart from each other (q2 << \uf04c ), so we only need to consider one peak: 1/4 2 2 2 tot( ) ( ) k q k − \uf04c \uf0e9 \uf0f9 \uf0eb \uf0fb − \uf0b5 + E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' (S10) The full width at half maximum of the peak given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S10 is 2 2 15 k q \uf064 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Note that the unit of the FT profiles in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 2 and 3 of the main text is 1/\uf072⊥ instead of k = 2\uf070/\uf072 ⊥, so the measured linewidth (W) should be scaled by a factor of 2\uf070, namely 2 15 / W q \uf070 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Therefore, we have: 1 2 (2 ) 15 / (2 ) P L q W \uf070 − = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' (S11) Based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S11, we converted the measured FT linewidths (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' S9a) into the propagation lengths at various excitation energies (squares in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Supplementary References 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Fei, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' et al.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Science 343, 1125-1129 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' Beal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' & Hughes, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFPT4oBgHgl3EQfADR0/content/2301.12980v1.pdf'} +page_content=' 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Montero de Hijes1, J. R. Espinosa2, C. Vega2, and C. Dellago1,∗ +1Faculty of Physics, University of Vienna, A-1090 Vienna, Austria and +2Departamento de Qu´ımica F´ısica, Facultad de Ciencias Qu´ımicas, +Universidad Complutense de Madrid, 28040 Madrid, Spain +Despite the importance of ice nucleation, this process has been barely explored at negative pres- +sures. Here, we study homogeneous ice nucleation in stretched water by means of Molecular Dy- +namics Seeding simulations using the TIP4P/Ice model. We observe that the critical nucleus size, +interfacial free energy, free energy barrier, and nucleation rate barely change between isobars from +-2600 to 500 bar when they are represented as a function of supercooling. This allows us to identify +universal empirical expressions for homogeneous ice nucleation in the pressure range from -2600 to +500 bar. We show that this universal behavior arises from the pressure dependence of the inter- +facial free energy which we compute by means of the mold integration technique finding a shallow +minimum around -2000 bar. Likewise, we show that the change in the interfacial free energy with +pressure is proportional to the excess entropy and the slope of the melting line, exhibiting the latter +a reentrant behavior also at the same negative pressure. Finally, we estimate the excess energy and +the excess entropy of the ice Ih-water interface. +∗christoph.dellago@univie.ac.at +I. +INTRODUCTION +Water crystallization is an essential phase transition +in nature and technology. However, in the cryopreser- +vation of biological samples [1, 2], ice formation can be +disastrous. The low temperature preserves the biological +material but causes the water within the sample to +be in a metastable state subject to crystallization [3]. +Interestingly, by keeping the sample under high pressure, +ice nuclei are less likely to form keeping water liquid for +a longer time [4–6]. The frequency of the nucleation pro- +cess is mainly determined by the thermodynamic driving +force and the cost of creating the interface between the +emerging nucleus and the metastable liquid. The reason +why high pressure slows down the nucleation process +is that the difference in chemical potential between ice +and water, ∆µ, which represents the thermodynamic +driving force, barely changes between isobars whereas +the cost of creating the interface notably increases [7]. +The interfacial free energy is the variable that quantifies +this cost. +At coexistence, through a planar interface, +the interfacial free energy γm differs from the value for a +critical nucleus γ due to the curvature of the surface [8]. +Nevertheless, both notably increase at high pressure [7]. +Homogeneous ice nucleation at standard and high +pressure has been extensively explored [7, 9–19]. How- +ever, ice nucleation in water under negative pressure, i.e. +stretched water, has caught less attention [20–22]. This +process is relevant in porous media containing water +solutions [23] and also in water transpiration inside +plants [24] where negative pressure occurs. +Creating +and maintaining negative pressure over a sample is +non-trivial in experiments [25]. This is because a liquid +at negative pressure is metastable [26, 27]. In general, +this metastability is considered with respect to the vapor +phase [28, 29], although at certain conditions, it can +be also metastable with respect to ice. Some ingenious +approaches to create negative pressure in metastable +water include the use of a Berthelot tube [30], cen- +trifugation [31], and more recently the use of acoustic +waves [25, 29, 32]. +In contrast, in computer simula- +tions is straightforward to work under negative pressures. +In this work, +we investigate how ice nucleation +properties are affected by negative pressure at different +degrees of supercooling. In fact, we find little effect when +pressure changes from strongly negative to moderately +positive. We investigate the role of the interfacial free +energy since it is a key property in determining the phase +behavior of water at high pressure [7]. +We find that +the slope of the melting line is crucial to describe the +change with pressure of the interfacial free energy which +displays a shallow minimum at negative pressure. Our +study is based on molecular dynamics simulations with +the TIP4P/Ice model [33] which has been extensively +used to describe ice nucleation [7, 9, 12, 21] and growth +[34, 35] as well as in supercooled water [36, 37]. +In +particular, we employ the seeding technique [9, 38] to +study nucleation and the mold integration technique [39] +to measure the interfacial free energy at coexistence. +II. +SIMULATION METHODS +All simulations have been done with the GROMACS +package (4.6.7-version in double precision) with the +TIP4P/Ice water model. The simulations are performed +in the isothermal-isobaric (NpT) ensemble with a time +step of 2 fs using the Noose-Hoover thermostat [40, 41] +and the Parrinello-Rahman barostat [42] both with a +relaxation time of 0.5 ps. Electrostatic interactions are + +2 +accounted for via the particle-mesh-Ewald summation +algorithm [43] with order 4 and a Fourier spacing of +0.1 nm. +The cutoff for the Lennard-Jones and the +Coulombic interactions is set to 0.9 nm and long-range +corrections to the Lennard-Jones part of the potential +are included in energy and pressure. +To study nucleation we use the seeding technique +[9, 44, 45] which involves the combination of molecular +dynamics simulations and Classical Nucleation Theory +(CNT) [46, 47]. This technique is based on the behavior +of a critical nucleus which has equal probability of grow- +ing and melting when surrounded by the metastable +phase at the critical pressure and temperature. +In +practice, one inserts a spherical ice-Ih seed in metastable +water and then keeps track of the time evolution of the +size of the cluster. One can vary T , p and the seed size +in order to find at which conditions a certain nucleus +size is critical (Nc). Once Nc is known, CNT is used to +find the interfacial free energy γ, the barrier height ∆Gc, +and the nucleation rate J. +Our system sizes ranged +between +80000 and +250000 water molecules in total. +The duration of the trajectories is between 40 and 115 ns. +It is important to note that in the Gibbsian descrip- +tion of interfaces, one has two bulk phases separated by +a dividing surface. However, there is some arbitrariness +in the location of the dividing surface which also affects +to the interfacial free energy γ when the interface has +curvature [48–51]. +Within the CNT framework, the +relevant dividing surface is the surface of tension [52, 53]. +In order to find the surface of tension we employ an +empirical approach that has been successfully applied +in crystal nucleation for a large variety of systems +[7, 21, 38, 39, 54, 55]. +In this approach, the averaged +Steinhardt bond order parameter [56], ¯q6(T, p) is used +in combination with the mislabelling criterion [9] to +identify ice-like and water-like molecules. +Within a +cutoff distance of 3.5 ˚A, we obtain ¯q6(T, p) for each +molecule. The molecules with ¯q6(T, p) above a certain +threshold +¯ +q6,t(T, p) are labeled as ice whereas those +below are labeled as liquid. +This threshold depends +weakly on the considered thermodynamic range covering +pressures from -2600 to -1000 bar and temperatures +from 250 to 270 K (see the supplementary material in +Ref. +[21] for the isothermal change in +¯ +q6,t(T, p) with +pressure). In this work, the value changes between 0.365 +for the highest temperature and pressure to 0.385 for +the lowest temperature and pressure. +Once Nc is known, we employ the CNT equations +[46, 47] to determine other important parameters. The +interfacial free energy γ is given as +γ = +�3Ncρ2 +ice|∆µ|3 +32π +�1/3 +, +(1) +where Nc is the size of the critical nucleus, ρice is the +number density of ice-Ih in the bulk at the metastable +conditions at which the nucleus is critical, and |∆µ| is +known as the driving force to nucleation, i.e. +the dif- +ference in chemical potential between the liquid and ice +phases at the conditions which cause the nucleus to be +critical. +This property can be obtained by thermody- +namic integration along an isobar [57], +���� +∆µ +kBT +���� = +���� +� T +Tm +1 +kBT 2 +�Hice +Nice +− Hw +Nw +� +dT +����, +(2) +where kB is the Boltzmann constant, Tm is the melting +temperature, and H the enthalpy, which can be obtained +from simulations of bulk ice-Ih and bulk water along the +isobar of interest. +Then, the free energy barrier is given as +∆Gc = +16πγ3 +3ρ2 +ice|∆µ|2 = Nc|∆µ| +2 +, +(3) +which allows us to obtain the nucleation rate J, the num- +ber of critical nuclei forming per unit of time and volume. +According to CNT, J is given as +J = ρw +� +|∆µ| +6πkBT Nc +f + exp +� +−∆Gc +kBT +� +, +(4) +where f + is the attachment rate which can be approxi- +mated through this expression [7, 21] +f + = 24DwN 2/3 +c +λ2 +, +(5) +where Dw is the diffusion coefficient of the metastable +liquid and λ is a characteristic length, the typical +distance that a water molecule covers in order to attach +into the nucleus, whose value is approximately 3.8 ˚A for +water [7, 21]. +To find the ice-Ih-water interfacial free energy at co- +existence for a planar interface, γm, we use the mold +integration technique [39], which consists in computing +the reversible work W that is necessary to form a crys- +tal slab within a liquid at coexistence. This work is re- +lated to the interfacial free energy at coexistence, γm, by +W = 2Aγm where A is the interfacial area and the num- +ber 2 accounts for the two interfaces of the slab. The +slab formation is induced by switching on an attractive +interaction between the mold of potential energy wells +and the particles of the initial liquid. The wells are ar- +ranged in the equilibrium positions of the oxygen atoms +in the ice facet under investigation at coexistence condi- +tions, i.e. for temperatures and pressures located along +the ice Ih-water equilibrium line for the TIP4P/Ice water +model. First, one has to obtain γrw, which is given as + +3 +γrw = 1 +2A +� +ǫwNw − +� ǫw +0 +⟨N(ǫ)⟩dǫ +� +, +(6) +where rw indicates the radius of the potential wells and ǫ +is their energy (with maximum depth equal to ǫw). Nw is +the number of wells in the mold and ⟨N(ǫ)⟩ is the average +number of occupied wells at a given potential depth ǫ. +The integration needs to be reversible. To ensure this, +thermodynamic integration is performed for wells whose +radius is larger than a certain value r0 +w. At r0 +w the slab is +fully formed and the stability no longer depends on the +mold-liquid interactions, hence, leading to potentially +irreversible ice growth. However, since this is the radius +that recovers the actual value of γm, thermodynamic +integration is repeated for several values of rw < r0 +w and +then γrw is extrapolated to its value at r0 +w giving γm [39]. +III. +RESULTS +A. +Universality in ice nucleation variables at +negative and moderate pressure +First, +we study nucleation along the isobars of +-2600, -2000, and -1000 bar by means of the seeding +approach. +For pressures below -3000 bar we observed +spontaneous cavitation occurring within the time scale +of the trajectories needed in the seeding method. +We +obtain the critical nucleus size Nc, the driving force to +nucleation |∆µ|, the interfacial free energy γ, the free +energy barrier to nucleation ∆Gc and the nucleation +rate J. These results are presented in Table I. As can +be seen, even though the pressure significantly differs, +the results are surprisingly similar for nuclei of similar +size for equivalent supercoolings. +This behaviour is +considerably different from what has been found when +comparing the nucleation scenario of normal vs. +high +pressure (i.e. 2000 bar; Ref. [7]), where the increase in +pressure brings down the ice nucleation rate. +To further understand this behavior, we connect our +results with those from previous works where nucleation +had been studied for the TIP4P/Ice model at different +pressures including negative, moderate, and high pres- +sure states [7, 21]. +In Fig. +1 a) we show the critical +nucleus size as a function of supercooling for several +isobars. We provide results at moderate supercoolings +at -2600, -2000, and -1000 bar. For these same isobars as +well as for the 1 bar isobar, we show the values reported +in Ref. +[21]. +For the 1 bar isobar, we also show the +values given in Ref. +[7], which also provides with the +values at the 2000 bar isobar. As can be seen, only the +points corresponding to the 2000 bar isobar [7] exhibit a +different trend. The isobars at -2600, -2000, and -1000 +bar from this work as well as from Ref. [21], and the 1 +bar isobar from both Refs. [7, 21] follow approximately +the same curve. +Notice that even a point at 450 bar +reported in Ref. [21] was included being in agreement +with this group of isobars. In fact, as shown in Fig. 1 b), +pressure hardly affects the nucleation free energy barrier +as a function of supercooling from -2600 bar to 450 bar. +Our results from Fig. +1 suggest that a similar +nucleation behaviour as a function of supercooling may +take place from -2600 to 450 bar. That is a strikingly +different behavior to the one observed when increasing +pressure to 2000 bar. +Thus, we propose universal +empirical expressions for the variation of different homo- +geneous ice nucleation properties with the supercooling +independently of the pressure as long as it is within +this regime. Nevertheless, we first need to confirm that +what was observed for Nc and ∆Gc also applies to J. +In Fig. 2 we show again a) Nc and b) ∆Gc as well as +c) γ and d) log10 J. This time, for each magnitude, we +include a common fit to data from moderately positive +to deeply negative pressure (including our own data +and those from Refs. +[7, 21]) along a separate fit at +high pressure [7]. In c) we show γ which exhibits higher +variance. +Finally, in d), we show how very different +pressures (from largely negative to moderately positive) +lead to approximately the same nucleation rate J as a +function of supercooling, ∆T = Tm − T . +The values +of Tm are given in Table II. Hence, we can use the +respective common fit as universal empirical expressions +to describe the change with supercooling within this +broad range of pressures. +For Nc we obtain +Nc(∆T ) = 1.2 · 107 · +�∆T +T0 +�−2.8 +(7) +and for ∆Gc (in kJ/mol) +∆Gc(∆T ) = 2.3 · 105 · +�∆T +T0 +�−2.1 +, +(8) +where T0 equals 1 K for correctness of units. For γ in +mJ/m2, we obtain +γ(∆T ) = 26.6 − 0.174 · ∆T, +(9) +and, finally, for J in m−3 s−1, one should use Eq. 4 along +with Eq. +8 (after converting into in kBT units), and +1036 m−3s−1 as the prefactor (ρw +� +|∆µ|/(6πkBT Nc)f +). +The results shown in Fig. +2 have interesting con- +sequences. +First, taking into account that Nc and γ +(panels a) and c) respectively) are roughly independent +of p when it goes from largely negative to moderately +positive pressures, the isobaric Tolman length which +determines the change in γ with the inverse of the radius +of curvature of the cluster along an isobar [8, 58, 59] is + +4 +Nc +T [K] +∆T [K] +p [bar] +ρw [g/cm3] +ρice [g/cm3] +|∆µ| [kJ/mol] +γ [mJ/m2] +∆Gc [kJ/mol] +log10(J [m−3s−1]) +1650 +255 +23 +-1000 +0.9208 +0.8999 +0.367 +21.62 +303 +-24 +7450 +264 +14 +-1000 +0.9285 +0.8985 +0.237 +23.03 +883 +-136 +1750 +255 +25 +-2000 +0.8855 +0.8916 +0.367 +21.87 +321 +-28 +7600 +266 +14 +-2000 +0.8876 +0.8894 +0.224 +21.74 +850 +-128 +1950 +255 +24 +-2600 +0.8674 +0.8866 +0.340 +20.89 +332 +-30 +TABLE I. Seeding results in tabular form. Nc is the critical nucleus size, T and p are the thermodynamic conditions that make +such nucleus size to be critical, and ∆T is the supercooling, Tm − T . The densities of water ρw and ice ρice are also shown, +as well as the interfacial free energy at nucleation γ, the barrier height ∆Gc, and the base-10 logarithm of the nucleation rate +log10(J). +10 +20 +30 +40 +50 +∆T [K] +0 +2000 +4000 +6000 +8000 +10000 +Nc +2000 bar [Espinosa et al. PRL 2016] +1 bar [Espinosa et al. PRL 2016] +450 bar [Bianco et al. PRL 2021] +1 bar [Bianco et al. PRL 2021] +-1000 bar [Bianco et al. PRL 2021] +-2000 bar [Bianco et al. PRL 2021] +-2600 bar [Bianco et al. PRL 2021] +-1000 bar [This work] +-2000 bar [This work] +-2600 bar [This work] +a) +10 +20 +30 +40 +50 +∆T [K] +0 +200 +400 +600 +800 +1000 +1200 +1400 +∆Gc [kJ/mol] +b) +FIG. 1. a) Critical nucleus size and b) free energy barrier to undergo nucleation against supercooling. The same legend applies +in both panels. Numerical details can be seen in Table I. The color indicates the pressure, whereas solid symbols correspond +to simulations performed in this work, and empty symbols correspond to data obtained from previous work as indicated in the +legend. For the same pressure but different work we use different symbols. The lines are power law fits to points sharing the +same pressure independently on the work in which they were obtained. +roughly constant too and equal to 0.24(5) nm, where the +parenthesis indicates uncertainty in the last digit. This +result is in agreement with previous work [54]. Second, +in panel d) one can see that from strongly negative +to moderately positive pressure we obtain the same +nucleation rate with respect to the supercooling which +means that the homogeneous nucleation line (HNL) +should be at a constant distance to the melting line in +this regime as predicted recently for this water model +[21] as well as for the mW model [60] in Ref. +[22]. +In Fig. 3 we show the estimates for the model [7, 21] +assuming that the HNL corresponds to an iso-nucleation +rate of log10 J/(m−3s−1) = 15 and we compare it to the +experimental HNL [4]. Also, the coexistence lines of the +model [21] and the experimental one [20] are presented +showing how the distance between the coexistence line +and the HNL is roughly constant until pressure increases +enough such that the required supercooling to reach +log10 J = 15 becomes larger. However, even though this +result might be useful, a physical explanation is still +missing. +In order to answer this question, we look at +the pressure-induced deceleration of ice nucleation. In +2016, Espinosa et al. [7] showed that the origin of this +phenomenon arises from the increase with pressure of +the interfacial free energy both at coexistence γm and +for nucleation (γ at a given supercooling ∆T ) while the +difference in chemical potential ∆µ does not change so +much with ∆T . Thus, one needs a larger ∆T to obtain +the same J at high pressure. In this work, we observe +approximately the same J as a function of ∆T from +strongly negative to moderately positive pressure. +Since we obtain roughly the same γ as a function +of ∆T at different negative pressures, we expect also +γm to barely change with p. +The term γm refers to +a planar interface between ice and water at certain +conditions along the coexistence line whereas the term +γ refers to a curved interface between a critical nucleus +of ice and water at a certain supercooling ∆T along +an isobar. +In both cases, thermodynamic equilibrium +holds. However, when the interface is planar then the +pressure is equal in both phases while in a spherical +interface the pressure changes between phases following +the Young-Laplace equation. +Then, we compute γm + +5 +10 +20 +30 +40 +50 +∆T [K] +0 +2000 +4000 +6000 +8000 +10000 +Nc +-2600 < p < 450 bar +2000 bar +a) +10 +20 +30 +40 +50 +∆T [K] +0 +200 +400 +600 +800 +1000 +1200 +1400 +∆Gc [kJ/mol] +b) +0 +10 +20 +30 +40 +50 +∆T [K] +15 +20 +25 +30 +35 +40 +γ [mJ/m +2] +c) +20 +30 +40 +50 +60 +∆T [K] +-300 +-240 +-180 +-120 +-60 +0 +60 +log10(J[m +-3s +-1]) +HNL +d) +FIG. 2. a) Critical nucleus size, b) nucleation free energy barrier, c) interfacial free energy, and d) log10 J against supercooling. +The same legend applies to all panels. The color indicates the pressure regime according to the legend. Points obtained in this +work are shown as solid symbols whereas results from Refs. [7, 21] as empty symbols. Black solid symbols are restricted to +pressures between -2600 and -1000 bar, whereas black empty symbols cover from -2600 up to 450 bar. Cyan empty symbols +correspond to 2000 bar. For each magnitude, a common fit to our data and those of Refs. [7, 21] is included. For panels a) +and b) a power law fit is used as given by Eq. 7 and Eq. 8 respectively, whereas for panel c) we use a linear fit (Eq. 9) and for +panel d) we use a CNT-based fit. HNL in panel d) is the iso-nucleation line of log10(J[m−3s−1]) = 15. +for several points. In addition to the negative pressure +isobars, we compute two points at 1000 bar and 2000 +bar respectively. We study only the basal plane as we +do not expect severe anisotropy (as much as ∼ 10%) +with the prismatic ones [39, 61–63]. +The results are +presented in Table II and in Fig. 4. As shown, γm barely +changes along the coexistence line when p varies from +strongly negative to moderately positive. Interestingly, +γm displays a shallow minimum. Thus, as long as ∆µ +does not change significantly with ∆T at negative p, +one can explain why in Fig. +2, Nc, ∆Gc, γ, and J +seem to be independent of p against the supercooling +when p is negative or moderate. +In order to confirm +this, we evaluate the effect of p on ∆µ as a function +of supercooling ∆T by comparing with the value at 1 +bar. To do so, we compute (∆µp − ∆µ1)/∆µ1 for the +different isobars p = -2600, -2000, -1000, 1, 2000 bar +as a function of ∆T (for 1 and 2000 bar we use the +data from Ref. [7]). As can be seen in Fig. 5 a), the +2000 bar isobar is very similar to the -1000 bar one in +terms of ∆µ with respect to ∆µ1, and the -2600 bar is +the one that deviates the most with up to 18%. This +deviation is however compensated in γ which is rather +dispersed and in the end Nc, ∆Gc, and J are very well +described by universal empirical expressions. Moreover, +in Fig. +5 b), we show ∆G obtained as Nc|∆µ|/2 by +setting Nc to the common fit of Eq. 7 and changing ∆µ +to that of the different isobars. As can be seen, from +strongly negative to moderately positive pressure, the +change in ∆µ does not significantly affect the free energy +barrier for isobars between -2600 to 450 bar. Thus, we +confirm that the universality in nucleation properties +presented in Fig. +1 and Fig. +2 is the consequence of +the small variation with p of the difference in chemical +potential ∆µ as well as in the interfacial free energy both +at coexistence γm and for the nucleation γ at a given ∆T . + +6 +180 +200 +220 +240 +260 +280 +Temperature [K] +-2000 +-1000 +0 +1000 +2000 +3000 +Pressure [bar] +THN TIP4P/Ice [Espinosa et al. PRL 2016] +THN TIP4P/Ice [Bianco at al. PRL 2021] +Tm TIP4P/Ice [Bianco et al. PRL 2021] +THN Exp +Tm Exp +∆T = 57.5 K +∆T = 36 K +FIG. 3. In solid lines, the coexistence lines Tm where blue +is experimental [20] and red is for the TIP4P/Ice [21]. The +dashed blue line corresponds to the experimental HNL [4]. +Empty red symbols correspond to simulation estimates for the +TIP4P/Ice of the HNL for log10 J/(m−3s−1) = 15 (the dashed +red line is a guide connecting these points). The turning point +of the melting curve of TIP4P/Ice occurs at 280K and -2000 +bar. +pm [bar] +Tm [K] +γm [mJ/m2] +-2600 +279.0 +27.1(1.5) +-2000 +280.0 +26.5(1.5) +-1000 +278.0 +25.6(1.5) +1 +270.0 +27.2 (0.8) +1000 +260.0 +29.0(1.5) +2000 +246.5 +37.2(1.5) +TABLE II. Interfacial free energy γm at different T −p points +of the coexistence line for the basal plane. Value at 1 bar is +from Ref. [39]. +-3000 +-2000 +-1000 +0 +1000 +2000 +Pressure [bar] +25 +30 +35 +40 +γm [mJ/m +2] +Espinosa et al. JPCC 2016 +This work +FIG. 4. Ice Ih-water interfacial free energy at coexistence for +the basal plane for the TIP4P/Ice model. +10 +15 +20 +25 +30 +35 +40 +45 +∆T [K] +5 +10 +15 +20 +25 +(∆µ1-∆µp)/∆µ1 (%) +2000 bar [Espinosa et al. PRL 2016] +-1000 bar +-2000 bar +-2600 bar +a) +10 +20 +30 +40 +50 +∆T [K] +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +∆Gc [kJ/mol] +2000 bar [Espinosa et al. PRL 2016] +(Nc∆µ-2600)/2 +(Nc∆µ-2000)/2 +(Nc∆µ-1000)/2 +(Nc∆µ1)/2 +∆Gc Common fit - Eq. 7 +b) +FIG. 5. a) Deviation in ∆µ at different isobars (-2600, -2000, +-1000, and 2000 bar) with respect to the one at 1 bar. b) In +dashed black (for 1 bar), green (for -1000 bar), red (for -2000 +bar), and blue (for -2600 bar) lines, we present free energy +barriers ∆Gc = Nc · ∆µ/2 where Nc(∆T ) is given by the +common fit of Eq. 7 and for ∆µ we use the corresponding +values for each isobar. In solid black line the common fit for +∆Gc proposed in Eq. 8 and in turquoise the fit for 2000 bar +from Ref. [7]. As black circles we show the data in the -2600 +bar < p < 450 bar regime, where solid circles are computed +in this work and empty come from Refs. [7, 21]. +B. +Interfacial free energy and melting line of the +ice Ih-water interface +We now understand the small variability with pressure +of the nucleation properties as a function of supercooling +at negative and moderate pressure. In order to under- +stand why γm displays a shallow minimum, we use the +thermodynamic formalism of Gibbs for interfaces [49, 64]. +The interfacial Gibbs-Duhem relation is given by, +dγm = −Γdµm − ηγdTm, +(10) +where Γ = Nγ/A is the surface excess density, also called +adsorption, and ηγ = Sγ/A is the excess contribution to + +7 +the entropy. Since the location of the dividing surface +is arbitrary, excess functions depend also on this choice +with the exception of γm. For a planar interface, γm does +not change with the location of the dividing surface un- +like in the case of curved interfaces, where γ does change +with its location [49, 51, 53]. The choice that most sim- +plifies the thermodynamic treatment in our case is the +equimolar dividing surface, usually denoted as the Gibbs +dividing surface, where the excess components Nγ is zero, +and so is Γ (see Appendix for a general dividing surface +treatment). Hence, we can write +dγm +dTm += −ηe +γ, +(11) +where the superscript e denotes the equimolar dividing +surface. Equation 11 provides us with the temperature +dependence of the interfacial free energy. +It is crucial +to note that this derivative must be taken along the co- +existence line so that p is not constant. In fact, we can +change Eq. 11 to describe the change of γm with pressure +along the melting line pm as, +dγm +dpm += −ηe +γ +dTm +dpm +. +(12) +In our case, Eq. +12 is more convenient due to the +reentrant behavior of the melting curve, i.e. +for each +Tm one has two values of pm whereas for each pm +there is only one value of Tm (see solid red curve in +Fig. 3). From Eq. 12, one can see that the change in +γm with pm is determined by the slope of the melting +line and the value of the excess entropy per area at +the equimolar dividing surface, ηe +γ. +This means that +if there is reentrant behavior for the melting point, +there must be reentrant behavior also for γm as a +function of pressure exactly at the same pm, because +ηe +γ must be finite. In fact, Bianco et al. [21] reported +reentrant behavior in the ice Ih-liquid coexistence line +of TIP4P/Ice, whose turning point occurred at -2000 bar. +Next, we want to confirm that the maximum in +the melting line Tm(pm) is consistent with the min- +imum in γm(pm) that we have obtained from the +mold integration technique. +Thus, we fit the data +for γm(pm) from mold integration with a quadratic +fit with the constraint of having the vertex at the +same pm (-2000 bar) as the quadratic fit for Tm(pm). +The latter, Tm(p) += +aTmp2 + bTmp + cTm has the +parameters cTm =271 K, bTm = −8.5 · 10−3 K/bar, +and aTm = −2 · 10−6 K/bar2. In this way, we assume +that ηe +γ is constant. The result is shown in Fig. 6. In +the left panel, we show the melting line with points +from the direct coexistence simulations of Ref. +[21] +and the quadratic fit. On the right panel, we show the +points of γm from mold integration from this work and +Ref. [39] as well as the quadratic fit. As can be seen +in the right panel, the fit is fairly good even though +we impose constant ηe +γ and quadratic fits with the +constraint of having the vertex at the same p. Therefore, +assuming that ηe +γ is constant seems to be a reasonable +approximation. +At this level of approximation, ηe +γ is found to be 0.32 +mJ/m2K. Notice that ηe +γ > 0 as expected from Eq. +12. For instance, from 1 bar to 2000 bar, Tm decreases +from 270 K to 246.5 K, and γm increases from 27.2 +mJ/m2 to 37.2 mJ/m2. Therefore, dγm/dpm > 0 and +dTm/dpm < 0, which means that ηe +γ is positive. On the +other side of the vertex, from -2600 bar to -2000 bar, +Tm increases from 279 K to 280K while γm decreases +from 27.1 mJ/m2 to 26.5 mJ/m2. Hence, dγm/dpm < +0 and dTm/dpm > 0 so that the same sign in ηe +γ holds. +Notice that Eq. +11 is only valid for planar interfaces +along the melting line. If one tries to apply this equation +away from of this line as was done in previous works +[7, 14, 65], probably one should incorporate terms that +account for the change in γ due to curvature. +Notice +also that the empirical relation proposed by Turnbull +which states that γm is proportional to the change in +melting enthalpy ∆Hm does not describe γm well at +high pressure. From 1 to 2000 bar, ∆Hm decreases from +1.44 kcal/mol to almost 1 kcal/mol in experiments [66] +and from 1.29 kcal/mol to approximately 1 kcal/mol [67] +for the TIP4P/Ice model. Thus, the Turnbull relation +predicts a decreasing γm, which is not supported by our +direct calculations via the mold integration technique. +As can be seen in Fig. 6, the knowledge of the equi- +librium melting curve, and the assumption of a constant +value for the interfacial excess entropy is sufficient to un- +derstand the complex variation of γm along the melting +line. Another relevant excess variable which depends on +γm, Tm, and ηe +γ is the excess energy ee +γ, +ee +γ = γm + Tmηe +γ. +(13) +The excess energy ee +γ is the difference in energy be- +tween the actual system having an interface and a vir- +tual system where the two phases remain unchanged up +to the dividing surface (the equimolar one in this case). +As a result of Eq. 12, the following relation holds, +dee +γ +dpm += Tm +dηe +γ +dpm +, +(14) +so that if ηe +γ is constant, then ee +γ must be constant as +well. If we approximate ηe +γ as constant with the value of +0.32 mJ/m2K, we find ee +γ = 115 mJ/m2. +IV. +CONCLUSIONS +In conclusion, we perform seeding simulations to study +ice nucleation at negative pressures. +Such conditions + +8 +-3000 +-1500 +0 +1500 +Pressure [bar] +25 +30 +35 +40 +γm [mJ/m +2] +fixed-vertex fit +-3000-1500 +0 +1500 +220 +230 +240 +250 +260 +270 +280 +Tm [K] +free fit +FIG. 6. Left: Melting temperature as a function of pressure. +Empty circles are from Ref. [21]. The line is a quadratic fit. +Right: Interfacial free energy as a function of pressure. Solid +points are from this work and empty points are from Ref. +[39]. The line is a quadratic fit constrained to have the vertex +at the same pressure (-2000 bar) than the quadratic fit of the +left panel. +can be relevant in porous media and water transport in +plants, where supercooled water can be at negative pres- +sure. By comparing with previous results, we show that +universal empirical expressions describe Nc, ∆Gc, γ, and +J, as a function of supercooling for isobars in the regime +from strongly negative (-2600 bar) to moderately pos- +itive pressures (500 bar). +Only when pressure is high +(2000 bar), these relations break down. In the regime +where pressure hardly plays any role, the isobaric Tolman +length is predicted to be positive and roughly constant +with the value of 0.24 nm. Also, our results suggest that +the homogeneous nucleation line should be parallel to the +coexistence line when pressure is below approximately +500 bar (while at higher pressure they are not). We ex- +plain this result by inspecting how the interfacial free +energy at coexistence changes with pressure. We evalu- +ate the interfacial free energy at coexistence at different +states from strongly negative to high pressure by means +of the mold integration technique. We show that the in- +terfacial free energy at coexistence barely changes with +pressure as long as the system is below 500 bar. In fact, +a shallow minimum is reported at negative pressure sug- +gesting that the minimum interfacial free energy between +ice Ih and water is around 26 ± 1 mJ/m2 for the basal +plane expanding for a broad range of pressure centered +around -2000 bar. Then, we use the Gibbsian formal- +ism to explain that this minimum in the interfacial free +energy is connected to a maximum in the melting tem- +perature as a function of pressure. In particular, we show +that the change in the interfacial free energy with pres- +sure is proportional to the excess entropy and to the slope +of the melting line. Thus, the reentrance in the interfacial +free energy occurs because of the reentrance in the melt- +ing line, which happens due to the cross-over in density +between ice and water. Finally, we estimate the excess +entropy and the excess energy of the ice Ih-water inter- +face. We suggest that a constant value of 0.32mJ/m2K +and 115 mJ/m2 respectively is enough to provide a good +description of the thermodynamics of the ice Ih-water +interface. +V. +ACKNOWLEDGMENTS +The authors thank Eduardo Sanz and Salvatore Ro- +mano for fruitful discussions. PMdH ackowledges sup- +port from the SFB TACO (project nr. F81-N) funded by +the Austrian Science Fund. JRE acknowledges funding +from the Oppenheimer Fellowship, the Roger Ekins Fel- +lowship from Emmanuel College, and a Ramon y Cajal +Fellowship (RYC2021-030937-I). CV acknowledges sup- +port from project PID2019-105898GB-C21 of the Minis- +terio de Educacion y Cultura. This work has been per- +formed using resources provided by the Spanish Super- +computing Network (RES), the Vienna Scientific Cluster +(VSC), and the Cambridge Tier-2 system operated by the +University of Cambridge Research Computing Service +(http://www.hpc.cam.ac.uk) funded by EPSRC Tier-2 +capital grant EP/P020259/1. +VI. +AUTHOR DECLARATIONS +A. +Conflict of Interest +The authors have no conflicts to disclose. +B. +Data availability +The data that support the findings of this study are +available from the corresponding author upon reasonable +request. +VII. +APPENDIX: INTERFACIAL FREE +ENERGY ALONG THE MELTING LINE FOR A +GENERAL DIVIDING SURFACE +In this work we used the equimolar dividing surface for +simplicity. However, Eqs. 11 and 12 can be generalized +for any choice of the dividing surface. To do so, it is nec- +essary to involve not only the interfacial Gibbs-Duhem +relation (10), but also the ice and liquid Gibbs-Duhem +relations. Respectively, these are, +dµm − vidpm + sidTm = 0, +(15) +dµm − vwdpm + swdTm = 0, +(16) + +9 +where v is the volume per molecule (the inverse of the +number density) and s is the entropy per molecule. Since +phase equilibrium holds, dµm, dpm, and dTm are common +in all phases. Notice that from Eq. 15 and Eq. 16, one +can obtain the Clausius-Clapeyron relation that explains +the slope of the melting line. +dTm +dpm += vw − vi +sw − si +. +(17) +By including also Eq. 10 in the relation, one can ob- +tain the temperature and pressure dependence of the in- +terfacial free energy without imposing a specific dividing +surface. 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Vega, Physical Chemistry +Chemical Physics 11, 556 (2009). + +-4000 +-2000 +0 +2000 +Pressure [bar] +0.0045 +0.005 +0.0055 +0.006 +0.0065 +Γ [nm +-2] +1 +10 +Time [ns] +1000 +1500 +2000 +2500 +3000 +3500 +Nice +Pressure: -2000 bar +Temperature: 255 K +0 +6 +12 +18 +24 +30 +Time [ns] +5000 +5500 +6000 +6500 +7000 +7500 +8000 +Nice +265 K +268 K +Pressure: -2000 bar +220 +230 +240 +250 +260 +Temperature [K] +-3000 +-2000 +-1000 +0 +1000 +2000 +3000 +Pressure [bar] +Ice Ih - Water Coexistence line +ρice(T,p) = ρw(T,p) +ρice(T,p) > ρw(T,p) +ρice(T,p) < ρw(T,p) +10 +15 +20 +25 +30 +35 +40 +45 +∆T [K] +0 +100 +200 +300 +400 +500 +∆Gc [kBT] +10 +20 +30 +40 +∆T [K] +0 +100 +200 +300 +400 +500 +600 +700 +∆Gc [kBT] +220 +230 +240 +250 +260 +270 +Temperature [K] +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +∆µ[kBT] +-1000 bar +-2000 bar +0 +10 +20 +30 +40 +∆T [K] +0 +0.1 +0.2 +0.3 +0.4 +∆µ[kBT] +-1000 bar +-2000 bar +0 +20 +40 +∆T [K] +0 +0.2 +0.4 +0.6 +0.8 +∆µ [kJ/mol] +2000 bar [Espinosa et al. PRL 2016] +1 bar [Espinosa et al. PRL 2016] +-1000 bar +-2000 bar +-2600 bar +-3000 +-2000 +-1000 +0 +1000 +Pressure [bar] +0 +0.04 +0.08 +0.12 +0.16 +|ρw - ρice| [g/cm +3] +230 K +240 K +255 K +15 +20 +25 +30 +35 +∆T [K] +0.8 +0.85 +0.9 +0.95 +1 +1.05 +1.1 +1.15 +1.2 +∆µp/∆µ1bar +-1000 bar +-2000 bar +-2600 bar +2000 bar [Espinosa et al. PRL 2016] +200 +210 +220 +230 +240 +250 +T [K] +10 +100 +1000 +10000 +Nc +2000 bar [Espinosa et al. PRL 2016] +1 bar [Espinosa et al. PRL 2016] +-1000 bar +-2000 bar +230 +240 +250 +260 +270 +Temperature [K] +-5 +-4 +-3 +-2 +-1 +0 +1 +µ - µcoex [kBT] +Pressure: -1000 bar +0 +20 +40 +60 +Time [ns] +0 +1000 +2000 +3000 +Nice +Pressure: -1000 bar +Temperature: 255 K +220 230 240 250 260 270 280 +Temperature [K] +0.89 +0.9 +0.91 +0.92 +0.93 +0.94 +0.95 +Density [g/cm +3] +Pressure: -1000 bar +220 +230 +240 +250 +260 +270 +Temperature [K] +0.0001 +0.001 +0.01 +0.1 +1 +D [ns +2/ns] +-1000 bar +-2000 bar +240 +250 +260 +270 +Temperature [K] +-5 +-4 +-3 +-2 +-1 +0 +1 +µ - µcoex [kBT] +Pressure: -2000 bar +240 +250 +260 +270 +280 +Temperature [K] +0.882 +0.884 +0.886 +0.888 +0.89 +0.892 +0.894 +0.896 +Density [g/cm +3] +Pressure: -2000 bar +0 +8 +16 +24 +Time [ns] +5000 +6000 +7000 +8000 +9000 +10000 +Nice +262 K +265 K +266 K +Pressure: -1000 bar +0 +8 +16 +24 +Time [ns] +0 +4000 +8000 +12000 +Nice +Pressure: -1000 bar +5 +10 +15 +20 +25 +30 +35 +40 +45 +∆T [K] +0.6 +0.7 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +∆µ +∆µ +∆µ +∆µ +γall +-2000 +-1000 +0 +1000 +Pressure [bar] +0 +50 +100 +150 +200 +250 +300 +350 +Excess energy [J/m +2] +γ2 +nd +γ3 +rd +γ4 +th +0.0036 +0.0038 +0.004 +0.0042 +1/Tm [K +-1] +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +eγ [J/m +2] +η(Tm) +constant η +220 +240 +260 +280 +Temperature [K] +-35 +-30 +-25 +-20 +Enthalpy [NkBT] +Pressure: -1000 bar +220 +240 +260 +280 +Temperature [K] +-36 +-32 +-28 +-24 +-20 +Enthalpy [NkBT] +Water +Ice +Pressure: -2000 bar +-0.02 +-0.01 +dTm/dp [K/bar] +-0.02 +0 +0.02 +0.04 +0.06 +dγ/dp [nm] +ηγ = 323.31 µJ/m +2K +This figure "etas.png" is available in "png"� format from: +http://arxiv.org/ps/2301.00178v1 + +-3000 +-2000 +-1000 +0 +1000 +Pressure [bar] +0 +0.25 +0.5 +0.75 +1 +Excess entropy [mJ/Km +2] +γ2 +nd +γ3 +rd +γ4 +th +-3000 +-1500 +0 +1500 +Pressure [bar] +25 +30 +35 +40 +γm [mJ/m +2] +-3000-1500 0 +1500 +-2000 +-1000 +0 +1000 +Pressure [bar] +25 +30 +35 +40 +γm [mJ/m +2] +2 +nd +3 +rd +4 +th +0 +10 +20 +30 +40 +∆T [K] +15 +20 +25 +30 +35 +40 +γ [mJ/m +2] +0.6 +0.8 +1 +1.2 +1.4 +kT/ε +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +γσ +2/ε +(100) Laird et al. JCP 2009 +(110) Laird et al. JCP 2009 +(111) Laird et al. JCP 2009 +LJ/Sol-Liq: η +300 +350 +400 +450 +500 +550 +T [K] +0 +20 +40 +60 +80 +100 +γ [mJ/m +2] +Experimental +TIP4P/Ice +Water/Liq-Vap: +230 +240 +250 +260 +270 +Temperature [K] +-2000 +-1000 +0 +1000 +2000 +3000 +Pressure [bar] +NNP +TIP4P/Ice (Bianco et al. PRL 2021) +-3000 +-2000 +-1000 +0 +1000 +Pressure [bar] +25 +30 +35 +40 +γm [mJ/m +2] +fixed-vertex fit +integration from -2600 bar + " -2000 bar + " -1000 bar + " 1 bar + " 1000 bar + " 2000 bar +This figure "integracion.png" is available in "png"� format from: +http://arxiv.org/ps/2301.00178v1 + +-3000 +-2000 +-1000 +0 +1000 +Pressure [bar] +25 +30 +35 +40 +γm [mJ/m +2] +Eq. (17) fit +integration from -2600 bar + " -2000 bar + " -1000 bar + " 1 bar + " 1000 bar + " 2000 bar +-4 +-2 +0 +2 +Area Deformation (%) +240 +250 +260 +270 +T (K) +This figure "skema.png" is available in "png"� format from: +http://arxiv.org/ps/2301.00178v1 + +This figure "sketi.png" is available in "png"� format from: +http://arxiv.org/ps/2301.00178v1 + +This figure "sketii.png" is available in "png"� format from: +http://arxiv.org/ps/2301.00178v1 + diff --git a/bdAyT4oBgHgl3EQfXPdL/content/tmp_files/load_file.txt b/bdAyT4oBgHgl3EQfXPdL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2e512a391ca878ebc833cb03f010a253a15be94c --- /dev/null +++ b/bdAyT4oBgHgl3EQfXPdL/content/tmp_files/load_file.txt @@ -0,0 +1,887 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf,len=886 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='00178v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='soft] 31 Dec 2022 Minimum in the pressure dependence of the interfacial free energy between ice Ih and water P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Montero de Hijes1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Espinosa2, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Vega2, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Dellago1,∗ 1Faculty of Physics, University of Vienna, A-1090 Vienna, Austria and 2Departamento de Qu´ımica F´ısica, Facultad de Ciencias Qu´ımicas, Universidad Complutense de Madrid, 28040 Madrid, Spain Despite the importance of ice nucleation, this process has been barely explored at negative pres- sures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Here, we study homogeneous ice nucleation in stretched water by means of Molecular Dy- namics Seeding simulations using the TIP4P/Ice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' We observe that the critical nucleus size, interfacial free energy, free energy barrier, and nucleation rate barely change between isobars from 2600 to 500 bar when they are represented as a function of supercooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' This allows us to identify universal empirical expressions for homogeneous ice nucleation in the pressure range from -2600 to 500 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' We show that this universal behavior arises from the pressure dependence of the inter- facial free energy which we compute by means of the mold integration technique finding a shallow minimum around -2000 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Likewise, we show that the change in the interfacial free energy with pressure is proportional to the excess entropy and the slope of the melting line, exhibiting the latter a reentrant behavior also at the same negative pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Finally, we estimate the excess energy and the excess entropy of the ice Ih-water interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' ∗christoph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='dellago@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='at I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' INTRODUCTION Water crystallization is an essential phase transition in nature and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' However, in the cryopreser- vation of biological samples [1, 2], ice formation can be disastrous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The low temperature preserves the biological material but causes the water within the sample to be in a metastable state subject to crystallization [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Interestingly, by keeping the sample under high pressure, ice nuclei are less likely to form keeping water liquid for a longer time [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The frequency of the nucleation pro- cess is mainly determined by the thermodynamic driving force and the cost of creating the interface between the emerging nucleus and the metastable liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The reason why high pressure slows down the nucleation process is that the difference in chemical potential between ice and water, ∆µ, which represents the thermodynamic driving force, barely changes between isobars whereas the cost of creating the interface notably increases [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The interfacial free energy is the variable that quantifies this cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' At coexistence, through a planar interface, the interfacial free energy γm differs from the value for a critical nucleus γ due to the curvature of the surface [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Nevertheless, both notably increase at high pressure [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Homogeneous ice nucleation at standard and high pressure has been extensively explored [7, 9–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' How- ever, ice nucleation in water under negative pressure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' stretched water, has caught less attention [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' This process is relevant in porous media containing water solutions [23] and also in water transpiration inside plants [24] where negative pressure occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Creating and maintaining negative pressure over a sample is non-trivial in experiments [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' This is because a liquid at negative pressure is metastable [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In general, this metastability is considered with respect to the vapor phase [28, 29], although at certain conditions, it can be also metastable with respect to ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Some ingenious approaches to create negative pressure in metastable water include the use of a Berthelot tube [30], cen- trifugation [31], and more recently the use of acoustic waves [25, 29, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In contrast, in computer simula- tions is straightforward to work under negative pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In this work, we investigate how ice nucleation properties are affected by negative pressure at different degrees of supercooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In fact, we find little effect when pressure changes from strongly negative to moderately positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' We investigate the role of the interfacial free energy since it is a key property in determining the phase behavior of water at high pressure [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' We find that the slope of the melting line is crucial to describe the change with pressure of the interfacial free energy which displays a shallow minimum at negative pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Our study is based on molecular dynamics simulations with the TIP4P/Ice model [33] which has been extensively used to describe ice nucleation [7, 9, 12, 21] and growth [34, 35] as well as in supercooled water [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In particular, we employ the seeding technique [9, 38] to study nucleation and the mold integration technique [39] to measure the interfacial free energy at coexistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' SIMULATION METHODS All simulations have been done with the GROMACS package (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='7-version in double precision) with the TIP4P/Ice water model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The simulations are performed in the isothermal-isobaric (NpT) ensemble with a time step of 2 fs using the Noose-Hoover thermostat [40, 41] and the Parrinello-Rahman barostat [42] both with a relaxation time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Electrostatic interactions are 2 accounted for via the particle-mesh-Ewald summation algorithm [43] with order 4 and a Fourier spacing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The cutoff for the Lennard-Jones and the Coulombic interactions is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='9 nm and long-range corrections to the Lennard-Jones part of the potential are included in energy and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' To study nucleation we use the seeding technique [9, 44, 45] which involves the combination of molecular dynamics simulations and Classical Nucleation Theory (CNT) [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' This technique is based on the behavior of a critical nucleus which has equal probability of grow- ing and melting when surrounded by the metastable phase at the critical pressure and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In practice, one inserts a spherical ice-Ih seed in metastable water and then keeps track of the time evolution of the size of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' One can vary T , p and the seed size in order to find at which conditions a certain nucleus size is critical (Nc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Once Nc is known, CNT is used to find the interfacial free energy γ, the barrier height ∆Gc, and the nucleation rate J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Our system sizes ranged between 80000 and 250000 water molecules in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The duration of the trajectories is between 40 and 115 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' It is important to note that in the Gibbsian descrip- tion of interfaces, one has two bulk phases separated by a dividing surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' However, there is some arbitrariness in the location of the dividing surface which also affects to the interfacial free energy γ when the interface has curvature [48–51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Within the CNT framework, the relevant dividing surface is the surface of tension [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In order to find the surface of tension we employ an empirical approach that has been successfully applied in crystal nucleation for a large variety of systems [7, 21, 38, 39, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In this approach, the averaged Steinhardt bond order parameter [56], ¯q6(T, p) is used in combination with the mislabelling criterion [9] to identify ice-like and water-like molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Within a cutoff distance of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5 ˚A, we obtain ¯q6(T, p) for each molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The molecules with ¯q6(T, p) above a certain threshold ¯ q6,t(T, p) are labeled as ice whereas those below are labeled as liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' This threshold depends weakly on the considered thermodynamic range covering pressures from -2600 to -1000 bar and temperatures from 250 to 270 K (see the supplementary material in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [21] for the isothermal change in ¯ q6,t(T, p) with pressure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In this work, the value changes between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='365 for the highest temperature and pressure to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='385 for the lowest temperature and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Once Nc is known, we employ the CNT equations [46, 47] to determine other important parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The interfacial free energy γ is given as γ = �3Ncρ2 ice|∆µ|3 32π �1/3 , (1) where Nc is the size of the critical nucleus, ρice is the number density of ice-Ih in the bulk at the metastable conditions at which the nucleus is critical, and |∆µ| is known as the driving force to nucleation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' the dif- ference in chemical potential between the liquid and ice phases at the conditions which cause the nucleus to be critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' This property can be obtained by thermody- namic integration along an isobar [57], ���� ∆µ kBT ���� = ���� � T Tm 1 kBT 2 �Hice Nice − Hw Nw � dT ����, (2) where kB is the Boltzmann constant, Tm is the melting temperature, and H the enthalpy, which can be obtained from simulations of bulk ice-Ih and bulk water along the isobar of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Then, the free energy barrier is given as ∆Gc = 16πγ3 3ρ2 ice|∆µ|2 = Nc|∆µ| 2 , (3) which allows us to obtain the nucleation rate J, the num- ber of critical nuclei forming per unit of time and volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' According to CNT, J is given as J = ρw � |∆µ| 6πkBT Nc f + exp � −∆Gc kBT � , (4) where f + is the attachment rate which can be approxi- mated through this expression [7, 21] f + = 24DwN 2/3 c λ2 , (5) where Dw is the diffusion coefficient of the metastable liquid and λ is a characteristic length, the typical distance that a water molecule covers in order to attach into the nucleus, whose value is approximately 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8 ˚A for water [7, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' To find the ice-Ih-water interfacial free energy at co- existence for a planar interface, γm, we use the mold integration technique [39], which consists in computing the reversible work W that is necessary to form a crys- tal slab within a liquid at coexistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' This work is re- lated to the interfacial free energy at coexistence, γm, by W = 2Aγm where A is the interfacial area and the num- ber 2 accounts for the two interfaces of the slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The slab formation is induced by switching on an attractive interaction between the mold of potential energy wells and the particles of the initial liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The wells are ar- ranged in the equilibrium positions of the oxygen atoms in the ice facet under investigation at coexistence condi- tions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' for temperatures and pressures located along the ice Ih-water equilibrium line for the TIP4P/Ice water model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' First, one has to obtain γrw, which is given as 3 γrw = 1 2A � ǫwNw − � ǫw 0 ⟨N(ǫ)⟩dǫ � , (6) where rw indicates the radius of the potential wells and ǫ is their energy (with maximum depth equal to ǫw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Nw is the number of wells in the mold and ⟨N(ǫ)⟩ is the average number of occupied wells at a given potential depth ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The integration needs to be reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' To ensure this, thermodynamic integration is performed for wells whose radius is larger than a certain value r0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' At r0 w the slab is fully formed and the stability no longer depends on the mold-liquid interactions, hence, leading to potentially irreversible ice growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' However, since this is the radius that recovers the actual value of γm, thermodynamic integration is repeated for several values of rw < r0 w and then γrw is extrapolated to its value at r0 w giving γm [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Universality in ice nucleation variables at negative and moderate pressure First, we study nucleation along the isobars of 2600, -2000, and -1000 bar by means of the seeding approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' For pressures below -3000 bar we observed spontaneous cavitation occurring within the time scale of the trajectories needed in the seeding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' We obtain the critical nucleus size Nc, the driving force to nucleation |∆µ|, the interfacial free energy γ, the free energy barrier to nucleation ∆Gc and the nucleation rate J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' These results are presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' As can be seen, even though the pressure significantly differs, the results are surprisingly similar for nuclei of similar size for equivalent supercoolings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' This behaviour is considerably different from what has been found when comparing the nucleation scenario of normal vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' high pressure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 2000 bar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [7]), where the increase in pressure brings down the ice nucleation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' To further understand this behavior, we connect our results with those from previous works where nucleation had been studied for the TIP4P/Ice model at different pressures including negative, moderate, and high pres- sure states [7, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 1 a) we show the critical nucleus size as a function of supercooling for several isobars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' We provide results at moderate supercoolings at -2600, -2000, and -1000 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' For these same isobars as well as for the 1 bar isobar, we show the values reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' For the 1 bar isobar, we also show the values given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [7], which also provides with the values at the 2000 bar isobar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' As can be seen, only the points corresponding to the 2000 bar isobar [7] exhibit a different trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The isobars at -2600, -2000, and -1000 bar from this work as well as from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [21], and the 1 bar isobar from both Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [7, 21] follow approximately the same curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Notice that even a point at 450 bar reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [21] was included being in agreement with this group of isobars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In fact, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 1 b), pressure hardly affects the nucleation free energy barrier as a function of supercooling from -2600 bar to 450 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Our results from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 1 suggest that a similar nucleation behaviour as a function of supercooling may take place from -2600 to 450 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' That is a strikingly different behavior to the one observed when increasing pressure to 2000 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Thus, we propose universal empirical expressions for the variation of different homo- geneous ice nucleation properties with the supercooling independently of the pressure as long as it is within this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Nevertheless, we first need to confirm that what was observed for Nc and ∆Gc also applies to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 2 we show again a) Nc and b) ∆Gc as well as c) γ and d) log10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' This time, for each magnitude, we include a common fit to data from moderately positive to deeply negative pressure (including our own data and those from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [7, 21]) along a separate fit at high pressure [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In c) we show γ which exhibits higher variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Finally, in d), we show how very different pressures (from largely negative to moderately positive) lead to approximately the same nucleation rate J as a function of supercooling, ∆T = Tm − T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The values of Tm are given in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Hence, we can use the respective common fit as universal empirical expressions to describe the change with supercooling within this broad range of pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' For Nc we obtain Nc(∆T ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='2 · 107 · �∆T T0 �−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8 (7) and for ∆Gc (in kJ/mol) ∆Gc(∆T ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='3 · 105 · �∆T T0 �−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='1 , (8) where T0 equals 1 K for correctness of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' For γ in mJ/m2, we obtain γ(∆T ) = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='6 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='174 · ∆T, (9) and, finally, for J in m−3 s−1, one should use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 4 along with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 8 (after converting into in kBT units), and 1036 m−3s−1 as the prefactor (ρw � |∆µ|/(6πkBT Nc)f +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 2 have interesting con- sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' First, taking into account that Nc and γ (panels a) and c) respectively) are roughly independent of p when it goes from largely negative to moderately positive pressures, the isobaric Tolman length which determines the change in γ with the inverse of the radius of curvature of the cluster along an isobar [8, 58, 59] is 4 Nc T [K] ∆T [K] p [bar] ρw [g/cm3] ρice [g/cm3] |∆µ| [kJ/mol] γ [mJ/m2] ∆Gc [kJ/mol] log10(J [m−3s−1]) 1650 255 23 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='9208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='367 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='62 303 24 7450 264 14 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='9285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='237 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='03 883 136 1750 255 25 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8855 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='367 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='87 321 28 7600 266 14 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8876 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='224 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='74 850 128 1950 255 24 2600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='340 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='89 332 30 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Seeding results in tabular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Nc is the critical nucleus size, T and p are the thermodynamic conditions that make such nucleus size to be critical, and ∆T is the supercooling, Tm − T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The densities of water ρw and ice ρice are also shown, as well as the interfacial free energy at nucleation γ, the barrier height ∆Gc, and the base-10 logarithm of the nucleation rate log10(J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 10 20 30 40 50 ∆T [K] 0 2000 4000 6000 8000 10000 Nc 2000 bar [Espinosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2016] 1 bar [Espinosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2016] 450 bar [Bianco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2021] 1 bar [Bianco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2021] 1000 bar [Bianco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2021] 2000 bar [Bianco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2021] 2600 bar [Bianco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2021] 1000 bar [This work] 2000 bar [This work] 2600 bar [This work] a) 10 20 30 40 50 ∆T [K] 0 200 400 600 800 1000 1200 1400 ∆Gc [kJ/mol] b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' a) Critical nucleus size and b) free energy barrier to undergo nucleation against supercooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The same legend applies in both panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Numerical details can be seen in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The color indicates the pressure, whereas solid symbols correspond to simulations performed in this work, and empty symbols correspond to data obtained from previous work as indicated in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' For the same pressure but different work we use different symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The lines are power law fits to points sharing the same pressure independently on the work in which they were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' roughly constant too and equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='24(5) nm, where the parenthesis indicates uncertainty in the last digit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' This result is in agreement with previous work [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Second, in panel d) one can see that from strongly negative to moderately positive pressure we obtain the same nucleation rate with respect to the supercooling which means that the homogeneous nucleation line (HNL) should be at a constant distance to the melting line in this regime as predicted recently for this water model [21] as well as for the mW model [60] in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 3 we show the estimates for the model [7, 21] assuming that the HNL corresponds to an iso-nucleation rate of log10 J/(m−3s−1) = 15 and we compare it to the experimental HNL [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Also, the coexistence lines of the model [21] and the experimental one [20] are presented showing how the distance between the coexistence line and the HNL is roughly constant until pressure increases enough such that the required supercooling to reach log10 J = 15 becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' However, even though this result might be useful, a physical explanation is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In order to answer this question, we look at the pressure-induced deceleration of ice nucleation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In 2016, Espinosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [7] showed that the origin of this phenomenon arises from the increase with pressure of the interfacial free energy both at coexistence γm and for nucleation (γ at a given supercooling ∆T ) while the difference in chemical potential ∆µ does not change so much with ∆T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Thus, one needs a larger ∆T to obtain the same J at high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In this work, we observe approximately the same J as a function of ∆T from strongly negative to moderately positive pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Since we obtain roughly the same γ as a function of ∆T at different negative pressures, we expect also γm to barely change with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The term γm refers to a planar interface between ice and water at certain conditions along the coexistence line whereas the term γ refers to a curved interface between a critical nucleus of ice and water at a certain supercooling ∆T along an isobar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In both cases, thermodynamic equilibrium holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' However, when the interface is planar then the pressure is equal in both phases while in a spherical interface the pressure changes between phases following the Young-Laplace equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Then, we compute γm 5 10 20 30 40 50 ∆T [K] 0 2000 4000 6000 8000 10000 Nc 2600 < p < 450 bar 2000 bar a) 10 20 30 40 50 ∆T [K] 0 200 400 600 800 1000 1200 1400 ∆Gc [kJ/mol] b) 0 10 20 30 40 50 ∆T [K] 15 20 25 30 35 40 γ [mJ/m 2] c) 20 30 40 50 60 ∆T [K] 300 240 180 120 60 0 60 log10(J[m 3s 1]) HNL d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' a) Critical nucleus size, b) nucleation free energy barrier, c) interfacial free energy, and d) log10 J against supercooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The same legend applies to all panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The color indicates the pressure regime according to the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Points obtained in this work are shown as solid symbols whereas results from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [7, 21] as empty symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Black solid symbols are restricted to pressures between -2600 and -1000 bar, whereas black empty symbols cover from -2600 up to 450 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Cyan empty symbols correspond to 2000 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' For each magnitude, a common fit to our data and those of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [7, 21] is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' For panels a) and b) a power law fit is used as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 7 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 8 respectively, whereas for panel c) we use a linear fit (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 9) and for panel d) we use a CNT-based fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' HNL in panel d) is the iso-nucleation line of log10(J[m−3s−1]) = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' for several points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In addition to the negative pressure isobars, we compute two points at 1000 bar and 2000 bar respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' We study only the basal plane as we do not expect severe anisotropy (as much as ∼ 10%) with the prismatic ones [39, 61–63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The results are presented in Table II and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' As shown, γm barely changes along the coexistence line when p varies from strongly negative to moderately positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Interestingly, γm displays a shallow minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Thus, as long as ∆µ does not change significantly with ∆T at negative p, one can explain why in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 2, Nc, ∆Gc, γ, and J seem to be independent of p against the supercooling when p is negative or moderate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In order to confirm this, we evaluate the effect of p on ∆µ as a function of supercooling ∆T by comparing with the value at 1 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' To do so, we compute (∆µp − ∆µ1)/∆µ1 for the different isobars p = -2600, -2000, -1000, 1, 2000 bar as a function of ∆T (for 1 and 2000 bar we use the data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 5 a), the 2000 bar isobar is very similar to the -1000 bar one in terms of ∆µ with respect to ∆µ1, and the -2600 bar is the one that deviates the most with up to 18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' This deviation is however compensated in γ which is rather dispersed and in the end Nc, ∆Gc, and J are very well described by universal empirical expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Moreover, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 5 b), we show ∆G obtained as Nc|∆µ|/2 by setting Nc to the common fit of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 7 and changing ∆µ to that of the different isobars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' As can be seen, from strongly negative to moderately positive pressure, the change in ∆µ does not significantly affect the free energy barrier for isobars between -2600 to 450 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Thus, we confirm that the universality in nucleation properties presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 2 is the consequence of the small variation with p of the difference in chemical potential ∆µ as well as in the interfacial free energy both at coexistence γm and for the nucleation γ at a given ∆T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 6 180 200 220 240 260 280 Temperature [K] 2000 1000 0 1000 2000 3000 Pressure [bar] THN TIP4P/Ice [Espinosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2016] THN TIP4P/Ice [Bianco at al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2021] Tm TIP4P/Ice [Bianco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2021] THN Exp Tm Exp ∆T = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5 K ∆T = 36 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In solid lines, the coexistence lines Tm where blue is experimental [20] and red is for the TIP4P/Ice [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The dashed blue line corresponds to the experimental HNL [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Empty red symbols correspond to simulation estimates for the TIP4P/Ice of the HNL for log10 J/(m−3s−1) = 15 (the dashed red line is a guide connecting these points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The turning point of the melting curve of TIP4P/Ice occurs at 280K and -2000 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' pm [bar] Tm [K] γm [mJ/m2] 2600 279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='1(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5) 2000 280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5) 1000 278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='6(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5) 1 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8) 1000 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='0(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5) 2000 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='2(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5) TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Interfacial free energy γm at different T −p points of the coexistence line for the basal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Value at 1 bar is from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 3000 2000 1000 0 1000 2000 Pressure [bar] 25 30 35 40 γm [mJ/m 2] Espinosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' JPCC 2016 This work FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Ice Ih-water interfacial free energy at coexistence for the basal plane for the TIP4P/Ice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 10 15 20 25 30 35 40 45 ∆T [K] 5 10 15 20 25 (∆µ1-∆µp)/∆µ1 (%) 2000 bar [Espinosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2016] 1000 bar 2000 bar 2600 bar a) 10 20 30 40 50 ∆T [K] 0 200 400 600 800 1000 1200 1400 1600 ∆Gc [kJ/mol] 2000 bar [Espinosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2016] (Nc∆µ-2600)/2 (Nc∆µ-2000)/2 (Nc∆µ-1000)/2 (Nc∆µ1)/2 ∆Gc Common fit - Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 7 b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' a) Deviation in ∆µ at different isobars (-2600, -2000, 1000, and 2000 bar) with respect to the one at 1 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' b) In dashed black (for 1 bar), green (for -1000 bar), red (for -2000 bar), and blue (for -2600 bar) lines, we present free energy barriers ∆Gc = Nc · ∆µ/2 where Nc(∆T ) is given by the common fit of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 7 and for ∆µ we use the corresponding values for each isobar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In solid black line the common fit for ∆Gc proposed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 8 and in turquoise the fit for 2000 bar from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' As black circles we show the data in the -2600 bar < p < 450 bar regime, where solid circles are computed in this work and empty come from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [7, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Interfacial free energy and melting line of the ice Ih-water interface We now understand the small variability with pressure of the nucleation properties as a function of supercooling at negative and moderate pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In order to under- stand why γm displays a shallow minimum, we use the thermodynamic formalism of Gibbs for interfaces [49, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The interfacial Gibbs-Duhem relation is given by, dγm = −Γdµm − ηγdTm, (10) where Γ = Nγ/A is the surface excess density, also called adsorption, and ηγ = Sγ/A is the excess contribution to 7 the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Since the location of the dividing surface is arbitrary, excess functions depend also on this choice with the exception of γm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' For a planar interface, γm does not change with the location of the dividing surface un- like in the case of curved interfaces, where γ does change with its location [49, 51, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The choice that most sim- plifies the thermodynamic treatment in our case is the equimolar dividing surface, usually denoted as the Gibbs dividing surface, where the excess components Nγ is zero, and so is Γ (see Appendix for a general dividing surface treatment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Hence, we can write dγm dTm = −ηe γ, (11) where the superscript e denotes the equimolar dividing surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Equation 11 provides us with the temperature dependence of the interfacial free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' It is crucial to note that this derivative must be taken along the co- existence line so that p is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In fact, we can change Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 11 to describe the change of γm with pressure along the melting line pm as, dγm dpm = −ηe γ dTm dpm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' (12) In our case, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 12 is more convenient due to the reentrant behavior of the melting curve, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' for each Tm one has two values of pm whereas for each pm there is only one value of Tm (see solid red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 12, one can see that the change in γm with pm is determined by the slope of the melting line and the value of the excess entropy per area at the equimolar dividing surface, ηe γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' This means that if there is reentrant behavior for the melting point, there must be reentrant behavior also for γm as a function of pressure exactly at the same pm, because ηe γ must be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In fact, Bianco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [21] reported reentrant behavior in the ice Ih-liquid coexistence line of TIP4P/Ice, whose turning point occurred at -2000 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Next, we want to confirm that the maximum in the melting line Tm(pm) is consistent with the min- imum in γm(pm) that we have obtained from the mold integration technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Thus, we fit the data for γm(pm) from mold integration with a quadratic fit with the constraint of having the vertex at the same pm (-2000 bar) as the quadratic fit for Tm(pm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The latter, Tm(p) = aTmp2 + bTmp + cTm has the parameters cTm =271 K, bTm = −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5 · 10−3 K/bar, and aTm = −2 · 10−6 K/bar2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In this way, we assume that ηe γ is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In the left panel, we show the melting line with points from the direct coexistence simulations of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [21] and the quadratic fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' On the right panel, we show the points of γm from mold integration from this work and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [39] as well as the quadratic fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' As can be seen in the right panel, the fit is fairly good even though we impose constant ηe γ and quadratic fits with the constraint of having the vertex at the same p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Therefore, assuming that ηe γ is constant seems to be a reasonable approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' At this level of approximation, ηe γ is found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='32 mJ/m2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Notice that ηe γ > 0 as expected from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' For instance, from 1 bar to 2000 bar, Tm decreases from 270 K to 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5 K, and γm increases from 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='2 mJ/m2 to 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='2 mJ/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Therefore, dγm/dpm > 0 and dTm/dpm < 0, which means that ηe γ is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' On the other side of the vertex, from -2600 bar to -2000 bar, Tm increases from 279 K to 280K while γm decreases from 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='1 mJ/m2 to 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5 mJ/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Hence, dγm/dpm < 0 and dTm/dpm > 0 so that the same sign in ηe γ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Notice that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 11 is only valid for planar interfaces along the melting line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' If one tries to apply this equation away from of this line as was done in previous works [7, 14, 65], probably one should incorporate terms that account for the change in γ due to curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Notice also that the empirical relation proposed by Turnbull which states that γm is proportional to the change in melting enthalpy ∆Hm does not describe γm well at high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' From 1 to 2000 bar, ∆Hm decreases from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='44 kcal/mol to almost 1 kcal/mol in experiments [66] and from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='29 kcal/mol to approximately 1 kcal/mol [67] for the TIP4P/Ice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Thus, the Turnbull relation predicts a decreasing γm, which is not supported by our direct calculations via the mold integration technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 6, the knowledge of the equi- librium melting curve, and the assumption of a constant value for the interfacial excess entropy is sufficient to un- derstand the complex variation of γm along the melting line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Another relevant excess variable which depends on γm, Tm, and ηe γ is the excess energy ee γ, ee γ = γm + Tmηe γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' (13) The excess energy ee γ is the difference in energy be- tween the actual system having an interface and a vir- tual system where the two phases remain unchanged up to the dividing surface (the equimolar one in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' As a result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 12, the following relation holds, dee γ dpm = Tm dηe γ dpm , (14) so that if ηe γ is constant, then ee γ must be constant as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' If we approximate ηe γ as constant with the value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='32 mJ/m2K, we find ee γ = 115 mJ/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' CONCLUSIONS In conclusion, we perform seeding simulations to study ice nucleation at negative pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Such conditions 8 3000 1500 0 1500 Pressure [bar] 25 30 35 40 γm [mJ/m 2] fixed-vertex fit 3000-1500 0 1500 220 230 240 250 260 270 280 Tm [K] free fit FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Left: Melting temperature as a function of pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Empty circles are from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The line is a quadratic fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Right: Interfacial free energy as a function of pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Solid points are from this work and empty points are from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' The line is a quadratic fit constrained to have the vertex at the same pressure (-2000 bar) than the quadratic fit of the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' can be relevant in porous media and water transport in plants, where supercooled water can be at negative pres- sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' By comparing with previous results, we show that universal empirical expressions describe Nc, ∆Gc, γ, and J, as a function of supercooling for isobars in the regime from strongly negative (-2600 bar) to moderately pos- itive pressures (500 bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Only when pressure is high (2000 bar), these relations break down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In the regime where pressure hardly plays any role, the isobaric Tolman length is predicted to be positive and roughly constant with the value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='24 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Also, our results suggest that the homogeneous nucleation line should be parallel to the coexistence line when pressure is below approximately 500 bar (while at higher pressure they are not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' We ex- plain this result by inspecting how the interfacial free energy at coexistence changes with pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' We evalu- ate the interfacial free energy at coexistence at different states from strongly negative to high pressure by means of the mold integration technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' We show that the in- terfacial free energy at coexistence barely changes with pressure as long as the system is below 500 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In fact, a shallow minimum is reported at negative pressure sug- gesting that the minimum interfacial free energy between ice Ih and water is around 26 ± 1 mJ/m2 for the basal plane expanding for a broad range of pressure centered around -2000 bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Then, we use the Gibbsian formal- ism to explain that this minimum in the interfacial free energy is connected to a maximum in the melting tem- perature as a function of pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' In particular, we show that the change in the interfacial free energy with pres- sure is proportional to the excess entropy and to the slope of the melting line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Thus, the reentrance in the interfacial free energy occurs because of the reentrance in the melt- ing line, which happens due to the cross-over in density between ice and water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Finally, we estimate the excess entropy and the excess energy of the ice Ih-water inter- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' We suggest that a constant value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='32mJ/m2K and 115 mJ/m2 respectively is enough to provide a good description of the thermodynamics of the ice Ih-water interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank Eduardo Sanz and Salvatore Ro- mano for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PMdH ackowledges sup- port from the SFB TACO (project nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' F81-N) funded by the Austrian Science Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' JRE acknowledges funding from the Oppenheimer Fellowship, the Roger Ekins Fel- lowship from Emmanuel College, and a Ramon y Cajal Fellowship (RYC2021-030937-I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' CV acknowledges sup- port from project PID2019-105898GB-C21 of the Minis- terio de Educacion y Cultura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' This work has been per- formed using resources provided by the Spanish Super- computing Network (RES), the Vienna Scientific Cluster (VSC), and the Cambridge Tier-2 system operated by the University of Cambridge Research Computing Service (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='hpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='uk) funded by EPSRC Tier-2 capital grant EP/P020259/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' AUTHOR DECLARATIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Conflict of Interest The authors have no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Data availability The data that support the findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' APPENDIX: INTERFACIAL FREE ENERGY ALONG THE MELTING LINE FOR A GENERAL DIVIDING SURFACE In this work we used the equimolar dividing surface for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' However, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 11 and 12 can be generalized for any choice of the dividing surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' To do so, it is nec- essary to involve not only the interfacial Gibbs-Duhem relation (10), but also the ice and liquid Gibbs-Duhem relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Respectively, these are, dµm − vidpm + sidTm = 0, (15) dµm − vwdpm + swdTm = 0, (16) 9 where v is the volume per molecule (the inverse of the number density) and s is the entropy per molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Since phase equilibrium holds, dµm, dpm, and dTm are common in all phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Notice that from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 15 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 16, one can obtain the Clausius-Clapeyron relation that explains the slope of the melting line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' dTm dpm = vw − vi sw − si .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' (17) By including also Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 10 in the relation, one can ob- tain the temperature and pressure dependence of the in- terfacial free energy without imposing a specific dividing surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' For the temperature, one obtains, dγm dTm = � Γvwsi − visw vw − vi − ηγ � , (18) whereas for the pressure, one finds dγm dpm = � Γvwsi − visw vw − vi − ηγ � dTm dpm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' (19) As can be seen, at the equimolar dividing surface where Γ = 0, one recovers Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 11 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 12 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' These expressions are relevant when nucleation data are extrapolated to coexistence because the relevant dividing surface in nucleation is usually the surface of tension for which Γ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Geidobler and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Winter, European Journal of Phar- maceutics and Biopharmaceutics 85, 214 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' [2] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Xue, H.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Sanz, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' Vega, Physical Chemistry Chemical Physics 11, 556 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' 4000 2000 0 2000 Pressure [bar] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='0045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='0055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='0065 Γ [nm 2] 1 10 Time [ns] 1000 1500 2000 2500 3000 3500 Nice Pressure: -2000 bar Temperature: 255 K 0 6 12 18 24 30 Time [ns] 5000 5500 6000 6500 7000 7500 8000 Nice 265 K 268 K Pressure: -2000 bar 220 230 240 250 260 Temperature [K] 3000 2000 1000 0 1000 2000 3000 Pressure [bar] Ice Ih - Water Coexistence line ρice(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='p) = ρw(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='p) ρice(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='p) > ρw(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='p) ρice(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='p) < ρw(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='p) 10 15 20 25 30 35 40 45 ∆T [K] 0 100 200 300 400 500 ∆Gc [kBT] 10 20 30 40 ∆T [K] 0 100 200 300 400 500 600 700 ∆Gc [kBT] 220 230 240 250 260 270 Temperature [K] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='4 ∆µ[kBT] 1000 bar 2000 bar 0 10 20 30 40 ∆T [K] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='4 ∆µ[kBT] 1000 bar 2000 bar 0 20 40 ∆T [K] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8 ∆µ [kJ/mol] 2000 bar [Espinosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2016] 1 bar [Espinosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2016] 1000 bar 2000 bar 2600 bar 3000 2000 1000 0 1000 Pressure [bar] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='16 |ρw - ρice| [g/cm 3] 230 K 240 K 255 K 15 20 25 30 35 ∆T [K] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='2 ∆µp/∆µ1bar 1000 bar 2000 bar 2600 bar 2000 bar [Espinosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2016] 200 210 220 230 240 250 T [K] 10 100 1000 10000 Nc 2000 bar [Espinosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2016] 1 bar [Espinosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2016] 1000 bar 2000 bar 230 240 250 260 270 Temperature [K] 5 4 3 2 1 0 1 µ - µcoex [kBT] Pressure: -1000 bar 0 20 40 60 Time [ns] 0 1000 2000 3000 Nice Pressure: -1000 bar Temperature: 255 K 220 230 240 250 260 270 280 Temperature [K] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='95 Density [g/cm 3] Pressure: -1000 bar 220 230 240 250 260 270 Temperature [K] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='1 1 D [ns 2/ns] 1000 bar 2000 bar 240 250 260 270 Temperature [K] 5 4 3 2 1 0 1 µ - µcoex [kBT] Pressure: -2000 bar 240 250 260 270 280 Temperature [K] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='886 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='896 Density [g/cm 3] Pressure: -2000 bar 0 8 16 24 Time [ns] 5000 6000 7000 8000 9000 10000 Nice 262 K 265 K 266 K Pressure: -1000 bar 0 8 16 24 Time [ns] 0 4000 8000 12000 Nice Pressure: -1000 bar 5 10 15 20 25 30 35 40 45 ∆T [K] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='4 ∆µ ∆µ ∆µ ∆µ γall 2000 1000 0 1000 Pressure [bar] 0 50 100 150 200 250 300 350 Excess energy [J/m 2] γ2 nd γ3 rd γ4 th 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='0036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='0038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='0042 1/Tm [K 1] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='35 eγ [J/m 2] η(Tm) constant η 220 240 260 280 Temperature [K] 35 30 25 20 Enthalpy [NkBT] Pressure: -1000 bar 220 240 260 280 Temperature [K] 36 32 28 24 20 Enthalpy [NkBT] Water Ice Pressure: -2000 bar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='01 dTm/dp [K/bar] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='06 dγ/dp [nm] ηγ = 323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='31 µJ/m 2K This figure "etas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='png" is available in "png"� format from: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='org/ps/2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='00178v1 3000 2000 1000 0 1000 Pressure [bar] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='75 1 Excess entropy [mJ/Km 2] γ2 nd γ3 rd γ4 th 3000 1500 0 1500 Pressure [bar] 25 30 35 40 γm [mJ/m 2] 3000-1500 0 1500 2000 1000 0 1000 Pressure [bar] 25 30 35 40 γm [mJ/m 2] 2 nd 3 rd 4 th 0 10 20 30 40 ∆T [K] 15 20 25 30 35 40 γ [mJ/m 2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='4 kT/ε 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='9 1 γσ 2/ε (100) Laird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' JCP 2009 (110) Laird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' JCP 2009 (111) Laird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' JCP 2009 LJ/Sol-Liq: η 300 350 400 450 500 550 T [K] 0 20 40 60 80 100 γ [mJ/m 2] Experimental TIP4P/Ice Water/Liq-Vap: 230 240 250 260 270 Temperature [K] 2000 1000 0 1000 2000 3000 Pressure [bar] NNP TIP4P/Ice (Bianco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content=' PRL 2021) 3000 2000 1000 0 1000 Pressure [bar] 25 30 35 40 γm [mJ/m 2] fixed-vertex fit integration from -2600 bar " -2000 bar " -1000 bar " 1 bar " 1000 bar " 2000 bar This figure "integracion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfXPdL/content/2301.00178v1.pdf'} +page_content='png" is available in "png"� format from: http://arxiv.' metadata={'source': 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All rights reserved. +Contents lists available at ScienceDirect +Computerized Medical Imaging and Graphics +journal homepage: www.elsevier.com/locate/compmedimag +Integrating features from lymph node stations for metastatic lymph node +detection +Chaoyi Wu a,1, Feng Chang a,1, Xiao Su c, Zhihan Wu d, Yanfeng Wang a,b, Ling Zhu c,∗, +Ya Zhang a,b,∗ +a Cooperative Medianet Innovation Center, Shanghai Jiao Tong University, Shanghai 200240, China +b Shanghai AI Laboratory, Shanghai 200232, China +c Department of Radiology, School of Medicine, Shanghai Ninth People’s Hospital, Shanghai Jiao Tong University, Shanghai 200011, China +d School of Medicine, Shanghai Jiao Tong University, Shanghai 200025, China +A R T I C L E +I N F O +Keywords: +Metastatic lymph node detection +Lymph node stations +Graph convolutional network +Radiotherapy +A B S T R A C T +Metastasis on lymph nodes (LNs), the most common way of spread for primary tumor cells, is a sign of increased +mortality. However, metastatic LNs are time-consuming and challenging to detect even for professional +radiologists due to their small sizes, high sparsity, and ambiguity in appearance. It is desired to leverage recent +development in deep learning to automatically detect metastatic LNs. Besides a two-stage detection network, +we here introduce an additional branch to leverage information about LN stations, an important reference for +radiologists during metastatic LN diagnosis, as supplementary information for metastatic LN detection. The +branch targets to solve a closely related task on the LN station level, i.e., classifying whether an LN station +contains metastatic LN or not, so as to learn representations for LN stations. Considering that a metastatic LN +station is expected to significantly affect the nearby ones, a GCN-based structure is adopted by the branch to +model the relationship among different LN stations. At the classification stage of metastatic LN detection, the +above learned LN station features, as well as the features reflecting the distance between the LN candidate +and the LN stations, are integrated with the LN features. We validate our method on a dataset containing 114 +intravenous contrast-enhanced Computed Tomography (CT) images of oral squamous cell carcinoma (OSCC) +patients and show that it outperforms several state-of-the-art methods on the mFROC, maxF1, and AUC scores, +respectively. +1. Introduction +Metastatic lymph nodes (LNs) play a pivotal part in the spread of +original tumors. Detecting metastatic LNs in time is of great importance +both in cancer prevention and treatment planning. However, metastatic +LN detection is extremely challenging and time-consuming even for +an experienced radiologist due to the following characteristics. Firstly, +the sizes of metastatic LNs are extremely small compared to other +organs and tissues. As shown in Fig. 1(a), most cases have a diameter +of less than 20 mm. Secondly, metastatic LNs are usually sparsely +distributed, with mostly 1–2 metastatic LNs per patient in our task (see +Fig. 1(b)). Thirdly, the appearance of metastatic LNs is ambiguous, i.e., +the intra-class differences of metastatic LNs are large while the inter- +class differences between metastatic LNs and normal LNs are small, as +shown in Fig. 2. Radiologists must judge whether an LN candidate is +metastatic by simultaneously examining its texture, shape, size, as well +as its neighboring tissues and organs. +∗ Corresponding authors. +E-mail addresses: zhul1757@2m9h.net (L. Zhu), ya_zhang@sjtu.edu.cn (Y. Zhang). +1 C. Wu and F. Chang contribute equally to this work. +Several previous studies start with an easier task of detecting en- +larged LNs (Barbu et al., 2011; Bouget et al., 2019; Nogues et al., 2016; +Oda et al., 2018). The enlarged LNs are highly likely to be metastatic, +but not vice versa. Such methods are thus limited in coverage as nearly +70% of metastatic LNs are within 10 mm in size, as shown in Fig. 1(a). +Some subsequent studies consider small metastatic LNs and leverage +PET modality or tumor location to provide additional information to +assist the detection of metastatic LNs and achieve promising perfor- +mance (Chao et al., 2020; Zhu et al., 2020a,c). However, in practice, +patients do not always have PET modality available due to its high cost, +or the primary tumors because they may have been surgically removed, +which makes the above methods inapplicable. This paper thus explores +leveraging other more accessible auxiliary information to help detect +metastatic LNs, as an important supplement to the above methods. +In this paper, we propose an efficient deep learning based method to +leverage information about LN stations to detect metastatic LNs. The LN +https://doi.org/10.1016/j.compmedimag.2022.102108 +Received 19 April 2022; Received in revised form 8 July 2022; Accepted 28 July 2022 + +Computerized +Medical Imaging +andGraphicsEISEVIERComputerized Medical Imaging and Graphics 101 (2022) 102108 +2 +C. Wu et al. +Fig. 1. Statistics of metastatic LNs. (a) Distribution of metastatic LNs sizes (short axis +length). (b) Distribution of the number of metastatic LNs per patient. +Fig. 2. Left two metastatic LNs, (a) and (b), are quite different in appearance, showing +the large intra-class difference. On the right, a normal LN (c) and a metastatic LN (d) +appear quite similar, showing the small inter-class difference. +Fig. 3. Examples of LN stations. Each row is a patient and the LN stations are marked +in different colors. +station is a marking system that describes the area where LNs are sited, +usually with edges of various physiological structures (such as muscles +and blood vessels) as the boundary (Guo et al., 2021), as shown in +Fig. 3. LNs in the same LN station share similar physiological properties. +A metastatic LN indicates the other LNs in the same LN station are much +more likely to be metastatic. Accordingly, experienced radiologists +would focus more on one LN station if they find a metastatic LN in it +to improve diagnosis accuracy. In addition, LN stations are connected +by lymphatic drainage directly. Therefore, there are strong correlations +between neighboring LN stations. If an LN station has been confirmed +to contain metastatic LN, the neighboring LN stations are considered +more vulnerable. +Compared with other additional information, such as the PET +modality and tumor locations, LN stations reflect the metastatic pattern +more directly. They can be auto-delineated precisely because their +locations are stable, making them more suitable for metastatic LN +detection. An LN station segmentation network is trained with a small +set of LN segmentation data to locate the LN stations automatically. +Fig. 4 presents the overall framework of our method, where an +additional LN station classification branch is introduced to the common +detection branch. For the LN station classification branch, a new task is +introduced to discriminate whether an LN station contains metastatic +LNs. To model the interaction between LN stations, a Graph Convolu- +tion Network (GCN)-based structure is integrated into the LN station +feature encoder. The features extracted this way, on the one hand, can +indicate which stations are riskier and help the network focus more on +the LNs in these stations so that metastatic LNs will not be missed due to +small sizes. On the other hand, the relationship information learned by +this structure can supplement the visual features and tackle the prob- +lem of ambiguous appearance. For the metastatic LN detection branch, +following the methods in Zhu et al. (2020b, 2019), we train a metastatic +LN candidate generator at high sensitivity. There are massive false- +positive cases in these candidates because of the sparsity of metastatic +LNs. We feed the LN station features embedding extracted from the LN +station classification branch together with the LN candidate into the LN +candidate classifier to decrease the false alarm ratio via leveraging LN +station information. +To validate the effectiveness of the proposed method, we collected +a dataset containing 114 intravenous contrast-enhanced CT scans of +oral squamous cell carcinoma (OSCC) patients. Extensive experiments +have shown that the proposed method improves the mean free response +operating characteristic (mFROC) by 4.77%, maxF1 by 0.0230, and +AUC by 0.0303 compared to the state-of-the-art 3D medical detection +method (Roth et al., 2015), demonstrating the effectiveness of our +method. +The contribution of the paper can be summarized as follows. +• We propose an efficient deep learning based method to leverage +the information of LN stations for metastatic LN detection. +• To learn representations for LN stations, we define a closely +related auxiliary task to assist metastatic LN detection by discrim- +inating an LN station is metastatic or not. +• To model the mutual influence among different LN stations, we +introduce a GCN-based structure for LN station classification. +2. Related work +2.1. General object detection in medical images +General object detection is viewed as a classic and typical task +to be studied for decades in the computer vision community (Zhao +et al., 2019). From the perspective of medical image analysis, it is also +promisingly valuable. Many medical issues like lesion detection (Yan +et al., 2019) and lung module detection (Teramoto et al., 2016) can +all be concluded into this task. The popular detection methods can be +divided into two types: end-to-end methods (Baumgartner et al., 2021; +Li et al., 2019; Luo et al., 2021; Yan et al., 2018, 2019; Zlocha et al., +2019) and two-stage methods (Sun et al., 2019; Ding et al., 2017; Duan +et al., 2019; Zhu et al., 2020c). End-to-end methods prefer to dealing +the detection task as a whole complete flowchart. These methods have +been widely used in universal detection tasks due to their few hyper- +parameters and guaranteed performance. nnDetection +(Baumgartner +et al., 2021) is one of the latest and most representative end-to-end +detection methods. It is self-configuring and can achieve encouraging +performance in various medical detection tasks. + +- +- +- +1 +- +T +- +1 +- +1 +1 +- +- +1 +- +0.1 +1 +1 +- +1 +1 +1 +- +1 +- +1 +- +0.08 +1 +- +Y +- +- +1 +- +- +- +- +1 +- +Densi +0.06 +- +1 +- +1 +- +- +1 +1 +- +- +1 +- +1 +- +1 +- +0.04 +L +L +- +1 +- +- +1 +- +- +- +- +0.02 +-L +L +- +L +1 +1 +1 +1 +- +1 +1 +- +1 +- +0 +10 +20 +30 +40 +50 +Diameterofshortaxis(mm)0.5 +1 +1 +- +1 +1 +I +- +1 +1 +1 +- +- +1 +1 +- +- +1 +-- +1 +- +- +- +1 +- +- +1 +0.4 +-- +- +-- +-- +- +- +: +-- +--- +-- +- +-- +1 +- +- +1 +- +1 +- +- +. +! +- +1 +- +- +- +1 +- +- +1 +1 +1 +4 +- +- +- +- +- +-1- +---- +- +- +-- +- +- +1 +- +- +- +1 +- +---- +--- +1 +1 +- +- +. +1 +- +1 +D +0.2 +r-- +1 +- +1 +1 +- +- +-- +r +- +- +1 +- +- +-- +- +- +-- +- +- +- +1 +- +- +1 +1 +- +- +1 +- +-- +1 +- +-- +-- +- +1 +1 +1 +0.1 +-- +- +-l- +- +- +-- +-- +1 +1 +- +- +--- +- +1 +1 +- +-- +! +- +-- +1 +1 +0 +1 +2 +3 +6 +8 +10 +11 +MetastaticLNs +per +patientComputerized Medical Imaging and Graphics 101 (2022) 102108 +3 +C. Wu et al. +Two-stage methods decouple the detection task into two sub-tasks: +proposal extractor and FP reducing stage. In the first proposal extrac- +tor stage, the generator will give many coarse candidate proposals +with high sensitivity and high false alarm ratio. The second stage is +trained to classify the candidate proposals and dismiss the false positive +cases while maintaining an acceptably high sensitivity score. In many +sparse-distributed and small object detection task, like lung module +detection (Teramoto et al., 2016) and metastatic LN detection task (Zhu +et al., 2020c), two-stage methods are proved to be more suitable +compared with end-to-end methods +(Chao et al., 2020; Zhu et al., +2020a). By separating the raw task into two easier parts, more effective +strategies can be designed for different parts. We adopt this design +paradigm in our task as well and prove the necessity by comparing our +method with the nnDetection (Baumgartner et al., 2021). +2.2. Metastatic lymph node detection +Many former works focus on detecting the enlarged LN whose short +axis is greater than 10 mm instead of metastatic LNs (Bouget et al., +2019; Barbu et al., 2011; Nogues et al., 2016; Oda et al., 2018) because +the detection of enlarged LNs is much easier and enlarged LNs are con- +sidered to be highly likely metastatic medically. Conventional detection +algorithm (Barbu et al., 2011; Feulner et al., 2013; Kitasaka et al., 2007) +based on morphological prior knowledge will design extraction of the +visual features carefully. Recent works based on deep learning (Nogues +et al., 2016; Oda et al., 2018) bring more possibilities to this field, +improving the performance impressively. The success encourages the +community to consider the detection of metastatic LNs regardless of +their sizes, which is more practical and challenging. Some works have +made successful attempts on this task (Chao et al., 2020; Zhu et al., +2020a,c). Zhu et al. (2020a,c) divide the network into two branches +to extract features separately via leveraging the distance between LN +candidates and the primary tumor. Chao et al. (2020) models the +relationship between LN candidates and uses the information of the +primary tumor as well. PET modality is also proved to be useful. +However, the locations of the primary tumor for the patients after +operation is often unknown. PET modality is also costly to collect and +in many practical cases, the patients lack the PET/CT for radiologists to +refer. Previous methods will fail when these two important information +are missing. Therefore, we hope to use the information of LN station +to help the detection of metastatic LNs as another optional auxiliary +information, so that a good detection accuracy can still be obtained +when the above information is missing. LN stations have been proved +to be very important reference in the detection of metastatic LNs +clinically (Grégoire et al., 2003; Naruke et al., 1978), but, as far as we +know, no prior work has considered how to add this important prior +knowledge to the automatic detection algorithm. We are the first to +leverage the information of LN stations assisting the detection task. +2.3. Graph convolution network +Graph neural network (GNN) is first proposed in Scarselli et al. +(2008). It combines the strong expressive power of the graph structure +with the neural network (Jie et al., 2020). It can be used to model +the relationship of the non-Euclidean structure data in many areas +like social network (Jin et al., 2019) and protein–protein interaction +networks (Fout, 2017). In Kipf and Welling (2016), graph convolution +network (GCN) is first proposed by adding the convolution operation +into the original GNN structure. GCN has received a lot of attention +due to its simplicity, easy understanding, and strong performance (He +et al., 2020; Lu et al., 2021). +In medical image analysis, GCN has great potential to be applied +in different tasks since it can directly model the relationship between +different physiological tissues which cannot be modeled by the regular +convolution neural network (CNN) (Chao et al., 2020; Kazmierski and +Haibe-Kains, 2021; Mao et al., 2019; Zhao et al., 2020). Kazmierski and +Fig. 4. Flowchart of the proposed method, which consists of the LN station classifi- +cation branch and Metastatic LN Detection branch. The former is leveraged to encode +useful features of the LN stations and the latter combines different types of information +to make the final decision. +Haibe-Kains (2021), Zhao et al. (2020) both use the GCN to combine +the information from the different volumes of interest (VOI). When +processing images in medical images, images are often sliced and this +structure can integrate global information well. Chao et al. (2020) +inspires our work a lot. It builds the graph on the LN candidate level +and assumes the metastatic LNs may affect each other related to their +physical distance. However, we build the graph on the LN station level +instead. We argue that due to the uncertainty of the individual, the +interaction between the LNs is not so accurate especially when LN +candidates have strong sparsity, and building the relationship on a +general level will be more suitable. +3. Method +LN stations play an important role in identifying metastatic LNs. +LNs in the same LN station share similar locations and properties. Due +to the similarity of LNs in a certain LN station, metastatic LNs usually +appear like a cluster. Certain LN stations are more likely to contain +metastatic LNs, e.g., station Ib and station II have much more metastatic +LNs than others (Fig. 5(a)). There is also a strong correlation between +LN stations. An abnormal LN station may reflect that other LN stations +near it have more potential to contain metastatic LNs as metastasis may +spread to farther LN stations from this abnormal one. Fig. 5(b) shows +the correlations between LN stations in terms of metastatic LNs. Some +LN stations, such as II_R and III_R, demonstrate strong correlation. As +a result, in clinical practice, experienced radiologists typically inspect +LN stations first in searching for the metastatic LNs. +Considering the importance of LN stations, we here explore to in- +corporate information about LN stations into the detection of metastatic +LNs, by introducing an additional LN station classification branch to the +standard detection branch, as shown in Fig. 4. The features from the +two branches are integrated for the final LN candidate classification. +3.1. Ln station classification branch +To learn representations for LN stations, the LN station classifi- +cation branch is trained with a closely related auxiliary task, i.e., to +discriminate whether an LN station has metastatic LNs. If an LN station +region contains the center point of any metastasis LN, it is considered to +have metastatic LNs; otherwise, it is considered not to have metastatic +LNs. In this way, the ground truth annotations for the LN station +classification can be directly inferred from the annotations of metastasis +LNs. With the above task setup, the goal of the LN station classification +branch is to learn representations that can best discriminate the LN +stations with and without metastasis LNs. To model the relationships +among LN stations, a GCN-based classification network is adopted by +the LN station classification branch. The overall architecture of the +classification network is shown in Fig. 6. + +LNStation +eatureEncode +LNCondidalComputerized Medical Imaging and Graphics 101 (2022) 102108 +4 +C. Wu et al. +Fig. 5. +Statistics of LN stations in the case of OSSC. ‘L’ and ‘R’ represent the left and +right independent parts of the LN station respectively. (a) Statistics of the metastatic +LNs contained in each LN station. (b) Spearman’s correlation of LN stations regarding +to metastatic LNs. +3.1.1. Data preparation +Let  = {𝑥1, 𝑥2, … , 𝑥𝑁 +} represents a set of 3D CT volumes, where +𝑥𝑖 ∈ 𝑅𝐻×𝑊 ×𝐷, the subscript 𝑖 stands for the 𝑖th patient, and 𝐻, 𝑊 , 𝐷 +are the height, width, and depth of the 3D CT volumes, respectively. +To get the LN station segmentation masks, an nnU-Net (Isensee et al., +2018) based segmentation network is trained on a small dataset of +coarsely annotated LN stations.2 Without loss of generality, we consider +a specific 3D CT volume 𝑥 to simplify the notations. Denote the LN +station region masks obtained by the segmentation network as 𝑆 = +𝑆𝑒𝑔𝑆(𝑥). Based on the segmentation result, the raw CT volume is +divided into 12 LN station patches. Noted that since some LN stations, +like II station and VI station, are symmetrical and independent with +each other, we divide them into left and right two parts based on the +central axis and treat them separately. Therefore though we only have +7 station classes in segmentation results, 12 LN station patches are +obtained after post-processing, denoted as 𝑆 = {𝑠1, 𝑠2, … , 𝑠12 +}. +3.1.2. Gcn-based classification network +To model the complex reflux relationship among LN stations, a GCN- +based classification network is adopted, as shown in Fig. 6. A graph is +formed where each LN station is treated as a vertex in a fully connected +graph with self-connections. A shared 3D CNN encoder 𝐶𝑆(⋅) is first +used to get the feature embedding for each LN station separately. +ℎ𝑗 = 𝐶𝑆(𝑠𝑗), +𝑗 ∈ {1, … , 12} . +(1) +Here ℎ𝑗 represents the encoder output of the 𝑗th LN stations, which is +further fed into a two-layer graph neural network. +𝑓 𝑆 +𝑗 = 𝜎2(𝑟𝑒𝑙𝑢(𝜎1(ℎ𝑗))), +𝑗 ∈ {1, … , 12} +𝜎𝑙(ℎ𝑗) = 𝛴𝑘∈{1,…,12}𝑎𝑘𝑗𝜙(ℎ𝑘), +𝑙 ∈ {1, 2} , +(2) +where 𝜎1(⋅) and 𝜎2(⋅) are the two layers of the graph neural network, +𝑎𝑘𝑗 is the edge weight between the 𝑘th station and the 𝑗th station, and +𝜙 is a learnable function. In the common setting of GCN, 𝑎 ∈ [0, 1] is set +based on some similarity metric before the operation of GCN. While in +our case, it is hard to define a reasonable similarity between LN stations +due to the complexity of the human circulatory system. Therefore we +also set 𝑎 as a learnable parameter so that the network can learn to +model the reflux relationship automatically. +For each LN stations, a shared 2-layer MLP is used to get the final +classification results: +𝑝𝑗 = 𝑀𝐿𝑃 (𝑓 𝑆 +𝑗 ), +𝑗 ∈ {1, … , 12} , +(3) +where 𝑝𝑗 is the final prediction result of the 𝑗th LN station. +2 The traditional registration methods (Avants et al., 2009) can also be used +here to obtain the segmentation of the LN station, but the segmentation results +are relatively poor. In order not to affect the effect of our subsequent methods, +we choose to use the neural network for its better performance. +Considering that metastatic LNs are quite sparse, which causes +severe class imbalance problem, focal loss (Lin et al., 2017) is adopted +as the training loss: +𝐿𝑆 = +{ +−𝛼(1 − 𝑝𝑗)𝛾𝑙𝑜𝑔(𝑝𝑗) +𝑦𝑗 = 1 +−(1 − 𝛼)(𝑝𝑗)𝛾𝑙𝑜𝑔(1 − 𝑝𝑗) +𝑦𝑗 = 0 , +(4) +where 𝛼 and 𝛾 are hyper-parameters, and 𝑦𝑗 ∈ {0, 1} is the ground truth +with 1 indicating metastasis and 0 otherwise. +3.2. Metastatic LN detection branch +We follow typical two-stage detection methods and divide this +task into two stages: extracting LN candidates and classifying the LN +candidates. +3.2.1. LN candidate generation +Following the common setting of LN detection tasks (Chao et al., +2020; Zhu et al., 2020c), we employ an LN candidate generator with +high sensitivity to generate LN candidates. Specifically, an nnU-Net +(Isensee et al., 2018) based segmentation network is trained with +pixel-wise metastatic LN annotations. The segmentation results of the +network are leveraged to generate LN candidates. We then locate the +center point of each connected component and crop out a cube of the +size 48 × 48 × 48 centering around the center point as LN candidates. +The cube size is expected to be large enough to cover all LN candidates. +This coarse result is expected to lead to a high sensitivity score but +severely suffer from the problem of false alarms as shown by previous +studies (Chao et al., 2020; Roth et al., 2015). Therefore, in the next +part, we mainly focus on classifying the metastatic LN candidates to +reduce the false alarm ratio. +3.2.2. LN candidate classification +After getting the LN candidates, the goal of the LN candidate +classifier is to judge whether an LN candidate is metastatic or not. The +network architecture is shown in Fig. 7. Without loss of generality, we +consider a specific LN candidate 𝑐 ∈ 𝑅48×48×48. First, a simple 3D CNN +network 𝐶𝐿 is used to get the visual information of the LN candidate: +𝑓 𝐿 = 𝐶𝐿(𝑐). +(5) +The appearance features 𝑓 𝐿 are expected to provide visual details of +the LN candidate patches, such as morphology, texture, and intensity. +In addition to the appearance features of the LN candidate 𝑐, we try +to integrate its corresponding LN station features in classifying the LN +candidates to reduce false alarms. The LN station features contain the +global information indicating whether there are metastatic LNs in this +LN station. In detail, we calculate the Euclidean distance between the +LN candidate 𝑐 and the center of the 12 LN stations and find the nearest +LN station. +𝑘 = arg +min +𝑗∈{1,…,12} 𝑑(𝑐, 𝑠𝑗). +(6) +The 𝑘th station is thus considered where the LN candidate 𝑐 belongs to +and its feature embedding 𝑓 𝑆 +𝑘 is then concatenated to the visual feature +embedding 𝑓 𝐿. +Some identified LN candidates might not belong to any LN station +regions, and the LN station features shall not be used in this case, +so we add distance features to prompt the network further when to +focus on the LN stations features. Specifically, the distance to each LN +station is represented as 12-dimensional features 𝑓 𝐷, which is further +concatenated into the above features. Adding the distance features can +also help the network reject the LN candidates that appear in places +far away from LN stations, such as the top of the head, which are +impossible to contain LNs. The final features are expressed as follow: +𝐹 = 𝑓 𝐿‖𝑓 𝑆 +𝑘 ‖𝑓 𝐷, +(7) +where ∥ denotes the concatenation operation. + +70 +60 +50 +Count +40 +30 +20 +10 +0 +la +IbLIbR IILIIR IIILIIIRIVLIVRVLVR +LN Stations1.0 +IbL +0.8 +IbR += +0.6 +IIR +0.4 +IIIR +0.2 +IVL +VLIVR +0.0 +VR +0.2 +Ia IbLIbR IIL IIR IIIL IIIRIVLIVRVL VRComputerized Medical Imaging and Graphics 101 (2022) 102108 +5 +C. Wu et al. +Fig. 6. Architecture of the metastatic LN station classification network, consisting of a 3D CNN encoder, a 2-layer GCN, and a 2-layer MLP. +Fig. 7. Overall framework of the proposed metastatic LNs classification method. For +each LN candidate, a CNN encoder is used to extract its appearance features, which is +further integrated with the distance features (marked in yellow) and the features of its +nearest LN station (marked in green) before feeding into the MLP for final classification. +Finally we use a 2-layer MLP to get the final classification results. +𝑜 = 𝑀𝐿𝑃 (𝐹). +(8) +The metastatic LN classification task faces the class imbalance prob- +lem due to the sparsity of metastatic LNs. The focal loss (Lin et al., +2017) is employed as the training loss: +𝐿𝐿 = +{ +−𝛽(1 − 𝑜)𝜃𝑙𝑜𝑔(𝑜) +𝑦𝐿 = 1 +− (1 − 𝛽)𝑜𝜃𝑙𝑜𝑔(1 − 𝑜) +𝑦𝐿 = 0 , +(9) +where 𝛽 and 𝜃 are hyper-parameters, and 𝑦𝐿 ∈ {0, 1} is the ground +truth label of the LN candidate, with 1 representing the metastatic LN +candidate and 0 otherwise. +3.3. Discussion +It is worth emphasizing that although we use additional LN station +annotations information in the training progress, in inference time +we do not need extra LN station annotations because the LN station +segmentation network automatically generates masks for each LN sta- +tion. Our method can improve the detection of metastatic LNs only if +the patient can provide contrast-enhanced CT images. This makes our +solution more in line with actual needs. +4. Experiments +4.1. Dataset +4.1.1. Lymph node station segmentation +To train the LN station segmentation model, a pixel-level labeled +dataset is collected in advance. The dataset contains 82 head & neck +intravenous contrast-enhanced CT scans. For each CT scan, the 3D +segmentation masks of 7 LN stations of the head & neck are pro- +vided by radiologists. To verify segmentation results, we randomly split +this dataset into 60%, 20%, 20% for training, validation, and testing +respectively. +4.1.2. Metastatic lymph nodes detection +We collected 114 intravenous contrast-enhanced CTs of oral OSCC +patients. The median age of these patients is 61 years old and interquar- +tile range is 50 ∼ 70. There are 80 males and 34 females. Patients in +this dataset are all under radiotherapy treatments, which means some +patients may have undergone surgery, and some parts of their head & +neck such as tumors have been manually processed. The metastatic LNs +of these patients have been pathologically confirmed. All CT scans are +with a slice thickness of 0.625 mm. For evaluation, we randomly split +the 114 CT scans into 80 for training and 34 for testing. +4.2. Implementation details +In our experiments, the Hounsfield unit value of the CT is clipped +to be within [−218, 600]. We implement the whole framework stage-by- +stage. +4.2.1. LN station segmentation and LN candidate segmentation +A U-Net (Ronneberger et al., 2015) based model is trained to +automatically segment the lymph node stations (7 classes) and LN can- +didates (1 class), respectively. Patches with a shape of 208 × 238 × 196 +are randomly cropped to be trained. We further apply random scaling, +random gamma, and random mirroring to augment the training data. +The 3d U-Net is trained on one GeForce RTX 3090 GPU with a batch +size of 2 for 200 epochs, each epoch containing 250 batches. The SGD +optimizer with a learning rate of 0.0003 is used with a momentum of +0.99 and a weight decay of 0.00005. +4.2.2. LN station classification +Based on the LN station segmentation masks, we clip the bounding +box of each LN station and resize it to 96 × 96 × 96. As some LN +stations are symmetrical and independent of each other, we divide them +into left and right two parts based on the central axis and treat them +separately. Thus, we can get 12 cubes with a size of 96 × 96 × 96 for +one CT scan. We treat the 12 cubes as 12 nodes in one graph and use +ResNet10 (He et al., 2016) as our feature extractor. 2 graph convolution +blocks containing graph convolution layer and Relu activation function +are used to exploit the mutual relationship. A fully connected layer is +inserted to make the final decision. +4.2.3. LN candidate classification +Every connected component obtained by LN candidate segmenta- +tion is treated as an LN candidate and the connected region whose +𝑖𝑜𝑢 with the ground truth mask greater than 0.1 is considered as a +metastatic case. The center point of each connected component is +located, centered around which a patch of 48 × 48 × 48 is cropped. +Both LN station classification and LN candidate classification use the +same training setup. Random affine and random elastic deformation are +applied in this stage. We trained the model with a batch size of 128 +for 100 epochs on one GeForce RTX 3090 GPU. The SGD optimizer +with a learning rate of 0.01 is used with a momentum of 0.9 and +a weight decay of 0.001. The learning rate changes to 1/10 of the +original when the epoch reaches 30, 50, and 70. To handle the category +imbalance, we supervise our network using the focal loss parameterized +by 𝛼 = 0.25, 𝛾 = 2. + +Computerized Medical Imaging and Graphics 101 (2022) 102108 +6 +C. Wu et al. +Table 1 +Comparison between LN station segmentation by radiologists and neural network in terms of dice and center point distance. Segmentation results are collected +from 5 radiologists, and those of the most experienced radiologists are used as ground truth. DSC↑ \CPD↓ are reported. +Annotator +LN station +Ia +Ib +II +III +IV +V +VIa +Average +Radiologist1 +69.18\8.31 +75.90\12.28 +73.71\6.52 +76.86\7.84 +70.31\8.72 +70.59\9.68 +39.10\14.30 +67.95\9.66 +Radiologist2 +65.50\3.40 +72.78\11.54 +60.93\11.17 +80.76\4.35 +64.61\10.48 +65.26\18.37 +49.66\9.16 +65.64\9.78 +Radiologist3 +75.52\4.88 +68.92\11.54 +73.97\4.63 +75.04\5.50 +56.44\16.28 +49.19\32.00 +31.57\10.17 +61.52\12.14 +Radiologist4 +74.52\4.71 +76.78\12.30 +74.61\6.72 +71.21\9.19 +70.53\7.33 +65.98\13.94 +56.57\6.70 +70.03\8.70 +Manual Delineation Average +71.18\5.33 +73.59\11.92 +70.81\7.21 +75.97\6.72 +65.47\10.70 +62.76\18.50 +44.23\10.08 +66.29\10.06 +Neural Network +82.56\5.23 +87.28\3.68 +87.69\3.82 +84.82\4.92 +84.51\4.82 +81.60\8.63 +75.88\4.88 +83.48\5.14 +aThere is a long contour line near the esophagus in the mask of the station VI. Since it is very narrow, the criteria for whether to outline this line differs greatly +between different people. Therefore, the results of humans on the LN station VI are much worse than the results automatic labeled by the machine. +Table 2 +Comparison with SOTA metastatic ln detection methods. the mean ± standard deviation are reported. +Method +FROC@1(%) +FROC@2(%) +FROC@3(%) +FROC@4(%) +mFROC(%) ↑ +Max F1↑ +AUC ↑ +nnDetection (Baumgartner et al., 2021) +43.33±6.47 +52.78±5.05 +58.06±5.08 +59.72±4.73 +53.47±4.97 +0.5220±0.370 +0.6166±0.0188 +single-net(3d classifier)a (Zhu et al., 2020a) +33.64±2.83 +53.83±3.63 +64.49±2.13 +74.58±3.26 +56.64±1.65 +0.5206±0.0150 +0.6673±0.0082 +single-net(2.5d classifier) (Roth et al., 2015) +34.20±1.27 +53.64±2.18 +68.41±1.09 +77.94±3.10 +58.55±1.50 +0.5337±0.0142 +0.6879±0.0103 +LN-level GCN (Chao et al., 2020) +23.92±1.40 +38.69±4.49 +50.47±6.11 +63.92±5.17 +44.24±3.02 +0.4968±0.0114 +0.5788±0.0257 +ours +41.12±1.74 +60.12±2.50 +71.34±1.62 +80.69±0.40 +63.32±1.04 +0.5571±0.0051 +0.7182±0.0050 +aMany clinical works +(Koizumi et al., 2020; Li et al., 2020; Zhang and Ren, 2019; Zhou et al., 2019; Zheng et al., 2021) carry out different experiments for +different body positions or different image modalities. Due to the differences of datasets, we cannot directly compare with them, but these methods can be unified +into a typical 3D CNN structure, so here we use single-net(3d classifier) to uniformly represent the effects of these works. +4.3. Evaluation metric +DSC: To evaluate the performance of segmentation tasks, we adopt +Dice Similarity Coefficient (DSC) as most medical image segmentation +tasks. DSC measures the overlap rate of the prediction mask 𝑝 and +ground truth mask 𝑔. +𝐷𝑆𝐶 = 2|𝑝 ∩ 𝑔| +|𝑝| + |𝑔| . +(10) +CPD: To evaluate the LN station segmentation results, we calculate +the center point distance (CPD) between two segmentation masks. CPD +measures the distance error between the prediction mask 𝑝 and ground +truth mask 𝑔. Let 𝐶𝑖 ∈ 𝑅3 denotes the center point coordinate of mask +𝑖. Voxel euclidean distance is adopted here to measure the distance: +𝐶𝑃 𝐷 = ‖𝐶𝑝 − 𝐶𝑔‖2. +(11) +mFROC: To evaluate the performance of metastatic LN detection +tasks, we use the free response operating characteristic (FROC) as the +metric, which measures the recall against different numbers of FPs +allowed per patient. We report the mFROC, i.e., the average sensitivity +at 1,2,3,4 FPs per patient. Since most of the previous work (Chao et al., +2020; Zhu et al., 2020a,c) similar to ours used mFROC as the main +reporting indicator, our analysis of method performance also focused +on this indicator mainly. +maxF1: Besides mFROC, we use maxF1 as a supplementary metric +for classification tasks. By adjusting the threshold that decides whether +an LN candidate is metastatic, an LN candidate with low confidence +to be metastatic can be considered as metastatic with the decrease of +the threshold, which leads to different F1-scores with the change of the +threshold, formulated as: +𝐹1 = 2 × 𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 × 𝑟𝑒𝑐𝑎𝑙𝑙 +𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 + 𝑟𝑒𝑐𝑎𝑙𝑙 +𝑚𝑎𝑥𝐹1 = max +𝑡∈[0,1] 𝐹1𝑡 +. +(12) +AUC: AUC represents the area under the receiver operating char- +acteristic (ROC) curve, which is a very common indicator in detection +and binary classification tasks. +4.4. Quantitative results and discussions +4.4.1. Lymph node station segmentation +To show that LN stations can be precisely auto-delineated, we com- +pare the segmentation results between humans and the segmentation +network. The quantitative results are shown in Table 1. On contrast- +enhanced CT images, it is difficult to find a border that is totally +consistent with the definition of LN stations in medical theory, and +the delineation of LN stations depends more on the experience of +radiologists. Thus, we collect segmentation masks of 5 radiologists and +use the mask of the most experienced radiologist as ground truth to +evaluate others’ performance. As shown in Table 1, the results of neural +network are even better than manual delineation and are sufficient for +metastatic LN detection task to crop LN station patches for follow-up +stages. This shows LN stations are much less costly to collect compared +with other additional information. Fig. 8 shows visualization results of +different radiologists and the neural network. +4.4.2. Metastatic LN detection +To show the effectiveness of our method for detecting metastatic +LNs, we compare with a set of metastatic LN detection methods, includ- +ing nnDetection (Baumgartner et al., 2021), a self-configuring frame- +work for 3D medical object detection tasks which has been demon- +strated effective on several public benchmarks, as well as several +methods especially designed for metastatic LNs detection (Chao et al., +2020; Roth et al., 2015; Zhu et al., 2020a), all of which are based +on two-stage framework. For a fair comparison, we use the same LN +candidate extractor for the later three methods (Chao et al., 2020; Roth +et al., 2015; Zhu et al., 2020a) as our method. +Table 2 outlines the quantitative comparison of different methods +for metastatic LN detection tasks, demonstrating the effectiveness of +our method. With mFROC of 0.6332, maxF1 of 0.5571, and AUC of +0.7182, our method exceeds nnDetection by 9.85% in mFROC, 0.0351 +in maxF1 and 0.1016 in AUC. Our method achieves better results in +every metric except FROC@1. nnDetection gives higher confidence to +apparent metastatic LNs so it gets a better result at FROC@1. But its +drawback is that it tends to overlook small and hard candidates so +it cannot achieve a high FROC score even when FP rate allowed per +patient is large. Our method outperforms it since we adopt a two-stage +methodology to handle small and sparse objects. +Our method achieves a gain of 4.77%, 0.0234, and 0.0303 over +the second best method (Roth et al., 2015) in terms of mFROC score, +maxF1 score, and AUC score respectively. Compared with Roth et al. +(2015) and Zhu et al. (2020a), our method introduces supplementary +features on LN stations as well as the distance to LN stations, which + +Computerized Medical Imaging and Graphics 101 (2022) 102108 +7 +C. Wu et al. +Fig. 8. Visualization of LN stations segmentation of a patient. Each column is either from one radiologist or the neural network. The results from radiologists are rough because +the masks are delineated slice by slice. The neural network instead generates more smooth results. +Fig. 9. Left: Comparison of our method to SOTA metastatic LNs detection methods in FROC. Middle: ROC of LN station classification with and without GCN. Right: Comparison +of the methods in Ablation study in FROC. +are both shown to contribute to a more powerful classifier and effec- +tively reduce the false alarm ratio caused by data imbalance. LN-level +GCN (Chao et al., 2020), which treats each LN candidate instead of +LN station as a vertex in graph, with the distance between the LN +candidates as the edge weights, performs worst among the methods. A +possible reason for this phenomenon is that the true metastatic LNs are +quite sparse and most LN candidates are actually false positives which +provide misguided information. So, it is more reasonable to leverage +the relationship information on a more general aspect, such as on the +LN station level, to exclude the effect of some individual mistakes. +Fig. 9 (Left) shows the FROC curves of these methods. There exists +an upper bound of 0.9065, because the recall rate for the LN candidate +detection is 0.9065 and the missing metastatic LNs cannot be recovered +at the classification stage. As a result, the best performance of LN +candidate classification cannot exceed this recall rate. +4.4.3. Stratified analysis for different LN sizes +In clinical practice, LNs with short axis diameter greater than 10 mm +are considered enlarged LNs. While many relevant studies focus on +the enlarged LNs +(Roth et al., 2015), the majority of metastatic LNs +are actually small metastatic LNs. To validate the effectiveness of our +method for LNs of different sizes, we separate the metastatic LNs into +two groups (i.e., Large and Small) based on their short axis length, using +10 mm as the cut-off threshold. The FROC curves of different methods +are separately presented for the above two size groups, as shown in +Fig. 11. For all methods, the maxF1, mFROC and AUC scores on the +large LNs are better than those on the small LNs, suggesting that it +is more challenging to detect metastatic LNs of small sizes. It can be +seen that for large LNs, our method outperforms other methods at most +sampling points. For small LNs, our method improves the sensitivity by +a large margin at different operating points such as 2 FP∕patient and 3 +FP∕patient. Table 3 shows quantitative results of these methods for the +two groups. The nnDetection (Baumgartner et al., 2021) performs well +for large LNs but terribly on small ones, showing that general detection +methods fail to handle the small LNs. For large LNs, our method +achieves mFROC of 71.23%, maxF1 of 0.5436 and AUC of 0.8555. +For small LNs, our method outperforms the comparative methods by a +large margin, i.e., an improvement of 3.72%, 0.0133 and 0.0057 over +the second best method, single-net(2.5d classifier), in terms of mFROC, +maxF1 and AUC. The above result confirms that introducing the LN +station level feature is effective for detecting small metastatic LNs. + +(Senesitivity +0.9 +0.9065 +0.8 +TruePositiveRate( +0.6 +0.5 +ours +w/odistance +0.4 +wlostationfeatures +w/oGCN +0.3 +w/oGcNanddistance +-highbar +0.2 +2 +3 +4 +5 +6 +7 +FaslePositiveRateperPatient(Senesitivity +AUC=0.7908 +0.8 +AUC=0.7314 +TruePositiveRate( +0.6 +0.4 +0.2 +ours +W/oGCN +-random +0 +0 +0.2 +0.4 +0.6 +0.8 +FaslePositiveRateSenesitivity +0.9 +0.9065 +0.8 +TruePositiveRate( +0.7 +0.6 +0.5 +ours +nnDetection +0.4 +2.5dclassifier +3dclassifier +0.3 +nodelevelGCN +-highbar +0.2 +1 +2 +3 +4 +5 +6 +7 +FaslePositiveRateperPatientComputerized Medical Imaging and Graphics 101 (2022) 102108 +8 +C. Wu et al. +Fig. 10. t-SNE visualization of LN station feature distribution in 2D space. +Table 3 +Improvement on LNs of different sizes. Large lns and small lns are separated based on the short axis (10 mm). +Method +Large LNs +Small LNs +mFROC (%)↑ +maxF1↑ +AUC↑ +mFROC (%)↑ +maxF1↑ +AUC↑ +nnDetection (Baumgartner et al., 2021) +68.86±5.37 +0.5361±0.0112 +0.8304±0.0545 +41.58±4.10 +0.3458±0.0334 +0.5262±0.0347 +single-net(3d classifier) (Zhu et al., 2020a) +65.63±0.63 +0.4882±0.0017 +0.7855±0.0096 +46.76±3.06 +0.3657±0.0066 +0.5608±0.0228 +single-net(2.5d classifier) (Roth et al., 2015) +65.71±2.28 +0.4832±0.0115 +0.7980±0.0173 +50.69±2.53 +0.3700±0.0099 +0.5887±0.0189 +LN-level GCN (Chao et al., 2020) +57.59±1.09 +0.4371±0.0100 +0.7337±0.0149 +30.69±3.83 +0.3370±0.0039 +0.5501±0.0270 +ours +71.23±0.56 +0.5436±0.0058 +0.8555±0.0014 +54.41±1.60 +0.3833±0.0082 +0.5944±0.0085 +Table 4 +Ablation analysis of different components. GCN and w/o GCN represent with and without the gcn-based structure, respectively. distance means distance features. +the mean ± standard deviation are reported. +Method +FROC@1(%)↑ +FROC@2(%)↑ +FROC@3(%)↑ +FROC@4(%)↑ +mFROC(%)↑ +maxF1↑ +AUC↑ +Distance +LN station features +w/o GCN +GCN +33.64±2.83 +53.83±3.63 +64.49±2.13 +74.58±3.26 +56.64±1.65 +0.5206±0.0150 +0.6673±0.0082 +✓ +29.53±2.99 +52.71±2.10 +66.54±2.53 +74.77±1.18 +55.89±1.74 +0.5194 ±0.0101 +0.6534±0.0117 +✓ +38.13±2.32 +58.50±2.01 +69.35±3.15 +77.94±2.26 +60.98±1.31 +0.5462 ±0.0160 +0.7060±0.0090 +✓ +37.94±4.93 +58.69±2.73 +70.09±1.96 +79.77±1.37 +61.49±0.73 +0.5443 ±0.0063 +0.7008±0.0118 +✓ +✓ +32.52±2.73 +52.81±1.73 +66.36±3.07 +76.26±1.73 +56.96±1.31 +0.5225 ±0.0097 +0.6668±0.0096 +✓ +✓ +41.12±1.74 +60.12±2.50 +71.34±1.62 +80.69±0.40 +63.32±1.04 +0.5571±0.0051 +0.7182±0.0050 +Fig. 11. Comparison to SOTA metastatic LNs detection methods in FROC on different +size groups. Left: the large metastatic LN group, right: the small one. +4.5. Ablation study +For ablation study, we mainly focus on evaluating the effectiveness +of the three most significant parts of our method: distance features, LN +station features, and GCN structure. Table 4 and Fig. 9 (Right) provide +the quantitative comparison among different settings. +4.5.1. Distance features +As can be seen from Table 4, introducing the distance as features +leads to clear gains in metastatic LN classification, with 4.84% (line +4 vs. line 1), 1.07% (line 5 vs. line 2), and 2.34% (line 6 vs. line 3) +improvement in mFROC score, respectively, suggesting that providing +the global spatial position of the LN candidate as distance features +is indeed useful. A possible explanation is that metastatic LNs are +definitely not possible to be present in certain areas of the head & neck +such as the brain, and the distance features make the network capture +this kind of knowledge. The distance features, designed to indicate +which LN stations are useful for the classification of LN candidates, +are also shown to be complementary to the LN station features, as +manifested by the gains after involving LN station features. +4.5.2. LN station features +Comparing the results of line 1 with those of line 3 in the Table 4, +adding the LN station features can improve the mFROC score by 4.34%, +suggesting that it is beneficial to leverage the information on the LN +station levels. LN station features express the global visual information +of a wide range of related regions, which is designed to imitate the +habit of radiologists to search for risky LN stations first. However, +modeling the relationship between LN stations is very vital to capture +the LN station features. Fig. 9 (Middle) shows that without GCN, the +performance of LN station classification drops significantly. We also +visualize the LN station features in 2D space using the classical t- +SNE method (Statistics, 2011). As shown in Fig. 10, without GCN, the +features between metastatic LN stations and normal ones are not clearly +separated. Both results suggest that the relationship among LN stations +is vital to reflect whether an LN station is metastatic. Consequently, as +shown in Table 4, adding the LN station features without GCN even +lead to a performance drop (Line 1 vs. Line 2, and Line 4 vs. Line 5). + +True Positive Rate (Sensitivity) +0.9 +(Sensitivity) +0.8 +0.8 +0.7 +Rate +0.6 +0.6 +ours +Positive +0.4 +single-net (3d) +ours +single-net (3d) +single-net (2.5d) +0.5 +0.2 +single-net (2.5d) +LN-levelGCN +LN-levelGCN +True +nnDetection +nnDetection +0.4 +0 +1 +2 +3 +4 +5 +9 +1 +2 +3 +4 +5 +6 +FalsePositiveRateperPatient +FalsePositiveRateperPatient15 +MetastaticLNstations +10.0 +MetastaticLNstations +NormalLNstations +NormalLNstations +10 +7.5 +5.0 +5 +2.5 +0.0 +0 +2.5 +-5 +-5.0 +7.5 +-10 ++ +++ +2.0 +1.5 +-1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +1.00-0.75-0.50-0.250.000.250.500.751.00Computerized Medical Imaging and Graphics 101 (2022) 102108 +9 +C. Wu et al. +Fig. 12. CAM analysis of the result. Light purple represents the positive LN candidate +and orange represents the negative LN candidate. The stripes bar on the right is the +CAM of the final concatenated features. The first 256-dim features are the LN candidate +features. The middle 16-dim features are distance features and the final 64-dim features +are the LN station features. The brighter the bar means the greater the impact the +features have over the result. +4.5.3. GCN structure +To show the effectiveness of the GCN structure, we experiment +with ResNet to directly encode the LN station feature (denoted as ’w/o +GCN’), i.e., removing the GCN structure in the GCN-based classification +network in Fig. 6. We first compare the two methods for LN station clas- +sification. As shown in Fig. 9 (Middle), with GCN structure, the method +improves the AUC by 0.0594, suggesting that the features extracted by +the GCN-based network are much more reliable to indicate whether +an LN station is metastatic. We further compare the performance of +the two methods on LN classification. As shown in Table 4 (Line 2 +vs. Line 3, and Line 5 vs. Line 6), adding the GCN structure always +improves the performance of the methods, showing a gain of 6.36% +and 5.09%, for with and without distance features, respectively. On +the other hand, without the GCN structure, the LN station feature even +leads to a performance drop, which also indicates the importance of +GCN structure. +4.6. Class activation map +To show how different types of features impact the results, we +draw the Class Activation Map (CAM) (Zhou et al., 2016) on the final +combined features. We follow the algorithm introduced in Zhou et al. +(2016) while we calculate heat-map on the final combined features +instead of the original 3D images. Fig. 12 provides two examples, where +the left shows an LN candidate within the LN station, and the right +shows an LN candidate not belonging to any LN station. The CAM bar +is put below the images. The abscissa of the CAM bar ranges from 0 to +331. The first 256 dims represent the LN candidate features and the +next 12 dims represent the distance features. The final 64 dims are +the LN station features. A brighter color on the bar means that the +corresponding features have a greater impact on the results. It can be +seen that most parts of the CAM map are equally bright for the left +case. This shows when a candidate is in an LN station, every type of +features is useful for the network and proves the LN station features and +distance features are closely related to judging whether LN candidates +are metastatic. In the right case, the distance features are much brighter +than other parts, indicating the network mainly focuses on the distance +features. This shows when the candidate is far away from LN stations +(where LNs should not exist), the distance features can be directly +used to rule out the candidates. This verifies the importance of keeping +distance features to reduce the difficulty of classification tasks and to +guide the network when to use other features. +5. Conclusion +This paper proposes a new multi-stage metastatic LN detection +framework combining the prior information on the LN stations. A +new task is defined to identify metastatic LN stations by imitating the +behavior of radiologists who examine the risky LN station region first +before searching for metastatic LNs. Through solving this task, features +of LN stations that are expected to benefit the downstream task, are +learned. To model the complex relationship among LN stations, a GCN- +based classification model is adopted, which refines the features for +LN stations so that they obeys the medical principle that a metastatic +LN station affects other LN stations through the circulatory system. +Based on the two-stage detection framework, we separate metastatic +LN detection task into generating LN candidates and classifying LN +candidates. On the classification network, LN station features are com- +bined to decrease the false alarm ratio further. Experiments on a +contrast-enhanced CTs dataset of 114 cases with OSSC have shown that +our method improves over the current SOTA metastatic LN detection +method (Roth et al., 2015) by 4.77%, 0.0234 on the mFROC and maxF1 +respectively, demonstrating its effectiveness. Satisfactory performance +when handling small metastatic LNs proves the clinical significance of +our method. +CRediT authorship contribution statement +Chaoyi Wu: Methodology, Software, Validation, Writing – origi- +nal draft. Feng Chang: Methodology, Software, Validation, Writing – +original draft. Xiao Su: Data curation, Investigation. Zhihan Wu: Data +curation, Investigation. Yanfeng Wang: Conceptualization, Writing – +review & editing. Ling Zhu: Conceptualization, Investigation, Writing – +review & editing. Ya Zhang: Conceptualization, Methodology, Writing +– review & editing. +Declaration of competing interest +The authors declare that they have no known competing finan- +cial interests or personal relationships that could have appeared to +influence the work reported in this paper. +Acknowledgments +This work is supported partially by National Key R&D Program +of China (No. 2019YFB1804304), SHEITC (No. 2018-RGZN-02046), +111 plan, China (No. BP0719010), and STCSM (No. 18DZ2270700), +the ‘‘Clinical +Plan’’ Project of the Shanghai Ninth People’s Hospital, +Shanghai Jiao Tong University School of Medicine (JYLJ201918) and +‘‘Technology transfer and promotion’’ Project of the Shanghai Jiao Tong +University School of Medicine, China (ZT202108). +References +Avants, B.B., Tustison, N., Song, G., et al., 2009. Advanced normalization tools (ANTS). +Insight J 2 (365), 1–35. +Barbu, A., Suehling, M., Xu, X., Liu, D., Zhou, S.K., Comaniciu, D., 2011. 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Springer, pp. 402–410. + diff --git a/dtE1T4oBgHgl3EQfeASf/content/tmp_files/load_file.txt b/dtE1T4oBgHgl3EQfeASf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..66c84a3bb3be84ee2b6c825301d06878f853e88a --- /dev/null +++ b/dtE1T4oBgHgl3EQfeASf/content/tmp_files/load_file.txt @@ -0,0 +1,1661 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf,len=1660 +page_content='Computerized Medical Imaging and Graphics 101 (2022) 102108 Available online 13 August 2022 0895-6111/© 2022 Elsevier Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Contents lists available at ScienceDirect Computerized Medical Imaging and Graphics journal homepage: www.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhihan Wu d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Yanfeng Wang a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Ling Zhu c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Ya Zhang a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='∗ a Cooperative Medianet Innovation Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Shanghai Jiao Tong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Shanghai 200240,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' China b Shanghai AI Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Shanghai 200232,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' China c Department of Radiology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' School of Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Shanghai Ninth People’s Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Shanghai Jiao Tong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Shanghai 200011,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' China d School of Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Shanghai Jiao Tong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Shanghai 200025,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' China A R T I C L E I N F O Keywords: Metastatic lymph node detection Lymph node stations Graph convolutional network Radiotherapy A B S T R A C T Metastasis on lymph nodes (LNs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' the most common way of spread for primary tumor cells,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' is a sign of increased mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' However, metastatic LNs are time-consuming and challenging to detect even for professional radiologists due to their small sizes, high sparsity, and ambiguity in appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' It is desired to leverage recent development in deep learning to automatically detect metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Besides a two-stage detection network, we here introduce an additional branch to leverage information about LN stations, an important reference for radiologists during metastatic LN diagnosis, as supplementary information for metastatic LN detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The branch targets to solve a closely related task on the LN station level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', classifying whether an LN station contains metastatic LN or not, so as to learn representations for LN stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Considering that a metastatic LN station is expected to significantly affect the nearby ones, a GCN-based structure is adopted by the branch to model the relationship among different LN stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' At the classification stage of metastatic LN detection, the above learned LN station features, as well as the features reflecting the distance between the LN candidate and the LN stations, are integrated with the LN features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We validate our method on a dataset containing 114 intravenous contrast-enhanced Computed Tomography (CT) images of oral squamous cell carcinoma (OSCC) patients and show that it outperforms several state-of-the-art methods on the mFROC, maxF1, and AUC scores, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Introduction Metastatic lymph nodes (LNs) play a pivotal part in the spread of original tumors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Detecting metastatic LNs in time is of great importance both in cancer prevention and treatment planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' However, metastatic LN detection is extremely challenging and time-consuming even for an experienced radiologist due to the following characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Firstly, the sizes of metastatic LNs are extremely small compared to other organs and tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 1(a), most cases have a diameter of less than 20 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Secondly, metastatic LNs are usually sparsely distributed, with mostly 1–2 metastatic LNs per patient in our task (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Thirdly, the appearance of metastatic LNs is ambiguous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', the intra-class differences of metastatic LNs are large while the inter- class differences between metastatic LNs and normal LNs are small, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Radiologists must judge whether an LN candidate is metastatic by simultaneously examining its texture, shape, size, as well as its neighboring tissues and organs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' ∗ Corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' E-mail addresses: zhul1757@2m9h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='net (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhu), ya_zhang@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='cn (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhang).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 1 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Wu and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Chang contribute equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Several previous studies start with an easier task of detecting en- larged LNs (Barbu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Bouget et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Nogues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Oda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The enlarged LNs are highly likely to be metastatic, but not vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Such methods are thus limited in coverage as nearly 70% of metastatic LNs are within 10 mm in size, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Some subsequent studies consider small metastatic LNs and leverage PET modality or tumor location to provide additional information to assist the detection of metastatic LNs and achieve promising perfor- mance (Chao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020a,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' However, in practice, patients do not always have PET modality available due to its high cost, or the primary tumors because they may have been surgically removed, which makes the above methods inapplicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' This paper thus explores leveraging other more accessible auxiliary information to help detect metastatic LNs, as an important supplement to the above methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' In this paper, we propose an efficient deep learning based method to leverage information about LN stations to detect metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The LN https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='compmedimag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='102108 Received 19 April 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Received in revised form 8 July 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Accepted 28 July 2022 Computerized Medical Imaging andGraphicsEISEVIERComputerized Medical Imaging and Graphics 101 (2022) 102108 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Statistics of metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (a) Distribution of metastatic LNs sizes (short axis length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (b) Distribution of the number of metastatic LNs per patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Left two metastatic LNs, (a) and (b), are quite different in appearance, showing the large intra-class difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' On the right, a normal LN (c) and a metastatic LN (d) appear quite similar, showing the small inter-class difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Examples of LN stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Each row is a patient and the LN stations are marked in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' station is a marking system that describes the area where LNs are sited, usually with edges of various physiological structures (such as muscles and blood vessels) as the boundary (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2021), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' LNs in the same LN station share similar physiological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' A metastatic LN indicates the other LNs in the same LN station are much more likely to be metastatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Accordingly, experienced radiologists would focus more on one LN station if they find a metastatic LN in it to improve diagnosis accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' In addition, LN stations are connected by lymphatic drainage directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Therefore, there are strong correlations between neighboring LN stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' If an LN station has been confirmed to contain metastatic LN, the neighboring LN stations are considered more vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Compared with other additional information, such as the PET modality and tumor locations, LN stations reflect the metastatic pattern more directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' They can be auto-delineated precisely because their locations are stable, making them more suitable for metastatic LN detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' An LN station segmentation network is trained with a small set of LN segmentation data to locate the LN stations automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4 presents the overall framework of our method, where an additional LN station classification branch is introduced to the common detection branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' For the LN station classification branch, a new task is introduced to discriminate whether an LN station contains metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' To model the interaction between LN stations, a Graph Convolu- tion Network (GCN)-based structure is integrated into the LN station feature encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The features extracted this way, on the one hand, can indicate which stations are riskier and help the network focus more on the LNs in these stations so that metastatic LNs will not be missed due to small sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' On the other hand, the relationship information learned by this structure can supplement the visual features and tackle the prob- lem of ambiguous appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' For the metastatic LN detection branch, following the methods in Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (2020b, 2019), we train a metastatic LN candidate generator at high sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' There are massive false- positive cases in these candidates because of the sparsity of metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We feed the LN station features embedding extracted from the LN station classification branch together with the LN candidate into the LN candidate classifier to decrease the false alarm ratio via leveraging LN station information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' To validate the effectiveness of the proposed method, we collected a dataset containing 114 intravenous contrast-enhanced CT scans of oral squamous cell carcinoma (OSCC) patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Extensive experiments have shown that the proposed method improves the mean free response operating characteristic (mFROC) by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='77%, maxF1 by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0230, and AUC by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0303 compared to the state-of-the-art 3D medical detection method (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2015), demonstrating the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The contribution of the paper can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We propose an efficient deep learning based method to leverage the information of LN stations for metastatic LN detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' To learn representations for LN stations, we define a closely related auxiliary task to assist metastatic LN detection by discrim- inating an LN station is metastatic or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' To model the mutual influence among different LN stations, we introduce a GCN-based structure for LN station classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Related work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' General object detection in medical images General object detection is viewed as a classic and typical task to be studied for decades in the computer vision community (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' From the perspective of medical image analysis, it is also promisingly valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Many medical issues like lesion detection (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2019) and lung module detection (Teramoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2016) can all be concluded into this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The popular detection methods can be divided into two types: end-to-end methods (Baumgartner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2018, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zlocha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2019) and two-stage methods (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' End-to-end methods prefer to dealing the detection task as a whole complete flowchart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' These methods have been widely used in universal detection tasks due to their few hyper- parameters and guaranteed performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' nnDetection (Baumgartner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2021) is one of the latest and most representative end-to-end detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' It is self-configuring and can achieve encouraging performance in various medical detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 1 T 1 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1 1 1 1 1 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='08 1 Y 1 1 Densi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='06 1 1 1 1 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='04 L L 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='02 L L L 1 1 1 1 1 1 1 0 10 20 30 40 50 Diameterofshortaxis(mm)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5 1 1 1 1 I 1 1 1 1 1 1 -- 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='4 -- -- -- : -- --- -- -- 1 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 1 1 1 1 1 4 1- ---- -- 1 1 ---- --- 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 1 1 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2 r-- 1 1 1 -- r 1 -- -- 1 1 1 1 -- 1 -- -- 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1 -- l- -- -- 1 1 --- 1 1 -- !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' -- 1 1 0 1 2 3 6 8 10 11 MetastaticLNs per patientComputerized Medical Imaging and Graphics 101 (2022) 102108 3 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Two-stage methods decouple the detection task into two sub-tasks: proposal extractor and FP reducing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' In the first proposal extrac- tor stage, the generator will give many coarse candidate proposals with high sensitivity and high false alarm ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The second stage is trained to classify the candidate proposals and dismiss the false positive cases while maintaining an acceptably high sensitivity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' In many sparse-distributed and small object detection task, like lung module detection (Teramoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2016) and metastatic LN detection task (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020c), two-stage methods are proved to be more suitable compared with end-to-end methods (Chao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' By separating the raw task into two easier parts, more effective strategies can be designed for different parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We adopt this design paradigm in our task as well and prove the necessity by comparing our method with the nnDetection (Baumgartner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Metastatic lymph node detection Many former works focus on detecting the enlarged LN whose short axis is greater than 10 mm instead of metastatic LNs (Bouget et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Barbu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Nogues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Oda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2018) because the detection of enlarged LNs is much easier and enlarged LNs are con- sidered to be highly likely metastatic medically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Conventional detection algorithm (Barbu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Feulner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Kitasaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2007) based on morphological prior knowledge will design extraction of the visual features carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Recent works based on deep learning (Nogues et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Oda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2018) bring more possibilities to this field, improving the performance impressively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The success encourages the community to consider the detection of metastatic LNs regardless of their sizes, which is more practical and challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Some works have made successful attempts on this task (Chao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020a,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (2020a,c) divide the network into two branches to extract features separately via leveraging the distance between LN candidates and the primary tumor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Chao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (2020) models the relationship between LN candidates and uses the information of the primary tumor as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' PET modality is also proved to be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' However, the locations of the primary tumor for the patients after operation is often unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' PET modality is also costly to collect and in many practical cases, the patients lack the PET/CT for radiologists to refer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Previous methods will fail when these two important information are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Therefore, we hope to use the information of LN station to help the detection of metastatic LNs as another optional auxiliary information, so that a good detection accuracy can still be obtained when the above information is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' LN stations have been proved to be very important reference in the detection of metastatic LNs clinically (Grégoire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Naruke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 1978), but, as far as we know, no prior work has considered how to add this important prior knowledge to the automatic detection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We are the first to leverage the information of LN stations assisting the detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Graph convolution network Graph neural network (GNN) is first proposed in Scarselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' It combines the strong expressive power of the graph structure with the neural network (Jie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' It can be used to model the relationship of the non-Euclidean structure data in many areas like social network (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2019) and protein–protein interaction networks (Fout, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' In Kipf and Welling (2016), graph convolution network (GCN) is first proposed by adding the convolution operation into the original GNN structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' GCN has received a lot of attention due to its simplicity, easy understanding, and strong performance (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' In medical image analysis, GCN has great potential to be applied in different tasks since it can directly model the relationship between different physiological tissues which cannot be modeled by the regular convolution neural network (CNN) (Chao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Kazmierski and Haibe-Kains, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Kazmierski and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Flowchart of the proposed method, which consists of the LN station classifi- cation branch and Metastatic LN Detection branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The former is leveraged to encode useful features of the LN stations and the latter combines different types of information to make the final decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Haibe-Kains (2021), Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (2020) both use the GCN to combine the information from the different volumes of interest (VOI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' When processing images in medical images, images are often sliced and this structure can integrate global information well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Chao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (2020) inspires our work a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' It builds the graph on the LN candidate level and assumes the metastatic LNs may affect each other related to their physical distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' However, we build the graph on the LN station level instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We argue that due to the uncertainty of the individual, the interaction between the LNs is not so accurate especially when LN candidates have strong sparsity, and building the relationship on a general level will be more suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Method LN stations play an important role in identifying metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' LNs in the same LN station share similar locations and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Due to the similarity of LNs in a certain LN station, metastatic LNs usually appear like a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Certain LN stations are more likely to contain metastatic LNs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', station Ib and station II have much more metastatic LNs than others (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 5(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' There is also a strong correlation between LN stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' An abnormal LN station may reflect that other LN stations near it have more potential to contain metastatic LNs as metastasis may spread to farther LN stations from this abnormal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 5(b) shows the correlations between LN stations in terms of metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Some LN stations, such as II_R and III_R, demonstrate strong correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' As a result, in clinical practice, experienced radiologists typically inspect LN stations first in searching for the metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Considering the importance of LN stations, we here explore to in- corporate information about LN stations into the detection of metastatic LNs, by introducing an additional LN station classification branch to the standard detection branch, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The features from the two branches are integrated for the final LN candidate classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Ln station classification branch To learn representations for LN stations, the LN station classifi- cation branch is trained with a closely related auxiliary task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', to discriminate whether an LN station has metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' If an LN station region contains the center point of any metastasis LN, it is considered to have metastatic LNs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' otherwise, it is considered not to have metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' In this way, the ground truth annotations for the LN station classification can be directly inferred from the annotations of metastasis LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' With the above task setup, the goal of the LN station classification branch is to learn representations that can best discriminate the LN stations with and without metastasis LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' To model the relationships among LN stations, a GCN-based classification network is adopted by the LN station classification branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The overall architecture of the classification network is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' LNStation eatureEncode LNCondidalComputerized Medical Imaging and Graphics 101 (2022) 102108 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Statistics of LN stations in the case of OSSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' ‘L’ and ‘R’ represent the left and right independent parts of the LN station respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (a) Statistics of the metastatic LNs contained in each LN station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (b) Spearman’s correlation of LN stations regarding to metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Data preparation Let \ue230 = {𝑥1, 𝑥2, … , 𝑥𝑁 } represents a set of 3D CT volumes, where 𝑥𝑖 ∈ 𝑅𝐻×𝑊 ×𝐷, the subscript 𝑖 stands for the 𝑖th patient, and 𝐻, 𝑊 , 𝐷 are the height, width, and depth of the 3D CT volumes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' To get the LN station segmentation masks, an nnU-Net (Isensee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2018) based segmentation network is trained on a small dataset of coarsely annotated LN stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2 Without loss of generality, we consider a specific 3D CT volume 𝑥 to simplify the notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Denote the LN station region masks obtained by the segmentation network as 𝑆 = 𝑆𝑒𝑔𝑆(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Based on the segmentation result, the raw CT volume is divided into 12 LN station patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Noted that since some LN stations, like II station and VI station, are symmetrical and independent with each other, we divide them into left and right two parts based on the central axis and treat them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Therefore though we only have 7 station classes in segmentation results, 12 LN station patches are obtained after post-processing, denoted as 𝑆 = {𝑠1, 𝑠2, … , 𝑠12 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Gcn-based classification network To model the complex reflux relationship among LN stations, a GCN- based classification network is adopted, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' A graph is formed where each LN station is treated as a vertex in a fully connected graph with self-connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' A shared 3D CNN encoder 𝐶𝑆(⋅) is first used to get the feature embedding for each LN station separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' ℎ𝑗 = 𝐶𝑆(𝑠𝑗), 𝑗 ∈ {1, … , 12} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (1) Here ℎ𝑗 represents the encoder output of the 𝑗th LN stations, which is further fed into a two-layer graph neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 𝑓 𝑆 𝑗 = 𝜎2(𝑟𝑒𝑙𝑢(𝜎1(ℎ𝑗))), 𝑗 ∈ {1, … , 12} 𝜎𝑙(ℎ𝑗) = 𝛴𝑘∈{1,…,12}𝑎𝑘𝑗𝜙(ℎ𝑘), 𝑙 ∈ {1, 2} , (2) where 𝜎1(⋅) and 𝜎2(⋅) are the two layers of the graph neural network, 𝑎𝑘𝑗 is the edge weight between the 𝑘th station and the 𝑗th station, and 𝜙 is a learnable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' In the common setting of GCN, 𝑎 ∈ [0, 1] is set based on some similarity metric before the operation of GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' While in our case, it is hard to define a reasonable similarity between LN stations due to the complexity of the human circulatory system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Therefore we also set 𝑎 as a learnable parameter so that the network can learn to model the reflux relationship automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' For each LN stations, a shared 2-layer MLP is used to get the final classification results: 𝑝𝑗 = 𝑀𝐿𝑃 (𝑓 𝑆 𝑗 ), 𝑗 ∈ {1, … , 12} , (3) where 𝑝𝑗 is the final prediction result of the 𝑗th LN station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 2 The traditional registration methods (Avants et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2009) can also be used here to obtain the segmentation of the LN station, but the segmentation results are relatively poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' In order not to affect the effect of our subsequent methods, we choose to use the neural network for its better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Considering that metastatic LNs are quite sparse, which causes severe class imbalance problem, focal loss (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2017) is adopted as the training loss: 𝐿𝑆 = { −𝛼(1 − 𝑝𝑗)𝛾𝑙𝑜𝑔(𝑝𝑗) 𝑦𝑗 = 1 −(1 − 𝛼)(𝑝𝑗)𝛾𝑙𝑜𝑔(1 − 𝑝𝑗) 𝑦𝑗 = 0 , (4) where 𝛼 and 𝛾 are hyper-parameters, and 𝑦𝑗 ∈ {0, 1} is the ground truth with 1 indicating metastasis and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Metastatic LN detection branch We follow typical two-stage detection methods and divide this task into two stages: extracting LN candidates and classifying the LN candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' LN candidate generation Following the common setting of LN detection tasks (Chao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020c), we employ an LN candidate generator with high sensitivity to generate LN candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Specifically, an nnU-Net (Isensee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2018) based segmentation network is trained with pixel-wise metastatic LN annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The segmentation results of the network are leveraged to generate LN candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We then locate the center point of each connected component and crop out a cube of the size 48 × 48 × 48 centering around the center point as LN candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The cube size is expected to be large enough to cover all LN candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' This coarse result is expected to lead to a high sensitivity score but severely suffer from the problem of false alarms as shown by previous studies (Chao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Therefore, in the next part, we mainly focus on classifying the metastatic LN candidates to reduce the false alarm ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' LN candidate classification After getting the LN candidates, the goal of the LN candidate classifier is to judge whether an LN candidate is metastatic or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The network architecture is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Without loss of generality, we consider a specific LN candidate 𝑐 ∈ 𝑅48×48×48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' First, a simple 3D CNN network 𝐶𝐿 is used to get the visual information of the LN candidate: 𝑓 𝐿 = 𝐶𝐿(𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (5) The appearance features 𝑓 𝐿 are expected to provide visual details of the LN candidate patches, such as morphology, texture, and intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' In addition to the appearance features of the LN candidate 𝑐, we try to integrate its corresponding LN station features in classifying the LN candidates to reduce false alarms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The LN station features contain the global information indicating whether there are metastatic LNs in this LN station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' In detail, we calculate the Euclidean distance between the LN candidate 𝑐 and the center of the 12 LN stations and find the nearest LN station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 𝑘 = arg min 𝑗∈{1,…,12} 𝑑(𝑐, 𝑠𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (6) The 𝑘th station is thus considered where the LN candidate 𝑐 belongs to and its feature embedding 𝑓 𝑆 𝑘 is then concatenated to the visual feature embedding 𝑓 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Some identified LN candidates might not belong to any LN station regions, and the LN station features shall not be used in this case, so we add distance features to prompt the network further when to focus on the LN stations features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Specifically, the distance to each LN station is represented as 12-dimensional features 𝑓 𝐷, which is further concatenated into the above features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Adding the distance features can also help the network reject the LN candidates that appear in places far away from LN stations, such as the top of the head, which are impossible to contain LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The final features are expressed as follow: 𝐹 = 𝑓 𝐿‖𝑓 𝑆 𝑘 ‖𝑓 𝐷, (7) where ∥ denotes the concatenation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 70 60 50 Count 40 30 20 10 0 la IbLIbR IILIIR IIILIIIRIVLIVRVLVR LN Stations1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0 IbL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='8 IbR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='6 IIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='4 IIIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2 IVL VLIVR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0 VR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2 Ia IbLIbR IIL IIR IIIL IIIRIVLIVRVL VRComputerized Medical Imaging and Graphics 101 (2022) 102108 5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Architecture of the metastatic LN station classification network, consisting of a 3D CNN encoder, a 2-layer GCN, and a 2-layer MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Overall framework of the proposed metastatic LNs classification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' For each LN candidate, a CNN encoder is used to extract its appearance features, which is further integrated with the distance features (marked in yellow) and the features of its nearest LN station (marked in green) before feeding into the MLP for final classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Finally we use a 2-layer MLP to get the final classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 𝑜 = 𝑀𝐿𝑃 (𝐹).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (8) The metastatic LN classification task faces the class imbalance prob- lem due to the sparsity of metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The focal loss (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2017) is employed as the training loss: 𝐿𝐿 = { −𝛽(1 − 𝑜)𝜃𝑙𝑜𝑔(𝑜) 𝑦𝐿 = 1 − (1 − 𝛽)𝑜𝜃𝑙𝑜𝑔(1 − 𝑜) 𝑦𝐿 = 0 , (9) where 𝛽 and 𝜃 are hyper-parameters, and 𝑦𝐿 ∈ {0, 1} is the ground truth label of the LN candidate, with 1 representing the metastatic LN candidate and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Discussion It is worth emphasizing that although we use additional LN station annotations information in the training progress, in inference time we do not need extra LN station annotations because the LN station segmentation network automatically generates masks for each LN sta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Our method can improve the detection of metastatic LNs only if the patient can provide contrast-enhanced CT images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' This makes our solution more in line with actual needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Dataset 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Lymph node station segmentation To train the LN station segmentation model, a pixel-level labeled dataset is collected in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The dataset contains 82 head & neck intravenous contrast-enhanced CT scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' For each CT scan, the 3D segmentation masks of 7 LN stations of the head & neck are pro- vided by radiologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' To verify segmentation results, we randomly split this dataset into 60%, 20%, 20% for training, validation, and testing respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Metastatic lymph nodes detection We collected 114 intravenous contrast-enhanced CTs of oral OSCC patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The median age of these patients is 61 years old and interquar- tile range is 50 ∼ 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' There are 80 males and 34 females.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Patients in this dataset are all under radiotherapy treatments, which means some patients may have undergone surgery, and some parts of their head & neck such as tumors have been manually processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The metastatic LNs of these patients have been pathologically confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' All CT scans are with a slice thickness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='625 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' For evaluation, we randomly split the 114 CT scans into 80 for training and 34 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Implementation details In our experiments, the Hounsfield unit value of the CT is clipped to be within [−218, 600].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We implement the whole framework stage-by- stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' LN station segmentation and LN candidate segmentation A U-Net (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2015) based model is trained to automatically segment the lymph node stations (7 classes) and LN can- didates (1 class), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Patches with a shape of 208 × 238 × 196 are randomly cropped to be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We further apply random scaling, random gamma, and random mirroring to augment the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The 3d U-Net is trained on one GeForce RTX 3090 GPU with a batch size of 2 for 200 epochs, each epoch containing 250 batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The SGD optimizer with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0003 is used with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='99 and a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='00005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' LN station classification Based on the LN station segmentation masks, we clip the bounding box of each LN station and resize it to 96 × 96 × 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' As some LN stations are symmetrical and independent of each other, we divide them into left and right two parts based on the central axis and treat them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Thus, we can get 12 cubes with a size of 96 × 96 × 96 for one CT scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We treat the 12 cubes as 12 nodes in one graph and use ResNet10 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2016) as our feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 2 graph convolution blocks containing graph convolution layer and Relu activation function are used to exploit the mutual relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' A fully connected layer is inserted to make the final decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' LN candidate classification Every connected component obtained by LN candidate segmenta- tion is treated as an LN candidate and the connected region whose 𝑖𝑜𝑢 with the ground truth mask greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1 is considered as a metastatic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The center point of each connected component is located, centered around which a patch of 48 × 48 × 48 is cropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Both LN station classification and LN candidate classification use the same training setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Random affine and random elastic deformation are applied in this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We trained the model with a batch size of 128 for 100 epochs on one GeForce RTX 3090 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The SGD optimizer with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='01 is used with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='9 and a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The learning rate changes to 1/10 of the original when the epoch reaches 30, 50, and 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' To handle the category imbalance, we supervise our network using the focal loss parameterized by 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='25, 𝛾 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Computerized Medical Imaging and Graphics 101 (2022) 102108 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Table 1 Comparison between LN station segmentation by radiologists and neural network in terms of dice and center point distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Segmentation results are collected from 5 radiologists, and those of the most experienced radiologists are used as ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' DSC↑ \\CPD↓ are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Annotator LN station Ia Ib II III IV V VIa Average Radiologist1 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='18\\8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='31 75.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='88 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='48\\5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='14 aThere is a long contour line near the esophagus in the mask of the station VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Since it is very narrow, the criteria for whether to outline this line differs greatly between different people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Therefore, the results of humans on the LN station VI are much worse than the results automatic labeled by the machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Table 2 Comparison with SOTA metastatic ln detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' the mean ± standard deviation are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Method FROC@1(%) FROC@2(%) FROC@3(%) FROC@4(%) mFROC(%) ↑ Max F1↑ AUC ↑ nnDetection (Baumgartner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2021) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='33±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5220±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='370 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='6166±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0188 single-net(3d classifier)a (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020a) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='64±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5206±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='6673±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0082 single-net(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5d classifier) (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2015) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='20±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='27 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='64±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='18 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='41±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='09 77.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='7182±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0050 aMany clinical works (Koizumi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhang and Ren, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2021) carry out different experiments for different body positions or different image modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Due to the differences of datasets, we cannot directly compare with them, but these methods can be unified into a typical 3D CNN structure, so here we use single-net(3d classifier) to uniformly represent the effects of these works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Evaluation metric DSC: To evaluate the performance of segmentation tasks, we adopt Dice Similarity Coefficient (DSC) as most medical image segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' DSC measures the overlap rate of the prediction mask 𝑝 and ground truth mask 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 𝐷𝑆𝐶 = 2|𝑝 ∩ 𝑔| |𝑝| + |𝑔| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (10) CPD: To evaluate the LN station segmentation results, we calculate the center point distance (CPD) between two segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' CPD measures the distance error between the prediction mask 𝑝 and ground truth mask 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Let 𝐶𝑖 ∈ 𝑅3 denotes the center point coordinate of mask 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Voxel euclidean distance is adopted here to measure the distance: 𝐶𝑃 𝐷 = ‖𝐶𝑝 − 𝐶𝑔‖2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (11) mFROC: To evaluate the performance of metastatic LN detection tasks, we use the free response operating characteristic (FROC) as the metric, which measures the recall against different numbers of FPs allowed per patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We report the mFROC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', the average sensitivity at 1,2,3,4 FPs per patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Since most of the previous work (Chao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020a,c) similar to ours used mFROC as the main reporting indicator, our analysis of method performance also focused on this indicator mainly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' maxF1: Besides mFROC, we use maxF1 as a supplementary metric for classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' By adjusting the threshold that decides whether an LN candidate is metastatic, an LN candidate with low confidence to be metastatic can be considered as metastatic with the decrease of the threshold, which leads to different F1-scores with the change of the threshold, formulated as: 𝐹1 = 2 × 𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 × 𝑟𝑒𝑐𝑎𝑙𝑙 𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 + 𝑟𝑒𝑐𝑎𝑙𝑙 𝑚𝑎𝑥𝐹1 = max 𝑡∈[0,1] 𝐹1𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (12) AUC: AUC represents the area under the receiver operating char- acteristic (ROC) curve, which is a very common indicator in detection and binary classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Quantitative results and discussions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Lymph node station segmentation To show that LN stations can be precisely auto-delineated, we com- pare the segmentation results between humans and the segmentation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The quantitative results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' On contrast- enhanced CT images, it is difficult to find a border that is totally consistent with the definition of LN stations in medical theory, and the delineation of LN stations depends more on the experience of radiologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Thus, we collect segmentation masks of 5 radiologists and use the mask of the most experienced radiologist as ground truth to evaluate others’ performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' As shown in Table 1, the results of neural network are even better than manual delineation and are sufficient for metastatic LN detection task to crop LN station patches for follow-up stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' This shows LN stations are much less costly to collect compared with other additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 8 shows visualization results of different radiologists and the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Metastatic LN detection To show the effectiveness of our method for detecting metastatic LNs, we compare with a set of metastatic LN detection methods, includ- ing nnDetection (Baumgartner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2021), a self-configuring frame- work for 3D medical object detection tasks which has been demon- strated effective on several public benchmarks, as well as several methods especially designed for metastatic LNs detection (Chao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020a), all of which are based on two-stage framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' For a fair comparison, we use the same LN candidate extractor for the later three methods (Chao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020a) as our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Table 2 outlines the quantitative comparison of different methods for metastatic LN detection tasks, demonstrating the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' With mFROC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='6332, maxF1 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5571, and AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='7182, our method exceeds nnDetection by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='85% in mFROC, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0351 in maxF1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1016 in AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Our method achieves better results in every metric except FROC@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' nnDetection gives higher confidence to apparent metastatic LNs so it gets a better result at FROC@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' But its drawback is that it tends to overlook small and hard candidates so it cannot achieve a high FROC score even when FP rate allowed per patient is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Our method outperforms it since we adopt a two-stage methodology to handle small and sparse objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Our method achieves a gain of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='77%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0234, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0303 over the second best method (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2015) in terms of mFROC score, maxF1 score, and AUC score respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Compared with Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (2015) and Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (2020a), our method introduces supplementary features on LN stations as well as the distance to LN stations, which Computerized Medical Imaging and Graphics 101 (2022) 102108 7 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Visualization of LN stations segmentation of a patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Each column is either from one radiologist or the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The results from radiologists are rough because the masks are delineated slice by slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The neural network instead generates more smooth results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Left: Comparison of our method to SOTA metastatic LNs detection methods in FROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Middle: ROC of LN station classification with and without GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Right: Comparison of the methods in Ablation study in FROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' are both shown to contribute to a more powerful classifier and effec- tively reduce the false alarm ratio caused by data imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' LN-level GCN (Chao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2020), which treats each LN candidate instead of LN station as a vertex in graph, with the distance between the LN candidates as the edge weights, performs worst among the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' A possible reason for this phenomenon is that the true metastatic LNs are quite sparse and most LN candidates are actually false positives which provide misguided information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' So, it is more reasonable to leverage the relationship information on a more general aspect, such as on the LN station level, to exclude the effect of some individual mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 9 (Left) shows the FROC curves of these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' There exists an upper bound of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='9065, because the recall rate for the LN candidate detection is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='9065 and the missing metastatic LNs cannot be recovered at the classification stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' As a result, the best performance of LN candidate classification cannot exceed this recall rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Stratified analysis for different LN sizes In clinical practice, LNs with short axis diameter greater than 10 mm are considered enlarged LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' While many relevant studies focus on the enlarged LNs (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2015), the majority of metastatic LNs are actually small metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' To validate the effectiveness of our method for LNs of different sizes, we separate the metastatic LNs into two groups (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', Large and Small) based on their short axis length, using 10 mm as the cut-off threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The FROC curves of different methods are separately presented for the above two size groups, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' For all methods, the maxF1, mFROC and AUC scores on the large LNs are better than those on the small LNs, suggesting that it is more challenging to detect metastatic LNs of small sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' It can be seen that for large LNs, our method outperforms other methods at most sampling points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' For small LNs, our method improves the sensitivity by a large margin at different operating points such as 2 FP∕patient and 3 FP∕patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Table 3 shows quantitative results of these methods for the two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The nnDetection (Baumgartner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2021) performs well for large LNs but terribly on small ones, showing that general detection methods fail to handle the small LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' For large LNs, our method achieves mFROC of 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='23%, maxF1 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5436 and AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='8555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' For small LNs, our method outperforms the comparative methods by a large margin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', an improvement of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='72%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0133 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0057 over the second best method, single-net(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5d classifier), in terms of mFROC, maxF1 and AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The above result confirms that introducing the LN station level feature is effective for detecting small metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (Senesitivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='9065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='8 TruePositiveRate( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5 ours w/odistance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='4 wlostationfeatures w/oGCN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='3 w/oGcNanddistance highbar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2 2 3 4 5 6 7 FaslePositiveRateperPatient(Senesitivity AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='7908 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='8 AUC=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='7314 TruePositiveRate( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2 ours W/oGCN random 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='8 FaslePositiveRateSenesitivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='9065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='8 TruePositiveRate( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5 ours nnDetection 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5dclassifier 3dclassifier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='3 nodelevelGCN highbar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2 1 2 3 4 5 6 7 FaslePositiveRateperPatientComputerized Medical Imaging and Graphics 101 (2022) 102108 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' t-SNE visualization of LN station feature distribution in 2D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Table 3 Improvement on LNs of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Large lns and small lns are separated based on the short axis (10 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Method Large LNs Small LNs mFROC (%)↑ maxF1↑ AUC↑ mFROC (%)↑ maxF1↑ AUC↑ nnDetection (Baumgartner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2021) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='86±5.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5608±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0228 single-net(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5d classifier) (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2015) 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='71±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0085 Table 4 Ablation analysis of different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' GCN and w/o GCN represent with and without the gcn-based structure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' distance means distance features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' the mean ± standard deviation are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Method FROC@1(%)↑ FROC@2(%)↑ FROC@3(%)↑ FROC@4(%)↑ mFROC(%)↑ maxF1↑ AUC↑ Distance LN station features w/o GCN GCN 33.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Comparison to SOTA metastatic LNs detection methods in FROC on different size groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Left: the large metastatic LN group, right: the small one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Ablation study For ablation study, we mainly focus on evaluating the effectiveness of the three most significant parts of our method: distance features, LN station features, and GCN structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Table 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 9 (Right) provide the quantitative comparison among different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Distance features As can be seen from Table 4, introducing the distance as features leads to clear gains in metastatic LN classification, with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='84% (line 4 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' line 1), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='07% (line 5 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' line 2), and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='34% (line 6 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' line 3) improvement in mFROC score, respectively, suggesting that providing the global spatial position of the LN candidate as distance features is indeed useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' A possible explanation is that metastatic LNs are definitely not possible to be present in certain areas of the head & neck such as the brain, and the distance features make the network capture this kind of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The distance features, designed to indicate which LN stations are useful for the classification of LN candidates, are also shown to be complementary to the LN station features, as manifested by the gains after involving LN station features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' LN station features Comparing the results of line 1 with those of line 3 in the Table 4, adding the LN station features can improve the mFROC score by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='34%, suggesting that it is beneficial to leverage the information on the LN station levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' LN station features express the global visual information of a wide range of related regions, which is designed to imitate the habit of radiologists to search for risky LN stations first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' However, modeling the relationship between LN stations is very vital to capture the LN station features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 9 (Middle) shows that without GCN, the performance of LN station classification drops significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We also visualize the LN station features in 2D space using the classical t- SNE method (Statistics, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 10, without GCN, the features between metastatic LN stations and normal ones are not clearly separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Both results suggest that the relationship among LN stations is vital to reflect whether an LN station is metastatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Consequently, as shown in Table 4, adding the LN station features without GCN even lead to a performance drop (Line 1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Line 2, and Line 4 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' True Positive Rate (Sensitivity) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='9 (Sensitivity) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='7 Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='6 ours Positive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='4 single-net (3d) ours single-net (3d) single-net (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='2 single-net (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5d) LN-levelGCN LN-levelGCN True nnDetection nnDetection 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='4 0 1 2 3 4 5 9 1 2 3 4 5 6 FalsePositiveRateperPatient FalsePositiveRateperPatient15 MetastaticLNstations 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0 MetastaticLNstations NormalLNstations NormalLNstations 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5 10 + ++ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='00-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='75-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='50-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='00Computerized Medical Imaging and Graphics 101 (2022) 102108 9 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' CAM analysis of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Light purple represents the positive LN candidate and orange represents the negative LN candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The stripes bar on the right is the CAM of the final concatenated features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The first 256-dim features are the LN candidate features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The middle 16-dim features are distance features and the final 64-dim features are the LN station features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The brighter the bar means the greater the impact the features have over the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' GCN structure To show the effectiveness of the GCN structure, we experiment with ResNet to directly encode the LN station feature (denoted as ’w/o GCN’), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', removing the GCN structure in the GCN-based classification network in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We first compare the two methods for LN station clas- sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 9 (Middle), with GCN structure, the method improves the AUC by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0594, suggesting that the features extracted by the GCN-based network are much more reliable to indicate whether an LN station is metastatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We further compare the performance of the two methods on LN classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' As shown in Table 4 (Line 2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Line 3, and Line 5 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Line 6), adding the GCN structure always improves the performance of the methods, showing a gain of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='36% and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='09%, for with and without distance features, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' On the other hand, without the GCN structure, the LN station feature even leads to a performance drop, which also indicates the importance of GCN structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Class activation map To show how different types of features impact the results, we draw the Class Activation Map (CAM) (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2016) on the final combined features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' We follow the algorithm introduced in Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' (2016) while we calculate heat-map on the final combined features instead of the original 3D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 12 provides two examples, where the left shows an LN candidate within the LN station, and the right shows an LN candidate not belonging to any LN station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The CAM bar is put below the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The abscissa of the CAM bar ranges from 0 to 331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The first 256 dims represent the LN candidate features and the next 12 dims represent the distance features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' The final 64 dims are the LN station features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' A brighter color on the bar means that the corresponding features have a greater impact on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' It can be seen that most parts of the CAM map are equally bright for the left case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' This shows when a candidate is in an LN station, every type of features is useful for the network and proves the LN station features and distance features are closely related to judging whether LN candidates are metastatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' In the right case, the distance features are much brighter than other parts, indicating the network mainly focuses on the distance features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' This shows when the candidate is far away from LN stations (where LNs should not exist), the distance features can be directly used to rule out the candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' This verifies the importance of keeping distance features to reduce the difficulty of classification tasks and to guide the network when to use other features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Conclusion This paper proposes a new multi-stage metastatic LN detection framework combining the prior information on the LN stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' A new task is defined to identify metastatic LN stations by imitating the behavior of radiologists who examine the risky LN station region first before searching for metastatic LNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Through solving this task, features of LN stations that are expected to benefit the downstream task, are learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' To model the complex relationship among LN stations, a GCN- based classification model is adopted, which refines the features for LN stations so that they obeys the medical principle that a metastatic LN station affects other LN stations through the circulatory system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Based on the two-stage detection framework, we separate metastatic LN detection task into generating LN candidates and classifying LN candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' On the classification network, LN station features are com- bined to decrease the false alarm ratio further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Experiments on a contrast-enhanced CTs dataset of 114 cases with OSSC have shown that our method improves over the current SOTA metastatic LN detection method (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=', 2015) by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='77%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content='0234 on the mFROC and maxF1 respectively, demonstrating its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Satisfactory performance when handling small metastatic LNs proves the clinical significance of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' CRediT authorship contribution statement Chaoyi Wu: Methodology, Software, Validation, Writing – origi- nal draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Feng Chang: Methodology, Software, Validation, Writing – original draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Xiao Su: Data curation, Investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Zhihan Wu: Data curation, Investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Yanfeng Wang: Conceptualization, Writing – review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Ling Zhu: Conceptualization, Investigation, Writing – review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Ya Zhang: Conceptualization, Methodology, Writing – review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Declaration of competing interest The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Acknowledgments This work is supported partially by National Key R&D Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 2019YFB1804304), SHEITC (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 2018-RGZN-02046), 111 plan, China (No.' metadata={'source': 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RetinaNet for CT lesion detection with dense masks from weak RECIST labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' In: International Conference on Medical Image Computing and Computer-Assisted Intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} +page_content=' 402–410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfeASf/content/2301.03202v1.pdf'} diff --git a/eNFQT4oBgHgl3EQfjzbM/content/tmp_files/2301.13356v1.pdf.txt b/eNFQT4oBgHgl3EQfjzbM/content/tmp_files/2301.13356v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cab68cb98c2b9118e8039f8fb7ad6fd7eec06276 --- /dev/null +++ b/eNFQT4oBgHgl3EQfjzbM/content/tmp_files/2301.13356v1.pdf.txt @@ -0,0 +1,1873 @@ +Inference Time Evidences of Adversarial Attacks for Forensic on Transformers +Hugo Lemarchant,1 Liangzi Li,1 Yiming Qian,1 Yuta Nakashima,1 Hajime Nagahara +1Osaka University +hugo@is.ids.osaka-u.ac.jp, li@ids.osaka-u.ac.jp, yimingqian@ids.osaka-u.ac.jp, n-yuta@ids.osaka-u.ac.jp, +nagahara@ids.osaka-u.ac.jp +Abstract +Vision Transformers (ViTs) are becoming a very popular +paradigm for vision tasks as they achieve state-of-the-art per- +formance on image classification. However, although early +works implied that this network structure had increased ro- +bustness against adversarial attacks, some works argue ViTs +are still vulnerable. This paper presents our first attempt to- +ward detecting adversarial attacks during inference time us- +ing the network’s input and outputs as well as latent features. +We design four quantifications (or derivatives) of input, out- +put, and latent vectors of ViT-based models that provide a +signature of the inference, which could be beneficial for the +attack detection, and empirically study their behavior over +clean samples and adversarial samples. The results demon- +strate that the quantifications from input (images) and out- +put (posterior probabilities) are promising for distinguishing +clean and adversarial samples, while latent vectors offer less +discriminative power, though they give some insights on how +adversarial perturbations work. +Introduction +All deep learning models used nowadays are vulnerable to +adversarial attacks (Szegedy et al. 2013; Naseer et al. 2021), +especially in the domain of image classification. This is in- +herent to the fact that gradient computation is required for +training (Kiefer and Wolfowitz 1952), which can also be +used to optimize adversarial noise instead of model weights. +Current research efforts around adversarial attacks are often +directed toward exploring why such attacks exist, where the +vulnerability comes from (Szegedy et al. 2013), and even +how to make the attack more efficient (Biggio and Roli +2018). +The situation is the same for recently-emerged Vision +Transformers (ViTs) (Dosovitskiy et al. 2010). By incor- +porating the self-attention mechanism, some portions of the +network became more understandable, but the vulnerability +inherent to a stack of a simple but large number of arithmetic +computations can still be exploited for adversarial attacks. +Many papers ask “Could ViTs be more robust than CNN?” +or “To which attack are they more robust?” (Benz et al. +2021; Bai et al. 2021; Raghu et al. 2021; Shao et al. 2021; +Paul and Chen 2022; Mahmood, Mahmood, and Van Dijk +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +2021). These studies gave us some insights into how ViTs +perform against attacks, sometimes compared to other fam- +ilies of networks. Yet, whether or not ViTs can offer funda- +mental ways to avoid vulnerability is still an open question. +For now, we may need to acknowledge that getting rid of +this vulnerability is a hard problem. +Adversarial attacks may have been an armchair prob- +lem, but the circumstance may be changing rapidly. Deep- +fakes, for example, have become an actual threat by coun- +terfeiting authorities’ statements (Suwajanakorn, Seitz, and +Kemelmacher-Shlizerman 2017), whereas researchers are +working toward detecting such deepfakes using neural net- +works (Albahar and Almalki 2019), playing a cat-and- +mouse game. Adversarial attacks may have the potential to +become a game-changer by facilitating the deepfake camp +to fool deepfake detectors. +A potential alternative to inventing a mechanism to ro- +bustify ViTs is providing some ways to detect adversarial +attacks. Closely related studies (Raghu et al. 2021; Cintas +et al. 2021) presented for the first time feature representation +(Kornblith et al. 2019) and attention profile analysis (Raghu +et al. 2021) on ViTs, applied for comparisons against CNNs +and between pre-trained and trained-from-scratch ViTs. +We propose to extend these studies exploring inference +time signatures of adversarial attacks that potentially imply +their presence in the input, to refine our understanding of +the reactive part of a network. This work is a preliminary to +a forensic exercise, where we wish to collect evidence of an +attack given a suspicious input. +More specifically, for vanilla ViTs and tokens-to-token +ViTs (T2T-ViTs) (Dosovitskiy et al. 2010; Yuan et al. 2021) +under various attacks, we show whether the input and the +latent vectors (including self-attentions) have any signatures +of the attack at multiple levels, including (i) the input itself +and its frequency decomposition, (ii) output vector, (iii) at- +tention profile (Raghu et al. 2021), (iv) the feature similar- +ity given by centered kernel alignment (CKA) (Raghu et al. +2021; Kornblith et al. 2019). +Our findings are summarized as follows: +• Input and output of the model tell a lot. Low- +frequency components in adversarialy perturbed images +carry more energy. Also, the entropy of posterior proba- +bility over all classes (for classification tasks) increases. +These results support the results in (Uelwer, Michels, and +arXiv:2301.13356v1 [cs.CV] 31 Jan 2023 + +Candido 2021; Gupta, Dasgupta, and Akhtar 2020) and +show that adversarial attacks are relatively visible outside +the model, which may make designing a detector feasi- +ble. +• Latent vectors are not discriminative but explanatory. +Some of our quantifications are based on the model’s la- +tent vectors, and they have lower discrimination power +to detect attacks. However, they provide more complete +information about which features are more sensitive and +how they react to attacks. This explainability makes these +quantifications valuable. +• Perturbations may spot the acquired structure of the +model. It is known that ViTs have some inductive bias, +which renders block-like structure in the CKA similarity +matrix. We empirically show that the latent vectors at the +“boundaries” of this structure react to adversarial pertur- +bations. This fact can facilitate an attack detector tailored +for a certain model. +This makes our contribution two-fold: First, we present +and evaluate the first draft of quantifications that potentially +show signatures of adversarial attacks on ViTs. Second, we +present intriguing internal behavior that may realize a more +robust network. +Related Work +Many recent works demonstrate the robustness of the ViTs +and their derivatives, such as (Benz et al. 2021; Bai et al. +2021; Shao et al. 2021; Paul and Chen 2022; Mao et al. +2022). However, these works focus only on evaluating how +much robustness gain can be obtained from ViTs compared +to CNNs (Mahmood, Mahmood, and Van Dijk 2021), or +how to robustify the architecture (Mao et al. 2022). We will +present in the following sections the literature review on the +robustness of ViTs in image classification tasks and on the +feature representation within the Transformers to rationalize +our choice of working on it to find signatures of attacks. +Adversarial attacks +The idea of adversarial attacks emerged (Szegedy et al. +2013) soon after the emergence of deep neural networks and +has now become a well-known research topic with many +variations to fool different architectures (Fu et al. 2022) and +knowledge levels, such as white-box and black-box (Biggio +and Roli 2018) attacks, warning the potential vulnerability +of the current machine learning trend. +On the defense side, most of the efforts towards better ro- +bustness improvement are made for architectural modifica- +tions (Mao et al. 2022) and input scrambling (Zantedeschi, +Nicolae, and Rawat 2017). Evaluating the robustness of a +specific model is often limited to input analysis (Carlini and +Wagner 2017a; Hendrycks and Gimpel 2016), with limited +attention given to the network behavior during inference. +Other techniques are using pre-processing to reduce the im- +pact of the following network basing their strategy on trans- +formations (Li et al. 2021), bringing back any perturbation +into the training distribution, or blending/blurring the lo- +cal information to reduce the chance of successful attacks. +While those techniques have proven to be successful, their +application is often limited to several architecture families +or attack families, and they may impair the accuracy of the +models. +Not knowing the internal representations made during in- +ference for sure hurts the possibilities to detect any attack. +This idea motivates us to explore the internal structure of +models in the inference time. +Robustness of ViTs +ViTs introduced in (Dosovitskiy et al. 2010) use the original +design of the Transformer (Vaswani et al. 2017; Devlin et al. +2018) but decompose the picture into patches as a sentence is +decomposed into words. One drawback of ViTs is the com- +putationally demanding pre-training stage required to max- +imize the generalization of the network, but it is addressed +with various architectural tricks such as T2T (Yuan et al. +2021) with no significant drawback in clean accuracy (Shao +et al. 2021), but in their experiment they show up to 40% of +adversarial accuracy loss. This architecture scored state of +the art on image classification benchmark and in real-world +applications (Han et al. 2022). The self attention mechanism +at heart of the ViT family seems to helps in the robustness +to out-of-distribution samples (Wang et al. 2022; Shao et al. +2021). However those papers and attack specific ones such +as (Fu et al. 2022) show that the ViTs actually trade compa- +rable results depending on the task at hand. We are currently +heavily evaluating the adversarial robustness on standard at- +tack sets, comprised of traditional attacks mainly developed +on Fully Connected NNs and CNNs. As shown in (Fu et al. +2022), it is possible to specifically attack self attention ba- +sic operation with success. In sight of that, we think that +acknowledging the robustness of ViTs in all cases is danger- +ously optimistic, and studying their adversarial vulnerability +and behavior should be also seriously considered. +Feature Representations in ViTs +The authors of (Raghu et al. 2021) discovered two character- +istics in the features of the ViTs that differs from the CNN. +They firstly present an inductive bias learned by the net- +works undergoing long pretraining and they show that those +networks learn how to attend to global and local features, +while those simply trained from scratch are limited to global +features. The ViTs ”learn” how to attend to local features, +when the CNNs have this implemented in the architecture. +Then they proceed in showing the appearance more mono- +lithic of the CKA matrix computed on ViTs compared to +the one of the CNNs. Indeed the later one did show two +blocks, while the ViT had only one, meaning the features ex- +tracted from the beginning were forwarded to the next block +in a very consistent way. The stability of the feature rep- +resentation throughout the network is promising for attack +detection. Additionally, self-supervised applications of the +ViTs such as (Caron et al. 2021) show that certain condition +of training favor the emergence of semantically meaningful +features right out of the network. This echoes to the final +observation of (Raghu et al. 2021) where they expose the +self-location properties of the ViTs heads. While we do not +study self-supervised trained models, one can ask if the at- + +tacks affect the capability of a Self-Attention head to focus +on locally relevant information. +Inference Time Signatures of Attacks +Due to the complexity of deep neural networks, identifying +signatures of adversarial attacks is rather exploratory. We in- +troduce here five derivatives of an input, a batch of inputs, +or their latent vectors that potentially exhibit reactions of a +Transformer network to adversarial attacks. These deriva- +tives are not our new proposal, but we re-purpose the orig- +inal intent of the derivatives for single-sample adversarial +attack detection. +Signatures in the frequency domain +Previous studies have paid attention to the energy spectrum +of the adversarial noises demonstrating that ViTs require a +wider-spread spectrum for successful attacks (Shao et al. +2021) while CNNs are more vulnerable to high-frequency +although they are still crucial for the attack on ViT (Paul and +Chen 2022). These results suggest that adversarial attacks +tailored for ViTs affect a wider portion of the input in the +frequency domain. Although the original image and adver- +sarial noise are inseparable in our inference time detection +scenario, the energy spectrum may provide some clues on +the pattern of attacks. +We thus compute the discrete cosine transform (DCT) of +the input. According to the previous results, we will be par- +ticularly attentive to the frequency variations in input sample +x, either a clean image x = o or attacked image x = o + η, +where o is the original image and η is the adversarial noise, +that is, attacked images may have more lower-frequency +components carrying more energy compared to the origi- +nal images. For quantification, we propose the frequency +ratio, which gives the ratio of high-frequency components +over the low-frequency components of the energy spectrum. +Let DCTij(x) denote the (i, j)-th energy component of x’s +DCT. Frequency ratio FR is defined as: +FR(x) = +� +(i,j)∈HF DCTij(x) +� +(i,j)∈LF DCTij(x) , +(1) +where HF and LF are the sets of indexes of high- and low- +frequency components, given using threshold φ by HF = +{(i, j)|i + j ≥ φ} and LF = {(i, j)|i + j < φ}. +Signatures in posterior probabilities +The output f(x) (after softmax) of any classifier is usu- +ally interpreted as the posterior probabilities for respective +classes of the task. Adversarial attacks, in turn, try to re- +duce the probability of the ground-truth class (or increase the +probabilities of the other classes). A naive deduction from +this argument is that the posterior entropy PH can be a basic +signature of adversarial attacks. +PH(x) = − +K +� +k=1 +fk(x) log fk(x), +(2) +where fk(x) is the k-th element of f(x) for k = 1, . . . , K. +2 +0 +2 +4 +6 +8 +10 +12 +Blocks (negative for T2T Blocks) +0 +20 +40 +60 +80 +100 +120 +140 +Attention Distance in pixels +Attention profile for Clean Samples +ViT-S-16 +T2T-ViT-t +Figure 1: Example attention profiles of two models ViT-S- +16 and T2T-ViT-t evaluated over the ImageNet test dataset. +Each block has multiple dots, which are AD(x) for respec- +tive self-attention heads, averaged over all x’s in the dataset. +The idea behind this quantification is that adversarial +noises well-tuned for the model and the visual quality may +hit the almost same probability masses for the ground-truth +class and the second-top class, leading to higher PH. +Signatures in attention distances +CNNs have embedded inductive bias in their structure that +constrains possible dependency among pixels only to their +neighbors. Literature (Raghu et al. 2021) has shown that +vanilla ViTs learn to attend neighboring patches when ex- +tensively pre-trained with a large dataset, such as JFT-300M +(Sun et al. 2017), but still ViTs have an ability to attend +wherever necessary for the classification task. Meanwhile, +ViTs are typically more robust to adversarial attacks com- +pared to CNNs. This contrast can imply that texture fea- +tures (or local dependency) is prone to successful attacks +while shape features (or global dependency) give better ro- +bustness. We argue that the locality of dependency can be a +key factor for robustness and are curious about the validity +of this statement under adversarial attacks. +For ViTs, the locality of dependency (beyond patches) +is encoded in the self-attention mechanism; that is, atten- +tion from a certain patch to only its neighbors suggests local +dependency, while the attention spanning all over the input +image suggests global dependency. We thus define the atten- +tion distance as the average of the distances in pixel between +a certain patch and any other patch weighted by the attention +value associated to this pair of patches. +Specifically, let Aij(x) be the attention between patches +i and j of a certain Transformer block and a certain self- +attention head for input x, and dij the distance between the +centers of patches i and j. Attention distance AD for this +self-attention head is given by +AD(x) = +� +ij Aij(x) · dij +� +ij Aij(x) +, +(3) +where the summations are computed for all patch pairs. + +We collectively refer to AD(x)’s for all Transformer +blocks and self-attention heads as the attention profile (AP). +Figure 1 shows example APs for ViT and T2T-ViT models, +evaluated over clean input samples (i.e., x = o). For read- +ability, we arrange them in a block-wise manner, where the +negative block indexes are for the T2T blocks. APs can be +computed for a set of samples by averaging AD(x) over x +in the set. +An AP consists of multiple ADs, but it may be convenient +to show a single-value summary of AP for a consistent dis- +play with other quantifications. Let ¯ +ADref +bh and ADbh(x) be +the mean AD computed for the set of clean samples and AD +computed for input x. We define summary SAP(x) as the +sum of their absolute differences, i.e., +SAP(x) = +� +b,h +| ¯ +ADref +bh − ADbh(x)|, +(4) +where the summation is computed over all blocks and heads +of the model. +We can generate the attention profile for a trained model +over a set of clean input samples, which gives reference dis- +tributions of attention distances. Comparison between these +reference distributions and a (set of) suspicious input may +provide some ideas on the presence of adversarial attacks. +Structures in latent representations +CKA (Raghu et al. 2021) is used for studying the represen- +tation structure in a model, by giving layer-to-layer similar- +ities in that model. Analyzing latent representations within +a model allows for identifying different general functions +throughout its architecture. Adversarial attacks may break +such functions as they exploit some patterns in the learned +parameters, which results in unusual responses. It should be +noted that, although we aim to discover inference time sig- +natures of adversarial attacks, CKA is a statistic and is com- +puted only for a set of samples but not for a single sample; +therefore, this signature is not always usable. +CKA takes as input Hi ∈ Rm×c and Hj ∈ Rm×c′, which +are latent vectors (or activation) at the i-th and j-th layers +(i, j = 1, . . . , l) with the dimensionalities of ci and cj, re- +spectively, obtained for the same m input samples. Letting +Ki = HiH⊤ +i and Kj = HjH⊤ +j denote the Gram matrices, +CKA is given by: +CKAij = +HSIC(Ki, Kj) +� +HSIC(Ki, Ki) HSIC(Kj, Kj) +(5) +where HSIC is the Hilbert-Schmidt independence criterion. +Given the centering matrix1 Hc = Ic − 1 +c1c with Ic being +the c × c diagonal matrix and 1c being the c × c matrix +whose elements are all 1, the centered Gram matrix for K +is computed by K′ = HcKHc. This operation is applied to +both Ki and Kj to obtain K′ +i and K′ +j. HSIC is given as the +similarity between these centered Gram matrices, computed +by +HSIC(Ki, Kj) = vec(K′ +i)⊤vec(K′ +j)/(m − 1)2, +(6) +1Here we omit the indices i and j for notation simplicity. +where vec(·) is the vectorization of a matrix. CKA is usu- +ally computed over the entire test set because of the statis- +tical nature of the metric. Hence, we compute CKA for the +minimum batch size (i.e. m = 4). +CKA is a similarity between an arbitrary pair of layers, +resulting in a similarity matrix. But this is not always handy +for our purpose of finding signatures of adversarial attacks. +We reduce the similarity matrix down to a single metric for +compacting the observations. Also to highlight the reactions +to adversarial attacks, we compute the difference between +CKAs computed for clean and attacked samples. +Specifically, let M ref ∈ Rl×l denote the reference CKA +similarity matrix computed for clean samples in the entire +test set (i.e., the (i, j)-th element of M ref is CKAij computed +for clean samples). Also, let ¯ +M denote the mean of CKA +similarity matrices of the same model computed for random +mini-batches with the minimum batch size and of samples +under the attack of interest (or clean samples), i.e., +¯ +M = 1 +|B| +� +B∈B +M(B), +(7) +where B is the set of aforementioned mini-batches and +M(B) is the CKA similarity matrix computed for B ∈ B. +To highlight the changes, we compute the absolute differ- +ence D between M ref and ¯ +M obtained with samples under +the attack of interest, where its (i, j)-th element Dij is given +by +Dij = |M ref +ij − ¯ +Mij|. +(8) +The summary SCKA is defined as the sum of these differ- +ences, i.e., +SCKA = +� +ij +Dij. +(9) +Experimental Setup +Models +We choose two major architectural families that cover vari- +ous paradigms of ViTs. The first family is the vanilla ViT +(Dosovitskiy et al. 2010), which comes with 3 variants: +ViT/T-16, ViT/B-32, and ViT/B-16. This allows us to draw +more general conclusions as they consist only of Trans- +former blocks, which are the main building block of the ma- +jority of the entire ViT family. We also investigate T2T-ViTs +(Yuan et al. 2021) with two variants: T2T-ViT performer +(Chen et al. 2022; Choromanski et al. 2020) and the origi- +nal T2T-ViT transformer. The T2T-ViT family addresses the +need for extensive pre-training on large datasets, which is +inevitable for the vanilla ViTs. By introducing a CNN-like +structure, this family also happens addressing the inductive +bias (Raghu et al. 2021), bringing a T2T-ViT model trained +from scratch over ImageNet with an attention profile com- +parable to fully pre-trained ViTs (Figure 1). Evaluation on +this family provides some insights about adversarial attacks +over architectural variations like hierarchical designs in ViT +(Liu et al. 2021) and variations without pre-training (Tou- +vron et al. 2021). + +Model +Encoding +Depth +# heads +# params. +ViT-T/16 +Conv. +12 +3 +5.7M +ViT-S/32 +Conv. +12 +6 +22M +ViT-S/16 +Conv. +12 +6 +22.6M +T2T-ViT-p +Performer +14 +6 +21.5M +T2T-ViT-t +Transformer +14 +6 +21.5M +Table 1: Summary of ViT and T2T-ViT models used in our +experiments. +Table 1 shows a summary of models we used. For vanilla +ViT variants, we used models pre-trained on JFT-300M.2 All +models are trained (or fine-tuned) on the ImageNet training +set (Steiner et al. 2021). For legibility, we present the results +for ViT-S/16 and T2T-ViT-t; the other models’ results are +presented in the supplementary material. +Dataset +Our study is conducted over the ImageNet dataset, which is +large enough to train the aforementioned models. The size +of input images is 224 × 224, which allows us to optimize +adversarial perturbations in a reasonable time. To perform +our attacks, we will be using the ImageNet-1K test set of 50k +samples. To unload the computational burden, we sampled +one out of five samples from the set, giving us 10k samples, +which is still representative of the dataset. +Adversarial Attacks +We evaluated the robustness under popular adversarial at- +tacks, i.e., the fast gradient sign method (FSGM) (Goodfel- +low, Shlens, and Szegedy 2014), projected gradient descent +(PGD) (Madry et al. 2017), and Carlini and Wagner (C&W) +(Carlini and Wagner 2017b). +FSGM and PGD are l∞ attacks, where attacked image x +for original image o is given by +x = argmax +x′ +L(x′, y) s.t. ∥x′ − o∥∞ ≤ ϵ, +(10) +where L is the loss function for the task (typically, softmax +cross-entropy) and y is the ground-truth label. This is an un- +targeted attack, in which the attack is successful when the +perturbation changes the model’s prediction. The above opti- +mization problem is intractable because of the l∞ constraint. +FGSM approximates the optimization problem by +x = o + ϵ sign(∇oL(o, y)), +(11) +where ∇oL is the gradient of L at o. This is a quick but +rough approximation. +PGD is one of FGSM’s extensions, which iteratively per- +forms FGSM. Letting xt denote the attacked image after t +iterations, PGD computes xt+1 by +xt+1 = clip[o−ϵ,o+ϵ](xt + α sign(∇xL(xt, y))), +(12) +where clip[o−ϵ,o+ϵ](·) is to clip the value within [o−ϵ, o+ϵ], +α is a predefined small value, and x0 = o. This iterative +2https://github.com/rwightman/pytorch-image-models and https://github.com/ +yitu-opensource/T2T-ViT +Table 2: Accuracy of ViT and T2T models under various +attacks. +Top-1 +Top-5 +Attack +ViT-S/16 +T2T-ViT-t +ViT-S/16 +T2T-ViT-t +Clean +81.38 +81.40 +96.13 +95.72 +C&W c=1×10−4 +24.86 +26.21 +94.68 +92.99 +PGD ϵ=1×10−3 +51.36 +44.37 +86.99 +80.79 +PGD ϵ=3×10−3 +20.47 +22.98 +71.65 +60.40 +PGD ϵ=5×10−3 +11.36 +17.30 +60.75 +49.68 +PGD ϵ=1×10−2 +6.50 +12.54 +46.08 +37.05 +FGSM ϵ=0.031 +8.94 +30.84 +36.31 +52.80 +FGSM ϵ=0.062 +10.62 +30.28 +36.78 +51.60 +method takes advantage of multiple small steps, controlled +by α, to refine the attack. Iteration enables the attack to be +more efficient than FGSM, so that a much smaller ϵ radius +enables the same level of perturbation as FGSM. However, +PDG is more computationally demanding; the computation +cost is linear with the number of iterations. We will use PGD +for 40 iterations with α = 0.025. +In our experiments, we use ϵ ∈ {0.031, 0.062} for FGSM +and ϵ ∈ {0.001, 0.003, 0.005, 0.01} for PGD. +C&W jointly optimizes the distortion introduced by the +perturbation in the l2 distance and the performance of the +attack. The method quantifies the performance of the attack +by +Q(x) = max +� +fy(x) − max +y′̸=y fy′(x), −κ +� +, +(13) +where we use the implementation3 of the untargeted version; +fy and fy′ denote the prediction confidences for ground- +truth label y and another label y′. A negative value of Q(x) +means that a certain class other than the ground truth gets +the highest confidence, though the confidence does not go +beyond the margin κ. +C&W parameterizes attacked image x by w, i.e., +x(w) = tanh(w) + 1 +2 +, +(14) +which guarantees x in the range of [0, 1] during the optimiza- +tion.4. The attacked image x of C&W is given by x(w∗), +where +w∗ = argmin +w +∥x(w) − o∥2 + c Q(x(w)) +(15) +with c = 1 × 10−4. +Metric for statistical signatures +Although we aim to identify inference time signatures of at- +tacks, it is also interesting to explore the statistical signatures +that our quantifications may exhibit over sets of clean and at- +tacked samples. We thus employ the discrete version of the +Bhattacharyya coefficient (BC) (Bhattacharyya 1946) for +scalar-valued quantification, i.e., RF, PH, AD’s, and SCKA. +3https://github.com/Harry24k/adversarial-attacks-pytorch +4We normalize original image o to be in [0, 1] as well + +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0 +500 +Sample count +ViT-S/16 +C&W c=0.0001 (1.00) +PGD =0.001 (1.00) +PGD =0.003 (1.00) +PGD =0.005 (0.99) +PGD =0.01 (0.99) +FGSM =0.031 (0.95) +FGSM =0.062 (0.74) +Clean +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Input Frequency Ratio +0 +500 +Sample count +T2T-ViT-t +C&W c=0.0001 (1.00) +PGD =0.001 (1.00) +PGD =0.003 (1.00) +PGD =0.005 (0.99) +PGD =0.01 (0.99) +FGSM =0.031 (0.90) +FGSM =0.062 (0.58) +Clean +Figure 2: Distributions of FR for representative two models. +The numbers in Parentheses are BCs computed between the +clean distribution and respective distributions with perturba- +tions. +Let h and h′ denote the empirical distributions (or his- +tograms) of one of our quantifications with 100 bins com- +puted for the sets of clean samples and attacked samples, +respectively. Their i-th bin, hi and h′ +i, give the numbers of +samples that fall into that bin, normalized to sum up to one, +where the bins are configured to cover the minimum and +maximum values of the quantification. BC is defined as +BC(h, h′) = +� +i +� +hih′ +i. +(16) +The range of BC is [0, 1] where BC = 1 means the most +similar. For better discrimination of clean and adversarial +samples, we hope for a value as close as possible to 0, which +gives a significant signature of adversarial attacks. However, +we realistically expect the value to be rather high for typical +(and thus “interesting”) configurations of methods for adver- +sarial attacks, as only a small perturbation is introduced. +Results +Frequency ratio +Figure 2 shows the empirical distributions of FR for clean +samples and attacked samples by the configurations pre- +sented earlier. We can see the distributions are different only +for the configurations with the largest perturbations (namely +FGSM with ϵ being 0.031 and 0.062), for which FR con- +sistently drifts towards 1. This means that the energy of the +perturbation η is relatively concentrated in lower frequen- +cies. This confirms that ViTs are also vulnerable to low fre- +quency when given enough perturbation budget (Paul and +Chen 2022). +The distributions for smaller perturbations are almost the +same as the clean one, which is also shown in the high BCs +computed for the attacked distributions; therefore, FR in it- +self cannot be reliable for attack detection. We also note at +this stage that the distribution shift is related to the pertur- +bation strength and not to the success rate of the attack in +Table 2. We will observe that same behavior on the other +signatures and discuss this problem in the conclusion. We +0 +1 +2 +3 +4 +5 +6 +7 +0 +2000 +Sample count +ViT-S/16 +C&W c=0.0001 (0.95) +PGD =0.001 (0.96) +PGD =0.003 (0.94) +PGD =0.005 (0.96) +PGD =0.01 (0.97) +FGSM =0.031 (0.90) +FGSM =0.062 (0.85) +Clean +0 +2 +4 +6 +8 +Entropy of the Output vector +0 +500 +Sample count +T2T-ViT-t +C&W c=0.0001 (0.95) +PGD =0.001 (0.90) +PGD =0.003 (0.92) +PGD =0.005 (0.92) +PGD =0.01 (0.91) +FGSM =0.031 (0.79) +FGSM =0.062 (0.73) +Clean +Figure 3: Distributions of PH for the two models. +know from (Harder et al. 2021) that it is possible to train de- +tector models on Fourier Transform of an input to detect to +presence of adversarial attack on CNNs, hence giving hint +for similar results with ViTs if the same technique was to be +used. Analysis in the frequency domain can provide some +signatures of attacks if the perturbation is large enough but +is not useful otherwise. +Posterior probability +The distributions of PH are shown in Figure 3. We see very +different behaviors from one model to another. The T2T-ViT +model displays noticeably more varied distributions com- +pared to the vanilla ViT model. We cannot explain that dif- +ference considering that on average the T2T outperforms the +ViT in the classification task. For that matter the ViT even +have better top-5 accuracy (96.13%) on clean samples com- +pared to the T2T (95.72%). This result demonstrates that PH +may be used in an attack detection pipeline for T2T-ViTs, +though we would have to make heavy compromises in the +accuracy or in the false positive rate if the quantification is +used by itself. It has to be noted that the significance of this +quantification is also proportional to the intensity of the at- +tack rather than the success rate of this one. +Attention distances and profiles +In order to analyze the attention distance, we base our ob- +servations on the way the attention profile is computed in +(Raghu et al. 2021): the profile is computed for the entire +test set, regardless of the accuracy of the detection. This base +gives us the reference of the nominal behavior of our model +for in-distribution inputs. +Figure 1 shows the distributions of SAP(x). Note that we +can still compute the distribution for the set of clean sam- +ples, which shows the difference between individual sam- +ples and their mean. +We observe that SAP(x) behaves in a similar way as the +previous metrics, showing a marginal separation (BC = +0.97) for the largest perturbation on both models. Since we +measure the absolute difference between the reference and +observed profiles, SAP(x) gives the gist but very little infor- +mation on the actual direction of the change. We can ob- +serve finer behavior when looking in detail at the level of + +300 +400 +500 +600 +700 +800 +900 +1000 +1100 +0 +250 +Sample count +ViT-S/16 +C&W c=0.0001 (1.00) +PGD =0.001 (1.00) +PGD =0.003 (1.00) +PGD =0.005 (1.00) +PGD =0.01 (0.99) +FGSM =0.031 (0.99) +FGSM =0.062 (0.97) +Clean +300 +400 +500 +600 +700 +800 +900 +1000 +1100 +Cumulated Head Attention Distance from Reference Profile +0 +250 +Sample count +T2T-ViT-t +C&W c=0.0001 (1.00) +PGD =0.001 (1.00) +PGD =0.003 (0.99) +PGD =0.005 (0.99) +PGD =0.01 (0.99) +FGSM =0.031 (0.99) +FGSM =0.062 (0.96) +Clean +Figure 4: Distributions of SAP(x). +0 +2 +4 +6 +8 +10 +Blocks (negative for T2T Blocks) +0 +20 +40 +60 +80 +100 +120 +140 +Attention Distance in pixels +Average of Attention Distribution per Blocks +for ViT-S-16 Pretrained +PGD eps:0.062 +PGD eps:0.031 +PGD eps:0.01 +PGD eps:0.005 +PGD eps:0.003 +PGD eps:0.001 +Clean +Figure 5: ADs per head observed on ViT-S/16 for PGD at- +tacks. The different attacks are sorted so that the configura- +tion that gives the lowest accuracy comes on the left within +each head. Each point is the mean over the 10k samples. +attention heads. In Figure 5, we show that perturbations ei- +ther diversify or shrink ADs on several heads, mostly in the +first blocks, and the trend of changes in ADs is constant for +them, i.e., either diversified or shrunk, while all the others re- +main unaffected. For instance, the head with the largest AD +in block 1 is diversified along with the amount of perturba- +tion, while the one with the second largest AD in the same +block shrinks. We note that diversification mostly occurs in +the lower half (i.e., smaller block indices) of the network +while shrinkage happens throughout it. +One simple explanation in which the discrepancy in the +attention results in a successful attack is that the perturba- +tion distracts the attention of all patches to some regions that +have no features leading to the ground-truth class. The at- +tack may be successful if the features from irrelevant patches +overwrite the features from the residual connection. From +this argument, we hypothesized that heads with attention +discrepancy are the sweet spot for adversarial attacks. Veri- +fication of this hypothesis is up to further investigation. +We think those volatile heads are a good hook to build +an attack detector in the future. We designed SAP(x) to be +Table 3: BCs computed for the most volatile heads in terms +of attention distances. +Attacks +ViT-S/16 +T2T-ViT-t +C&W c = 1 × 10−4 +1.00 (+0.00) +1.00 (+0.00) +PGD ϵ = 1 × 10−3 +1.00 (+0.00) +1.00 (+0.00) +PGD ϵ = 3 × 10−3 +1.00 (+0.00) +1.00 (+0.00) +PGD ϵ = 5 × 10−3 +1.00 (+0.00) +1.00 (+0.00) +PGD ϵ = 1 × 10−2 +0.99 (+0.00) +0.99 (+0.00) +FGSM ϵ = 0.031 +0.97 (+0.02) +0.98 (+0.01) +FGSM ϵ = 0.062 +0.85 (+0.12) +0.96 (+0.01) +36 +38 +40 +42 +44 +46 +48 +50 +52 +0 +500 +Sample count +ViT-S/16 +C&W c=0.0001 (0.99) +PGD =0.001 (0.99) +PGD =0.003 (0.99) +PGD =0.005 (0.99) +PGD =0.01 (0.98) +FGSM =0.031 (0.97) +FGSM =0.062 (0.94) +Clean +55 +60 +65 +70 +75 +80 +85 +Cumulated Feature Similarity Difference from Reference Profile +0 +250 +Sample count +T2T-ViT-t +C&W c=0.0001 (0.99) +PGD =0.001 (0.99) +PGD =0.003 (0.99) +PGD =0.005 (0.99) +PGD =0.01 (0.98) +FGSM =0.031 (0.98) +FGSM =0.062 (0.95) +Clean +Figure 6: Difference between the observed CKA similarity +and the reference one. +a summary of all heads in all blocks, so the distributions in +Figure 4 may be too vague. Focusing on a few heads that +exhibit large discrepancies can give a clearer distinction be- +tween clean and adversarial samples. Table 3 shows the BC +values computed over AD(x) for a cherry-picked head of +each model and each configuration of attacks and the corre- +sponding improvement of BC values. +CKA Similarity +The distributions of SCKA in 6 show even worse separation +capabilities compared to the other quantifications, where +even the largest perturbations seem to have only slightly +shifted the curves from the original clean data. In a simi- +lar fashion to AP, we take a step back from SCKA and look +at the layer-wise difference Dij of the reference CKA simi- +larity matrix and that of attacked samples, because the sum- +mary by SCKA does not provide sufficient information on the +presence of any attacks. +In Figure 7, we observe clear patterns appearing with +the increase of attack intensity. The patterns are presum- +ably related to the layers in Transformer blocks, which are +comprised of layers after multi-head attention, after projec- +tion, and after multi-layer perceptron (MLP). One can point +that the latent vectors after the attention heads are remark- +ably stable, and only the first layer (or two in the case of +T2T-ViT) are affected when using the largest perturbations. + +Figure 7: Left-most column: Reference CKA similarity ma- +trix M ref. Second left column: D computed for clean sam- +ples. Third left to right-most columns: D’s computed for +samples under the attack of interest. From top to bottom: +ViT-S/16’s attention heads, projections, and MLPs; T2T- +ViT-t’s attention heads, projections, and MLPs. +Table 4: BCs computed for the most reactive latent vectors +in terms of the CKA similarity. +Attacks +ViT-S/16 +T2T-ViT-t +C&W c = 1 × 10−4 +0.99 (+0.00) +0.99 (+0.00) +PGD ϵ = 1 × 10−3 +0.99 (+0.00) +0.99 (+0.00) +PGD ϵ3 × 10−3 +0.99 (+0.00) +0.98 (+0.01) +PGD ϵ = 5 × 10−3 +0.99 (+0.00) +0.98 (+0.01) +PGD ϵ = 1 × 10−2 +0.99 (-0.01) +0.98 (+0.00) +FGSM ϵ = 0.031 +0.97 (+0.00) +0.96 (+0.02) +FGSM ϵ = 0.062 +0.93 (+0.01) +0.91 (+0.04) +Ones after the projection layers are mildly affected, diffus- +ing for all layers. A remarkable impact is observed for our +largest FGSM attack (ϵ = 0.062), where the first layer reacts +strongly. The last latent vectors are the one after MLPs of +each attention block. These vectors are affected throughout +the whole spectrum of attacks, and the effect is noticeable +around one specific block, that is, the 7-th one for ViT-S/16 +and also the 7-th, which is the 5-th if we count in the ViT +backbone only, for the T2T-ViT-t. The perturbation seems to +locally spread from this block with the intensity increased, +as well as the perturbation appearing in the first blocks when +using strong FGSM attacks. +We show in Table 4 the fine-grained BCs for cherry- +picked layers (i.e. the most reactive layers) for each model. +The table shows that targeting specific layers can slightly +help separate attacked samples from clean ones. However, +the separation is still marginal, and the majority of the sam- +ples are confused with the clean distribution. +Conclusion +We presented in this paper four different types of quantifica- +tions that potentially exhibit some signatures of adversarial +attacks in the inference time. The biggest challenge in ad- +versarial attack detection comes from the great variability +in models’ reactions, and our experimental results generally +show the difficulty in building a simple criterion to distin- +guish an attacked sample from clean ones. We empirically +demonstrated that the input frequency ratio and the posterior +entropy are promising out-of-the-box. We also showed that +quantifications based on the latent vectors, such as attention +profiles and CKA similarity allow separating only the largest +attacks. However, these quantifications render some patterns +specific to each model, which may collectively serve as a +stronger signature. The results also showed that these quan- +tifications mostly react to the amplitude of perturbations but +not to the effectiveness of the attack (Table 2). This implies +that fine-tuned adversarial attacks such as C&W and PGD +with a very small perturbation budget are both very efficient +and undetectable. The current state of inference time signa- +ture is very rough, and the very little information does not +seem worth the cost of the computation. However we have +great hope in the finer analysis of the latent vector to get bet- +ter understanding in the future of where in the network will +we have better chance of detecting an attack. Future work +includes exploring interactions between quantifications as +stated above. +Acknowledgements +This work was partly supported by JST CREST Grant No. +JPMJCR20D3. +References +Albahar, M.; and Almalki, J. 2019. Deepfakes: Threats and +countermeasures systematic review. Journal of Theoretical +and Applied Information Technology, 97(22): 3242–3250. +Bai, Y.; Mei, J.; Yuille, A. L.; and Xie, C. 2021. Are Trans- +formers more robust than CNNs? 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Tokens-to-token +ViT: Training vision transformers from scratch on imagenet. +In Proc. IEEE/CVF International Conference on Computer +Vision, 558–567. +Zantedeschi, V.; Nicolae, M.-I.; and Rawat, A. 2017. Ef- +ficient defenses against adversarial attacks. In Proc. 10th +ACM Workshop on Artificial Intelligence and Security, 39– +49. + +Supplementary Material +Frequency ratio +We present in Figure 8 the distribution over 3 additional net- +work variations as presented in Table 1. We clearly see a +direct extension of the result observed on ViT-S/16 repro- +duced in both the smaller ViT-T/16 and the ViT-S/32 with +larger patch size. We note a small variation of the BC on the +ViTs for the largest FGSM attack, but we think it is within +the margin of error. As for the T2T family the results here are +exactly similar, that is in complete agreement with the per- +former being faster equivalent operation of the transformer. +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0 +500 +Sample count +ViT-T/16 +C&W c=0.0001 (1.00) +PGD =0.001 (1.00) +PGD =0.003 (1.00) +PGD =0.005 (0.99) +PGD =0.01 (0.99) +FGSM =0.031 (0.95) +FGSM =0.062 (0.76) +Clean +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0 +500 +Sample count +ViT-S/16 +C&W c=0.0001 (1.00) +PGD =0.001 (1.00) +PGD =0.003 (1.00) +PGD =0.005 (0.99) +PGD =0.01 (0.99) +FGSM =0.031 (0.95) +FGSM =0.062 (0.74) +Clean +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0 +500 +Sample count +ViT-S/32 +C&W c=0.0001 (1.00) +PGD =0.001 (1.00) +PGD =0.003 (1.00) +PGD =0.005 (0.99) +PGD =0.01 (0.99) +FGSM =0.031 (0.92) +FGSM =0.062 (0.70) +Clean +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0 +500 +Sample count +T2T-ViT-t +C&W c=0.0001 (1.00) +PGD =0.001 (1.00) +PGD =0.003 (1.00) +PGD =0.005 (0.99) +PGD =0.01 (0.99) +FGSM =0.031 (0.90) +FGSM =0.062 (0.58) +Clean +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Input Frequency Ratio +0 +500 +Sample count +T2T-ViT-p +C&W c=0.0001 (1.00) +PGD =0.001 (1.00) +PGD =0.003 (1.00) +PGD =0.005 (0.99) +PGD =0.01 (0.99) +FGSM =0.031 (0.90) +FGSM =0.062 (0.58) +Clean +Figure 8: Distributions of FR for representative two models. +The numbers in Parentheses are BCs computed between the +clean distribution and respective distributions with perturba- +tions. +Posterior probability +In Figure 9 we extend the evaluation to the 3 other models. +We show here that the behavior display by the ViT-S/16 and +the T2T-ViT-t are completely representative of their family. +We find interesting here that the T2T-ViTs in general have +entropy of approx. 2 while the ViTs are stuck at 0. The fact +that the ViTs learned to output exactly one very high value as +prediction is maintained in the adversarial setups, while the +Table 5: BCs computed for the most volatile heads in terms +of attention distances. +Attacks +ViT-T/16 +ViT-S/16 +ViT-S/32 +T2T-ViT-t +T2T-ViT-p +C&W c = 1 × 10−4 +1.00 (+0.00) +1.00 (+0.00) +1.00 (+0.00) +1.00 (+0.00) +1.00 (+0.00) +PGD ϵ = 1 × 10−3 +1.00 (+0.00) +1.00 (+0.00) +1.00 (+0.00) +1.00 (+0.00) +1.00 (+0.00) +PGD ϵ = 3 × 10−3 +0.99 (+0.01) +1.00 (+0.00) +1.00 (-0.01) +1.00 (+0.00) +1.00 (-0.01) +PGD ϵ = 5 × 10−3 +0.99 (+0.00) +1.00 (+0.00) +0.99 (+0.00) +1.00 (+0.00) +1.00 (-0.01) +PGD ϵ = 1 × 10−2 +0.99 (+0.00) +0.99 (+0.00) +0.99 (+0.00) +0.99 (+0.00) +1.00 (-0.01) +FGSM ϵ = 0.031 +0.99 (+0.00) +0.97 (+0.02) +0.98 (+0.01) +0.98 (+0.01) +0.99 (+0.00) +FGSM ϵ = 0.062 +0.98 (+0.00) +0.85 (+0.12) +0.99 (-0.05) +0.96 (+0.01) +0.97 (+0.03) +T2Ts who usually are more varied have even less decisive +answers when attacked. However we note that this behavior +from the T2T family does not impact the accuracy and is +even outperforming the ViTs. (Table 2). +0 +1 +2 +3 +4 +5 +6 +7 +0 +1000 +Sample count +ViT-T/16 +C&W c=0.0001 (0.95) +PGD =0.001 (0.97) +PGD =0.003 (0.95) +PGD =0.005 (0.96) +PGD =0.01 (0.97) +FGSM =0.031 (0.89) +FGSM =0.062 (0.80) +Clean +0 +1 +2 +3 +4 +5 +6 +7 +0 +2000 +Sample count +ViT-S/16 +C&W c=0.0001 (0.95) +PGD =0.001 (0.96) +PGD =0.003 (0.94) +PGD =0.005 (0.96) +PGD =0.01 (0.97) +FGSM =0.031 (0.90) +FGSM =0.062 (0.85) +Clean +0 +1 +2 +3 +4 +5 +6 +7 +8 +0 +1000 +Sample count +ViT-S/32 +C&W c=0.0001 (0.97) +PGD =0.001 (0.98) +PGD =0.003 (0.95) +PGD =0.005 (0.95) +PGD =0.01 (0.97) +FGSM =0.031 (0.90) +FGSM =0.062 (0.79) +Clean +0 +2 +4 +6 +8 +0 +500 +Sample count +T2T-ViT-t +C&W c=0.0001 (0.95) +PGD =0.001 (0.90) +PGD =0.003 (0.92) +PGD =0.005 (0.92) +PGD =0.01 (0.91) +FGSM =0.031 (0.79) +FGSM =0.062 (0.73) +Clean +0 +2 +4 +6 +8 +Entropy of the Output vector +0 +500 +Sample count +T2T-ViT-p +C&W c=0.0001 (0.95) +PGD =0.001 (0.89) +PGD =0.003 (0.93) +PGD =0.005 (0.94) +PGD =0.01 (0.93) +FGSM =0.031 (0.81) +FGSM =0.062 (0.73) +Clean +Figure 9: Distributions of PH for the two models. + +Attention distances and profiles +We also expend here the experiments of Section and show +here varied behaviors. At first the BC computed for the ViTs +increase in the order presented here with ViT-T having the +lowest separability on the attention distance due to the low +number of heads to compare. The other ViT-T/32 shows bet- +ter BC for the largest attacks. We attribute this to the fact +that this network has by design a larger attention scope, and +that making it diverge, only from one patch, is much more +noticeable. The T2Ts also present interesting changes, with +the T2T-ViT-p showing very poor BC. When we compare +Figure 14 and Figure 15 we see a very clear difference in +the attention of the T2T block where most of the drift under +attacks were measured. +We also try to refine those scores with the same technique +and observe very similar to the ones in their respective fam- +ily again. +100 +200 +300 +400 +500 +0 +250 +Sample count +ViT-T/16 +C&W c=0.0001 (1.00) +PGD =0.001 (1.00) +PGD =0.003 (1.00) +PGD =0.005 (0.99) +PGD =0.01 (0.99) +FGSM =0.031 (0.99) +FGSM =0.062 (0.98) +Clean +300 +400 +500 +600 +700 +800 +900 +1000 +1100 +0 +250 +Sample count +ViT-S/16 +C&W c=0.0001 (1.00) +PGD =0.001 (1.00) +PGD =0.003 (1.00) +PGD =0.005 (1.00) +PGD =0.01 (0.99) +FGSM =0.031 (0.99) +FGSM =0.062 (0.97) +Clean +300 +400 +500 +600 +700 +800 +900 +1000 +1100 +0 +250 +Sample count +ViT-S/32 +C&W c=0.0001 (1.00) +PGD =0.001 (1.00) +PGD =0.003 (0.99) +PGD =0.005 (0.99) +PGD =0.01 (0.99) +FGSM =0.031 (0.99) +FGSM =0.062 (0.94) +Clean +300 +400 +500 +600 +700 +800 +900 +1000 +1100 +0 +250 +Sample count +T2T-ViT-t +C&W c=0.0001 (1.00) +PGD =0.001 (1.00) +PGD =0.003 (0.99) +PGD =0.005 (0.99) +PGD =0.01 (0.99) +FGSM =0.031 (0.99) +FGSM =0.062 (0.96) +Clean +300 +400 +500 +600 +700 +800 +900 +1000 +1100 +Cumulated Head Attention Distance from Reference Profile +0 +200 +Sample count +T2T-ViT-p +C&W c=0.0001 (1.00) +PGD =0.001 (1.00) +PGD =0.003 (0.99) +PGD =0.005 (0.99) +PGD =0.01 (0.99) +FGSM =0.031 (0.99) +FGSM =0.062 (1.00) +Clean +Figure 10: ADs per head observed on ViT-S/16 for PGD at- +tacks. The different attacks are sorted so that the configura- +tion that gives the lowest accuracy comes on the left within +each head. Each point is the mean over the 10k samples. +Stability of the Attention distances and profiles +We show in this section some testimonies proving that the at- +tention drifts reported in Section are consistent. We mainly +show here that when the distributions are drifting, they do so +in a very consistent way with no huge distribution spread or +grouping (with the exception of the attention head -1 on Fig- +ure 14), meaning that making observations on the average is +indeed representative of the drift measured. + +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Blocks (negative for T2T Blocks) +0 +20 +40 +60 +80 +100 +120 +140 +Attention Distance in pixels +First 3 heads of ViT-T/16 +FGSM =0.062 +FGSM =0.031 +PGD =0.01 +PGD =0.005 +PGD =0.003 +PGD =0.001 +C&W c=0.0001 +Clean +Figure 11: Distribution of the ADs over the 10k samples for +the first 3 heads observed on ViT-T/16 for our set of attacks. +The different attacks are sorted for each block in the same +order as the legend and in Figure 5. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Blocks (negative for T2T Blocks) +0 +20 +40 +60 +80 +100 +120 +140 +Attention Distance in pixels +First 3 heads of ViT-S/16 +FGSM =0.062 +FGSM =0.031 +PGD =0.01 +PGD =0.005 +PGD =0.003 +PGD =0.001 +C&W c=0.0001 +Clean +Figure 12: Distribution of the ADs over the 10k samples for +the first 3 heads observed on ViT-S/16 for our set of attacks. +The different attacks are sorted for each block in the same +order as the legend and in Figure 5. + +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Blocks (negative for T2T Blocks) +0 +20 +40 +60 +80 +100 +120 +140 +Attention Distance in pixels +First 3 heads of ViT-S/32 +FGSM =0.062 +FGSM =0.031 +PGD =0.01 +PGD =0.005 +PGD =0.003 +PGD =0.001 +C&W c=0.0001 +Clean +Figure 13: Distribution of the ADs over the 10k samples for +the first 3 heads observed on ViT-S/32 for our set of attacks. +The different attacks are sorted for each block in the same +order as the legend and in Figure 5. +-2 +-1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +Blocks (negative for T2T Blocks) +0 +20 +40 +60 +80 +100 +120 +140 +Attention Distance in pixels +First 3 heads of T2T-ViT-t +FGSM =0.062 +FGSM =0.031 +PGD =0.01 +PGD =0.005 +PGD =0.003 +PGD =0.001 +C&W c=0.0001 +Clean +Figure 14: Distribution of the ADs over the 10k samples for +the first 3 heads observed on T2T-ViT-t for our set of attacks. +The different attacks are sorted for each block in the same +order as the legend and in Figure 5. + +-2 +-1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +Blocks (negative for T2T Blocks) +0 +20 +40 +60 +80 +100 +120 +140 +Attention Distance in pixels +First 3 heads of T2T-ViT-p +FGSM =0.062 +FGSM =0.031 +PGD =0.01 +PGD =0.005 +PGD =0.003 +PGD =0.001 +C&W c=0.0001 +Clean +Figure 15: Distribution of the ADs over the 10k samples for +the first 3 heads observed on T2T-ViT-p for our set of at- +tacks. The different attacks are sorted for each block in the +same order as the legend and in Figure 5. + diff --git a/eNFQT4oBgHgl3EQfjzbM/content/tmp_files/load_file.txt b/eNFQT4oBgHgl3EQfjzbM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..51cb3ba122bee4fd91aa6d7b8d82f8e847f135fc --- /dev/null +++ b/eNFQT4oBgHgl3EQfjzbM/content/tmp_files/load_file.txt @@ -0,0 +1,1561 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf,len=1560 +page_content='Inference Time Evidences of Adversarial Attacks for Forensic on Transformers Hugo Lemarchant,1 Liangzi Li,1 Yiming Qian,1 Yuta Nakashima,1 Hajime Nagahara 1Osaka University hugo@is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='ids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} 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+page_content='jp Abstract Vision Transformers (ViTs) are becoming a very popular paradigm for vision tasks as they achieve state-of-the-art per- formance on image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' However, although early works implied that this network structure had increased ro- bustness against adversarial attacks, some works argue ViTs are still vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This paper presents our first attempt to- ward detecting adversarial attacks during inference time us- ing the network’s input and outputs as well as latent features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We design four quantifications (or derivatives) of input, out- put, and latent vectors of ViT-based models that provide a signature of the inference, which could be beneficial for the attack detection, and empirically study their behavior over clean samples and adversarial samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The results demon- strate that the quantifications from input (images) and out- put (posterior probabilities) are promising for distinguishing clean and adversarial samples, while latent vectors offer less discriminative power, though they give some insights on how adversarial perturbations work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Introduction All deep learning models used nowadays are vulnerable to adversarial attacks (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Naseer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021), especially in the domain of image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This is in- herent to the fact that gradient computation is required for training (Kiefer and Wolfowitz 1952), which can also be used to optimize adversarial noise instead of model weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Current research efforts around adversarial attacks are often directed toward exploring why such attacks exist, where the vulnerability comes from (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2013), and even how to make the attack more efficient (Biggio and Roli 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The situation is the same for recently-emerged Vision Transformers (ViTs) (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' By incor- porating the self-attention mechanism, some portions of the network became more understandable, but the vulnerability inherent to a stack of a simple but large number of arithmetic computations can still be exploited for adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Many papers ask “Could ViTs be more robust than CNN?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' or “To which attack are they more robust?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' (Benz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Raghu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Paul and Chen 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Mahmood, Mahmood, and Van Dijk Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' These studies gave us some insights into how ViTs perform against attacks, sometimes compared to other fam- ilies of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Yet, whether or not ViTs can offer funda- mental ways to avoid vulnerability is still an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' For now, we may need to acknowledge that getting rid of this vulnerability is a hard problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Adversarial attacks may have been an armchair prob- lem, but the circumstance may be changing rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Deep- fakes, for example, have become an actual threat by coun- terfeiting authorities’ statements (Suwajanakorn, Seitz, and Kemelmacher-Shlizerman 2017), whereas researchers are working toward detecting such deepfakes using neural net- works (Albahar and Almalki 2019), playing a cat-and- mouse game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Adversarial attacks may have the potential to become a game-changer by facilitating the deepfake camp to fool deepfake detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' A potential alternative to inventing a mechanism to ro- bustify ViTs is providing some ways to detect adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Closely related studies (Raghu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Cintas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021) presented for the first time feature representation (Kornblith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2019) and attention profile analysis (Raghu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021) on ViTs, applied for comparisons against CNNs and between pre-trained and trained-from-scratch ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We propose to extend these studies exploring inference time signatures of adversarial attacks that potentially imply their presence in the input, to refine our understanding of the reactive part of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This work is a preliminary to a forensic exercise, where we wish to collect evidence of an attack given a suspicious input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' More specifically, for vanilla ViTs and tokens-to-token ViTs (T2T-ViTs) (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021) under various attacks, we show whether the input and the latent vectors (including self-attentions) have any signatures of the attack at multiple levels, including (i) the input itself and its frequency decomposition, (ii) output vector, (iii) at- tention profile (Raghu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021), (iv) the feature similar- ity given by centered kernel alignment (CKA) (Raghu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Kornblith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Our findings are summarized as follows: Input and output of the model tell a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Low- frequency components in adversarialy perturbed images carry more energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Also, the entropy of posterior proba- bility over all classes (for classification tasks) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' These results support the results in (Uelwer, Michels, and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='13356v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='CV] 31 Jan 2023 Candido 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Gupta, Dasgupta, and Akhtar 2020) and show that adversarial attacks are relatively visible outside the model, which may make designing a detector feasi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Latent vectors are not discriminative but explanatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Some of our quantifications are based on the model’s la- tent vectors, and they have lower discrimination power to detect attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' However, they provide more complete information about which features are more sensitive and how they react to attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This explainability makes these quantifications valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Perturbations may spot the acquired structure of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' It is known that ViTs have some inductive bias, which renders block-like structure in the CKA similarity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We empirically show that the latent vectors at the “boundaries” of this structure react to adversarial pertur- bations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This fact can facilitate an attack detector tailored for a certain model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This makes our contribution two-fold: First, we present and evaluate the first draft of quantifications that potentially show signatures of adversarial attacks on ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Second, we present intriguing internal behavior that may realize a more robust network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Related Work Many recent works demonstrate the robustness of the ViTs and their derivatives, such as (Benz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Paul and Chen 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' However, these works focus only on evaluating how much robustness gain can be obtained from ViTs compared to CNNs (Mahmood, Mahmood, and Van Dijk 2021), or how to robustify the architecture (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We will present in the following sections the literature review on the robustness of ViTs in image classification tasks and on the feature representation within the Transformers to rationalize our choice of working on it to find signatures of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Adversarial attacks The idea of adversarial attacks emerged (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2013) soon after the emergence of deep neural networks and has now become a well-known research topic with many variations to fool different architectures (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2022) and knowledge levels, such as white-box and black-box (Biggio and Roli 2018) attacks, warning the potential vulnerability of the current machine learning trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' On the defense side, most of the efforts towards better ro- bustness improvement are made for architectural modifica- tions (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2022) and input scrambling (Zantedeschi, Nicolae, and Rawat 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Evaluating the robustness of a specific model is often limited to input analysis (Carlini and Wagner 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Hendrycks and Gimpel 2016), with limited attention given to the network behavior during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Other techniques are using pre-processing to reduce the im- pact of the following network basing their strategy on trans- formations (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021), bringing back any perturbation into the training distribution, or blending/blurring the lo- cal information to reduce the chance of successful attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' While those techniques have proven to be successful, their application is often limited to several architecture families or attack families, and they may impair the accuracy of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Not knowing the internal representations made during in- ference for sure hurts the possibilities to detect any attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This idea motivates us to explore the internal structure of models in the inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Robustness of ViTs ViTs introduced in (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2010) use the original design of the Transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2018) but decompose the picture into patches as a sentence is decomposed into words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' One drawback of ViTs is the com- putationally demanding pre-training stage required to max- imize the generalization of the network, but it is addressed with various architectural tricks such as T2T (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021) with no significant drawback in clean accuracy (Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021), but in their experiment they show up to 40% of adversarial accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This architecture scored state of the art on image classification benchmark and in real-world applications (Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The self attention mechanism at heart of the ViT family seems to helps in the robustness to out-of-distribution samples (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' However those papers and attack specific ones such as (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2022) show that the ViTs actually trade compa- rable results depending on the task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We are currently heavily evaluating the adversarial robustness on standard at- tack sets, comprised of traditional attacks mainly developed on Fully Connected NNs and CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' As shown in (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2022), it is possible to specifically attack self attention ba- sic operation with success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' In sight of that, we think that acknowledging the robustness of ViTs in all cases is danger- ously optimistic, and studying their adversarial vulnerability and behavior should be also seriously considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Feature Representations in ViTs The authors of (Raghu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021) discovered two character- istics in the features of the ViTs that differs from the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' They firstly present an inductive bias learned by the net- works undergoing long pretraining and they show that those networks learn how to attend to global and local features, while those simply trained from scratch are limited to global features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The ViTs ”learn” how to attend to local features, when the CNNs have this implemented in the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Then they proceed in showing the appearance more mono- lithic of the CKA matrix computed on ViTs compared to the one of the CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Indeed the later one did show two blocks, while the ViT had only one, meaning the features ex- tracted from the beginning were forwarded to the next block in a very consistent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The stability of the feature rep- resentation throughout the network is promising for attack detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Additionally, self-supervised applications of the ViTs such as (Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021) show that certain condition of training favor the emergence of semantically meaningful features right out of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This echoes to the final observation of (Raghu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021) where they expose the self-location properties of the ViTs heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' While we do not study self-supervised trained models, one can ask if the at- tacks affect the capability of a Self-Attention head to focus on locally relevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Inference Time Signatures of Attacks Due to the complexity of deep neural networks, identifying signatures of adversarial attacks is rather exploratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We in- troduce here five derivatives of an input, a batch of inputs, or their latent vectors that potentially exhibit reactions of a Transformer network to adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' These deriva- tives are not our new proposal, but we re-purpose the orig- inal intent of the derivatives for single-sample adversarial attack detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Signatures in the frequency domain Previous studies have paid attention to the energy spectrum of the adversarial noises demonstrating that ViTs require a wider-spread spectrum for successful attacks (Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021) while CNNs are more vulnerable to high-frequency although they are still crucial for the attack on ViT (Paul and Chen 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' These results suggest that adversarial attacks tailored for ViTs affect a wider portion of the input in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Although the original image and adver- sarial noise are inseparable in our inference time detection scenario, the energy spectrum may provide some clues on the pattern of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We thus compute the discrete cosine transform (DCT) of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' According to the previous results, we will be par- ticularly attentive to the frequency variations in input sample x, either a clean image x = o or attacked image x = o + η, where o is the original image and η is the adversarial noise, that is, attacked images may have more lower-frequency components carrying more energy compared to the origi- nal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' For quantification, we propose the frequency ratio, which gives the ratio of high-frequency components over the low-frequency components of the energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Let DCTij(x) denote the (i, j)-th energy component of x’s DCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Frequency ratio FR is defined as: FR(x) = � (i,j)∈HF DCTij(x) � (i,j)∈LF DCTij(x) , (1) where HF and LF are the sets of indexes of high- and low- frequency components, given using threshold φ by HF = {(i, j)|i + j ≥ φ} and LF = {(i, j)|i + j < φ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Signatures in posterior probabilities The output f(x) (after softmax) of any classifier is usu- ally interpreted as the posterior probabilities for respective classes of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Adversarial attacks, in turn, try to re- duce the probability of the ground-truth class (or increase the probabilities of the other classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' A naive deduction from this argument is that the posterior entropy PH can be a basic signature of adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' PH(x) = − K � k=1 fk(x) log fk(x), (2) where fk(x) is the k-th element of f(x) for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2 0 2 4 6 8 10 12 Blocks (negative for T2T Blocks) 0 20 40 60 80 100 120 140 Attention Distance in pixels Attention profile for Clean Samples ViT-S-16 T2T-ViT-t Figure 1: Example attention profiles of two models ViT-S- 16 and T2T-ViT-t evaluated over the ImageNet test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Each block has multiple dots, which are AD(x) for respec- tive self-attention heads, averaged over all x’s in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The idea behind this quantification is that adversarial noises well-tuned for the model and the visual quality may hit the almost same probability masses for the ground-truth class and the second-top class, leading to higher PH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Signatures in attention distances CNNs have embedded inductive bias in their structure that constrains possible dependency among pixels only to their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Literature (Raghu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021) has shown that vanilla ViTs learn to attend neighboring patches when ex- tensively pre-trained with a large dataset, such as JFT-300M (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2017), but still ViTs have an ability to attend wherever necessary for the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Meanwhile, ViTs are typically more robust to adversarial attacks com- pared to CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This contrast can imply that texture fea- tures (or local dependency) is prone to successful attacks while shape features (or global dependency) give better ro- bustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We argue that the locality of dependency can be a key factor for robustness and are curious about the validity of this statement under adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' For ViTs, the locality of dependency (beyond patches) is encoded in the self-attention mechanism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' that is, atten- tion from a certain patch to only its neighbors suggests local dependency, while the attention spanning all over the input image suggests global dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We thus define the atten- tion distance as the average of the distances in pixel between a certain patch and any other patch weighted by the attention value associated to this pair of patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Specifically, let Aij(x) be the attention between patches i and j of a certain Transformer block and a certain self- attention head for input x, and dij the distance between the centers of patches i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Attention distance AD for this self-attention head is given by AD(x) = � ij Aij(x) · dij � ij Aij(x) , (3) where the summations are computed for all patch pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We collectively refer to AD(x)’s for all Transformer blocks and self-attention heads as the attention profile (AP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Figure 1 shows example APs for ViT and T2T-ViT models, evaluated over clean input samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=', x = o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' For read- ability, we arrange them in a block-wise manner, where the negative block indexes are for the T2T blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' APs can be computed for a set of samples by averaging AD(x) over x in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' An AP consists of multiple ADs, but it may be convenient to show a single-value summary of AP for a consistent dis- play with other quantifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Let ¯ ADref bh and ADbh(x) be the mean AD computed for the set of clean samples and AD computed for input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We define summary SAP(x) as the sum of their absolute differences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=', SAP(x) = � b,h | ¯ ADref bh − ADbh(x)|, (4) where the summation is computed over all blocks and heads of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We can generate the attention profile for a trained model over a set of clean input samples, which gives reference dis- tributions of attention distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Comparison between these reference distributions and a (set of) suspicious input may provide some ideas on the presence of adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Structures in latent representations CKA (Raghu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021) is used for studying the represen- tation structure in a model, by giving layer-to-layer similar- ities in that model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Analyzing latent representations within a model allows for identifying different general functions throughout its architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Adversarial attacks may break such functions as they exploit some patterns in the learned parameters, which results in unusual responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' It should be noted that, although we aim to discover inference time sig- natures of adversarial attacks, CKA is a statistic and is com- puted only for a set of samples but not for a single sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' therefore, this signature is not always usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' CKA takes as input Hi ∈ Rm×c and Hj ∈ Rm×c′, which are latent vectors (or activation) at the i-th and j-th layers (i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' , l) with the dimensionalities of ci and cj, re- spectively, obtained for the same m input samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Letting Ki = HiH⊤ i and Kj = HjH⊤ j denote the Gram matrices, CKA is given by: CKAij = HSIC(Ki, Kj) � HSIC(Ki, Ki) HSIC(Kj, Kj) (5) where HSIC is the Hilbert-Schmidt independence criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Given the centering matrix1 Hc = Ic − 1 c1c with Ic being the c × c diagonal matrix and 1c being the c × c matrix whose elements are all 1, the centered Gram matrix for K is computed by K′ = HcKHc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This operation is applied to both Ki and Kj to obtain K′ i and K′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' HSIC is given as the similarity between these centered Gram matrices, computed by HSIC(Ki, Kj) = vec(K′ i)⊤vec(K′ j)/(m − 1)2, (6) 1Here we omit the indices i and j for notation simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' where vec(·) is the vectorization of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' CKA is usu- ally computed over the entire test set because of the statis- tical nature of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Hence, we compute CKA for the minimum batch size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' m = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' CKA is a similarity between an arbitrary pair of layers, resulting in a similarity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' But this is not always handy for our purpose of finding signatures of adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We reduce the similarity matrix down to a single metric for compacting the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Also to highlight the reactions to adversarial attacks, we compute the difference between CKAs computed for clean and attacked samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Specifically, let M ref ∈ Rl×l denote the reference CKA similarity matrix computed for clean samples in the entire test set (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=', the (i, j)-th element of M ref is CKAij computed for clean samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Also, let ¯ M denote the mean of CKA similarity matrices of the same model computed for random mini-batches with the minimum batch size and of samples under the attack of interest (or clean samples), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=', ¯ M = 1 |B| � B∈B M(B), (7) where B is the set of aforementioned mini-batches and M(B) is the CKA similarity matrix computed for B ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' To highlight the changes, we compute the absolute differ- ence D between M ref and ¯ M obtained with samples under the attack of interest, where its (i, j)-th element Dij is given by Dij = |M ref ij − ¯ Mij|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' (8) The summary SCKA is defined as the sum of these differ- ences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=', SCKA = � ij Dij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' (9) Experimental Setup Models We choose two major architectural families that cover vari- ous paradigms of ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The first family is the vanilla ViT (Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2010), which comes with 3 variants: ViT/T-16, ViT/B-32, and ViT/B-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This allows us to draw more general conclusions as they consist only of Trans- former blocks, which are the main building block of the ma- jority of the entire ViT family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We also investigate T2T-ViTs (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021) with two variants: T2T-ViT performer (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Choromanski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2020) and the origi- nal T2T-ViT transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The T2T-ViT family addresses the need for extensive pre-training on large datasets, which is inevitable for the vanilla ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' By introducing a CNN-like structure, this family also happens addressing the inductive bias (Raghu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021), bringing a T2T-ViT model trained from scratch over ImageNet with an attention profile com- parable to fully pre-trained ViTs (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Evaluation on this family provides some insights about adversarial attacks over architectural variations like hierarchical designs in ViT (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021) and variations without pre-training (Tou- vron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Model Encoding Depth # heads # params.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' ViT-T/16 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 12 3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='7M ViT-S/32 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 12 6 22M ViT-S/16 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 12 6 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='6M T2T-ViT-p Performer 14 6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5M T2T-ViT-t Transformer 14 6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5M Table 1: Summary of ViT and T2T-ViT models used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Table 1 shows a summary of models we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' For vanilla ViT variants, we used models pre-trained on JFT-300M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='2 All models are trained (or fine-tuned) on the ImageNet training set (Steiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' For legibility, we present the results for ViT-S/16 and T2T-ViT-t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' the other models’ results are presented in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Dataset Our study is conducted over the ImageNet dataset, which is large enough to train the aforementioned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The size of input images is 224 × 224, which allows us to optimize adversarial perturbations in a reasonable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' To perform our attacks, we will be using the ImageNet-1K test set of 50k samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' To unload the computational burden, we sampled one out of five samples from the set, giving us 10k samples, which is still representative of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Adversarial Attacks We evaluated the robustness under popular adversarial at- tacks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=', the fast gradient sign method (FSGM) (Goodfel- low, Shlens, and Szegedy 2014), projected gradient descent (PGD) (Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2017), and Carlini and Wagner (C&W) (Carlini and Wagner 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' FSGM and PGD are l∞ attacks, where attacked image x for original image o is given by x = argmax x′ L(x′, y) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' ∥x′ − o∥∞ ≤ ϵ, (10) where L is the loss function for the task (typically, softmax cross-entropy) and y is the ground-truth label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This is an un- targeted attack, in which the attack is successful when the perturbation changes the model’s prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The above opti- mization problem is intractable because of the l∞ constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' FGSM approximates the optimization problem by x = o + ϵ sign(∇oL(o, y)), (11) where ∇oL is the gradient of L at o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This is a quick but rough approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' PGD is one of FGSM’s extensions, which iteratively per- forms FGSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Letting xt denote the attacked image after t iterations, PGD computes xt+1 by xt+1 = clip[o−ϵ,o+ϵ](xt + α sign(∇xL(xt, y))), (12) where clip[o−ϵ,o+ϵ](·) is to clip the value within [o−ϵ, o+ϵ], α is a predefined small value, and x0 = o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This iterative 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='com/rwightman/pytorch-image-models and https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='com/ yitu-opensource/T2T-ViT Table 2: Accuracy of ViT and T2T models under various attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Top-1 Top-5 Attack ViT-S/16 T2T-ViT-t ViT-S/16 T2T-ViT-t Clean 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='38 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='40 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='13 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='72 C&W c=1×10−4 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='86 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='21 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='68 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 PGD ϵ=1×10−3 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='36 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='37 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='79 PGD ϵ=3×10−3 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='47 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='98 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='65 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='40 PGD ϵ=5×10−3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='36 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='30 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='75 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='68 PGD ϵ=1×10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='50 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='54 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='08 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='05 FGSM ϵ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='94 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='84 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='31 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='80 FGSM ϵ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='62 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='28 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='78 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='60 method takes advantage of multiple small steps, controlled by α, to refine the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Iteration enables the attack to be more efficient than FGSM, so that a much smaller ϵ radius enables the same level of perturbation as FGSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' However, PDG is more computationally demanding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' the computation cost is linear with the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We will use PGD for 40 iterations with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' In our experiments, we use ϵ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062} for FGSM and ϵ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01} for PGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' C&W jointly optimizes the distortion introduced by the perturbation in the l2 distance and the performance of the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The method quantifies the performance of the attack by Q(x) = max � fy(x) − max y′̸=y fy′(x), −κ � , (13) where we use the implementation3 of the untargeted version;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' fy and fy′ denote the prediction confidences for ground- truth label y and another label y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' A negative value of Q(x) means that a certain class other than the ground truth gets the highest confidence, though the confidence does not go beyond the margin κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' C&W parameterizes attacked image x by w, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=', x(w) = tanh(w) + 1 2 , (14) which guarantees x in the range of [0, 1] during the optimiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The attacked image x of C&W is given by x(w∗), where w∗ = argmin w ∥x(w) − o∥2 + c Q(x(w)) (15) with c = 1 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Metric for statistical signatures Although we aim to identify inference time signatures of at- tacks, it is also interesting to explore the statistical signatures that our quantifications may exhibit over sets of clean and at- tacked samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We thus employ the discrete version of the Bhattacharyya coefficient (BC) (Bhattacharyya 1946) for scalar-valued quantification, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=', RF, PH, AD’s, and SCKA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='com/Harry24k/adversarial-attacks-pytorch 4We normalize original image o to be in [0, 1] as well 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 0 500 Sample count ViT-S/16 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='95) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='74) Clean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 Input Frequency Ratio 0 500 Sample count T2T-ViT-t C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='90) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='58) Clean Figure 2: Distributions of FR for representative two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The numbers in Parentheses are BCs computed between the clean distribution and respective distributions with perturba- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Let h and h′ denote the empirical distributions (or his- tograms) of one of our quantifications with 100 bins com- puted for the sets of clean samples and attacked samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Their i-th bin, hi and h′ i, give the numbers of samples that fall into that bin, normalized to sum up to one, where the bins are configured to cover the minimum and maximum values of the quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' BC is defined as BC(h, h′) = � i � hih′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' (16) The range of BC is [0, 1] where BC = 1 means the most similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' For better discrimination of clean and adversarial samples, we hope for a value as close as possible to 0, which gives a significant signature of adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' However, we realistically expect the value to be rather high for typical (and thus “interesting”) configurations of methods for adver- sarial attacks, as only a small perturbation is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Results Frequency ratio Figure 2 shows the empirical distributions of FR for clean samples and attacked samples by the configurations pre- sented earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We can see the distributions are different only for the configurations with the largest perturbations (namely FGSM with ϵ being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062), for which FR con- sistently drifts towards 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This means that the energy of the perturbation η is relatively concentrated in lower frequen- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This confirms that ViTs are also vulnerable to low fre- quency when given enough perturbation budget (Paul and Chen 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The distributions for smaller perturbations are almost the same as the clean one, which is also shown in the high BCs computed for the attacked distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' therefore, FR in it- self cannot be reliable for attack detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We also note at this stage that the distribution shift is related to the pertur- bation strength and not to the success rate of the attack in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We will observe that same behavior on the other signatures and discuss this problem in the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We 0 1 2 3 4 5 6 7 0 2000 Sample count ViT-S/16 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='95) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='96) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='94) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='96) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='97) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='90) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='85) Clean 0 2 4 6 8 Entropy of the Output vector 0 500 Sample count T2T-ViT-t C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='95) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='90) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='92) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='92) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='91) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='79) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='73) Clean Figure 3: Distributions of PH for the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' know from (Harder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021) that it is possible to train de- tector models on Fourier Transform of an input to detect to presence of adversarial attack on CNNs, hence giving hint for similar results with ViTs if the same technique was to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Analysis in the frequency domain can provide some signatures of attacks if the perturbation is large enough but is not useful otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Posterior probability The distributions of PH are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We see very different behaviors from one model to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The T2T-ViT model displays noticeably more varied distributions com- pared to the vanilla ViT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We cannot explain that dif- ference considering that on average the T2T outperforms the ViT in the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' For that matter the ViT even have better top-5 accuracy (96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='13%) on clean samples com- pared to the T2T (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='72%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This result demonstrates that PH may be used in an attack detection pipeline for T2T-ViTs, though we would have to make heavy compromises in the accuracy or in the false positive rate if the quantification is used by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' It has to be noted that the significance of this quantification is also proportional to the intensity of the at- tack rather than the success rate of this one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Attention distances and profiles In order to analyze the attention distance, we base our ob- servations on the way the attention profile is computed in (Raghu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2021): the profile is computed for the entire test set, regardless of the accuracy of the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This base gives us the reference of the nominal behavior of our model for in-distribution inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Figure 1 shows the distributions of SAP(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Note that we can still compute the distribution for the set of clean sam- ples, which shows the difference between individual sam- ples and their mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We observe that SAP(x) behaves in a similar way as the previous metrics, showing a marginal separation (BC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='97) for the largest perturbation on both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Since we measure the absolute difference between the reference and observed profiles, SAP(x) gives the gist but very little infor- mation on the actual direction of the change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We can ob- serve finer behavior when looking in detail at the level of 300 400 500 600 700 800 900 1000 1100 0 250 Sample count ViT-S/16 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='97) Clean 300 400 500 600 700 800 900 1000 1100 Cumulated Head Attention Distance from Reference Profile 0 250 Sample count T2T-ViT-t C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='96) Clean Figure 4: Distributions of SAP(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 0 2 4 6 8 10 Blocks (negative for T2T Blocks) 0 20 40 60 80 100 120 140 Attention Distance in pixels Average of Attention Distribution per Blocks for ViT-S-16 Pretrained PGD eps:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 PGD eps:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 PGD eps:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 PGD eps:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 PGD eps:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 PGD eps:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 Clean Figure 5: ADs per head observed on ViT-S/16 for PGD at- tacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The different attacks are sorted so that the configura- tion that gives the lowest accuracy comes on the left within each head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Each point is the mean over the 10k samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' In Figure 5, we show that perturbations ei- ther diversify or shrink ADs on several heads, mostly in the first blocks, and the trend of changes in ADs is constant for them, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=', either diversified or shrunk, while all the others re- main unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' For instance, the head with the largest AD in block 1 is diversified along with the amount of perturba- tion, while the one with the second largest AD in the same block shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We note that diversification mostly occurs in the lower half (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=', smaller block indices) of the network while shrinkage happens throughout it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' One simple explanation in which the discrepancy in the attention results in a successful attack is that the perturba- tion distracts the attention of all patches to some regions that have no features leading to the ground-truth class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The at- tack may be successful if the features from irrelevant patches overwrite the features from the residual connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' From this argument, we hypothesized that heads with attention discrepancy are the sweet spot for adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Veri- fication of this hypothesis is up to further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We think those volatile heads are a good hook to build an attack detector in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We designed SAP(x) to be Table 3: BCs computed for the most volatile heads in terms of attention distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Attacks ViT-S/16 T2T-ViT-t C&W c = 1 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD ϵ = 1 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD ϵ = 3 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD ϵ = 5 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD ϵ = 1 × 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) FGSM ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='97 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='02) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='98 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01) FGSM ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='85 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='96 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01) 36 38 40 42 44 46 48 50 52 0 500 Sample count ViT-S/16 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='98) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='97) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='94) Clean 55 60 65 70 75 80 85 Cumulated Feature Similarity Difference from Reference Profile 0 250 Sample count T2T-ViT-t C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='98) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='98) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='95) Clean Figure 6: Difference between the observed CKA similarity and the reference one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' a summary of all heads in all blocks, so the distributions in Figure 4 may be too vague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Focusing on a few heads that exhibit large discrepancies can give a clearer distinction be- tween clean and adversarial samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Table 3 shows the BC values computed over AD(x) for a cherry-picked head of each model and each configuration of attacks and the corre- sponding improvement of BC values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' CKA Similarity The distributions of SCKA in 6 show even worse separation capabilities compared to the other quantifications, where even the largest perturbations seem to have only slightly shifted the curves from the original clean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' In a simi- lar fashion to AP, we take a step back from SCKA and look at the layer-wise difference Dij of the reference CKA simi- larity matrix and that of attacked samples, because the sum- mary by SCKA does not provide sufficient information on the presence of any attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' In Figure 7, we observe clear patterns appearing with the increase of attack intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The patterns are presum- ably related to the layers in Transformer blocks, which are comprised of layers after multi-head attention, after projec- tion, and after multi-layer perceptron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' One can point that the latent vectors after the attention heads are remark- ably stable, and only the first layer (or two in the case of T2T-ViT) are affected when using the largest perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Figure 7: Left-most column: Reference CKA similarity ma- trix M ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Second left column: D computed for clean sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Third left to right-most columns: D’s computed for samples under the attack of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' From top to bottom: ViT-S/16’s attention heads, projections, and MLPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' T2T- ViT-t’s attention heads, projections, and MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Table 4: BCs computed for the most reactive latent vectors in terms of the CKA similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Attacks ViT-S/16 T2T-ViT-t C&W c = 1 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD ϵ = 1 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD ϵ3 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='98 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01) PGD ϵ = 5 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='98 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01) PGD ϵ = 1 × 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='98 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) FGSM ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='97 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='96 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='02) FGSM ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='93 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='91 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='04) Ones after the projection layers are mildly affected, diffus- ing for all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' A remarkable impact is observed for our largest FGSM attack (ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062), where the first layer reacts strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The last latent vectors are the one after MLPs of each attention block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' These vectors are affected throughout the whole spectrum of attacks, and the effect is noticeable around one specific block, that is, the 7-th one for ViT-S/16 and also the 7-th, which is the 5-th if we count in the ViT backbone only, for the T2T-ViT-t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The perturbation seems to locally spread from this block with the intensity increased, as well as the perturbation appearing in the first blocks when using strong FGSM attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We show in Table 4 the fine-grained BCs for cherry- picked layers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' the most reactive layers) for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The table shows that targeting specific layers can slightly help separate attacked samples from clean ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' However, the separation is still marginal, and the majority of the sam- ples are confused with the clean distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Conclusion We presented in this paper four different types of quantifica- tions that potentially exhibit some signatures of adversarial attacks in the inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The biggest challenge in ad- versarial attack detection comes from the great variability in models’ reactions, and our experimental results generally show the difficulty in building a simple criterion to distin- guish an attacked sample from clean ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We empirically demonstrated that the input frequency ratio and the posterior entropy are promising out-of-the-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We also showed that quantifications based on the latent vectors, such as attention profiles and CKA similarity allow separating only the largest attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' However, these quantifications render some patterns specific to each model, which may collectively serve as a stronger signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The results also showed that these quan- tifications mostly react to the amplitude of perturbations but not to the effectiveness of the attack (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' This implies that fine-tuned adversarial attacks such as C&W and PGD with a very small perturbation budget are both very efficient and undetectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The current state of inference time signa- ture is very rough, and the very little information does not seem worth the cost of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' However we have great hope in the finer analysis of the latent vector to get bet- ter understanding in the future of where in the network will we have better chance of detecting an attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Future work includes exploring interactions between quantifications as stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Acknowledgements This work was partly supported by JST CREST Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' JPMJCR20D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' References Albahar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' CVPR Workshop on Ad- versarial Machine Learning in Real-World Computer Vision Systems and Online Challenges (AML-CV), 21–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Bhattacharyya, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 1946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' On a measure of divergence be- tween two multinomial populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Sankhy¯a: the Indian Journal of Statistics, 401–406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Biggio, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' and Roli, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Wild patterns: Ten years after the rise of adversarial machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Pattern Recogni- tion, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 10th ACM workshop on artificial intelligence and se- curity, 3–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Carlini, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' and Wagner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2017b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Towards evaluating the robustness of neural networks.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Touvron, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Misra, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' J´egou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Mairal, J.' 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from scratch on imagenet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' IEEE/CVF International Conference on Computer Vision, 558–567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Zantedeschi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Nicolae, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' and Rawat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Ef- ficient defenses against adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 10th ACM Workshop on Artificial Intelligence and Security, 39– 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Supplementary Material Frequency ratio We present in Figure 8 the distribution over 3 additional net- work variations as presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We clearly see a direct extension of the result observed on ViT-S/16 repro- duced in both the smaller ViT-T/16 and the ViT-S/32 with larger patch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We note a small variation of the BC on the ViTs for the largest FGSM attack, but we think it is within the margin of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' As for the T2T family the results here are exactly similar, that is in complete agreement with the per- former being faster equivalent operation of the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 0 500 Sample count ViT-T/16 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='95) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='76) Clean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 0 500 Sample count ViT-S/16 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='95) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='74) Clean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 0 500 Sample count ViT-S/32 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='92) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='70) Clean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 0 500 Sample count T2T-ViT-t C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='90) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='58) Clean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0 Input Frequency Ratio 0 500 Sample count T2T-ViT-p C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='90) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='58) Clean Figure 8: Distributions of FR for representative two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The numbers in Parentheses are BCs computed between the clean distribution and respective distributions with perturba- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Posterior probability In Figure 9 we extend the evaluation to the 3 other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We show here that the behavior display by the ViT-S/16 and the T2T-ViT-t are completely representative of their family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We find interesting here that the T2T-ViTs in general have entropy of approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2 while the ViTs are stuck at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The fact that the ViTs learned to output exactly one very high value as prediction is maintained in the adversarial setups, while the Table 5: BCs computed for the most volatile heads in terms of attention distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Attacks ViT-T/16 ViT-S/16 ViT-S/32 T2T-ViT-t T2T-ViT-p C&W c = 1 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD ϵ = 1 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD ϵ = 3 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01) PGD ϵ = 5 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01) PGD ϵ = 1 × 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01) FGSM ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='97 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='02) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='98 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='98 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) FGSM ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='98 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='85 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='05) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='96 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='97 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='03) T2Ts who usually are more varied have even less decisive answers when attacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' However we note that this behavior from the T2T family does not impact the accuracy and is even outperforming the ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 0 1000 Sample count ViT-T/16 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='95) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='97) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='95) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='96) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='97) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='89) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='80) Clean 0 1 2 3 4 5 6 7 0 2000 Sample count ViT-S/16 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='95) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='96) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='94) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='96) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='97) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='90) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='85) Clean 0 1 2 3 4 5 6 7 8 0 1000 Sample count ViT-S/32 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='97) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='98) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='95) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='95) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='97) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='90) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='79) Clean 0 2 4 6 8 0 500 Sample count T2T-ViT-t C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='95) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='90) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='92) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='92) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='91) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='79) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='73) Clean 0 2 4 6 8 Entropy of the Output vector 0 500 Sample count T2T-ViT-p C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='95) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='89) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='93) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='94) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='93) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='81) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='73) Clean Figure 9: Distributions of PH for the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Attention distances and profiles We also expend here the experiments of Section and show here varied behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' At first the BC computed for the ViTs increase in the order presented here with ViT-T having the lowest separability on the attention distance due to the low number of heads to compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The other ViT-T/32 shows bet- ter BC for the largest attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We attribute this to the fact that this network has by design a larger attention scope, and that making it diverge, only from one patch, is much more noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The T2Ts also present interesting changes, with the T2T-ViT-p showing very poor BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' When we compare Figure 14 and Figure 15 we see a very clear difference in the attention of the T2T block where most of the drift under attacks were measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We also try to refine those scores with the same technique and observe very similar to the ones in their respective fam- ily again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 100 200 300 400 500 0 250 Sample count ViT-T/16 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='98) Clean 300 400 500 600 700 800 900 1000 1100 0 250 Sample count ViT-S/16 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='97) Clean 300 400 500 600 700 800 900 1000 1100 0 250 Sample count ViT-S/32 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='94) Clean 300 400 500 600 700 800 900 1000 1100 0 250 Sample count T2T-ViT-t C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='96) Clean 300 400 500 600 700 800 900 1000 1100 Cumulated Head Attention Distance from Reference Profile 0 200 Sample count T2T-ViT-p C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='99) FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='00) Clean Figure 10: ADs per head observed on ViT-S/16 for PGD at- tacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The different attacks are sorted so that the configura- tion that gives the lowest accuracy comes on the left within each head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Each point is the mean over the 10k samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' Stability of the Attention distances and profiles We show in this section some testimonies proving that the at- tention drifts reported in Section are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' We mainly show here that when the distributions are drifting, they do so in a very consistent way with no huge distribution spread or grouping (with the exception of the attention head -1 on Fig- ure 14), meaning that making observations on the average is indeed representative of the drift measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 12 Blocks (negative for T2T Blocks) 0 20 40 60 80 100 120 140 Attention Distance in pixels First 3 heads of ViT-T/16 FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 Clean Figure 11: Distribution of the ADs over the 10k samples for the first 3 heads observed on ViT-T/16 for our set of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The different attacks are sorted for each block in the same order as the legend and in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 12 Blocks (negative for T2T Blocks) 0 20 40 60 80 100 120 140 Attention Distance in pixels First 3 heads of ViT-S/16 FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 Clean Figure 12: Distribution of the ADs over the 10k samples for the first 3 heads observed on ViT-S/16 for our set of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The different attacks are sorted for each block in the same order as the legend and in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 12 Blocks (negative for T2T Blocks) 0 20 40 60 80 100 120 140 Attention Distance in pixels First 3 heads of ViT-S/32 FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 Clean Figure 13: Distribution of the ADs over the 10k samples for the first 3 heads observed on ViT-S/32 for our set of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The different attacks are sorted for each block in the same order as the legend and in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Blocks (negative for T2T Blocks) 0 20 40 60 80 100 120 140 Attention Distance in pixels First 3 heads of T2T-ViT-t FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 Clean Figure 14: Distribution of the ADs over the 10k samples for the first 3 heads observed on T2T-ViT-t for our set of attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The different attacks are sorted for each block in the same order as the legend and in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Blocks (negative for T2T Blocks) 0 20 40 60 80 100 120 140 Attention Distance in pixels First 3 heads of T2T-ViT-p FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='062 FGSM =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='031 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='01 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='005 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='003 PGD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='001 C&W c=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content='0001 Clean Figure 15: Distribution of the ADs over the 10k samples for the first 3 heads observed on T2T-ViT-p for our set of at- tacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} +page_content=' The different attacks are sorted for each block in the same order as the legend and in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFQT4oBgHgl3EQfjzbM/content/2301.13356v1.pdf'} diff --git a/edA0T4oBgHgl3EQfHP-i/content/tmp_files/2301.02059v1.pdf.txt b/edA0T4oBgHgl3EQfHP-i/content/tmp_files/2301.02059v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..614dfd6db53b66e273e15bd395ad1f545afd8328 --- /dev/null +++ b/edA0T4oBgHgl3EQfHP-i/content/tmp_files/2301.02059v1.pdf.txt @@ -0,0 +1,1790 @@ +Zen: LSTM-based generation of individual spatiotemporal +cellular traffic with interactions +Anne Josiane Kouam +Inria +France +Aline Carneiro Viana +Inria +France +Alain Tchana +Grenoble INP +France +ABSTRACT +Domain-wide recognized by their high value in human presence +and activity studies, cellular network datasets (i.e., Charging Data +Records, named CdRs), however, present accessibility, usability, +and privacy issues, restricting their exploitation and research repro- +ducibility. This paper tackles such challenges by modeling Cdrs that +fulfill real-world data attributes. Our designed framework, named +Zen follows a four-fold methodology related to (i) the LTSM-based +modeling of users’ traffic behavior, (ii) the realistic and flexible +emulation of spatiotemporal mobility behavior, (iii) the structure +of lifelike cellular network infrastructure and social interactions, +and (iv) the combination of the three previous modules into realis- +tic Cdrs traces with an individual basis, realistically. Results show +that Zen’s first and third models accurately capture individual and +global distributions of a fully anonymized real-world Cdrs dataset, +while the second model is consistent with the literature’s revealed +features in human mobility. Finally, we validate Zen Cdrs ability of +reproducing daily cellular behaviors of the urban population and +its usefulness in practical networking applications such as dynamic +population tracing, Radio Access Network’s power savings, and +anomaly detection as compared to real-world CdRs. +CCS CONCEPTS +• Data and Communication Traffic → Charging Data Records; +• Cellular Traffic → Mobility and Network events; • Modeling → +LSTM. +KEYWORDS +Human mobility modeling, Data and Communication traffic model- +ing, Recurrent Neural Networks. +1 +INTRODUCTION +Charging Data Records are acknowledged as a common tool for +studying human mobility, infrastructure usage, and traffic behav- +ior [18]. We name such datasets as CdRs to distinguish them from +the standard Call Detail Records (CDRs), describing only call and +SMS cellular communication information. CdRs describe time-stamped +and geo-referenced event types (i.e., data, calls, SMS) generated +by each mobile device interacting with operator networks (cf. Ta- +ble 1). They comprise city-, region-, or country-wide areas and +usually cover long periods (months or years); no other technology +currently provides an equivalent per-device precise scope. As a +result, CdRs are exploited in different research domains and indus- +tries, such as sociology [39], epidemiology [7], transportation [38], +and networking [35]. For a quantitative appreciation of such CdRs’ +worth recognition, Fig. 1 identifies 14 different research domains +leveraging CdRs among 100 most relevant works (sorted by Google +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +Mobility research +Transportations +Big Data engineering +Tourism +Urban planning +Billing +Fraud and Anomaly detection +Crime investigation +Networking +Traffic analysis and research +Health research +Social analysis +Economics +Mobile security +2017 +2018 +2019 +2020 +2021 +Figure 1: Distribution by domain of the last 5-year most rel- +evant publications using CdRs. +Scholar) selected from 1022 last 5-year publications. This clearly +shows a great diversity of domains in this sample only (∼ 10%). +Yet, the exploitation of real-world CdRs for research faces many +limitations. First, accessibility: CdRs datasets are not publicly avail- +able, imposing strict mobile operators’ agreements. Second, usabil- +ity: CdRs are usually available in an aggregated form (i.e., grouped +mobility flows and coarse spatiotemporal information), limiting +related analyses’ preciseness. Third, privacy: even anonymized, non- +aggregated CdRs describe sensitive information of users’ habits, +which hardens their shareability [28]. This paper addresses such +limitations by enabling the autonomous generation of realistic and +privacy-compliant CdRs by scientific community, thus providing new +avenues for research advances. +Moreover, generated CdRs should conform to essential attributes, +namely, completeness, realisticness, fine-grained description, and pri- +vacy. Unfortunately, those attributes make the generation of realis- +tic CdRs challenging and complex. In particular, achieving complete- +ness requires (i) either real-world complete CdRs datasets (hard +to obtain) describing mobility, traffic, and pairwise users commu- +nications or (ii) to cope with the difficulty in modeling the intrin- +sic correlations between information describing users’ behaviors +in space, time, and social communication. Achieving realisticness +implies considering real-world cellular network complexities (ar- +chitecture and topology) at all levels of the generation process. +The fine-grained description achievement is impeded by the het- +erogeneity of users’ behaviors, especially in cellular traffic. Finally, +generated traces should be privacy-compliant to avoid backtrack- +ing real users’ identities, most often done through their mobility +information. +To the best of our knowledge, this is the first work in literature +producing realistic Charging Data Records (CdRs) that fulfill the +arXiv:2301.02059v1 [cs.NI] 5 Jan 2023 + +Kouam et al. +above-mentioned attributes. Our designed framework, named Zen +employs a four-fold methodology: +(1) Leveraging on a real-world fully anonymized CdRs describ- +ing users’ traffic behavior (i.e, events information on its type – data, +call, and SMS –, duration, pairwise information, etc), we propose the +first literature modeling that captures long-range and inter-CdRs +specificity correlations while addressing the population hetero- +geneity (§3). Our model captures population diverseness in the +reproduction of individual traffic behaviors. We use three separate +Long-Short-Term Memory neural networks (LSTM) to model event +types generation (i.e., what), the inter-event duration (i.e., when), +the social interactions (i.e., whom), and leverage statistical analysis +to model CdRs metrics such as calls duration (i.e., how). Overall, +Zen traffic modeling presents significant high performance values, +providing for 80% of users (i) more than 95% (for event-type) and +75% (for inter-event) of modeling accuracy, and (ii) less than 6.68% +(for inter-event) and 12.5% (for social) of Mean Absolute Error’s +maximum values. +(2) Mobility behaviors of individuals (§4) are emulated according +to the infrastructure of a real-world metropolitan city (here, the +Helsinki EU city), and resulting trajectories are coupled with the cor- +responding cell towers distribution of existing operator networks in +the same city [33]. Here, we leverage city planning, transportation +information [17] as well literature investigations on laws dictating +human mobility [1, 11, 30]. Such real-world information and realis- +tic human mobility modeling are then incorporated in the literature +Working Day Mobility (WDM) model [9] – extensively enhancing +it – to emulate urban daily-life mobility behaviors of individuals. +Moreover, we rely on the ONE simulator [20] to bring flexibility to +our model regarding population size, duration, and covered area. +(3) We then design a separate module (§5) to realistically re- +produce on top of generated mobility traces, a cellular network +organization with multiple operators and build social ties between +users. This enables the first-of-a-kind flexibility to produce CdRs +of numerous operators at the same period. +(4) We combine all the previous models to generate complete +CdRs describing individual mobility, traffic, pairwise communica- +tions following real traffic behavior (§6). +Note that the disjoint behaviors modeling of realistic emulated +mobility and of real-world traffic hides real individuals’ spatiotem- +poral daily-life habits in routine and leisure times (e.g., home/work, +nightlife, etc.), bringing the privacy-preserving capability to the +produced CdRs. +2 +ZEN OVERVIEW +In the following, we provide an overview of Zen architecture and +describe the different real-world datasets we leverage. +2.1 +Architecture +According to input parameters, we generate realistic CdRs (cf. Table +1) through four phases, each implemented in a module of the Zen +architecture (cf. Fig. 2). Zen architecture consists of (1) a traffic +module, (2) a mobility module; (3) a social-ties module, and (4) a +CdR-combiner, or merger module. +The traffic module (§3) leverages Long-Short-Term Memory neural +networks (LSTM) jointly with statistical analysis to model users’ + +, +CDR-combiner +Zen CdRs : +Zen real-world datasets Inputs +Social-ties +module +userID's phone +number & +correspondents list + +§5 +Mobility +module + +§4 +§6 + +§3 +Event-type +model +Traffic module +IET +model +Correspondent +model +Metric +model +Figure 2: Zen architecture. +traffic behavior from real-world CdRs. It provides answers to what, +when, with whom, and how to generate events. The mobility mod- +ule (§4) (i) emulates users temporal displacements on a real-world +geographical map over a selected period, and (ii) associates corre- +sponding users positions with a real-world cellular topology. This +dataset feeds the social-ties module (§5) that builds the network +social structure on top of which users’ communication interactions +occur by building users’ phonebooks, i.e., list of phone numbers +a user is likely to contact. Finally, the CdR-combiner module (§6) +combines the previous modules’ outputs to generate realistic CdRs +per network operator over a specified duration and particular urban +area. +2.2 +Real-world reference datasets +Zen models real-world datasets to produce realistic outputs. In +particular, as depicted on top of Fig. 2, Zen uses three real-world +reference datasets described in what follows. +RefCdRs are used by both the traffic and the social-ties modules. +RefCdRs refer to a fully-anonymized CdRs dataset collected by a +major mobile network operator. They describe 1-month (from 2018- +06-01 to 2018-06-30) per-user traffic resulting in about 3 million +timestamped events generated by 186,738 distinct phone numbers, +where about 17,000 are from the RefCdRs’ operator. RefCdRs are +incomplete; they lack mobility features and incoming-SMS traffic +type (i.e., only have outgoing SMS). Still, there is no information +on the size of sessions in the data traffic type. RefCdRs provide +each user’s operator network code. We leverage this information +to identify the list of operators appearing in the datasets. +On the other hand, the mobility module leverages the ChineseDB [1] +and Geolife [48] datasets, by extracting statistics describing real- +life mobility behavior of users. ChineseDB (non-public and fully +anonymized mobility CdRs) contains trajectories of 642K users +during two weeks. In particular, we did not have access to ChineseDB +but only to related statistics available in [1]. Geolife (public and +anonymized GPS dataset) contains trajectories of 182 users during +64 months. +2.3 +Zen CdRs attributes +We present hereafter the positioning of Zen generated CdRs with +respect to our goals: +Completeness: Complete CdRs comprise mobility and traffic fea- +tures and should, thus, include, in addition to user positions (i.e., + +Zen: LSTM-based generation of individual spatiotemporal cellular traffic with interactions +0 +10 +20 +Daily time (h) +0 +50 +100 +0 +10 +20 +Daily time (h) +0 +10 +20 +Daily time (h) +Figure 3: Temporal event sequences of 100 users for: (left) +a real-world, (center) a statistically- and (right) the Zen- +generated CdRs. +network cell Ids), all event types, namely data, call, and SMS. Here, +the limited access to complete real-world CdRs hardens the mod- +eling and reproduction of complete CdRs. Zen circumvents this +limitation and provides complete CdRs by jointly modeling sepa- +rate CdRs features to capture the implicit correlations between them +: e.g., the choice of whom to communicate with is generally time +(when) and event (what) dependent. Therefore, Table 2 shows Zen +yields complete CdRs compared to most state-of-the-art contribu- +tions which instead provide only one CdRs feature, either mobility +or an event type. +Realisticness: Zen modeling integrates a real telecom network +topology (inducing users’ cell-tower locations) and organization in +multi-operators, as a requirement to produce realistic CdRs. Con- +sidering the latter, Zen CdRs conveys inter-operator interaction +patterns, valuable for telecom fraud investigation for instance [22]. +Fine-grained description: This relates to the realistic reproduc- +tion of the individual users behaviors in terms of mobility and +traffic, beyond the global behavior of the population. In particular, +daily individuals’ cellular traffic presents a notable heterogeneity +that challenges its reproductions. Fig. 3(left) shows a daily traffic +(i.e., sequence of events per user) of 100 randomly selected users +from a real-world CdRs. We can see a great diversity of users regard- +ing events generation. Statistical approaches (see Fig. 3(center)), as +commonly used in the state-of-the-art [29, 32, 42], are limited in +reproducing such traffic dynamics as they do not allow per-user +modeling but per-user profile (i.e., group of users with similar be- +havior). Improving this result, the approach we use in Zen better +captures such individuals heterogeneity (see Fig. 3 right). +Privacy compliance: Zen leverages refCdRs with no geographical +information associated with traffic events. From such CdRs, Zen +uniquely captures and reproduces individuals’ traffic behavior in +time, which can be then associated with any modeling of individu- +als’ daily urban mobility. In Zen, mobility behavior is emulated as +realistically as possible. Such disjoint modeling hides real individ- +uals’ spatiotemporal daily-life habits in routine and leisure times +(e.g., home/work, nightlife, etc.), bringing the privacy-preserving +capability to the produced CdRs. +3 +THE TRAFFIC MODULE +We describe here the generative model used to reproduce CdRs traf- +fic behavior. Our generative model has enough expressive power +to capture inter-CdRs feature correlations while considering in- +dividual users’ behavior. In particular, we leverage an enhanced +Table 1: CdRs format. +CdR field +All +Phone number +IMEI +cell Id +Timestamp +Event-type (call/SMS/data) +Call +Call type (MO/MT/IMO/IMT) +Call duration +Phone number of the correspondent +SMS +SMS type (MO/MT/IMO/IMT) +Phone number of the correspondent +Data +Data session size +Table 2: Review of generated +CdRs completeness +Mobility +features +Traffic features +Call +SMS +data +[32] +✗ +✓ +✗ +✗ +[42] +✗ +✓ +✗ +✗ +[15] +✗ +✓ +✗ +✗ +[29] +✗ +✗ +✗ +✓ +[6] +✓ +✗ +✗ +✗ +[16] +✓ +✗ +✗ +✗ +[49] +✓ +✗ +✗ +✗ +[23] +✓ +✗ +✗ +✗ +Zen +✓ +✓ +✓ +✓ +recurrent neural network (RNN), named Long-Short-Term Memory +(LSTM), known for its ability to generate complex, realistic long- +range sequences. +Our model is trained from RefCdRs which report a set of times- +tamped events generated by several users. Each CdRs’ event (or +a line) includes the following information: start time, user id (i.e., +phone number), event-type (i.e., data, SMS, call), corresponding +user id (for calls and SMS), call duration (for calls only), and data +volume (for data only). +We organize RefCdRs by user: the set of events chronologically +generated by the user 𝑢 throughout the trace forms a sequence of +events (𝑒𝑢 +1,𝑒𝑢 +2,𝑒𝑢 +3, ...𝑒𝑢 +𝑁𝑢 ) of size 𝑁𝑢, which is the model basis. Hence, +data reproduction is done in a sequential order, i.e., from time step +1 to 𝑁𝑢. The generation of an event in a sequence is a four-stage +process, where each stage relies on the previous output. +Stage 1: at step 𝑡, we predict the next event-type 𝑒𝑢 +𝑡+1 a user will +perform, using the event-type model (cf. §3.1). +Stage 2: given the event-type, the inter-event time (IET) model gener- +ates the IET value used to deduce the starting time for the predicted +event-type 𝑒𝑢 +𝑡+1 (cf. §3.2). +Stage 3: the correspondent model predicts which of its correspon- +dent a user will interact with for the next event 𝑒𝑢 +𝑡+1 (§3.3). This +model is executed only if 𝑒𝑢 +𝑡+1 is a call or SMS, i.e., the only events +requiring correspondent interactions. +Stage 4: Finally, the metric model refers to how the events are gen- +erated: For call events, it generates its duration, while for for data +events, it produces the data volume (§3.4). Note that the temporal +information is not constant throughout the pipeline. From stages 1 +to 2, we use the temporal information of the event-type at step 𝑡 to +predict the one of the event-type at step 𝑡 + 1, then used in stage 3. +3.1 +Event-type modeling +The event-type model predicts the next event-type a user will gener- +ate from four types of events: data, local calls (uniquely outgoing), +international calls (outgoing or incoming), and local SMS (uniquely +outgoing). Local incoming calls and SMS are modeled here as they +are induced from outgoing calls and SMS during the generation. +Modeling international calls separately from local calls, rather than +having a unique "call" event-type and determining probalistically +if it is local or international, allows distinguishing different user +behaviors towards international calls. As shown in Fig. 3, some +users may not make international calls while others make them + +call +sms +inter-call +dataKouam et al. +frequently. Finally, we did not model international SMS event-type +because it is rare and not present in RefCdRs. +The event-type model. We model sequences of event-types us- +ing an LSTM. At step 𝑡, the LSTM takes as input a vector of fea- +tures 𝑥𝑡 and generates a vector of four scores, 𝑦𝑡 = (𝑦1 +𝑡 ,𝑦2 +𝑡 ,𝑦3 +𝑡 ,𝑦4 +𝑡 ). +These scores parameterize a multinomial distribution 𝑃𝑟 (�𝑒𝑢 +𝑡 |𝑦𝑡) for +the next event-type �𝑒𝑢 +𝑡+1, through a softmax function: 𝑃𝑟 (�𝑒𝑢 +𝑡 |𝑦𝑡) = +𝑒𝑥𝑝 (𝑦𝑘 +𝑡 ) +�4 +𝑘′=1 𝑒𝑥𝑝 (𝑦𝑘′ +𝑡 ) . +When training, the true previous event-types at step 𝑡 are encoded +as input for the next step. Network parameters’ training is done +according to the standard approach of minimizing the negative-log- +likelihood of the training data. We compute the gradient of this loss +with respect to our network parameters through backpropagation. +Features 𝑥𝑡. At step 𝑡, we distinguish four features for predicting +𝑒𝑢 +𝑡+1: the event-type at step 𝑡 (one-hot encoded) and its temporal +features, i.e., Day-of-Week (DOW, one-hot encoded), Hour-of-Day +(HOD, one-hot encoded), and Second-of-Day (SOD, cyclical en- +coded). A one-hot encoding represents the ith of N features using +a N-sized vector of all zeros, except for the ith element, which is +set to 1. A cyclical encoding maps a continuous inherently-cyclical +feature into two dimensions using a sine and cosine transformation. +The HOD and DOW features capture the seasonality and regularity +of mobile traffic (less activity at night and during weekends [5]). +The fine-grained encoding of time as SOD is used to capture the +very short temporal difference between consecutive events (e.g., +tens of seconds for data events). +3.2 +Inter-event time modeling +The IET model returns the possible time values between a sequence’s +events with a confidence interval. It works in two steps: first, we +use an LSTM to parameterize a multinomial distribution over a +discrete set of time bins. Then, we use statistical methods to sample +a continuous value inside a predicted time bin. In the following, +we present our considerations for discrete IET estimation, then +the detail of our LSTM network, and finally, our methodology for +sampling an IET value given an IET bin. +Discrete IET estimation. IET are divided into discrete bins,𝑏1, ..,𝑏𝐽 , +representing 𝐽 consecutive intervals of time. To determine the bin +boundaries, [24] recommends setting boundaries at evenly-spaced +quantiles of time in training data. We found that, in our case, such +a setting results in tiny intervals for the smallest values of IET due +to the IET’s heavy-tailed distribution. For instance, considering +the 4-quantiles, there are as many elements in [1𝑠 − 20𝑠[ as in +[20𝑠 − 72𝑠[. A division at the 20s could distort the model’s accuracy +while being acceptable for realistic CdRs. Thus, we chose the IET +bins empirically to make the model less complex and easier to train +without increasing the reconstruction error in mapping back to +continuous values. We, therefore, divide IET into three intervals: +[0𝑠 − 30𝑚𝑖𝑛] ]30𝑚𝑖𝑛 − 24ℎ], and > 24ℎ. +The IET LSTM model. The LSTM network takes at each step, 𝑡, +as input a feature vector, 𝑥𝑡 and generates as output a vector of +scores 𝑦𝑡, with one score for each possible IET bin. As with the +event-type model, these scores are used as logits in a softmax to get +a multinomial distribution over the time bins. To train the network +Table 3: IET distribution and parameters per bin +IET bin +Distribution +Parameters +[0𝑠 − 30𝑚𝑖𝑛] +Lognormal +𝜎 = 1.798, 𝜇 = 4.04,𝑥0 = 0.99 +]30𝑚𝑖𝑛 − 24ℎ] +Lognormal +𝜎 = 1.731, 𝜇 = 8.59,𝑥0 = 1749.08 +> 24ℎ +Exponential +𝜆 = 6.21𝑒 − 06,𝑥0 = 86401 +parameters, we minimize the negative-log-likelihood of the training +data. +Features 𝑥𝑡. At each step 𝑡, we consider as features, the temporal +information of 𝑒𝑢 +𝑡 (§3.1) as well as the predicted event-type 𝑒𝑢 +𝑡+1, +one-hot encoded. +Continuous estimation. Generating CdRs traffic requires know- +ing the precise starting time of the next event of the sequence, +which is used for further predictions. Therefore, we convert the +predicted discretized IET bins to real-values. We apply to each IET +bin the KS statistic test to estimate the distribution and related +parameters best fitting the corresponding empirical distribution in +RefCdRs. Table 3 shows the fitted distributions to sample an IET +value per bin. The model returns the median value and the confi- +dence interval of the values obtained after 𝑛 sampling (by default +𝑛 = 1). +3.3 +Correspondent modeling +The correspondent model applies only for event-types requiring +interaction with a correspondent (i.e., SMS and local or interna- +tional calls). We first define the notion of friendship degree (𝑓 𝑑), +intuitively capturing the friendship strength of a user with each of +its correspondents. Let 𝑢 be a user, with #𝑐𝑢 correspondents over +the considered period, we then call #𝑒𝑢𝑐 the number of events the +user 𝑢 had with his correspondent 𝑐. We increasingly order the +correspondents of 𝑢 according to their corresponding number of +events such that #𝑒𝑢 +1 ≤ #𝑒𝑢 +2 ≤ .. ≤ #𝑒𝑢 +𝑗 ≤ .. ≤ #𝑒𝑢 +#𝑐𝑢 . The friendship +degree of the correspondent 𝑐 of 𝑢 is the rank 𝑗 of 𝑐 in this order. +Hence, at step 𝑡, the correspondent model returns a predicted friend- +ship degree � +𝑓 𝑑 +𝑢 +𝑡 for the correspondent with whom the event 𝑒𝑢 +𝑡 is +done. +Correspondent LSTM model. The correspondent model is also a +LSTM network that takes as input at step 𝑡, a feature vector 𝑥𝑡 per +user. It generates as output the predicted friendship degree � +𝑓 𝑑 +𝑢 +𝑡 . The +network parameters training minimizes the Mean Absolute Error +(MAE) of the training data. +Features 𝑥𝑡. At step 𝑡, the features are: the temporal information +of 𝑒𝑢 +𝑡 (cf. §3.1) except the SOD, the one-hot encoded event-type 𝑒𝑢 +𝑡 , +and the number of correspondent of 𝑢, #𝑐𝑢. This later is constant +throughout a user sequence and is essential to help the model +captures that � +𝑓 𝑑 +𝑢 +𝑡 ≤ #𝑐𝑢. Accordingly, it is not encoded and is left +to its actual value. +3.4 +Metric modeling +This section presents the models used to generate the metrics (i.e., +a model per metric) associated with events generation, namely the +call duration and the data volume. + +Zen: LSTM-based generation of individual spatiotemporal cellular traffic with interactions +0 4 8 12 16 20 +Hour +0 +23 +46 +69 +92 +115 +138 +161 +184 +Frienship degree +500 +1000 +(a) +0 +500 +1000 +1500 +Call duration (seconds) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CDF +empirical +lognormal +(b) +Figure 4: (a) Avg call duration (s) per hour and friendship +degree (b) Call duration CDF for RefCdRs. +Call duration We use a statistical method to model the call dura- +tion. In fact, contrary to the previously modeled parameters, there +is no explicit features dependency or variability (and therefore, no +complexity) regarding call durations, which implies that a used RNN +could hardly train. This is confirmed in Fig. 4a, which shows the +variation of the average call duration per hour and per friendship +degree over the entire dataset. We can see that overall, call duration +does not vary much, and thus, there is no particular correlation +between these parameters. Moreover, the per-user behavior regard- +ing call duration (easily assessed through the average call duration +per user) closely depends on the number of calls each user makes +over the CdRs duration, which is opportunely already captured by +the IET model. Accordingly, the call duration model corresponds +to the estimation of the parameters of the continuous distribution +that best fits the empirical distribution of call duration, as shown +in Fig. 4b. From a statistical test, we found this distribution to be +Lognormal of parameters 𝜎 = 1.29, 𝜇 = 3.78,𝑥0 = −0.47. +Data volume. The data volume model returns a data volume value +for each data event. According to 3GPP standards, each data-typed +CdRs line corresponds to the generation of a data session by a user. +Unfortunately, as RefCdRs lack this information, we rely on the +study done in [29] to design the data volume model. To the best +of our knowledge, [29] is the only work that conducts a thorough +characterization of data volume usage per session and per user over +time extracted from real-world CdRs, as well as designs a generator +of realistic CdRs that conforms to these characterizations. +[29] profiled users’ data usage over time according to their gener- +ated amount of data (volume profile, i.e., Light, Medium, or Heavy) +and to how often they generate data sessions (frequency profile, i.e., +Occasional or Frequent). Besides, it extracted from real-world CdRs +the distributions of data session volume according to a user’s profile +and the day period (peak or off-peak hours) and the percentage of +users per profile. We use such percentages to first assign a volume +profile to each user in Zen. As the frequency profile could be incon- +sistent with the frequency of data event-type as predicted by the +event-type model, we attribute to each user, in Zen the Occasional +frequency profile. In fact, the distribution of the number of data +sessions per day and user from RefCdRs shows the majority of the +population to be of this latter profile. Finally, we sample from the +distributions found in [29] to get a data session volume. +4 +THE MOBILITY MODULE +The Zen’s mobility module produces realistic CdRs mobility traces +in three steps each covered by a sub-module. The mobility-generator +(§4.1) emulates a population urban mobility in a real-world city map, +with users displacements generated according to public sources [34] +and describing city planning and transportation information [17]. +Next, from the input city map, the topology-builder (§4.2) builds a +realistic cellular topology using cell towers’ positioning of mobile +operators deployed in the considered real-world city, gotten from +OpenCellID [33]. This topology is then used in the position-to-cellId +module (§4.3) to map the mobility traces produced by the mobility- +generator to the cell granularity. +4.1 +Mobility generator +Zen mobility-generator inherits the highly configurable capability +of the Opportunistic Network Environment (ONE) [20] simulator. +Besides, it enhances the Working Day Mobility model (WDM) [9] of +ONE into a model named En-WDM, and generates CdRs of format +. +Our motivation to use WDM as a basic building block of Zen is +twofold. First, contrary to related models [41, 44], WDM originality +comes from the combination of various mobility aspects present in +people daily life (e.g., home and workplaces, day periods). Second, +WDM closely reproduces wireless interactions (i.e., inter-contact +and contact time) distributions found in two real-world measure- +ment experiments (i.e., iMote and Dartmouth), asserting modeling +generality. +Nevertheless, WDM is limited in capturing some fine-grained real +mobility habits or fine-tuning. En-WDM tackles such limitations and +strengthens the model with additional literature’s intuitions on laws +dictating human mobility behavior, such as preferential attachment, +regular daily behavior, transportation-dependent shortest-path pref- +erences, and most importantly, uncertainty (i.e., novelty-seeking +behaviors) and heterogeneity. Next, we detail En-WDM. +WDM’s inherited functionalities. En-WDM models week work- +ing days’ movements into three activities and their transitions, i.e., +”home”, ”working”, and ”night activity”. The night activity corre- +sponds to leisure-related times spent in preferred spots of friends +groups. +Exploration profiling. Users in En-WDM emulation decide in a +probabilistic-way whether to go home or to a night activity. To +setup such probabilities, we rely on the exploration phenomenon +profiling conducted in [1] and define three mobility profiles: scouters +are more inclined to explore and discover new places to visit, rou- +tiners rarely explore and prefer to stay among their familiar and +few known places, and regulars constantly alternate between explo- +ration and routine. We then accordingly classify users given by the +ChineseDB dataset (cf. Sec 2.3) in these three profiles. Results de- +scribe a population with 20.27% of scouters, 54.75% of regulars, and +24.98% of routiners. After this classification, we assign to users in +each profile, the probabilities of “nightlife activity”: 0.8 for scouters, +0.5 for regulars, and 0.2 for routiners. +Neighborhood and popularity. Rather than considering home/office +locations’ (lat, lon) coordinates, En-WDM associates each location +coordinate to the center of a neighborhood of rectangular shape + +Kouam et al. +Table 4: Key parameters for En-WDM emulation +En-WDM Parameter Description +Value vs Default +Size of the office squared-shaped side +100 +Minimum size of a friends group for evening activities +1 +Maximum size of a friends group for evening activities +5 vs 3 +Minimum value for evening activities duration +1h vs 10s +Maximum value for evening activities duration +4h vs 2h +Probability for a user to own a car +0.19 vs 0.5 +(width, heigth) of the emulation area +(10000, 8000) +(width, heigth) of a home cluster +(250, 150) +(width, heigth) of a office cluster +(500, 300) +and configurable size. This allows to distinguish areas with high +housing density (e.g., residential areas, university campuses), areas +with high business density (business districts), and popular leisure +locations. A user is first assigned a home/office neighborhood and +then, chooses her exact home/office location randomly inside the +neighborhood. Moreover, we added the neighborhood popularity, +which represents the probability for a user to choose a given neigh- +borhood as a home/office neighborhood or, in the case of night +activity, the probability of choosing a spot for her evening activity. +Distance-based profiling. En-WDM enables the definition of cities’ +districts (hereafter, areas) to replicate the real world. Accordingly, +we associate each user to one of the three profiles representing +area displacements: profile 1 inside a single area, profile 2 among +two areas, and profile 3 in the whole map. To get the population +percentage to be considered in each profile, we profile Geolife’s +users resulting in: Profile 1 including 72% of users whose maximum +distance is less than 1/3 of the maximum observed distance 𝐷𝑚𝑎𝑥 +(≈ 2.49 × 103𝑘𝑚). Profile 2 with 19% of users with a maximum dis- +tance between 1/3 and 2/3 of 𝐷𝑚𝑎𝑥, and profile 3 including 9% of +users with a maximum distance greater than 2/3 of 𝐷𝑚𝑎𝑥. +Simple parameterization. We report here all the key configura- +tion parameters needed for En-WDM emulation. Table 4 summarizes +them. We use italic style for those we used the default value and +regular one for those we modified. We set the value of ProbOwnCar +to 0.19 based on transportation statistics in the city of Helsinki [17]. +Parameters in bold (homeRange and officeRange) are those we added +for clusters implementation. In particular, the ratio between the +worldSize, officeSize, and cluster sizes may vary depending on the +emulated city. These parameters values in Table 4 are adapted for a +emulation in the city of Helsinki. +4.2 +Topology builder +The topology-builder uses the geographical positions of base sta- +tions (BS) in the emulated area, as given by OpenCellId [33], and +performs a Voronoi tesselation. The tessellation produces a cellular +network topology with heterogeneous cell sizes close to reality, +containing each input BS. Each Voronoi cell defines the commu- +nication boundaries of an input BS. For generality and simplicity +reasons, we include all operators’ base stations given by OpenCellId +in a bigger architecture to derive the Voronoi topology. This unique +topology is assigned to all operators considered in Zen’s process. +In practice, sharing BSs between different operators is commonly +done for cost savings. +4.3 +Position-to-cellId module +The position-to-cellId module assembles the modeled users’ mo- +bility and the designed Voronoi cellular topology. For this, each +user’s geographical position given by the mobility-generator traces +is mapped to the corresponding OpenCellId’s BS identifier, i.e., +cellID, in the Voronoi topology. It outputs mobility CdRs in the +format describing users’ spatiotem- +poral daily mobility in a real city map and adapted to a real network +topology. Despite the realism given by such leveraged real-world +information, the generation of users’ mobility has a realistic and not +a real nature since no ground-truth information on users’ real-life +routine is available. This brings privacy benefits to Zen CdRs. +5 +THE SOCIAL-TIES MODULE +Zen CdRs generation lays on the social-ties module providing the +network social structure. This structure induces phone numbers +from users of the mobility CdRs and builds the network social graph +by creating per user’s phonebook, i.e., the users she can interacts +through calls or SMS. +Mobility users to phone numbers. From the number of network’s +operators and the users distribution per operator (taken as parame- +ter or induced from OpenCellId[33]), the social-ties module assigns +an operator per user and generates a phone number in the format + <5 random digits>, where MCC and MNC describe +the mobile code for country and the operator network code within +the country, respectively. +Network social graph. Reproducing the social graph of users’ in- +teractions implies answering the following three questions. The +term "correspondent" refers to a phone number in a user’s phone- +book. +(Q1) how many correspondents does each user have? To an- +swer this question, the social-ties module relies on the distribution +of correspondents per user from RefCdRs. Let 𝑢 ∈ 𝑈 be a user with +#𝑐𝑢 correspondents; we consider the non-parametric distribution +𝑃#𝑐 = 𝑃(#𝑐𝑢 = #𝑐) +∀ #𝑐 ∈ [1, 𝑀𝐴𝑋]. Thus, for each generated +user 𝑢′, its number of correspondents #𝑐𝑢′ is obtained with the +multinomial distribution of parameters 𝑃#𝑐. +We then define four disjoint categories of correspondents: interna- +tional correspondents (𝑐𝑖𝑛𝑡𝑒𝑟 ), outgoing local correspondents (𝑐𝑜𝑢𝑡), +incoming local correspondents (𝑐𝑖𝑛), and both outgoing and incom- +ing local correspondents (𝑐𝑏𝑜𝑡ℎ). Thus, ∀ 𝑢 ∈ 𝑈, #𝑐𝑢 = #𝑐𝑖𝑛𝑡𝑒𝑟,𝑢 + +#𝑐𝑜𝑢𝑡,𝑢 +#𝑐𝑖𝑛,𝑢 +#𝑐𝑏𝑜𝑡ℎ,𝑢 = (𝑥𝑖𝑛𝑡𝑒𝑟,𝑢 +𝑥𝑜𝑢𝑡,𝑢 +𝑥𝑖𝑛,𝑢 +𝑥𝑏𝑜𝑡ℎ,𝑢) ×#𝑐𝑢. +We export the average values 𝑥𝑐𝑎𝑡,𝑢 ∀ 𝑐𝑎𝑡 ∈ {𝑖𝑛𝑡𝑒𝑟, 𝑜𝑢𝑡, 𝑖𝑛, 𝑏𝑜𝑡ℎ}. +Then, we use the multinomial distribution of 𝑃 = 𝑥𝑐𝑎𝑡,𝑢 to induce +the number of correspondents, in each category, of each user. +(Q2) how do we choose these correspondents? We create user +phonebooks by implementing a variant of the configuration model +algorithm [45], which allows building a graph from given users de- +grees. We apply this algorithm by correspondents’ category so that +each user is an 𝑐𝑖𝑛 correspondent of its 𝑐𝑜𝑢𝑡 correspondents and +a 𝑐𝑏𝑜𝑡ℎ of its 𝑐𝑏𝑜𝑡ℎ correspondents. Moreover, we add a heuristic +to choose users’ correspondents based on their relationship type, +(i.e., neighbors, colleagues, or friends) extracted from the generated +mobility dataset (cf. §4) as follows. users located inside the same +home/work cluster between 1am to 4 am and 10 am to 2 pm, over the + +Zen: LSTM-based generation of individual spatiotemporal cellular traffic with interactions +whole dataset duration, are considered neighbors and colleagues, +respectively. As well, users in the same group for night activities, +when they occur, are considered friends. Hence, a user’s corre- +spondents are selected according to defined probabilities (taken as +parameters) from its list of neighbors, colleagues, friends, and other +users until we reach the fixed number of the user’s correspondents. +At last, the social-ties module outputs each user’s list of corre- +spondents organized in the categories 𝑐𝑜𝑢𝑡, 𝑐𝑏𝑜𝑡ℎ, and 𝑐𝑖𝑛𝑡𝑒𝑟 , while +𝑐𝑖𝑛 category is induced from the 𝑐𝑜𝑢𝑡 one. +(Q3) how does a user interact with all of its correspondents? +While (Q1) and (Q2) are tackled by the social-ties module, question +(Q3) is addressed through the correspondent model of the traffic +module detailed in section 3.3. +6 +THE CDR-COMBINER MODULE +Zen’s CdR-combiner module integrates outputs of previous modules +to produce realistic CdRs, as follows. +Using event-type and IET models from the traffic module, the CdR- +combiner generates timestamped sequences of events over the total +duration. Then, each sequence is associated with a correspondent +determined by the social-ties module, based on each user’s number +of correspondents per category, indicating which event-types the +user can generate. At this point, using the correspondent model, the +CdR-combiner predicts a correspondent friendship degree per user +event that is later associated with the corresponding phone number +from users’ phonebooks. +Next, we add complementary metrics to users’ events. For all +calls events, the call duration metric relates only to available cor- +respondents of users. We do not consider unavailable users’ cor- +respondents (i.e., already in an ongoing communication) at the +caller-callee association. Hence, for available correspondents, a call +duration value is sampled from the call duration model distribution. +This value is upper-bounded by the time to the closest scheduled +call. As well, for data events, the data volume metric is assigned +according to the data volume model. +Following, the CdR-combiner integrates CdRs spatial information, +i.e., corresponding users’ cell Ids at each event timestamp (resulting +from the mobility module). At last, based on users’ phone numbers, +the CdR-combiner infers CdRs traces produced by each operator. +Zen, therefore, generates a complete and realistic CdRs trace per +generated mobile operator in the format specified in Table 1. +7 +EVALUATIONS +This section confirms Zen’s validity by evaluating traffic and mo- +bility modeling separately, then their merging into CdRs. +7.1 +Traffic module +Hereafter, we evaluate the accuracy and the performance of pre- +dictions resulting from the traffic module’s stages. As there is no +similar contribution in the literature, we compare Zen’s models +to designed baseline predictors. Table 5 summarizes all compari- +son metrics and provides their distributions on the right of each +evaluation result. +7.1.1 +Experimental datasets. We train and evaluate our models +on RefCdRs after some data handling. First, we only consider events +of users subscribed to the operator network collecting RefCdRs. Then, +we filter out users having less than 3 generated events in the whole +period of 4 weeks and those with more than one event at the same +timestamp. Those manipulations result in the selection of nearly +6000 users totalizing 1,782,829 events or CdRs entries, i.e., 77.8% of +the RefCdRs’ initial size. We then use as training set the first two +weeks of the dataset, the 3rd week as validation set, and the 4th +week as the test set. Because our traffic predictions are user-based, +the non-filtered remaining users compose all the three previous +sets and only their event sequences varies according to the week +considered in each set. +7.1.2 +Models training and Hyper-parameters. We used a 2- +layer LSTM with 50 hidden units per layer for the event-type model +and 100 hidden units per layer for the two other models. To avoid +over-fitting the training dataset, we used a dropout regularization +with 𝑝 = 0.2. The LSTM losses are iteratively minimized using +mini-batch gradient descent with the Adam optimizer. Each mini- +batch contains 64 sequences of events (i.e., users). We chose event +sequences’ lengths of 302 for training, 157 for validation, 159 for test, +sampled from the distribution of the number of events generated +by users in each experimental set. Therefore, we pad all sequences +to the sequence length in each experimental set to homogenize +datasets and ease the training. We use a masking layer to tag added +values in each sequence to ignore them in the loss calculation. +Besides, we fixed a gradient clip value of 0.01 to avoid "exploding +gradients" prone to affect RNN. +7.1.3 +Event-type model. We compare our event-type model’s +predictions (cf. §3.1) to the ones of the following baselines: Uni- +form – each event-type is equally likely to occur at each time step; +Multinomial – each event-type probability is given by its empirical +count in training data; RepeatEvt – the next event-type is always +predicted to be the same as the previous one. We use the follow- +ing evaluation metrics: (NLL) Negative-log-likelihood of next-step +probabilities, and (Accuracy) next-step 1-best correct classification +rate (for this metric, the traditional Multinomial approach always +output the most frequent event-type). Results are presented in Table +5. Selecting event-type according to the Multinomial is significantly +more predictive than the Uniform, but worse than RepeatEvt. Our +Zen’s event-type model works the best. For both NLL and Accuracy, +Zen is significantly better than RepeatEvt, i.e., the most probable +event-type is not always the previous one. +7.1.4 +IET model. As before, we compare the acuteness of our +model in predicting the next IET Bin (cf. §3.2) with the correspond- +ing above-defined baselines. Table 5 shows that for both metrics, +NLL and Accuracy, the performance of Zen’s IET model is much +higher than RepeatBin (that simply repeats the previous IET Bin), +followed by the Uniform and the Multinomial baselines. Disregard- +ing the prediction approach, we compute the discretized proba- +bilities of IET Bins and map them to IET values in a continuous +domain: named Bin sampling mapping. To evaluate how efficient +Zen’s and baselines’ Bin sampling are, we compare them to the +Overall sampling mapping, both described next. +- Bin sampling: At each Bin, the IET value is obtained after averaging +𝑛 = 500 samplings of the corresponding continuous IET distri- +bution (see §3.2). We apply this approach to all the previously + +Kouam et al. +Table 5: Traffic LSTM models evaluation results. +Event-type model +Predictor type +NLL +Accur. +Uniform +0.27 +2.91% +Multinomial +0.21 +38.97 +RepeatEvt +N/A +43.27 +Zen +0.037 +91.82 +IET model +Predictor type +NLL +Accur. MAE +MAE +MAE +MAE +]0, 30 min] +(82.8%) +]30min, 24h] +(15.45%) +>24h +(1.75%) +Uniform +0.215 +64.56 +1097 +1033 +1120 +2319 +Multinomial +0.165 +64.56 +231 +68 +334 +2877 +RepeatBin +N/A +58.05 +239 +73 +347 +2871 +Lognormal +N/A +N/A +249 +78 +361 +2973 +Zen +0.118 +69.25 +185 +16 +295 +2904 +Correspondent model +Predictor type +MAE (time-based) +MAE (user-based) +All +[1,6] +]6,21] +>21 +All +[1,6] +(50%) +]6,21] +(30%) +>21 +(20%) +Ranged- +Uniform +25.12 +1.17 +5.04 +29.57 +26.38 +1.08 +4.53 +31.17 +Ranged- +Multinomial +15.78 +0.91 +3.87 +18.42 +17.28 +0.81 +3.41 +20.38 +Zen +11.81 +0.65 +3.02 +13.77 +13.23 +0.63 +2.57 +15.68 +Bin-based models, i.e., Zen’s IET model, Uniform, Multinomial, and +RepeatBin predictors. +- Overall sampling: We perform a fitting of the empirical IET distri- +bution (i.e., with no bins) and obtain a Lognormal distribution with +𝜎 = 2.67, 𝜇 = 4.97,𝑥0 = 1. Then, we straightly predict continuous +values by sampling the resulted fitted IET distribution. We name +this prediction Lognormal. +The Mean Absolute Error (MAE) of the IETs in minutes is used as +the comparison metric. It estimates the average distance between +actual and predicted IET. From Table 5, we can notice that the +Bin-sampling of Multinomial and RepeatBin have comparable MAE +performances, followed by the Overall-sampling Lognormal pre- +dictor. This behavior is also verified per Bin (three last columns). +Overall, Zen works the best. In the first bin ]0, 30𝑚𝑖𝑛], which is +the most sensitive, we note that except for the Zen, all models on +average predict an IET value outside the initial interval. +7.1.5 +Correspondent model. At last, we evaluate the correspon- +dent model (cf. §3.3) by comparing its predictions to the following +baselines: +- RangedUniform: Per user 𝑢, correspondents 𝑐𝑖, ∀𝑖 = 1, 2, ..., #𝑐𝑢 +are equally likely to be predicted at each sequence step. +- RangedMultinomial: Per user u, each correspondent 𝑐𝑖 is chosen +with a probability (𝑝𝑢 +𝑖 , 1 ≤ 𝑖 ≤ #𝑐𝑢) extracted from the procedure +as follows: +Let 𝑈 be the set of users and 𝑢 a user in 𝑈 . We recall that #𝑒𝑢𝑐𝑖 +refers to the number of events 𝑢 made with his correspondent 𝑐𝑖. +From this definition, we derive 𝑃𝑢𝑐𝑖 the proportion of events made +by 𝑢 with its correspondent 𝑐𝑖 : 𝑃𝑢𝑐𝑖 = #𝑒𝑢𝑐𝑖 /� +𝑖 #𝑒𝑢𝑐𝑖 . +For all 𝑖 = 1, 2, ..., 𝑀𝐴𝑋 (#𝑐𝑢) we extract the mean values 𝑃𝑐𝑖 = +𝑃𝑢𝑐𝑖 ∀𝑢 ∈ 𝑈 . Hence, for a user 𝑢, the probabilities (𝑝𝑢 +𝑖 ,𝑖 = 1, 2, ..#𝑐𝑢) +is obtained by normalizing the first #𝑐𝑢 mean values (𝑃𝑐𝑖,𝑖 = +1, 2, ..#𝑐𝑢) such that � +𝑖 𝑝𝑢 +𝑖 = 1. +The evaluation metric is the MAE of the predictions � +𝑓 𝑑𝑡 in the +test dataset. We found that as we train the correspondent model +with chronologically-separated experimental windows (defined in +§7.1.1), the MAE loss value continually increases in the validation +dataset. This is due to the fact that in the training period (i.e., first +two weeks), users only interact with some of their correspondents, +making it difficult for the model to generalize. To fix this issue, +we instead split training, validation, and test datasets by selecting +users traffic over the whole dataset period (4 weeks). The training +dataset includes 60% of the users, while the validation and test +datasets each represent 20%. Results in Table 5 show the Ranged- +Multinomial predictor has significantly better results compared to +the RangedUniform predictor. +Overall, Zen is the modeling that best performs, showing its +ability to capture users interaction with their correspondents. In +particular, the detailed distribution plots show Zen presents for 80% +of users (i) more than 95% and 75% of accuracy for respectively, the +event-type and IET models, and (ii) less than 6.68% and 12.5% of +MAE maximum values for respectively, the IET and correspondent +models. + +1.0 +0.8 +Uniform +0.6 +CDF +Multinomial +RepeatEvt +0.4 +Zen +0.2 +0.0 +0.00 +0.25 +0.50 +0.75 +1.00 +accuracy1.0 +0.8 +0.6 +CDF +Uniform +0.4 +Multinomial +RepeatBin +0.2 +Lognormal +Zen +0.0 +0 +50 +100 +150 +MAE (hour)1.0 +0.8 +0.6 +CDF +0.4 +Uniform +Multinomial +0.2 +RepeatBin +Zen +0.0 +0.00 +0.25 +0.50 +0.75 +1.00 +accuracyRangedUniform +RangedMultinomial +ZenRangedUniform +RangedMultinomial +ZenZen: LSTM-based generation of individual spatiotemporal cellular traffic with interactions +7.2 +Mobility module +We validate our En-WDM mobility model by comparing it to its +original version, the WDM [9]. We rely on WDM results closely +following real-world measurement datasets distributions (i.e., iMote +or Dartmouth). Since En-WDM adds new functionalities in modeling +mobility to WDM, we are not looking for identical results from +both models but for similarities in terms of distributions and curve +behaviors. +Fig. 5 shows well-known metrics for characterization of wire- +less networking meetings (inter-contact and contact time) and the +tendencies in human mobility, i.e., confinement (radius of gyra- +tion) and repetitiveness (probability to return to previously visited +places). As for WDM, we emulate a scenario with 1000 and 6000 +users, moving in the Helsinki city center area with roughly 7×8.5km +for 5.105𝑠 and with the same arrangement of home/work and POIs. +We use the same representation of results for comparison reasons. +We can see that En-WDM’s inter-contact time distribution (cf. +Fig. 5a) and the normalized number of contacts (Fig. 5b) closely +follows the ones of WDM, attesting the realistic modeling of such +metric at population scale and the capability of reproducing het- +erogeneity to mobility decisions. +At last, we evaluate the capability of the two models in reproducing +seminal literature analytical human mobility laws [1, 11, 30]. The +radius of gyration (Fig. 5c) estimates the area size mostly covered +by daily displacements of a user. In En-WDM, the radius of gyration +is globally smaller due to routiners and regulars (79.73% of the pop- +ulation) who have more confined displacements, consistent with +real-life mobility behavior [1]. Moreover, the average return prob- +ability (Fig. 5d) and per-cell repetitiveness ((Fig. 5e) results show +that users have a regular and periodical spatial mobility behavior +with a higher probability of returning to a previous small set of +visited locations, as shown in [11, 30]. +7.3 +Zen CdRs use cases +We evaluate the complete CdRs resulting from Zen framework as +compared to RefCdRs when applied to three use cases. As RefCdRs +lack ground-truth in mobility information, we enrich them with +Zen CdRs’ emulated user trajectories using Zen’s CdR-combiner +methodology (ref. §6), we name it M-RefCdRs. Based on the con- +firmed Zen performance in reproducing human mobility laws, we +focus our use-cases analysis on the reproduction of cellular traffic +behavior for which we have a ground-truth. We generate Zen CdRs +with 6000 users, corresponding to the same number of users in +RefCdRs (see §7.1.1) and consider a week-long period. +Dynamic urban tracking. Real-time population density tracking +is a key functionality to support adaptive urban and transport plan- +ning. As shown in [21], such density at time 𝑡 can be derived from +the corresponding network activity load at 𝑡 computed as the mean +number of network events (e.g., here ongoing calls, exchanged +SMS, and established data sessions) per individual. Following this +methodology, Fig. 6 shows the spatial distribution (values in the +color bar) of people presence in network cells of an Helsinki area +(2.2km × 3.6km), at four representative time hours of individuals’ +routine, obtained with M-RefCdRs and Zen CdRs. As in M-RefCdRs, +we see that people presence at the office period (8h-12h) is concen- +trated in specific zones corresponding to defined Helsinki business +neighborhoods. In contrast, the after-work period (18h-22h) in- +cludes displacements times and night activities not made at specific +spots (e.g., groups of users can walk down the streets for their +night activity), explaining people presence is spread over a broader +zone. Besides, we notice that people presence is captured equiva- +lently in M-RefCdRs and Zen’s CdRs, especially in working period +(8h-12h). We believe the resulting few dissimilarities, particularly +for the after-work period (18h-22h), are mainly due to the non- +deterministic association of user’s traffic to trajectories in Zen’s +CdR-combiner. +Data-Driven Micro BS Sleeping. Numerous works studied power +savings in Radio Access Networks (RAN). Inspired by [47], we +investigate how a traffic-aware Base Station (BS) on/off-switching +strategy [43] performs when informed with Zen CdRs compared to +M-RefCdRs. We assume an heterogeneous RAN deployment where +each cell is served by a separate micro BS, whereas macro BSs +provide umbrella coverage to a larger area. Specifically, we consider +a grid tessellation of 5X5 macro BSs in the considered zone. The +power needed to the operation of a BS at time 𝑡 is 𝑃(𝑡) = 𝑁𝑡𝑟𝑥 (𝑃0 + +Δ𝑝𝑃𝑚𝑎𝑥𝜌(𝑡)), 0 ≤ 𝜌(𝑡) ≤ 1, where 𝜌(𝑡) is the relative traffic load at +time 𝑡 with 𝑃0, 𝑁𝑡𝑟𝑥, 𝑃𝑚𝑎𝑥 and Δ𝑝 being constants defined for micro +and macro BSs in [47]. Then, if 𝜌(𝑡) ≤ 𝜌𝑚𝑖𝑛 = 0.37 as considered in +[8] the micro BS offloads its local traffic to the macro BS and goes +into sleep mode, where it consumes negligible power. Accordingly, +Fig. 7 shows the power consumption (𝑃(𝑡) values in the color bar) +of each cell’s micro BS at two hours in Helsinki (a zoomed-in area +of 2.2km × 1.6km) with and without such a strategy implemented. +We can see that comparable cells are kept on, while the strategy +brings similar energy savings. +Anomaly detection. Beyond global population-related applica- +tions, the fine-grained state of Zen CdRs allows for the investigation +of per-user spatiotemporal behavior for cellular anomaly detection. +Such anomalies can be unusual events possibly generated by some +security incidents (e.g., stolen account, malware device infection) +[10] or users with a fraudulent behavior profile. As an instance of +the latter, SIMBox fraud is a prevalent scam in telecommunication +networks consisting of "fake" user accounts re-injecting diverted in- +ternational calls as local calls to a country [22]. We assess the utility +of Zen CdRs for investigating such fraud by applying a user profil- +ing method where traffic or mobility users’ behaviors are leveraged +to classify a user as fraudulent or not. To this end, we apply for both +Zen and real ones, a DBSCAN clustering to a set of per-user traffic- +related features specific to detect SIMBox fraudulent behavior as +described in Table 1 of [40]. Results show a similarity between Zen +CdRs and real-world ones: while M-RefCdRs’ estimated number +of clusters and outliers are 10 and 1241, Zen CdRs’ confidence in- +tervals for these metrics are 9.1 ± 1.66 and 1122.3 ± 35.02 for 10 +samples of Zen CdRs’ call duration feature (ref. §3.4). +8 +RELATED LITERATURE +CdRs’ inaccessibility has pushed researchers to generate their own +synthetically, commonly using features modeling. This leads to +either mobility- or traffic-specific CdRs, often with grouped-based +analysis of individuals’ behavior. Zen tackles such lacks by empow- +ering the scientific community with the autonomy needed for the +generation of realistic, complete, precise, and flexible CdRs. + +Kouam et al. +10 +2 +10 +3 +10 +4 +10 +5 +ICT (s) +10 +4 +10 +3 +10 +2 +10 +1 +10 +0 +P [X > x] +WDM 1K +WDM 6K +En +WDM 1K +En +WDM 6K +(a) +40 +60 +80 +Time [h] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized Nb. contacts (%) +(b) +10 +0 +2 × 10 +0 +3 × 10 +0 +4 × 10 +0 +6 × 10 +0 +Radius of gyration (km) +0.0 +0.5 +1.0 +1.5 +PDF +(c) +0 +24 +48 +72 +96 +120 +144 +168 +Time [h] +0.0 +0.2 +0.4 +0.6 +Return probability +En +WDM 6K +(d) +0 +20 +40 +60 +80 +100 +Repetitiveness per cell +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +CDF +En +WDM 6k +(e) +Figure 5: Mobility metrics compared with initial WDM: (a) Inter-contact time CCDF (b) Normalized number of contacts per +hour (c) Radius of gyration CDF (d) Return probability (e) Per-cell repetitiveness CDF +. +Figure 6: Dynamic people presence estimated at four daily +time in Helsinki for real-world and Zen traffic. +(a) Always-active micro BS. +(b) Cell-sleeping strategy. +Figure 7: Power consumption per cell (a) for always-active +micro BS and (b) with a cell-sleeping strategy. +Traffic-related: Instead of aggregated network traffic generation +as done in [25, 46, 47], we focus here on per-individual CdRs gen- +eration that tackles different challenges. In this domain, literature’s +synthetic CdRs lack completeness in describing both call [15, 32, 42] +and mobile data [29] usages, and to the best of our knowledge, +pay no attention to SMS usage. Murtić et al. [32] used Social Net- +work Analysis to reproduce call behaviors’ features (i.e., temporal +likelihood of calls and call duration distribution) per user profile, +extracted from real-world CdRs. Nevertheless, the work did not +include any validation. In the same vein, Songailait˙e et al. [42] sta- +tistically model key parameters from real CdRs to produce realistic +CdRs. Calling behaviors is simulated based on the empirical fitting +of call duration, call count, call likelihood per hour, and weekdays +similarity in behaviors. However, simulation relies on a simplis- +tic and randomly-built network social structure leveraging static +parameters such as the maximum number of friends and acquain- +tances. Using a GAN generative model, Hughes et al. [15] show +the deep learning models’ capability to learn inherent and complex +distributions from real CdRs. Unfortunately, real and generated +CdRs included only two features: the starting call hour and du- +ration in minutes, revealing a limited extent of modeled features +compared to complete CdRs. Finally, Oliveira et al. [29] focused on +the data-traffic profiling, modeling, and generation from real CdRs. +Their model allows generating data usage’s timestamped records +per profiled user. Although providing flexible settings for profiles’ +granularity, this work also has the drawback of modeling only data +traffic features, lacking thus real-world CdRs’ completeness. +Mobility-related: Synthetically generated mobility traces are regu- +lar in literature and frequently extracted from models implemented +in ONE [20], BonnMotion [2], or SUMO [26] realistic simulators. +Several works on mobility modeling actually focus on the gener- +ation of synthetic traces that capture specific features in human +mobility that are often domain-specific: e.g., MANETS and DTNs +(e.g., inter-contact and contact time) [31, 41], Disaster Manage- +ment [3, 36] or Sociology [4]. Still, a few works such as [9, 12, 19, 37] +aim to model real-life mobility and propose more complex models, +valuable for more applications. This paper leverages the [9]’s origi- +nality in combining various mobility aspects and realistically mod- +eling them. Other strategies rely on recurrent neural networks [23] +or statistical generative models based on real mobility traces such +as Markov models [6], spatiotemporal empirical distributions [16] +or travel demand [49]. Yet, only a few works [13, 27] address the +privacy issues of generated mobility traces, which is however cru- +cial. +9 +CONCLUSION AND DISCUSSION +This paper presented Zen, the first framework allowing the au- +tonomous generation of complete and realistic CdRs in an individual +basis. To this end, we relied on a fully anonymized and incomplete +(only traffic-related) CdRs datasets and provide the first literature +modeling that captures long-range and inter-CdRs traffic features + +8h +12h +18h +22h +1.0 +60.170 +60.165 +Real +tituc +0.8 +60.160 +a +60.155 +0.6 +60.170 +0.4 +60.165 +titude +en +0.2 +N +60.160 +60.155 +24.92 24.94 24.96 +24.92 24.94 24.96 +24.92 24.94 24.96 +24.92 24.94 24.96 +Longitude +Longitude +Longitude +Longitude12h +22h +60.170 +60.165 +Latitude +keal +60.160 +60.155 +60.150 +60.170 +60.165 +Latitude +Len +60.160 +N +60.155 +60.150 +24.94 +24.95 +24.96 +24.97 +24.95 +24.96 +24.97 +Longitude +Longitude12h +22h +60.170 +140 +60.165 +ititude +120 +60.160 +Lat +-100 +60.155 +60.150 +80 +60.170 + 60 +savr +83.67 +60.165 +itude +40 +60.160 +Lati +20 +60.155 +60.150 +24.94 +24.95 +24.96 +24.97 +24.95 +24.96 +24.97 +Longitude +LongitudeZen: LSTM-based generation of individual spatiotemporal cellular traffic with interactions +correlation, individuals heterogeneity and social-ties in communi- +cation. The disjoint modeling of realistic emulated mobility and +captured real-world traffic behaviors hides real individuals’ daily- +life habits in routine and leisure times (e.g., home/work, nightlife, +etc.), bringing the privacy-preserving capability to the produced +Zen CdRs. Finally, we validate Zen Cdrs (i) realisticness in reproduc- +ing daily cellular behaviors of urban population and (ii) usefulness +in practical networking applications such as dynamic population +tracing, Radio Access Network’s power savings, and anomaly de- +tection as compared to real-world Cdrs. Next, we provide extra +discussions on possible alternatives and improvements. +Flexibility and generalization: All the contextual building blocks +feeding the Zen mobility modeling (e.g., Census information, bus +schedule, real city map, neighborhood popularity, etc.) bring gener- +ality and flexibility to the representation of city urban life, yielding +individuals’ cyclic behavior. On the other hand, though Zen provides +realistic traffic behavior models trained from a unique real-world +traffic CdRs, the modeling methodology of this paper is general and +can be applied to other CdRs with different cultural traffic habits. +Alternative modeling approaches: LSTMs are perhaps the sim- +plest network (in terms of manual tuning) that can reliably model +long-term dependencies and has the flexibility to be used jointly +with other more complex architectures. For example, a GAN [14, 47] +uses paired generator/discriminator networks to enable very realis- +tic output; our work provides the networks that can be used inside +the GAN. +Future improvements: As mentioned, Zen extensively enhances +the original WDM model. Nevertheless, as with any research con- +tribution, the mobility generation of Zen is still open for improve- +ments, such as the addition of complementary features in fine- +grained human mobility laws, cities’ contextual information (e.g., +friendship, popular leisure zones in the city, etc.), representing +weekend mobility and behaviors induced by teleworking or mi- +nor users profiles (e.g., unemployed), or modeling from real-world +mobility CdRs. +Privacy vs individual precision: Although presenting ground- +truth modeling and validation opportunities, individual-based mo- +bility modeling of real-world CdRs brings important privacy issues: +As users’ actual habits in mobility are captured in the model, the +generated CdRs have the weakness of revealing aspects in users’ +daily-life routine, such as important locations (e.g., home/work), reg- +ular trajectories (e.g., preferred places for leisure, etc). 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Procedia Computer Science +32 (12 2014), 802–807. https://doi.org/10.1016/j.procs.2014.05.494 + diff --git a/edA0T4oBgHgl3EQfHP-i/content/tmp_files/load_file.txt b/edA0T4oBgHgl3EQfHP-i/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..066543a5a96f3fb6a3b46af6156c78478bd0f17f --- /dev/null +++ b/edA0T4oBgHgl3EQfHP-i/content/tmp_files/load_file.txt @@ -0,0 +1,1338 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf,len=1337 +page_content='Zen: LSTM-based generation of individual spatiotemporal cellular traffic with interactions Anne Josiane Kouam Inria France Aline Carneiro Viana Inria France Alain Tchana Grenoble INP France ABSTRACT Domain-wide recognized by their high value in human presence and activity studies, cellular network datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=', Charging Data Records, named CdRs), however, present accessibility, usability, and privacy issues, restricting their exploitation and research repro- ducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' This paper tackles such challenges by modeling Cdrs that fulfill real-world data attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Our designed framework, named Zen follows a four-fold methodology related to (i) the LTSM-based modeling of users’ traffic behavior, (ii) the realistic and flexible emulation of spatiotemporal mobility behavior, (iii) the structure of lifelike cellular network infrastructure and social interactions, and (iv) the combination of the three previous modules into realis- tic Cdrs traces with an individual basis, realistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Results show that Zen’s first and third models accurately capture individual and global distributions of a fully anonymized real-world Cdrs dataset, while the second model is consistent with the literature’s revealed features in human mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Finally, we validate Zen Cdrs ability of reproducing daily cellular behaviors of the urban population and its usefulness in practical networking applications such as dynamic population tracing, Radio Access Network’s power savings, and anomaly detection as compared to real-world CdRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' CCS CONCEPTS Data and Communication Traffic → Charging Data Records;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Cellular Traffic → Mobility and Network events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' • Modeling → LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' KEYWORDS Human mobility modeling, Data and Communication traffic model- ing, Recurrent Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' 1 INTRODUCTION Charging Data Records are acknowledged as a common tool for studying human mobility, infrastructure usage, and traffic behav- ior [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' We name such datasets as CdRs to distinguish them from the standard Call Detail Records (CDRs), describing only call and SMS cellular communication information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' CdRs describe time-stamped and geo-referenced event types (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=', data, calls, SMS) generated by each mobile device interacting with operator networks (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Ta- ble 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' They comprise city-, region-, or country-wide areas and usually cover long periods (months or years);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' no other technology currently provides an equivalent per-device precise scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' As a result, CdRs are exploited in different research domains and indus- tries, such as sociology [39], epidemiology [7], transportation [38], and networking [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' For a quantitative appreciation of such CdRs’ worth recognition, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' 1 identifies 14 different research domains ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='leveraging CdRs among 100 most relevant works (sorted by Google ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Mobility research ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Transportations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Big Data engineering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Tourism ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Urban planning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Billing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Fraud and Anomaly detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Crime investigation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Networking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Traffic analysis and research ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Health research ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Social analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Economics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Mobile security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='2017 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='2018 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='2019 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='2020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='Figure 1: Distribution by domain of the last 5-year most rel- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='evant publications using CdRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Scholar) selected from 1022 last 5-year publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' This clearly shows a great diversity of domains in this sample only (∼ 10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Yet, the exploitation of real-world CdRs for research faces many limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' First, accessibility: CdRs datasets are not publicly avail- able, imposing strict mobile operators’ agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Second, usabil- ity: CdRs are usually available in an aggregated form (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=', grouped mobility flows and coarse spatiotemporal information), limiting related analyses’ preciseness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Third, privacy: even anonymized, non- aggregated CdRs describe sensitive information of users’ habits, which hardens their shareability [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' This paper addresses such limitations by enabling the autonomous generation of realistic and privacy-compliant CdRs by scientific community, thus providing new avenues for research advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Moreover, generated CdRs should conform to essential attributes, namely, completeness, realisticness, fine-grained description, and pri- vacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Unfortunately, those attributes make the generation of realis- tic CdRs challenging and complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' In particular, achieving complete- ness requires (i) either real-world complete CdRs datasets (hard to obtain) describing mobility, traffic, and pairwise users commu- nications or (ii) to cope with the difficulty in modeling the intrin- sic correlations between information describing users’ behaviors in space, time, and social communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Achieving realisticness implies considering real-world cellular network complexities (ar- chitecture and topology) at all levels of the generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' The fine-grained description achievement is impeded by the het- erogeneity of users’ behaviors, especially in cellular traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Finally, generated traces should be privacy-compliant to avoid backtrack- ing real users’ identities, most often done through their mobility information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' To the best of our knowledge, this is the first work in literature producing realistic Charging Data Records (CdRs) that fulfill the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='02059v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='NI] 5 Jan 2023 Kouam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' above-mentioned attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Our designed framework, named Zen employs a four-fold methodology: (1) Leveraging on a real-world fully anonymized CdRs describ- ing users’ traffic behavior (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='e, events information on its type – data, call, and SMS –, duration, pairwise information, etc), we propose the first literature modeling that captures long-range and inter-CdRs specificity correlations while addressing the population hetero- geneity (§3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Our model captures population diverseness in the reproduction of individual traffic behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' We use three separate Long-Short-Term Memory neural networks (LSTM) to model event types generation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=', what), the inter-event duration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=', when), the social interactions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=', whom), and leverage statistical analysis to model CdRs metrics such as calls duration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=', how).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Overall, Zen traffic modeling presents significant high performance values, providing for 80% of users (i) more than 95% (for event-type) and 75% (for inter-event) of modeling accuracy, and (ii) less than 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='68% (for inter-event) and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='5% (for social) of Mean Absolute Error’s maximum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' (2) Mobility behaviors of individuals (§4) are emulated according to the infrastructure of a real-world metropolitan city (here, the Helsinki EU city), and resulting trajectories are coupled with the cor- responding cell towers distribution of existing operator networks in the same city [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Here, we leverage city planning, transportation information [17] as well literature investigations on laws dictating human mobility [1, 11, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Such real-world information and realis- tic human mobility modeling are then incorporated in the literature Working Day Mobility (WDM) model [9] – extensively enhancing it – to emulate urban daily-life mobility behaviors of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Moreover, we rely on the ONE simulator [20] to bring flexibility to our model regarding population size, duration, and covered area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' (3) We then design a separate module (§5) to realistically re- produce on top of generated mobility traces, a cellular network organization with multiple operators and build social ties between users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' This enables the first-of-a-kind flexibility to produce CdRs of numerous operators at the same period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' (4) We combine all the previous models to generate complete CdRs describing individual mobility, traffic, pairwise communica- tions following real traffic behavior (§6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Note that the disjoint behaviors modeling of realistic emulated mobility and of real-world traffic hides real individuals’ spatiotem- poral daily-life habits in routine and leisure times (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=', home/work, nightlife, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' ), bringing the privacy-preserving capability to the produced CdRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' 2 ZEN OVERVIEW In the following, we provide an overview of Zen architecture and describe the different real-world datasets we leverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content='1 Architecture According to input parameters, we generate realistic CdRs (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Table 1) through four phases, each implemented in a module of the Zen architecture (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' Zen architecture consists of (1) a traffic module, (2) a mobility module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' (3) a social-ties module, and (4) a CdR-combiner, or merger module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' The traffic module (§3) leverages Long-Short-Term Memory neural networks (LSTM) jointly with statistical analysis to model users’ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edA0T4oBgHgl3EQfHP-i/content/2301.02059v1.pdf'} +page_content=' CDR-combiner Zen CdRs : β. Especially, we will use P instead of P(0). + +4 +Kaliyappan Vijaya, Gangadharan Murugusundaramoorthy and Hatun ¨Ozlem G¨uney +Theorem 1.2. ([6]) The function ˜p(z) = +1+τ 2z2 +1−τz−τ 2z2 belongs to the class P(β) with β = +√ +5/10 ≈ 0.2236. +Now we recall the following lemma which will be relevant for our study . +Lemma 1.3. ([11]) Let p ∈ P with p(z) = 1 + c1z + c2z2 + · · · , then +|cn| ≤ 2 +for +n ≥ 1. +(6) +In this present work, we introduce a new subclass of Σ associated with shell-like func- +tions connected with Fibonacci numbers and obtain the initial Taylor coefficients |a2| and +|a3| for this function class. Also, we give bounds for the Fekete-Szeg¨o functional |a3 − µa2 +2| +for this class. +2 +Bi-Univalent function class PSLλ +Σ(˜p(z)) +In this section, we introduce a new subclass of Σ associated with shell-like functions +connected with Fibonacci numbers and obtain the initial Taylor coefficients |a2| and |a3| +for the function class by subordination. +Firstly, let p(z) = 1+p1z+p2z2+· · · , and p ≺ ˜p. Then there exists an analytic function +u such that |u(z)| < 1 in U and p(z) = ˜p(u(z)). Therefore, the function +h(z) = 1 + u(z) +1 − u(z) = 1 + c1z + c2z2 + . . . +(1) +is in the class P(0). It follows that +u(z) = c1z +2 + +� +c2 − c2 +1 +2 +� z2 +2 + +� +c3 − c1c2 + c3 +1 +4 +� z3 +2 + · · · +(2) +and +˜p(u(z)) += +1 + ˜p1c1z +2 ++ +�1 +2 +� +c2 − c2 +1 +2 +� +˜p1 + c2 +1 +4 ˜p2 +� +z2 ++ +�1 +2 +� +c3 − c1c2 + c3 +1 +4 +� +˜p1 + 1 +2c1 +� +c2 − c2 +1 +2 +� +˜p2 + c3 +1 +8 ˜p3 +� +z3 + · · · . +(3) +And similarly, there exists an analytic function v such that |v(w)| < 1 in U and p(w) = +˜p(v(w)). Therefore, the function +k(w) = 1 + v(w) +1 − v(w) = 1 + d1w + d2w2 + . . . +(4) +is in the class P(0). It follows that +v(w) = d1w +2 ++ +� +d2 − d2 +1 +2 +� w2 +2 + +� +d3 − d1d2 + d3 +1 +4 +� w3 +2 + · · · +(5) + +On λ− Pseudo bi-starlike functions related with Fibonacci numbers +5 +and +˜p(v(w)) += +1 + ˜p1d1w +2 ++ +�1 +2 +� +d2 − d2 +1 +2 +� +˜p1 + d2 +1 +4 ˜p2 +� +w2 ++ +�1 +2 +� +d3 − d1d2 + d3 +1 +4 +� +˜p1 + 1 +2d1 +� +d2 − d2 +1 +2 +� +˜p2 + d3 +1 +8 ˜p3 +� +w3 + · · · . +(6) +The class Lλ(α) of λ-pseudo-starlike functions of order α was introduced and investi- +gated by Babalola [1]. A function f ∈ A is in the class Lλ(α) if it satisfies +ℜ +�z(f ′(z))λ +f(z) +� +> α, +(0 ≤ α < 1) +where λ ≥ 1, λ ∈ R and z ∈ U. In [1] it was showed that all pseudo-starlike functions are +Bazileviˇc functions of type (1 − 1/λ) and of order α1/λ and univalent in open unit disk U. +Recently Joshi et al. [5] defined the bi-pseudo-starlike functions class and obtained +the bounds for the initial coefficients |a2| and |a3|. In this paper we define a new class +PSLλ +Σ(˜p(z)),λ-bi-pseudo-starlike functions of Σ related to shell-like curves connected with +Fibonacci numbers and determine the bounds for the initial Taylor-Maclaurin coefficients +of |a2| and |a3|. Further we consider the Fekete-Szeg¨o problem in this class and the special +cases are stated as corollaries which are new and have not been studied so far. +Definition 2.1. For λ ≥ 1 and λ ∈ R, a function f ∈ Σ of the form (1) is said to be in the +class PSLλ +Σ(˜p(z)) if the following subordination hold: +z(f ′(z))λ +f(z) +≺ ˜p(z) = +1 + τ 2z2 +1 − τz − τ 2z2 +(7) +and +w(g′(w))λ +g(w) +≺ ˜p(w) = +1 + τ 2w2 +1 − τw − τ 2w2 +(8) +where τ = (1 − +√ +5)/2 ≈ −0.618 where z, w ∈ U and g is given by (2). +Specializing the parameter λ = 1 and λ = 2, we have the following remarks, respec- +tively: +Remark 2.2. [8] For λ = 1 a function f ∈ Σ is in the class PSL1 +Σ(˜p(z)) ≡ SLΣ(˜p(z)) if +the following conditions are satisfied: +zf ′(z) +f(z) ≺ ˜p(z) = +1 + τ 2z2 +1 − τz − τ 2z2 +(9) +and +wg′(w) +g(w) +≺ ˜p(w) = +1 + τ 2w2 +1 − τw − τ 2w2 +(10) +where τ = (1 − +√ +5)/2 ≈ −0.618 where z, w ∈ U and g is given by (2). + +6 +Kaliyappan Vijaya, Gangadharan Murugusundaramoorthy and Hatun ¨Ozlem G¨uney +Remark 2.3. For λ = 2 a function f ∈ Σ is in the class PSL2 +Σ(˜p(z)) ≡ GSLΣ(˜p(z)) if the +following conditions are satisfied: +� +f ′(z)zf ′(z) +f(z) +� +≺ ˜p(z) = +1 + τ 2z2 +1 − τz − τ 2z2 +(11) +and +� +g′(w)wg′(w) +g(w) +� +≺ ˜p(w) = +1 + τ 2w2 +1 − τw − τ 2w2 +(12) +where τ = (1 − +√ +5)/2 ≈ −0.618 where z, w ∈ U and g is given by (2). +In the following theorem we determine the initial Taylor coefficients |a2| and |a3| for the +function class PSLλ +Σ(˜p(z)). Later we will reduce these bounds to other classes for special +cases. +Theorem 2.4. Let f given by (1) be in the class PSLλ +Σ(˜p(z)), then +|a2| ≤ +|τ| +� +(2λ − 1)2 − (10λ2 − 11λ + 3)τ +(13) +and +|a3| ≤ +|τ| [(2λ − 1)2 − 2(5λ2 − 4λ + 1)τ] +(3λ − 1) [(2λ − 1)2 − (10λ2 − 11λ + 3)τ], +(14) +where λ ≥ 1. +Proof. Let f ∈ PSLλ +Σ(˜p(z)) and g = f −1. Considering (7) and (8), we have +z(f ′(z))λ +f(z) += ˜p(u(z)) +(15) +and +w(g′(w))λ +g(w) += ˜p(v(w)), +(16) +where τ = (1 − +√ +5)/2 ≈ −0.618 where z, w ∈ U and g is given by (2). Since +z(f ′(z))λ +f(z) += 1 + (2λ − 1)a2z + [(3λ − 1)a3 + +� +2λ2 − 4λ + 1 +� +a2 +2]z2 + . . . +and +w(g′(w))λ +g(w) += 1 − (2λ − 1)a2w + [ +� +2λ2 + 2λ − 1 +� +a2 +2 − (3λ − 1)a3]w2 + . . . . +Thus we have +1 + (2λ − 1)a2z + [(3λ − 1)a3 + +� +2λ2 − 4λ + 1 +� +a2 +2]z2 + . . . +(17) += +1 + ˜p1c1 +2 z + +�1 +2 +� +c2 − c2 +1 +2 +� +˜p1 + c2 +1 +4 ˜p2 +� +z2 ++ +�1 +2 +� +c3 − c1c2 + c3 +1 +4 +� +˜p1 + 1 +2c1 +� +c2 − c2 +1 +2 +� +˜p2 + c3 +1 +8 ˜p3 +� +z3 + · · · . +(18) + +On λ− Pseudo bi-starlike functions related with Fibonacci numbers +7 +and +1 − (2λ − 1)a2w + [ +� +2λ2 + 2λ − 1 +� +a2 +2 − (3λ − 1)a3]w2 + . . . , += +1 + ˜p1d1w +2 ++ +�1 +2 +� +d2 − d2 +1 +2 +� +˜p1 + d2 +1 +4 ˜p2 +� +w2 ++ +�1 +2 +� +d3 − d1d2 + d3 +1 +4 +� +˜p1 + 1 +2d1 +� +d2 − d2 +1 +2 +� +˜p2 + d3 +1 +8 ˜p3 +� +w3 + · · · . +(19) +It follows from (17) and (19) that +(2λ − 1)a2 = c1τ +2 , +(20) +(3λ − 1)a3 + +� +2λ2 − 4λ + 1 +� +a2 +2 = 1 +2 +� +c2 − c2 +1 +2 +� +τ + c2 +1 +4 3τ 2, +(21) +and +−(2λ − 1)a2 = d1τ +2 , +(22) +� +2λ2 + 2λ − 1 +� +a2 +2 − (3λ − 1)a3 = 1 +2 +� +d2 − d2 +1 +2 +� +τ + d2 +1 +4 3τ 2. +(23) +From (20) and (22), we have +c1 = −d1, +(24) +and +a2 +2 = (c2 +1 + d2 +1) +8(2λ − 1)2τ 2. +(25) +Hence +|a2| ≤ +|τ| +2λ − 1. +(26) +Now, by summing (21) and (23), we obtain +2(2λ − 1)2a2 +2 = 1 +2(c2 + d2)τ − 1 +4(c2 +1 + d2 +1)τ + 3 +4(c2 +1 + d2 +1)τ 2. +(27) +Substituting (25) in (27), we have +2 +� +(2λ − 1)2 − (10λ2 − 11λ + 3)τ +� +a2 +2 = 1 +2(c2 + d2)τ 2. +(28) +Therefore, using Lemma (1.3) we obtain +|a2| ≤ +|τ| +� +(2λ − 1)2 − (10λ2 − 11λ + 3)τ +. +(29) +Now, so as to find the bound on |a3|, let’s subtract from (21) and (23). So, we find +2(3λ − 1)a3 − 2(3λ − 1)a2 +2 = 1 +2 (c2 − d2) τ. +(30) + +8 +Kaliyappan Vijaya, Gangadharan Murugusundaramoorthy and Hatun ¨Ozlem G¨uney +Hence, we get +2(3λ − 1)|a3| ≤ 2|τ| + 2(3λ − 1)|a2|2. +(31) +Then, in view of (29), we obtain +|a3| ≤ +|τ| [(2λ − 1)2 − 2(5λ2 − 4λ + 1)τ] +(3λ − 1) [(2λ − 1)2 − (10λ2 − 11λ + 3)τ]. +By taking the parameter λ = 1 and λ = 2 in the above theorem, we have the fol- +lowing the initial Taylor coefficients |a2| and |a3| for the function classes SLΣ(˜p(z)) and +GSLΣ(˜p(z)), respectively. +Corollary 2.5. [8] Let f given by (1) be in the class SLΣ(˜p(z)), then +|a2| ≤ +|τ| +√1 − 2τ +(32) +and +|a3| ≤ |τ|(1 − 4τ) +2(1 − 2τ) . +(33) +Corollary 2.6. Let f given by (1) be in the class GSLΣ(˜p(z)), then +|a2| ≤ +|τ| +√9 − 21τ +(34) +and +|a3| ≤ |τ|(9 − 26τ) +5(9 − 21τ) . +(35) +3 +Fekete-Szeg¨o inequality for the function class PSLλ +Σ(˜p(z)) +Fekete and Szeg¨o [7] introduced the generalized functional |a3 − µa2 +2|, where µ is some +real number. Due to Zaprawa [??],in the following theorem we determine the Fekete-Szeg¨o +functional for f ∈ PSLλ +Σ(˜p(z)). +Theorem 3.1. Let f given by (1) be in the class PSLλ +Σ(˜p(z)) and µ ∈ R, then +|a3 − µa2 +2| ≤ +� +|τ| +4(3λ−1), +0 ≤ |h(µ)| ≤ +|τ| +4(3λ−1), +4|h(µ)|, +|h(µ)| ≥ +|τ| +4(3λ−1), +where +h(µ) = +(1 − µ)τ 2 +4 [(2λ − 1)2 − (10λ2 − 11λ + 3)τ]. +(1) + +On λ− Pseudo bi-starlike functions related with Fibonacci numbers +9 +Proof. From (28) and (30) we obtain +a3 − µa2 +2 = (1 − µ) +τ 2(c2 + d2) +4 [(2λ − 1)2 − (10λ2 − 11λ + 3)τ] + τ(c2 − d2) +4(3λ − 1) +(2) += +� +(1 − µ)τ 2 +4 [(2λ − 1)2 − (10λ2 − 11λ + 3)τ] + +τ +4(3λ − 1) +� +c2 ++ +� +(1 − µ)τ 2 +4 [(2λ − 1)2 − (10λ2 − 11λ + 3)τ] − +τ +4(3λ − 1) +� +d2. +So we have +a3 − µa2 +2 = +� +h(µ) + +|τ| +4(3λ − 1) +� +c2 + +� +h(µ) − +|τ| +4(3λ − 1) +� +d2, +(3) +where +h(µ) = +(1 − µ)τ 2 +4 [(2λ − 1)2 − (10λ2 − 11λ + 3)τ]. +(4) +Then, by taking modulus of (3), we conclude that +|a3 − µa2 +2| ≤ +� +|τ| +4(3λ−1), +0 ≤ |h(µ)| ≤ +|τ| +4(3λ−1), +4|h(µ)|, +|h(µ)| ≥ +|τ| +4(3λ−1). +Taking µ = 1, we have the following corollary. +Corollary 3.2. If f ∈ PSLλ +Σ(˜p(z)), then +|a3 − a2 +2| ≤ +|τ| +4(3λ − 1). +(5) +By specializing the parameter λ = 1 and λ = 2 in the above theorem, we have the fol- +lowing the Fekete-Szeg¨o inequalities for the function classes SLΣ(˜p(z)) and GSLΣ(˜p(z)), +respectively. +Corollary 3.3. [8] Let f given by (1) be in the class SLΣ(˜p(z)) and µ ∈ R,. then we have +|a3 − µa2 +2| ≤ +� +|τ| +8 , +0 ≤ |h(µ)| ≤ |τ| +8 , +4|h(µ)|, +|h(µ)| ≥ |τ| +8 , +where +h(µ) = (1 − µ)τ 2 +4 [1 − 2τ]. +(6) +Further if µ = 1 we get +|a3 − a2 +2| ≤ |τ| +8 . + +10 +Kaliyappan Vijaya, Gangadharan Murugusundaramoorthy and Hatun ¨Ozlem G¨uney +Corollary 3.4. Let f given by (1) be in the class GSLΣ(˜p(z)) and µ ∈ R, then we have +|a3 − µa2 +2| ≤ +� +|τ| +20, +0 ≤ |h(µ)| ≤ |τ| +20, +4|h(µ)|, +|h(µ)| ≥ |τ| +20, +where +h(µ) = (1 − µ)τ 2 +4 [9 − 21τ]. +(7) +Further if µ = 1 we get +|a3 − a2 +2| ≤ |τ| +20 . +Acknowledgements:The authors thank the referees of this paper for their insightful +suggestions and corrections to improve the paper in present form. +References +[1] Babalola K.O.: On λ-pseudo-starlike functions. J. Class. Anal. 3(2) (2013) 137–147. +[2] Brannan D.A., Clunie J. and Kirwan W.E.: Coefficient estimates for a class of starlike functions. +Canad. J. Math. 22 (1970) 476–485. +[3] Brannan D.A. and Taha T.S.: On some classes of bi-univalent functions . Studia +Univ.Babes-Bolyai Math. 31(2) (1986) 70–77. +[4] +Duren P.L.: Univalent Functions. In: Grundlehren der Mathematischen Wissenschaften,Band 259 +New York, Berlin, Heidelberg and Tokyo, Springer-Verlag (1983). +[5] Joshi S., Joshi S. and Pawar H. : On some subclasses of bi-univalent functions associated with +pseudo-starlike function . J.Egyptian Math.Soc. 24 (2016) 522–525. +[6] Dziok J., Raina R.K. and Sok´o�l J.: On α−convex functions related to a shell-like curve connected +with Fibonacci numbers . Appl. Math. Comp. 218 (2011) 996–1002. +[7] Fekete M. and Szeg¨o G.: Eine Bemerkung ¨uber ungerade schlichte Functionen . J. London Math. +Soc. 8 (1933) 85–89. +[8] G¨uney H.¨O., Murugusundaramoorthy G. and Sok´o�l J.: Subclasses of bi-univalent functions related +to shell-like curves connected with Fibonacci numbers. Commun. Fac. Sci. Univ. Ank. Ser. +A1-Math. Stat. 68(2) (2019) 1909–1921. +[9] Miller S.S. and Mocanu P.T. : Differential Subordinations Theory and Applications. P Series of +Monographs and Text Books in Pure and Applied Mathematics Marcel Dekker, New York (2000). +[10] Lewin M. : On a coefficient problem for bi-univalent functions. Proc. Amer.Math. Soc. 18 (1967) +63–68. +[11] Pommerenke Ch. : Univalent Functions. Math. Math, Lehrbucher, Vandenhoeck and Ruprecht, +G¨ottingen (1975). +[12] Raina R.K. and Sok´o�l J.: Fekete-Szeg¨o problem for some starlike functions related to shell-like +curves. Math. Slovaca 66 (2016) 135–140. +[13] Sok´o�l J.: On starlike functions connected with Fibonacci numbers. Folia Scient. Univ. Tech. +Resoviensis 175 (1999) 111–116. + +On λ− Pseudo bi-starlike functions related with Fibonacci numbers +11 +[14] Srivastava H.M., Mishra A.K. and Gochhayat P.: Certain subclasses of analytic and bi-univalent +functions . Appl. Math. Lett. 23(10) (2010) 1188–1192. +[15] +Xu Q.-H., Gui Y.-C. and Srivastava H.M.: Coefficient estimates for a certain subclass of analytic +and bi-univalent functions . Appl. Math. Lett. 25 (2012) 990–994. +[16] +Li X-F. and Wang A-P : Two new subclasses of bi-univalent functions. Inter. Math. Forum, 7(30) +(2012) 1495–1504. +[17] +Zaprawa P. : On the Fekete-Szeg¨o problem for classes of bi-univalent functions. Bull. Belg. Math. +Soc. Simon Stevin 21(1) (2014) 69–78. +Received: Received date +Accepted for publication: Accepted date +Communicated by: Handling Editor + diff --git a/edFKT4oBgHgl3EQfAS1-/content/tmp_files/load_file.txt b/edFKT4oBgHgl3EQfAS1-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..af45b2722d6aa3428fe9300163ab0e7bbcb20c43 --- /dev/null +++ b/edFKT4oBgHgl3EQfAS1-/content/tmp_files/load_file.txt @@ -0,0 +1,308 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf,len=307 +page_content='Communications in Mathematics n (2023), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' m, 00–11 DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='46298/cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='ABCD ©2023 Kaliyappan Vijaya, Gangadharan Murugusundaramoorthy and Hatun ¨Ozlem G¨uney This is an open access article licensed under the CC BY-SA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='0 1 On λ− Pseudo bi-starlike functions related with Fibonacci numbers Kaliyappan Vijaya, Gangadharan Murugusundaramoorthy and Hatun ¨Ozlem G¨uney Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' In this paper we define a new subclass λ-bi-pseudo-starlike functions of Σ related to shell-like curves connected with Fibonacci numbers and determine the initial Taylor-Maclaurin coefficients |a2| and |a3| for f ∈ PSLλ Σ(˜p(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Further we determine the Fekete-Szeg¨o result for the function class PSLλ Σ(˜p(z)) and for special cases, corollaries are stated which some of them are new and have not been studied so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' 1 Introduction Let A denote the class of functions f which are analytic in the open unit disk U = {z : z ∈ C and |z| < 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Also let S denote the class of functions in A which are univalent in U and normalized by the conditions f(0) = f ′(0) − 1 = 0 and are of the form: f (z) = z + ∞ � n=2 anzn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (1) The Koebe one quarter theorem [4] ensures that the image of U under every univalent function f ∈ A contains a disk of radius 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Thus every univalent function f has an inverse f −1 satisfying f −1(f(z)) = z, (z ∈ U) and f(f −1(w)) = w (|w| < r0(f), r0(f) ≥ 1 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' MSC 2020: AMS classification 30C45 (see https://mathscinet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='ams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='org/mathscinet/msc/msc2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='html) Keywords: Analytic functions, bi-univalent, shell-like curve, Fibonacci numbers, starlike functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Affiliation: K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='Vijaya– School of Advanced Sciences, Vellore Institute of Techlogy, Vellore-632014, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' E-mail: kvijaya@vit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='Murugusundaramoorthy– School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' E-mail: gmsmoorthy@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='com H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='¨Ozlem G¨uney– Dicle University, Faculty of Science, Department of Mathematics, Diyarbakır-T¨URK˙IYE E-mail: ozlemg@dicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='tr arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='11698v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='CV] 27 Jan 2023 =P sciences2 Kaliyappan Vijaya, Gangadharan Murugusundaramoorthy and Hatun ¨Ozlem G¨uney A function f ∈ A is said to be bi-univalent in U if both f and f −1 are univalent in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Let Σ denote the class of bi-univalent functions defined in the unit disk U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Since f ∈ Σ has the Maclaurian series given by (1), a computation shows that its inverse g = f −1 has the expansion g(w) = f −1(w) = w − a2w2 + (2a2 2 − a3)w3 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (2) One can see a short history and examples of functions in the class Σ in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Several authors have introduced and investigated subclasses of bi-univalent functions and obtained bounds for the initial coefficients (see [2],[3],[10],[14],[15],[16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' For a brief historical account and for several interesting examples of functions in the class Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' see the pioneering work on this subject by Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' [14], which actually revived the study of bi-univalent functions in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' From the work of Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' [14], we choose to recall the following examples of functions in the class Σ : z 1 − z, − log(1 − z) and 1 2 log �1 + z 1 − z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' We notice that the class Σ is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' However, the Koebe function is not a member of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' An analytic function f is subordinate to an analytic function F in U, written as f ≺ F (z ∈ U), provided there is an analytic function ω defined on U with ω(0) = 0 and |ω(z)| < 1 satisfying f(z) = F(ω(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' It follows from Schwarz Lemma that f(z) ≺ F(z) ⇐⇒ f(0) = F(0) and f(U) ⊂ F(U) , z ∈ U (for details see [4], [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' We recall important subclasses of S in geometric function theory such that if f ∈ A and zf ′(z) f(z) ≺ p(z) and 1 + zf ′′(z) f ′(z) ≺ p(z) where p(z) = 1+z 1−z, then we say that f is starlike and convex, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' These functions form known classes denoted by S∗ and C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Recently, in [13], Sok´o�l introduced the class SL of shell-like functions as the set of functions f ∈ A which is described in the following definition: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' The function f ∈ A belongs to the class SL if it satisfies the condition that zf ′(z) f(z) ≺ ˜p(z) with ˜p(z) = 1 + τ 2z2 1 − τz − τ 2z2, where τ = (1 − √ 5)/2 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' On λ− Pseudo bi-starlike functions related with Fibonacci numbers 3 It should be observed SL is a subclass of the starlike functions S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' The function ˜p is not univalent in U, but it is univalent in the disc |z| < (3 − √ 5)/2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' For example, ˜p(0) = ˜p(−1/2τ) = 1 and ˜p(e∓i arccos(1/4)) = √ 5/5, and it may also be noticed that 1 |τ| = |τ| 1 − |τ|, which shows that the number |τ| divides [0, 1] such that it fulfils the golden section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' The image of the unit circle |z| = 1 under ˜p is a curve described by the equation given by (10x − √ 5)y2 = ( √ 5 − 2x)( √ 5x − 1)2, which is translated and revolved trisectrix of Maclaurin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' The curve ˜p(reit) is a closed curve without any loops for 0 < r ≤ r0 = (3 − √ 5)/2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' For r0 < r < 1, it has a loop, and for r = 1, it has a vertical asymptote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Since τ satisfies the equation τ 2 = 1 + τ, this expression can be used to obtain higher powers τ n as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of τ and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' The resulting recurrence relationships yield Fibonacci numbers un: τ n = unτ + un−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' In [12] Raina and Sok´o�l showed that ˜p(z) = 1 + τ 2z2 1 − τz − τ 2z2 = � t + 1 t � t 1 − t − t2 = 1 √ 5 � t + 1 t � � 1 1 − (1 − τ)t − 1 1 − τt � = � t + 1 t � ∞ � n=1 untn = 1 + ∞ � n=1 (un−1 + un+1)τ nzn, (3) where un = (1 − τ)n − τ n √ 5 , τ = 1 − √ 5 2 (n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (4) This shows that the relevant connection of ˜p with the sequence of Fibonacci numbers un, such that u0 = 0, u1 = 1, un+2 = un + un+1 for n = 0, 1, 2, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' And they got ˜p(z) = 1 + ∞ � n=1 ˜pnzn = 1 + (u0 + u2)τz + (u1 + u3)τ 2z2 + ∞ � n=3 (un−3 + un−2 + un−1 + un)τ nzn = 1 + τz + 3τ 2z2 + 4τ 3z3 + 7τ 4z4 + 11τ 5z5 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (5) Let P(β), 0 ≤ β < 1, denote the class of analytic functions p in U with p(0) = 1 and Re{p(z)} > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Especially, we will use P instead of P(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' 4 Kaliyappan Vijaya, Gangadharan Murugusundaramoorthy and Hatun ¨Ozlem G¨uney Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' ([6]) The function ˜p(z) = 1+τ 2z2 1−τz−τ 2z2 belongs to the class P(β) with β = √ 5/10 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='2236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Now we recall the following lemma which will be relevant for our study .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' ([11]) Let p ∈ P with p(z) = 1 + c1z + c2z2 + · · · , then |cn| ≤ 2 for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (6) In this present work, we introduce a new subclass of Σ associated with shell-like func- tions connected with Fibonacci numbers and obtain the initial Taylor coefficients |a2| and |a3| for this function class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Also, we give bounds for the Fekete-Szeg¨o functional |a3 − µa2 2| for this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' 2 Bi-Univalent function class PSLλ Σ(˜p(z)) In this section, we introduce a new subclass of Σ associated with shell-like functions connected with Fibonacci numbers and obtain the initial Taylor coefficients |a2| and |a3| for the function class by subordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Firstly, let p(z) = 1+p1z+p2z2+· · · , and p ≺ ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Then there exists an analytic function u such that |u(z)| < 1 in U and p(z) = ˜p(u(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Therefore, the function h(z) = 1 + u(z) 1 − u(z) = 1 + c1z + c2z2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (1) is in the class P(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' It follows that u(z) = c1z 2 + � c2 − c2 1 2 � z2 2 + � c3 − c1c2 + c3 1 4 � z3 2 + · · · (2) and ˜p(u(z)) = 1 + ˜p1c1z 2 + �1 2 � c2 − c2 1 2 � ˜p1 + c2 1 4 ˜p2 � z2 + �1 2 � c3 − c1c2 + c3 1 4 � ˜p1 + 1 2c1 � c2 − c2 1 2 � ˜p2 + c3 1 8 ˜p3 � z3 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (3) And similarly, there exists an analytic function v such that |v(w)| < 1 in U and p(w) = ˜p(v(w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Therefore, the function k(w) = 1 + v(w) 1 − v(w) = 1 + d1w + d2w2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (4) is in the class P(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' It follows that v(w) = d1w 2 + � d2 − d2 1 2 � w2 2 + � d3 − d1d2 + d3 1 4 � w3 2 + · · · (5) On λ− Pseudo bi-starlike functions related with Fibonacci numbers 5 and ˜p(v(w)) = 1 + ˜p1d1w 2 + �1 2 � d2 − d2 1 2 � ˜p1 + d2 1 4 ˜p2 � w2 + �1 2 � d3 − d1d2 + d3 1 4 � ˜p1 + 1 2d1 � d2 − d2 1 2 � ˜p2 + d3 1 8 ˜p3 � w3 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (6) The class Lλ(α) of λ-pseudo-starlike functions of order α was introduced and investi- gated by Babalola [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' A function f ∈ A is in the class Lλ(α) if it satisfies ℜ �z(f ′(z))λ f(z) � > α, (0 ≤ α < 1) where λ ≥ 1, λ ∈ R and z ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' In [1] it was showed that all pseudo-starlike functions are Bazileviˇc functions of type (1 − 1/λ) and of order α1/λ and univalent in open unit disk U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Recently Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' [5] defined the bi-pseudo-starlike functions class and obtained the bounds for the initial coefficients |a2| and |a3|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' In this paper we define a new class PSLλ Σ(˜p(z)),λ-bi-pseudo-starlike functions of Σ related to shell-like curves connected with Fibonacci numbers and determine the bounds for the initial Taylor-Maclaurin coefficients of |a2| and |a3|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Further we consider the Fekete-Szeg¨o problem in this class and the special cases are stated as corollaries which are new and have not been studied so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' For λ ≥ 1 and λ ∈ R, a function f ∈ Σ of the form (1) is said to be in the class PSLλ Σ(˜p(z)) if the following subordination hold: z(f ′(z))λ f(z) ≺ ˜p(z) = 1 + τ 2z2 1 − τz − τ 2z2 (7) and w(g′(w))λ g(w) ≺ ˜p(w) = 1 + τ 2w2 1 − τw − τ 2w2 (8) where τ = (1 − √ 5)/2 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='618 where z, w ∈ U and g is given by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Specializing the parameter λ = 1 and λ = 2, we have the following remarks, respec- tively: Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' [8] For λ = 1 a function f ∈ Σ is in the class PSL1 Σ(˜p(z)) ≡ SLΣ(˜p(z)) if the following conditions are satisfied: zf ′(z) f(z) ≺ ˜p(z) = 1 + τ 2z2 1 − τz − τ 2z2 (9) and wg′(w) g(w) ≺ ˜p(w) = 1 + τ 2w2 1 − τw − τ 2w2 (10) where τ = (1 − √ 5)/2 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='618 where z, w ∈ U and g is given by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' 6 Kaliyappan Vijaya, Gangadharan Murugusundaramoorthy and Hatun ¨Ozlem G¨uney Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' For λ = 2 a function f ∈ Σ is in the class PSL2 Σ(˜p(z)) ≡ GSLΣ(˜p(z)) if the following conditions are satisfied: � f ′(z)zf ′(z) f(z) � ≺ ˜p(z) = 1 + τ 2z2 1 − τz − τ 2z2 (11) and � g′(w)wg′(w) g(w) � ≺ ˜p(w) = 1 + τ 2w2 1 − τw − τ 2w2 (12) where τ = (1 − √ 5)/2 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='618 where z, w ∈ U and g is given by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' In the following theorem we determine the initial Taylor coefficients |a2| and |a3| for the function class PSLλ Σ(˜p(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Later we will reduce these bounds to other classes for special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Let f given by (1) be in the class PSLλ Σ(˜p(z)), then |a2| ≤ |τ| � (2λ − 1)2 − (10λ2 − 11λ + 3)τ (13) and |a3| ≤ |τ| [(2λ − 1)2 − 2(5λ2 − 4λ + 1)τ] (3λ − 1) [(2λ − 1)2 − (10λ2 − 11λ + 3)τ], (14) where λ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Let f ∈ PSLλ Σ(˜p(z)) and g = f −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Considering (7) and (8), we have z(f ′(z))λ f(z) = ˜p(u(z)) (15) and w(g′(w))λ g(w) = ˜p(v(w)), (16) where τ = (1 − √ 5)/2 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='618 where z, w ∈ U and g is given by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Since z(f ′(z))λ f(z) = 1 + (2λ − 1)a2z + [(3λ − 1)a3 + � 2λ2 − 4λ + 1 � a2 2]z2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' and w(g′(w))λ g(w) = 1 − (2λ − 1)a2w + [ � 2λ2 + 2λ − 1 � a2 2 − (3λ − 1)a3]w2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Thus we have 1 + (2λ − 1)a2z + [(3λ − 1)a3 + � 2λ2 − 4λ + 1 � a2 2]z2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (17) = 1 + ˜p1c1 2 z + �1 2 � c2 − c2 1 2 � ˜p1 + c2 1 4 ˜p2 � z2 + �1 2 � c3 − c1c2 + c3 1 4 � ˜p1 + 1 2c1 � c2 − c2 1 2 � ˜p2 + c3 1 8 ˜p3 � z3 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (18) On λ− Pseudo bi-starlike functions related with Fibonacci numbers 7 and 1 − (2λ − 1)a2w + [ � 2λ2 + 2λ − 1 � a2 2 − (3λ − 1)a3]w2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' , = 1 + ˜p1d1w 2 + �1 2 � d2 − d2 1 2 � ˜p1 + d2 1 4 ˜p2 � w2 + �1 2 � d3 − d1d2 + d3 1 4 � ˜p1 + 1 2d1 � d2 − d2 1 2 � ˜p2 + d3 1 8 ˜p3 � w3 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (19) It follows from (17) and (19) that (2λ − 1)a2 = c1τ 2 , (20) (3λ − 1)a3 + � 2λ2 − 4λ + 1 � a2 2 = 1 2 � c2 − c2 1 2 � τ + c2 1 4 3τ 2, (21) and −(2λ − 1)a2 = d1τ 2 , (22) � 2λ2 + 2λ − 1 � a2 2 − (3λ − 1)a3 = 1 2 � d2 − d2 1 2 � τ + d2 1 4 3τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (23) From (20) and (22), we have c1 = −d1, (24) and a2 2 = (c2 1 + d2 1) 8(2λ − 1)2τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (25) Hence |a2| ≤ |τ| 2λ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (26) Now, by summing (21) and (23), we obtain 2(2λ − 1)2a2 2 = 1 2(c2 + d2)τ − 1 4(c2 1 + d2 1)τ + 3 4(c2 1 + d2 1)τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (27) Substituting (25) in (27), we have 2 � (2λ − 1)2 − (10λ2 − 11λ + 3)τ � a2 2 = 1 2(c2 + d2)τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (28) Therefore, using Lemma (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='3) we obtain |a2| ≤ |τ| � (2λ − 1)2 − (10λ2 − 11λ + 3)τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (29) Now, so as to find the bound on |a3|, let’s subtract from (21) and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' So, we find 2(3λ − 1)a3 − 2(3λ − 1)a2 2 = 1 2 (c2 − d2) τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (30) 8 Kaliyappan Vijaya, Gangadharan Murugusundaramoorthy and Hatun ¨Ozlem G¨uney Hence, we get 2(3λ − 1)|a3| ≤ 2|τ| + 2(3λ − 1)|a2|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (31) Then, in view of (29), we obtain |a3| ≤ |τ| [(2λ − 1)2 − 2(5λ2 − 4λ + 1)τ] (3λ − 1) [(2λ − 1)2 − (10λ2 − 11λ + 3)τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' By taking the parameter λ = 1 and λ = 2 in the above theorem, we have the fol- lowing the initial Taylor coefficients |a2| and |a3| for the function classes SLΣ(˜p(z)) and GSLΣ(˜p(z)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' [8] Let f given by (1) be in the class SLΣ(˜p(z)), then |a2| ≤ |τ| √1 − 2τ (32) and |a3| ≤ |τ|(1 − 4τ) 2(1 − 2τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (33) Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Let f given by (1) be in the class GSLΣ(˜p(z)), then |a2| ≤ |τ| √9 − 21τ (34) and |a3| ≤ |τ|(9 − 26τ) 5(9 − 21τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (35) 3 Fekete-Szeg¨o inequality for the function class PSLλ Σ(˜p(z)) Fekete and Szeg¨o [7] introduced the generalized functional |a3 − µa2 2|, where µ is some real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Due to Zaprawa [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' ],in the following theorem we determine the Fekete-Szeg¨o functional for f ∈ PSLλ Σ(˜p(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Let f given by (1) be in the class PSLλ Σ(˜p(z)) and µ ∈ R, then |a3 − µa2 2| ≤ � |τ| 4(3λ−1), 0 ≤ |h(µ)| ≤ |τ| 4(3λ−1), 4|h(µ)|, |h(µ)| ≥ |τ| 4(3λ−1), where h(µ) = (1 − µ)τ 2 4 [(2λ − 1)2 − (10λ2 − 11λ + 3)τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (1) On λ− Pseudo bi-starlike functions related with Fibonacci numbers 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' From (28) and (30) we obtain a3 − µa2 2 = (1 − µ) τ 2(c2 + d2) 4 [(2λ − 1)2 − (10λ2 − 11λ + 3)τ] + τ(c2 − d2) 4(3λ − 1) (2) = � (1 − µ)τ 2 4 [(2λ − 1)2 − (10λ2 − 11λ + 3)τ] + τ 4(3λ − 1) � c2 + � (1 − µ)τ 2 4 [(2λ − 1)2 − (10λ2 − 11λ + 3)τ] − τ 4(3λ − 1) � d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' So we have a3 − µa2 2 = � h(µ) + |τ| 4(3λ − 1) � c2 + � h(µ) − |τ| 4(3λ − 1) � d2, (3) where h(µ) = (1 − µ)τ 2 4 [(2λ − 1)2 − (10λ2 − 11λ + 3)τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (4) Then, by taking modulus of (3), we conclude that |a3 − µa2 2| ≤ � |τ| 4(3λ−1), 0 ≤ |h(µ)| ≤ |τ| 4(3λ−1), 4|h(µ)|, |h(µ)| ≥ |τ| 4(3λ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Taking µ = 1, we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' If f ∈ PSLλ Σ(˜p(z)), then |a3 − a2 2| ≤ |τ| 4(3λ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (5) By specializing the parameter λ = 1 and λ = 2 in the above theorem, we have the fol- lowing the Fekete-Szeg¨o inequalities for the function classes SLΣ(˜p(z)) and GSLΣ(˜p(z)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' [8] Let f given by (1) be in the class SLΣ(˜p(z)) and µ ∈ R,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' then we have |a3 − µa2 2| ≤ � |τ| 8 , 0 ≤ |h(µ)| ≤ |τ| 8 , 4|h(µ)|, |h(µ)| ≥ |τ| 8 , where h(µ) = (1 − µ)τ 2 4 [1 − 2τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (6) Further if µ = 1 we get |a3 − a2 2| ≤ |τ| 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' 10 Kaliyappan Vijaya, Gangadharan Murugusundaramoorthy and Hatun ¨Ozlem G¨uney Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Let f given by (1) be in the class GSLΣ(˜p(z)) and µ ∈ R, then we have |a3 − µa2 2| ≤ � |τ| 20, 0 ≤ |h(µ)| ≤ |τ| 20, 4|h(µ)|, |h(µ)| ≥ |τ| 20, where h(µ) = (1 − µ)τ 2 4 [9 − 21τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' (7) Further if µ = 1 we get |a3 − a2 2| ≤ |τ| 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Acknowledgements:The authors thank the referees of this paper for their insightful suggestions and corrections to improve the paper in present form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' References [1] Babalola K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' : On λ-pseudo-starlike functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} +page_content=' Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFKT4oBgHgl3EQfAS1-/content/2301.11698v1.pdf'} diff --git a/etE3T4oBgHgl3EQffAo5/vector_store/index.faiss b/etE3T4oBgHgl3EQffAo5/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..41ee5d968c9decc7ef1a09adce875cd9fc3f30eb --- /dev/null +++ b/etE3T4oBgHgl3EQffAo5/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6b4a6ad4eee524f18453d48aa2cc09f83cfad0360f3bc7e271920c180aa92a54 +size 7340077 diff --git a/fNFJT4oBgHgl3EQfTywB/content/tmp_files/2301.11505v1.pdf.txt b/fNFJT4oBgHgl3EQfTywB/content/tmp_files/2301.11505v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..adbb5f0f44268a518812a70cdf4120f383118cba --- /dev/null +++ b/fNFJT4oBgHgl3EQfTywB/content/tmp_files/2301.11505v1.pdf.txt @@ -0,0 +1,635 @@ +Design of an FPGA-based USB 3.0 device controller +Zhe Ninga,1,Yunhua Suna +a Institute of High Energy Physics, Chinese Academy of Sciences +Beijing 100049, China +E-mail: ningzhe@ihep.ac.cn +ABSTRACT: The traditional USB 3.0 communication based on FPGA uses an external chip as a +USB PHY or a USB controller including a USB PHY. This paper realizes a USB 3.0 controller +using FPGA resources, in which FPGA logic realizes a serial interface engine, and an FPGA +internal transceiver is a USB PHY. Used slices percent after implementation is 4.59% in Kintex- +7 325t. The test result shows that the speed of USB 3.0 is more than 320 MB/s bulk-in and bulk- +out transfers. +KEYWORDS: USB 3.0; FPGA; Transceivers. + +1. Introduction +Because high precision and high-speed components are used in instruments based on FPGA, +the requirement of high-speed transmission is necessary. The standard transmission scheme +includes 125 MB/s Ethernet and USB 3.0. A widely used example of 125 MB/s Ethernet is +SiTCP [1] which is realized based on a hardware protocol stack plus an external PHY. But USB +3.0 maximum speed is 625 MB/s, which is faster than 125 MB/s of Ethernet. Now the readout +electronics integrated with Photomultiplier tubes (PMT) [2,3] is more and more popular. Still, +the area of printed circuit boards (PCB) is limited by the size of PMT and power dissipation. +At the same time, if dark rates of PMT which is nearly 50 kHz [4], needed to be studied deeply, +the transmission bandwidth is very high. Because the window size of waveform sampling is +1000 points generally, the total size per second is 1000 points * 2 bytes/point * 50k = 100 MB/s +which is very close to the limit of 125 MB/s of Ethernet. USB transmission schemes look better. +But most USB schemes are realized using an external USB chip. +As shown in Figure 1, there are several typical schemes for USB in FPGA boards. In +Architecture A, PHY, a serial interface engine and drivers are packaged into one chip, such as +Cypress CY3014 [5] and FIDI FT600 [6], so that user logic in FPGA can send and receive data +from the chip quickly. These chips are also called USB-FIFO chips. But they need scalability +and cannot be a USB host. Architecture B, whose external chip includes a PHY, has more +scalability and is more complicated because the other layers, such as the serial interface engine +(SIE), are realized by FPGA logic. The most common external chip in architecture B is +TUSB1310A [7] from Texas Instruments. It is unlucky that this chip has stopped production, +and there is no replacement. This paper discusses Architecture C, which is the upgrade of +Architecture B and tries to use an FPGA internal transceiver as a USB PHY to replace +TUSB1310A. A smaller area of PCB for readout electronics could be adapted with PMT well. + +Figure 1 USB schemes for FPGA + +FPGA +Cypress CY3014 +User +USB +Architecture A +Driver +SIE +PHY +Logic +Interface +FPGA +TUSB1310A +User +USB +Architecture B +Driver +SIE +PHY +Logic +Interface +FPGA +User +USB +Architecture C +Driver +SIE +PHY +Logic +Interface2. System design +According to the document and support from Xilinx, Xilinx FPGA external transceivers +are not used as a USB 3.0 PHY [8]. The primary function of USB PHY should be realized +respectively. The standard USB 3.0 PHY includes a module transmitting Low-Frequency +Periodic Signaling (LFPS) in which gigabit transceivers are turned off. The other is an SIE +module communicating regular USB 3.0 data in which gigabit transceivers are turned on. +2.1 PHY modules +2.1.1 LFPS definition +USB 3.0 PHY is similar to the PHY of PCIe and SATA, which should transmit Low- +Frequency Periodic Signaling (LFPS) to initiate links or wake up the link partner in a low- +power link state. +As Figure 2 shown, there are three parameters for LFPS: tPeriod, tBurst, and tRepeat. +Because LFPS is a square wave, tPeriod is the period of a square wave whose minimum value +is 20 ns, and the maximum value is 100 ns; tBurst is a period occupied by the transmission of +continuous LFPS signal, and its value is dependent on LFPS types shown by Table 1 [9 ]; +tRepeat are composed of tPeriod and the time of electric idle states by keeping two transmission +differential wires at the same voltage, and its value is also depended on LFPS types. The most +common LFPS type is polling, whose normal tBurst is 1.0 μs, and normal tRepeat is 10.0 μs. + +Figure 2 LFPS signaling +Table 1 LFPS Transmitter Timing for SuperSpeed Design + +tBurst +tRepeat + +Minimum Normal Maximum +Minimum +Number of +LFPS +Cycles +Minimum Normal Maximum +Polling +0.6 μs +1.0 μs +1.4 μs + +6 μs +10 μs +14 μs +Ping +40 ns + +200 ns +2 +160 ms +200 ms +240 ms + +tPeriod +Electric Idle +tBurst +tRepeattReset +80 ms +100 ms +120 ms + + + + +U1 Exit +600 ns + +2 ms + + + + +U2 Exit +80 μs + +2 ms + + + + +U1 Wakeup +80 μs + +10 ms + + + + +2.1.2 Transmitting and receiving LFPS +As discussed, some cases, such as link initiations or waking up link partners, should transmit +LFPS by configuring some ports of transmitters shown in Figure 3. When initiating links, the +generation conditions of LFPS are txpd = 2'b0 && rxpd = 2'b0 && TXDETECTRX = 1 && +TXELECIDLE = 1 [10]; When waking up link partners, the generation conditions of LFPS are +txppd = 2'b1 && rxpd = 2'b1 && TXELECIDLE = 0. For the detection of LFPS receiving, +RXELECIDLE = 0 means LFPS detection, and RXELECIDLE = 1 means no LFPS detection +[11]. +First, a square wave should be generated repeatedly during the tPeriod stages by toggling +the wires between a differential '1' and '0' at a frequency. The data width of transceivers is 40 +bits, in which 8b10b encoding is enabled but bypassed where TX8B10BEN = 1'b1 and +TX8B10BBYPASS = 8'h0F. Due to the 625 MB/s speed of transceivers, the time occupied by +each bit is 0.2 ns, so when tx_data = 40'hFF_FFFF_FFFF, the time of high level for one cycle +is 0.2 × 40 = 8 ns, which is less than ten ns which is the half of one LFPS cycle, so the time of +high level should be doubled at least. Figure 4 from oscilloscopes shows the tPeriod is 32 ns +after doubling the time of the high level of square waves. If polling LFPS will be transmitted, +tBurst consists of 32 tPeriod. Second, after the completion of tBurst, wires will come into an +electric idle stage by asserting TXELECIDLE high. + +Figure 3 A transmitter structure + +125 MHz +125 MHz +5 GHz +TXDETECTRX +TX DATA +8 +10 +(32) +Tx +Parallel +8b/10b +Differential +Buffer +To +Encoding +Driver +32 to 8 +Serial +TXELECIDLE +TX DATAK +Figure 4 The tPeriod of LFPS signal captured by an oscilloscope +2.2 SIE design +As Figure 5 shown, SIE comprises three modules: the PIPE interface module, the link +module, and the protocol module. The PIPE interface module is used for scrambling and +descrambling data; The link module includes two parts: one is responsible for a link module +and management, which is used to generate link command management packets, such as +LGOOD and LCRD. The other one is Link Training Status State Machine (LTSSM), which is +responsible for the transfer management from one state to another state, such as U0 and U1; +the protocol module is used to generate tokens or data packets for the response of commands +from a USB host when as a device or generate token or data packets to a USB device when as +a host. For a USB device, an endpoints management module is needed to manage Endpoints +0, which is used for enumeration, Endpoint 1, and Endpoint 2, which is used for bulk-in and +out. For a USB host, an enumeration management module is necessary to generate commands +for enumerations after a device is inserted. A device driver module is also needed depending +on the type of devices attached to hosts. Resource utilization is shown in Figure 6, and the +percent of used slices in Kintex-7 325t is 4.59%. + +LaCroy +Measure +P1:pkpk(C1) +P2:rms(C1) +P3:--- +P4:- - . +P5-- +P6.-- +value +461mV +440.2 mV +mean +451.58 mV +436.806 mV +min +32 mV +328.2 mV +max +467mV +441.2mV +sdev +40.35 mV +10.973 mV +num +112 +112 +status +Thase +-175.6nsTriggerC30g +200mVId +200mVd +0.0mVofst +20.0ns/dv +2.5GS/s +Stop +368mV +-668.0 +-188mV +500S +Edge +Positme +30mV +218m +Figure 5 SIE design + +Figure 6 Resource utilization of the USB controller +2.3 Initiation process +As shown in Figure 7, in the Rx.Detect state, PHY should be told to begin a receiver detection +operation by asserting TXDETECTRX if the Signal phystatus is low. Signal phystatus is +asserted High during receiver detection to indicate receiver detection completion [12]. So the +state goes into a polling state. Transceivers should continue to transmit LFPS signals; On the +other side, transceivers should check if the received LFPS signal is a polling signal. If it is, the +transceivers are turned on and then transmit a TSEQ ordered set to the link partner for training +the equalizer for a certain period. Following this, TS1 and TS2 ordered sets are sent as a +handshake to finalize the link training and request info from the typical setup, such as a speedy +response. The state will go into a U0 state from the polling state when all these are done. In the +U0 state, at first, the USB controller will have a response for the enumeration request from a +host and then come into bulk-in or bulk-out status. + + +LTSSM +Transceivers +PIPE +Protocol +Endpoints +Link +Device SIE +LTSSM +Transceivers +PIPE +Protocol +Enumeration +Driver +Link +Host SIESlice LUTs +Slice Registers +F7 Muxes +Slice +LUT as Logic +LUT as Memory +Block RAM +DSPs +Bonded IOB +Bonded IPADs +Bonded OPADs +IBUFDS +GTXE2_CHANNEL +(203800) +(407600) +(101900) +(50950) +(203800) +(64000) +Tile (445) +(840) +(500) +(05) +(32) +(480) +(16) +5795 +6340 +6 +2338 +5665 +130 +9.5 +1 +4 +4 +2 +1 +1U1 +Inactive +Rx.Detect +Polling +UO +U2 +U3Figure 7 LTSSM High-Level States +3. Experiment results +3.1 Hardware setup +As shown in Figure 8, a KC705 board [13] is used for verification, and a mezzanine card named +HiTech USB [14] is used for adding a USB interface connecting with GTX transceivers of +Kintex-7 directly. More details about this connection are shown in Figure 9. It is noticeable that +there should be a 0.1 uF capacitor in an RX termination, which is very common in a TX +termination. The clock of the USB controller is from the output of the phase lock loop (PLL), +whose input is associated with the TXOUTCLK port from transceivers whose reference clock +is from a 200 MHz differential oscillator. So a FIFO is necessary between user logic and USB +controllers due to different clock domains. + +Figure 8 A hardware setup + +Figure 9 The FPGA USB 3.0 connector schematics +3.2 Speed tests +The speed tests include bulk-in and bulk-out transfer tests, which could be based on the upper +computer software such as the cypress stream program or the third-party software of USB +analyzers such as LeCroy adviser T3 [15]. For bulk-in tests shown in Figure 10 and Figure 11, +both of cypress stream program and LeCroy adviser T3 show that the USB speed is more than +320 MB/s. At the same time, the USB analyzer also shows no bit error after transferring more +than 3 TB of data, and then an estimated bit error rate is less than 10-13, which is less than 10-12 + +PCB +FPGA +USB +USB +Upper +User Logic +Driver +SIE +Transceivers +4 +Interfaces +Cables +Computers0.1 uF +USB 3.0 +FPGA +0.1 uF +0.1 uF +Interface +0.1 uFof the USB specification requirement. The bulk-out tests shown in Figure 12 and Figure 13 also +indicate a similar result. + +Figure 10 Bulk-in transfer test based on Cypress stream program + +Figure 11 Bulk-in transfer test based on LeCroy adviser T3 + +? C++ Streamer +口 +Cornmected Devices +(0x04B4 - 0x1003) + Cypress FX2LP StreamerExample Device +BULK IH, +16384 Bytes, 15 MaxBurst, +(0 - 0x81) +Endpoint +[32 +Packets per Xfer +Successes +2208 +Xfers to Queue +16 +Failures +0 +Timeout Per Xfer (ms) +1500 +Stop +Transfer Rate (KBps) +336500JSB4/3.2/2 /PD +(SN:12269) SuperSpeed Host L Data Payload Throughput & SuperSpeed Host R Data Payload Throughput +Q? +SN:12269 +USB 2.0 +Ch 0 +Ch 1 +Data Packets +00,000,000,000 +N/A +Data Bytes +00,000,000,000 +N/A +Total Bytes +00,000,000,000 +N/A +USB 3.2 +Endpoint Statistics (Address, Endpoint, Direction +7, 1, In +0, 0, Both +280 +Activity +Throughput 345.394 MB/s +0 B/s +260 +Bytes +3.015 TB +0 Bytes +ACK +2958221903 +227351 +240 +Retry +0 +NRDY +0 +0 +220 +ERDY +0 +0 +DP +2953077047 +0 +DP Error +0 +0 +200 +TP +2958310096 +227351 +TP Error +0 +0 +180 +Link Statistics +160 +Left +Right +LBAD +0 +CRC-5 +0 +CRC-16 +0 +0 +CRC-32 +0 +120 +RD +0 +Inv Sym +0 +2953344551 +0 +DP Err +DP Err % +000000'0 +0.000 +8 +TP +2958428521 +TP Err +0 +8 +02 : 29 : 15 +Restart + Save and Restart +Time From Start +导 +2 +24,434 +24,435 +24,436 +24,437 +24,438 +24,439 +24,440 +24,441 +24,442 +Time (s) +> +Figure 12 Bulk-out transfer test based on Cypress stream program. + +Figure 13 Bulk-out transfer test based on LeCroy adviser T3. +4. Summary +This paper proves that it is possible for USB 3.0 controllers, including PHY and SIE, based on +FPGA internal transceivers and FPGA logic, and the test result shows the speed of bulk-in and +bulk-out is more than 320 MB/s, and the bit error rate is less than 10-13. It is noticeable that the +solution provided by this paper can be well dealt with USB 3.0 communication with upper +computers. Still, if USB 2.0 compatibility is needed, an external USB 2.0 chip has to be added +to the circuit. +Acknowledgments +This project is supported by Beijing Natural Science Foundation (Grant No. 1214029). + +? C++ Streamer +口 +Connected Devices +(0x04B4 - 0x1003) Cypress FX2LP StreamerExample Device- +BULKOUT, +16384 Bytes, 15 MaxBurst (0 - 0x02) +Endpoint +[32 +Packets per Xfer +Successes +6115440 +16 +Xfers to Queue +Failures +0 +Timeout Per Xfer (ms) +1500 +Strrt +Transfer Rate (KBps) +339300Real-Time Statistics - [Teledyne LeCroy USB Protocol Suite - USB4 / 3.2 / 2 / PD] +(SN:12269) SuperSpeed Host L Data Payload Throughput & SuperSpeed Host R Data Payload Throughput +SN:12269 +USB 2.0 +Ch 0 +40 +Data Packets +00,000,000,000 +N/ +Data Bytes +00,000.000.000 +N/ +Total Bytes +00.000.000.000 +N/ +USB 3.2 +Endpoint Statistics (Address, Endpoint, Directic +250 +7, 2, Out +0, 0, Both +Activity +11 +Throughput 339.686 MB/s +82.498 KB/s +200 +Bytes +3.143 TB +840.303 MB +ACK +3069893184 +0 +Retry +0 +0 +150 +NRDY +0 +0 +ERDY +0 +0 +DP +3069072555 +820608 +DP Error +0 +0 +100 +TP +3069893184 +0 +TP Error +0 +0 + Link Statistics +8 +Left +Right +LBAD +0 +CRC-5 +0 +CRC-16 +49,440 +49,441 +49,442 +49,444 +Time (s) +0 +49.438 +49,439 +49,443 +49.445 +49.446 +CRC-32 +0 +> +, where ρ is an effective field responsible of the +strong interaction in the model used in Ref. [17]. +5 + +0 +100 +200 +300 +400 +500 +E [MeV] +10 +12 +10 +10 +10 +8 +10 +6 +10 +4 +10 +2 +100 +102 +104 +E2f (E )[MeV2] +k = 1 +e +e +, , +, +0 +100 +200 +300 +400 +500 +E [MeV] +10 +12 +10 +10 +10 +8 +10 +6 +10 +4 +10 +2 +100 +102 +104 +E2f (E )[MeV2] +k = 6 +e +e +, , +, +FIG. 1. +Energy distribution of electron (green) and muon/tau (black) neutrinos (solid) and antineutrinos +(dashed) as a function of the neutrino energy for shells k = 1 (left) and k = 6 (right) of Table I. +In Fig. 1, we have represented the energy distribution, E2 +νβf(Eνβ, µ∗ +νβ, T), for muon and tau neutrinos +(black) and for electron neutrinos (green solid) and antineutrinos (green dashed), as a function of the +neutrino energy for the first and sixth shells of the star (see Table I), on the left and right plots +respectively. +Since the chemical potential for muon/tau neutrinos and antineutrinos is zero, their +distribution functions coincide, fνµ,τ = f¯νµ,τ . However, for electron antineutrinos µ∗ +¯νe = −µ∗ +νe and, +therefore, in the inner shells, where the electron neutrino chemical potential reaches the highest values, +their distribution functions substantially differ, fνe ≫ f¯νe. This, in turn, leads to a negligible density +of electron antineutrinos in the inner shells, which suppresses the νe¯νe interactions. In the outer shells, +the electron neutrino and antineutrino chemical potentials significantly decrease in absolute value and +therefore fνe ∼ f¯νe ∼ fνµ,τ . Due to the high matter density and frequent collisions with the SN medium +inside the proto-NS, flavour conversions are expected to be suppressed. Neutrino oscillation becomes +possible only after neutrinos exit the core. +It should be pointed out that there are uncertainties in the duration of the neutrino signal from +SN 1987A, mainly associated with the low statistics of the observed neutrinos and the unknown relative +time offsets of the three detectors (Kamiokande II, IMB, and Baksan). Likewise, the SM prediction is +also subject to uncertainties, for example in the choice of the equation of state for nuclear matter. +The duration of the neutrino signal in the SM can be computed assuming that neutrino-nucleon +interactions are the leading contribution to the neutrino mean free path, λνβ. The neutrino diffusion +time can be computed as +c∆tνβ = +n +� +k=1 +� +R2 +k − R2 +k−1 +� � +1/λνβ +� +k , +(9) +where k = 1, ..., 6 are the different shells of Table I for a time after bounce of 1 second, Rk are the +radius of each shell, and +� +1/λνβ +� +k is the average of the inverse of the mean free path in each shell, +⟨1/λνβ⟩ = +� +dEνβf(Eνβ, µ∗ +νβ, T)E2 +νβλ−1 +νβ (Eνβ) +� +dEνβf(Eνβ, µ∗νβ, T)E2νβ +, +(10) +where Eνβ, EN, ⃗pνβ, ⃗pN are the energies and momenta of the incoming neutrino and nucleon, E′ +νβ, E′ +N, +⃗p′νβ, ⃗p′N are those of the outgoing states, and pνβ, pN, p′ +νβ, p′ +N are the corresponding four-momenta. +In this expression, β = e, µ, τ indicates the neutrino flavour, N = n, p refers to the nucleon states, i.e., +neutrons and protons, and +λ−1 +νβ (Eνβ) = +� +N +� +2 d3 ⃗ +pN +(2π)3 f(EN, µ∗ +N, T)| ⃗ +vνβ − ⃗vN|σνβ,NF(E′ +ν, E′ +N). +(11) +6 + +νβ +¯νβ +να +¯να +Z′ +νβ +¯να +νβ +¯να +Z′ +FIG. 2. +New physics contribution to neutrino-antineutrino scattering through a low-mass vector boson, Z′. +The diagram on the left represents the s-channel that is the leading contribution for the relevant range of +parameters. For the U(1)Lµ−Lτ model α, β = µ, τ while for U(1)B−L, α, β = e, µ, τ. +In this expression, | ⃗ +vνβ − ⃗vN| is the relative velocity between the neutrino and the target, σνβ,N is the +neutrino-nucleon scattering cross section, f(EN, µ∗ +N, T)) is the Fermi Dirac distribution function for +the incoming nucleon and F(E′ +ν, E′ +N) = (1 − f(E′ +ν, µ∗ +ν, T))(1 − f(E′ +N, µ∗ +N, T)) accounts for the Pauli +blocking of the outgoing states. +If we consider only SM interactions, we obtain ∆tνµ,τ +SM +∼ 1.3 s for muon and tau neutrinos and +∆tνe +SM ∼ 3 s for electron neutrinos. This leads to tE ∼ 10 s in Eq. (6), due to the fact that the stored +thermal energy in matter continues to be emitted in neutrinos even after the first neutrinos escape, +which is compatible with the observed duration of the neutrino signal from SN 1987A. +IV. +IMPACT OF RESONANT PRODUCTION OF NEW LIGHT MEDIATORS ON +NEUTRINO DIFFUSION +New physics in the neutrino sector can alter the neutrino-nucleus scattering cross section, thereby +leading to a longer neutrino burst. Compatibility with SN 1987A observation leads to upper bounds +on the relevant new physics couplings. However, as we explained in the Introduction, according to +Ref. [19] this does not apply to neutrino-neutrino or neutrino-antineutrino interactions. This argument +is valid as long as neutrinos follow an equilibrium distribution. However, this condition might not hold +if a new mediator is produced on-shell. In this section, we analyse two complementary regimes: when +the new gauge coupling is large (so that neutrino self-interactions are comparable or exceed the SM +contribution) and when it is small (so that the new mediator can travel a long distance inside the +proto-NS star or even exit before decaying back to neutrinos). +IV.1. +Large coupling regime +In this section, we discuss how on-shell Z′ production can affect the neutrino burst duration in +the relatively large coupling regime where the rate of neutrino-antineutrino interaction is comparable +or larger than the neutrino-nucleon interaction rate. +In Fig. 2, we show the neutrino-antineutrino +scattering diagrams via the new vector mediator, Z′. The cross section of this process can be greatly +enhanced through s-channel resonance when the mediator is produced on-shell2. This is very sensitive +to the energy distribution of neutrinos in the SN core of Fig. 1. +The t-channel contribution would be leading at very large values of the coupling (since it scales +with the fourth power of the coupling) and small mediator masses (due to collinear enhancement), +2 If the new gauge coupling is so large that Z′ reaches thermal equilibrium with the neutrino gas and the plasma, it +can in principle receive a share of the entropy content of the core. We may therefore wonder whether this significantly +reduces the temperature relative to the SM prediction. Since neutrinos below the so-called energy neutrinosphere are +in thermal equilibrium with the plasma, which act as a huge thermal energy source (i.e., Etot +th /Eν +th ≫ 1), the addition +of the three bosonic degrees of freedom associated with the three polarizations of the Z′ boson can only change the +temperature by a negligible amount suppressed by Eν +th/Etot +th +∼ 0.2. We therefore assume that around 1 s after the +bounce, the temperature profile of the core is similar to what is predicted within the SM even in the limit of large +coupling. +7 + +i.e., mZ′ ∼ 1 MeV for gµ−τ ∼ 0.1 or for gB−L ∼ 0.1, however, these regions of the parameter space +are generally ruled out by existing experimental and observational constraints. +We have explicitly +checked that, when the resonant condition is met, the main contribution to the neutrino-antineutrino +cross section is given by the s-channel diagram. Likewise, other processes such as neutrino-neutrino +scattering and neutrino-lepton scattering, both of which would proceed through t−channel, are sub- +leading. +The s-channel resonance in neutrino-antineutrino scattering is not relevant in the SM, since the +typical neutrino energies inside the proto-NS are not high enough to produce the Z boson on-shell. +Because of this, in the previous works where the impact of new neutrino interactions in SN dynamics +via light mediators was studied, such as Refs. [12, 14, 17], neutrino-antineutrino scattering was not +taken into account. However, in models with new gauge bosons in the MeV range, the resonance can +significantly increase the neutrino-antineutrino scattering cross section. The models that we consider +are lepton flavor conserving so νβ will fuse only with ¯νβ to produce an on-shell Z′. +The impact of scatterings of neutrinos off nucleons on the average diffusion time is quite different +from the effects of neutrino scattering off the background antineutrinos. Nucleons are heavy particles +with mass much larger than the temperature and therefore the energies of neutrinos in the SN core. +As a result, the scattering of a single neutrino (or antineutrino) off a nucleon changes its direction and +therefore prolongs its diffusion time as formulated in Eq. (6). However, when pairs of neutrino and an- +tineutrinos in a neutrino-antineutrino gas with isotropic velocity distribution scatter off each other, the +angular distribution of the velocities does not change. As shown in Ref. [19], this means that the diffu- +sion time will not be proportional to 1/λνβ ¯νβ→Z′→νβ ¯νβ ∼ Rνβ ¯νβ→Z′→νβ ¯νβ, where Rνβ ¯νβ→Z′→νβ ¯νβ is the +neutrino-antineutrino scattering rate, even when 1/λνβ ¯νβ→Z′→νβ ¯νβ ≫ 1/λνβ in which λνβ ¯νβ→Z′→νβ ¯νβ +and λνβ are respectively the mean free path of neutrinos due to scattering off antineutrinos and off +nucleons. However, in this limit, the neutrino self-scattering can in principle redistribute the energies +of neutrinos with a rate larger than that of scattering off the background electrons which brings the +distribution back to the thermal Fermi-Dirac distribution. +Let us consider a neutrino with energy +E1 ∼ T scattering off an antineutrino whose momentum makes an angle of θ with the direction of the +initial neutrino. In order for such a pair to produce an on-shell Z′, the antineutrino energy should be +E2 = +m2 +Z′ +2E1(1 − cos θ). +(12) +Simple kinematics3 show that the final neutrino and antineutrino from the Z′ decay will have flat +energy distribution in the range [(E1 + E2)(1 − vZ′)/2, (E1 + E2)(1 + vZ′)/2 in which vZ′ = (1 − +m2 +Z′/(E1 + E2)2)1/2. For E2 ∼ E1 ∼ T, the energies of the final particles will be of the same order as +those of the initial neutrinos so their diffusion time cannot be significantly altered. Let us consider the +limit that E2 ≫ E1. Considering that the scattering cross section of neutrinos off nuclei is proportional +to the square of the neutrino energy, this can in principle change the diffusion time. If the rate of +neutrino self-scattering is small, the impact will of course be tiny. On the other hand, in case the +rate of scattering off antineutrinos (or neutrinos) with energy E2 such that E2 > Elim ≫ T 4 is +larger than the rate of scattering off the electrons (which thermalizes back the energy distribution), +the higher energy tail of the neutrino distribution (with energies exceeding Elim) will be used up, +without having time to be replaced. Thus, independently of the value of the coupling, the fraction +of neutrinos or antineutrinos that come out of the thermal Fermi-Dirac distribution will be very tiny +and given by the ratio of the number density of antineutrinos with energy larger than Elim (nlim) +to the total number density of neutrinos. This can be understood with the following argument. The +dynamics of nlim can be described as ˙nlim = −ΓSM(nlim−neq +lim)−ΓNEW nlim where ΓSM is determined +by the rate of scattering off the electrons and neq +lim is the value of nlim computed with the Fermi- +Dirac distribution neq +lim = +� ∞ +Elim f(Eν, −µ∗ +ν, T)E2 +νdEν. The asymptotic (stable) value of nlim will be +given by nasym +lim += neq +limΓSM/ΓNEW . Let us denote the number density of neutrinos scattered up to +higher energies out of the Fermi-Dirac distribution by nout. The evolution of nout can be written as +3 If Z′ scatters multiple times before decay, it will reach thermal equilibrium with the neutrinos and this argument does +not therefore apply. Thermalization of Z′ however requires σ(Z′ν → Z′ν)nν/ΓZ′ > 1 which can be achieved only for +couplings larger than 1. +4 In this discussion, Elim is taken an arbitrary value which Elim ≫ T. +8 + +˙nout = −Γ′ +SMnout + ΓNEW nasym +lim . Thus, the asymptotic solution which shows the density of neutrinos +kicked out of equilibrium is nout = (ΓSM/Γ′ +SM)neq +lim. For Elim ≫ T this quantity is suppressed by a +factor +� ∞ +Elim f(Eν, µ∗ +ν, T)E2 +νdEν +� ∞ +0 +f(Eν, −µ∗ν, T)E2νdEν +≪ 1. +(13) +A less extreme scenario is the one where E1 = πT −∆ and E2 as in Eq. (12), where 0 < ∆ < πT is an +arbitrary energy. For example, let us consider muon and tau neutrinos with energies of E1 = 15 MeV, +interacting with antineutrinos with energies E2 = m2 +Z′/(30 MeV(1 − cos θ)). If the mediator mass is +mZ′ = 50 MeV, the antineutrino energies must be E2 ≳ 41.7 MeV in order to produce the mediator +on-shell. From the left panel of Fig. 1, we can see that, for these neutrino and antineutrino energies, +E2 +νfν(Eν) can be of order 1 and 100, respectively, in the first shell of the star. After the interaction, +ν1 will gain energy and therefore its interactions with nucleons will be stronger, since σν,N ∝ E2 +ν. As a +result, ν1 will be more bounded to stellar matter and stay for longer in the star. +This picture is consistent with what is shown in Ref.[49] for a scenario of relatively strong DM-nucleon +and DM-neutrino interactions inside SN. In this case, DM particles are thermalised with the stellar +material (as our neutrinos and antineutrinos are) and neutrino-DM interactions bound neutrinos to the +star out to larger radii and lower temperatures, increasing the neutrino diffusion time. Note that in +page 20 of Ref. [50], they comment about these results and argue that what Ref.[49] obtained is similar +to what would be expected from simply increasing the strength of neutrino interactions with regular +matter. Besides, they compare this scenario with the one where DM is not thermalised with the stellar +material and they state that in the latter case strong DM-neutrino interactions are similar to strong +neutrino self-interactions in SN since they both involve the emission of a strongly-coupled gas, and +hence strong DM-neutrino interactions do not significantly affect the cooling time for SN. +The lower energy tail of the spectrum can therefore be scattered up to higher energies via Z′ resonance +interaction. We should however remember that the fraction of muon or tau neutrinos with Eν < T +(Eν < T/3) is only 8 % (0.48%) of the whole number density. Thus, even if all of neutrinos in the low +energy tail scatter up, the impact on the diffusion time is not observable in SN 1987A data. A more +detailed analysis is needed to determine whether this can be a noticeable feature in the event of future +SN detection with a greatly improved statistics. +In the following, we would like to study the regions of the parameter space for the two well-motivated +U(1)Lµ−Lτ and U(1)B−L models where this effect could take place. +IV.1.1. +Implications for the parameter space of the U(1)Lµ−Lτ and U(1)B−L models +To find the regions where neutrino-antineutrino interactions are as frequent or more than neutrino- +nucleon interactions, we need to calculate the neutrino-antineutrino scattering rate via the new Z′ +mediator. The average of the neutrino-antineutrino scattering rate can be calculated by integrating the +scattering cross section as follows, +⟨R⟩νβ ¯νβ→Z′→να¯να = +� +dEνβf(Eνβ, µ∗ +νβ, T)E2 +νβRνβ ¯νβ→Z′→να¯να(Eνβ) +� +dEνβf(Eνβ, µ∗νβ, T)E2νβ +, +(14) +where β = e, µ, τ indicates the neutrino flavour and +Rνβ ¯νβ→Z′→να¯να(Eνβ) = +� d3⃗p¯νβ +(2π)3 f(E¯νβ, µ∗ +¯νβ, T)|⃗vνβ − ⃗v¯νβ|σνβ,¯νβ +(15) +is the neutrino-antineutrino scattering rate via Z′. Note that we are considering massless neutrinos +and therefore |⃗pνβ| = Eνβ. +For the νβ¯νβ → Z′ → να¯να process, the factor |⃗vνβ − ⃗v¯νβ|σνβ,¯νβ can be written as +|⃗vνβ − ⃗v¯νβ|σνβ,¯νβ = +� +d3⃗p′να +(2π)32E′να +� +d3⃗p′¯να +(2π)32E′ +¯να +(2π)4δ(4)(pνβ + p¯νβ − p′ +να + p′ +¯να) +|M|2 +νβ,¯νβ +4EνβE¯νβ +F(E′ +να, E′ +¯να), +(16) +9 + +where Eνβ, E¯νβ, ⃗pνβ, ⃗p¯νβ are the energies and momenta of the incoming particles, E′ +να, E′ +¯να, ⃗p′να, +⃗p′¯να are those of the outgoing states, and pνβ, p¯νβ, p′ +να, p′ +¯να are the corresponding four-momenta. +The factor F(E′ +να, E′ +¯να) = (1 − f(E′ +να, µ∗ +να, T))(1 − f(E′ +¯να, µ∗ +¯να, T)) accounts for Pauli blocking in the +outgoing states. +Since, in the parameter space under analysis, the resonant production of the Z′ boson is the leading +new physics process and ΓZ′→¯νβνβ/mZ′ ≪ 1 is fulfilled we can use the narrow width approximation +(NWA) in order to perform the calculation of the neutrino-antineutrino interaction rate for these new +physics interactions, Rνβ ¯νβ→Z′→να¯να(Eνβ). In this limit, it can be shown that +Rνβ ¯νβ→Z′→να¯να(Eνβ) = +1 +32π +� ∞ +Emin +¯νβ +dE¯νβ +f(E¯νβ, µ∗ +¯νβ, T)E¯νβ +E¯νβ + Eνβ +� +mZ′ +E¯νβEνβ +�2 +|M|2 +νβ ¯νβ→Z′ ΓZ′→να¯να +Γtot +Z′ +, (17) +with Emin +¯νβ += m2 +Z′/(4Eνβ) and Γtot +Z′ = � +α ΓZ′→¯νανα + ΓZ′→β+β−, where β = µ, τ for the U(1)Lµ−Lτ +model and β = e, µ, τ for the U(1)B−L model. Note that the decay into taus and mesons will not be +open for the values of the mediator masses considered in this work. The squared amplitudes can be +written as |M|2 +Z′→νβ ¯νβ = g2 +µ−τm2 +Z′/2 for the U(1)Lµ−Lτ model and |M|2 +Z′→νβ ¯νβ = g2 +B−Lm2 +Z′/2 for the +U(1)B−L model. Here we are neglecting Pauli blocking for neutrinos and antineutrinos. For muon and +tau neutrinos, since their chemical potential vanishes, it is clear that (1 − f(E′ +¯ν, µ∗ +¯ν, T)) ∼ 1. In the +case of electron neutrinos, for the relevant range of outgoing energies in the integration, Pauli blocking +does not have an important impact in our final results,5 except for the case νµ¯νµ → νe¯νe which is +totally Fermi blocked. For this reason, in our calculations, we do not take into account νµ¯νµ → νe¯νe +interactions in the first forth shells of the star. +In principle, one can think that the annihilation of neutrino-antineutrino into muons via s-channel +Z′ diagram may have an impact on the physics of the proto-NS. However, these processes generate the +same amount of muons and antimuons, not changing their chemical potential, and therefore we do not +expect the equation of state of the nuclear matter to change. Besides, even though muon neutrinos +(antineutrinos) can scatter on antimuons (muons), as the muon density and its scattering cross section +with muon neutrinos and antineutrinos are much smaller than that of nucleons, the mean free path +of scattering on muons (antimuons) is negligible compared to the scattering on nucleons. The only +effect we could expect comes from the decay of muons into muon neutrinos close to the neutrinosphere, +which can change equality between the muon neutrino and tau neutrino fluxes that come out of the +neutrinosphere, having consequences for collective neutrino oscillation [51–55]. +For completeness, we have also explicitly checked that processes at one loop, such as neutrino- +antineutrino annihilation into µ+µ− and the subsequent µ+µ− → ν¯ν are sub-leading for the calculation +of the neutrino-antineutrino scattering rate. +In this subsection, we show the parameter space from which the rate of neutrino-antineutrino inter- +action can exceed that of the neutrino scattering off nucleons for the two specific models, U(1)Lµ−Lτ +and U(1)B−L, one by one. As we mentioned above, these regions cannot be ruled out using SN 1987A +data, but they are indicative of the regions that might be explored when a better measurement of the +SN neutrino flux (perhaps with better statistics from future detectors) is available. We shall compare +them with the range ruled out with different observables and considerations. +• U(1)Lµ−Lτ +In this scenario, the new neutrino couplings to electrons and nucleons are extremely suppressed, +and only the contributions to νµ,τ scattering have to be included in Fig. 2. The mean free path +of electron neutrinos is not modified, and therefore charged-current β-processes do not need to +be taken into account. +We have represented in Fig. 3 the values of gµ−τ as a function of mZ′ for which the new physics +contribution to the neutrino scattering becomes as important as the SM one for each of the six +5 This is equivalent to what we observed in Ref. [17] when comparing SN bounds for the lepton number violating (where +the chemical potential vanishes and there is no Pauli blocking) and lepton number conserving models with scalar +mediators for the same incoming neutrino energies. +10 + +100 +101 +102 +103 +mZ′ [MeV] +10 +7 +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +g +Neff +White Dwarfs +(g +2) +CHARM-II +COHERENT CsI +COHERENT Ar +BaBar 4 +L +L +k=1 +k=2 +k=3 +k=4 +k=5 +k=6 +FIG. 3. Values of (gµ−τ, mZ′) for which the inverse neutrino mean-free path of Eq. (17) equals the SM value +in each of the proto-NS shells of Table I. The band in red shows the region in the parameter space consistent +with the (g − 2)µ measurement. Gray areas show different complementary bounds from Neff [13], Borexino +[56], white dwarf cooling [57], COHERENT CsI [58] and LAr [56], neutrino tridents (Charm-II) [29, 59] and +BaBar [60]. +shells described in Table I. We should emphasize that, contrary to what might be inferred from +Ref. [41] the region compatible with (g − 2)µ is not yet ruled out. +The inner shells of the proto-NS (k = 1 − 4) are characterised by high temperatures and nucleon +densities. +These regions give the largest contribution to the neutrino scattering cross section +both in the SM and when new physics is added. Due to the larger available neutrino energies, +in these inner regions the resonant condition for the light mediator holds for larger masses (the +tail of the neutrino distribution function - albeit very small - is still considerably larger than in +the outer regions as we showed in Fig. 1). From these lines, we see that a coupling of the order +of gµ−τ ∼ 10−4 − 10−5 is sufficient for the new physics diagrams to be comparable to the SM +contribution. A small kink at 210 MeV= 2mµ can be observed in all the lines. This is due to the +opening of the Z′ → µ+µ− decay channel, which slightly increases the total Z′ decay width in +Eq. (17), reducing the cross section. +In the outer shells (k = 5, 6) the nucleon density is considerably smaller (the neutrinosphere +is located at ∼ 20 km for muon and tau neutrinos). This leads to a large suppression of SM +processes (neutrino-neutron scattering) and it makes it easier for the new physics to dominate +(considerably reducing the values of gµ−τ at which this occurs). Also, the temperature is much +smaller in these regions, which means that the resonant condition can only be fulfilled for small +masses of mZ′. For this reason, the lines at which the new physics equals the SM contributions +move to lower values of gµ−τ and mZ′. However, the time scale of neutrino diffusion from these +two last layers up to the neutrinosphere is much smaller than one second, and therefore they +are not relevant for the diffusion time. What sets the time scale of cooling of the outer shells is +the graybody ”surface” cooling. Since the new interaction does not absorb νx, it cannot affect +the grayness. For comparison, we show the existing bounds on the parameter space from various +observations. +11 + +100 +101 +102 +103 +mZ′ [MeV] +10 +7 +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +gB +L +NA64 +E141 +CHARM +II +Orsay +BaBar +B +L, +, +k=1 +k=2 +k=3 +k=4 +k=5 +k=6 +100 +101 +102 +103 +mZ′ [MeV] +10 +7 +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +gB +L +NA64 +E141 +CHARM +II +Orsay +BaBar +B +L, +e +k=1 +k=2 +k=3 +k=4 +k=5 +k=6 +FIG. 4. The same as in Fig. 3, but for the U(1)B−L model. The plot on the left only considers muon and tau +neutrinos and the plot on the right only considers electron neutrinos. In grey we show complementary bounds +from different experiments [57]. +• U(1)B−L +In this scenario, electron neutrinos can also interact via the new force and the Z′ also mediates +neutrino-nucleon and neutrino-electron interactions. However, since the muon and tau neutrino +chemical potentials are zero, neutrino-antineutrino scattering is more efficient for muon/tau neu- +trinos. In particular, in the inner shells of the star the number density of electron antineutrinos is +negligible and therefore these interactions are significantly suppressed. Since the electron neutrino +chemical potential decreases with the radial distance to the centre of the star, electron neutrino- +antineutrino scatterings become more frequent in the outer shells of the star, where the rate of +νe¯νe scattering becomes comparable to the one for muon/tau neutrinos. +Besides, for ∼ 1 after the bounce, neutrino-electron interactions are smaller than the neutrino- +nucleon scatterings, both in the SM [45] and for our new physics interactions. Note that ne ≪ nn. +In Fig. 4, we show the values of gB−L as a function of the mediator mass for which the new +physics contribution to the neutrino scattering becomes as important as the SM one for each of +the six shells of the proto-NS. The plot on the left corresponds to muon and tau neutrinos and +it is analogous to the U(1)Lµ−Lτ case of Fig. 3. The plot on the right represents the results for +electron neutrinos. Since the electron antineutrino abundance in the inner shells of the proto-NS +is largely suppressed by its chemical potential, the new physics contribution is small in these +regions and a very large coupling is needed to match the SM value. +IV.2. +Small coupling regime +In this section, we focus on the range of parameter space with very small couplings, where the decay +length of Z′ can be comparable to or larger than the size of the proto-NS core. If Z′ decays back +only into ν¯ν, this would lead to neutrinos effectively escaping earlier, shortening the length of the SN +neutrino burst. Similarly to Eq. (17), the rate of the scattering of a single ν off any ¯ν in the medium +can be computed via the following relation, +Rνβ ¯νβ→Z′ = +1 +32π +� ∞ +Emin +¯νβ +dE¯νβ +f(E¯νβ, µ∗ +¯νβ, T)E¯νβ +E¯νβ + Eνβ +� +mZ′ +E¯νβEνβ +�2 +|M|2 +νβ ¯νβ→Z′. +(18) +The time scale of the interaction can be defined as τνβ ¯νβ→Z′ = c/Rνβ ¯νβ→Z′. +12 + +The decay length of Z′, ℓZ′, is given by lifetime multiplied by the relativistic boost factor, γZ′ = +EZ′/mZ′, averaged over the energy of Z′, which is set by the energies of the neutrino and antineutrino +producing the mediator on-shell, +ℓZ′ = +� d3⃗pνβ +(2π)3 f(Eνβ, µ∗ +νβ, T) +� d3⃗p¯νβ +(2π)3 f(E¯νβ, µ∗ +¯νβ, T)σνβ ¯νβ→Z′| ⃗ +vνβ − ⃗v¯νβ|γZ′vZ′ +ℏ +Γtot +Z′ +� d3⃗pνβ +(2π)3 f(Eνβ, µ∗νβ, T) +� d3⃗p¯νβ +(2π)3 f(E¯νβ, µ∗ +¯νβ, T)σνβ ¯νβ→Z′| ⃗ +vνβ − ⃗v¯νβ| +, +(19) +where vZ′ is the velocity of Z′ and σνβ ¯νβ→Z′ is the cross section for the νβ¯νβ → Z′ interaction. Since the +temperature and therefore the energy distribution of neutrinos and antineutrinos producing Z′ depend +on the distance from the center, ℓZ′ also varies with the distance from the center. We have computed +ℓZ′ for each shell described in Table I. +As we shall see, there is a range of couplings at which τν¯ν→Z′ for muon and tau neutrinos is much +shorter than the typical time scale of neutrino diffusion time while ℓZ′ is larger than (or of the same +order as) the radius of the neutrinosphere. Thus, neutrinos and antineutrinos can coalesce into an +on-shell Z′ as they diffuse out of the proto-NS. The Z′ travels unhindered outside the neutrinosphere +and decays back into a ν¯ν pair outside (or close to) the neutrinosphere, where ν and ¯ν can straightly +come out of the star without scattering. The outcome is that the average diffusion time for νµ and ντ +and their antiparticles deep inside the core can be reduced to τνµ,τ ¯νµ,τ →Z′, which is much smaller than +the typical diffusion time. At a snapshot of ∼ 1 s after the bounce (at the onset of the cooling phase), +the energy of the νµ, ¯νµ, ντ and ¯ντ gases altogether, Eνµ,τ +th +account for about 2% of the total thermal +energy Etot +th of the core. Likewise, the energy stored in the νe gas in the core, Eνe +th is about 10 % of the +total thermal energy. +Considering that neutrinos are in thermal equilibrium with each other as well as with matter inside +the core, the time scale over which the total binding energy can be transferred outside via Z′ can be +estimated as +τ µ,τ +trans = τνµ,τ ¯νµ,τ →Z′ Etot +th /Eνµ,τ +th +∼ 50 τνµ,τ ¯νµ,τ →Z′ , +(20) +τ e +trans = τνe¯νe→Z′ Etot +th /Eνe +th ∼ 10 τνe¯νe→Z′. +(21) +If τtrans is of the order of 10 s (the SM prediction of the neutrino burst), the time duration can be +significantly reduced. However, the change may be hidden within the uncertainties in the duration +measurement and prediction as already discussed in Section III. For τtrans ≪ 10 s, the predicted +duration will be so short that it can already be ruled out via the measurement of the SN 1987A +neutrino burst duration. +This has two important consequences. First, the energy spectrum of neutrinos reaching the Earth +will be harder than what expected in the SM. This feature has been discussed within the Majoron +model in Refs. [21, 31]. However, the total energy carried away via neutrinos will be the same as what +predicted in the SM which amounts to the binding energy of the star. Thus, the cooling argument (that +is setting luminosity in form of Z′ equal to neutrino luminosities in the SM) is therefore not applicable, +although it points towards the same part of parameter space as our burst duration argument does +[11, 61]. +The cooling argument is only valid for massless mediators which cannot decay back into +neutrinos [9, 10, 62, 63]. +Second, the flavor composition of neutrinos transferred by Z′ can be different from the SM prediction. +For example, in the U(1)Lµ−Lτ model, neutrinos reaching out of the neutrinosphere will be composed of +νµ, ντ, ¯νµ and ¯ντ with equal spectra and vanishing νe and ¯νe components, up to the corrections from the +contributions from the standard neutrino diffusion from the neutrinosphere. Of course, these neutrinos +will then go through flavor conversion traversing the outer shells of the star (and if ℓZ′ is larger than +the star size, in the vacuum between the star and the Earth). Fluxes of νe and ¯νe will be produced via +oscillation but the ratio between electron neutrino and muon and tau neutrino fluxes, i.e., Fνe/Fνµ,τ , +as well as the equivalent for electron antineutrinos, F¯νe/Fµ,τ, at Earth will be different from the SM +prediction. Within the U(1)B−L model, Z′ can decay into νe¯νe, νµ¯νµ and ντ ¯ντ in equal amounts so we +expect Fνe = F¯νe = Fνµ = F¯νµ = Fντ = F¯ντ both at a distance of ℓZ′ from the core center and at the +Earth. Notice that even oscillation (or matter effects in the star) cannot alter this flavor democracy +because of the unitarity of the PMNS matrix and the fact that � +β P(να → νβ) = � +β P(¯να → ¯νβ) = 1, +where P(να → νβ) and P(¯να → ¯νβ) are the conversion probabilities of neutrinos and antineutrinos of +flavour α to neutrinos and antineutrinos of flavour β . +13 + +10 +7 +10 +5 +10 +3 +10 +1 +101 +103 +Z′ [s] +mZ′ = 10 MeV +L +L +10 +10 +10 +9 +10 +8 +10 +7 +g +10 +3 +10 +1 +101 +103 +Z′ [km] +k=1 +k=2 +k=3 +k=4 +k=5 +k=6 +10 +7 +10 +5 +10 +3 +10 +1 +101 +103 +Z′ [s] +mZ′ = 50 MeV +10 +10 +10 +9 +10 +8 +10 +7 +g +10 +3 +10 +1 +101 +103 +Z′ [km] +10 +7 +10 +5 +10 +3 +10 +1 +101 +103 +Z′ [s] +mZ′ = 100 MeV +10 +10 +10 +9 +10 +8 +10 +7 +g +10 +3 +10 +1 +101 +103 +Z′ [km] +FIG. 5. +The time scale of the neutrino-antineutrino interaction producing the Z′ vector mediator on-shell and +the decay length of Z′ as a function of the coupling for the U(1)Lµ−Lτ model in the upper and lower panels, +respectively. We show in different colours the corresponding values for each shell of the star. Different masses +are depicted in the three columns showed. In particular, mZ′ = 10, 50, 100 MeV for the left, central and right +columns. For reference, the dotted lines correspond to the cases for which the timescale of the interaction is +0.02 s, and where the decay length of the mediator is equal to the neutrinosphere radius. +IV.2.1. +Implications for the parameter space of the U(1)Lµ−Lτ and U(1)B−L models +• U(1)Lµ−Lτ +In the upper panels of Fig. 5, we represent the time scale of νµ¯νµ → Z′ or ντ ¯ντ → Z′ inside +different shells of the proto-NS as a function of the gauge coupling, for various values of the +mediator mass (mZ′ = 10, 50, 100 MeV). The horizontal dotted line corresponds to τν¯ν→Z′ = +0.02 s, which is equivalent to τtrans = 1 s for muon and tau neutrinos. The lower panels show +ℓZ′ for the different shells and the same choice of mediator masses, and the horizontal dotted +line highlights the value of the neutrinosphere at 20 km. +From these plots we can infer the +range of values of the gauge coupling for which τν¯ν→Z′ ≤ 0.02 s and ℓZ′ is equal or larger to the +neutrinosphere radius6. +Notice that ℓZ′ is shorter in the outer shells because the Z′ produced with smaller temperatures +have a smaller boost factor, EZ′/mZ′. Also, the ν¯ν → Z′ interaction rate is maximised when the +temperature of the layer is similar to ∼ mZ′/2π (since the average of neutrino and antineutrino +energies for a given temperature of the star is ∼ πT) and consequently, τν¯ν→Z′ becomes smaller +in those shells. This behaviour can be observed in the three columns of Fig. 5. In particular, +the value of τν¯ν→Z′ in the outer layer increases with mZ′, since the temperature is lower and Z′ +production is suppressed by the Boltzmann factor. +The shaded region in Fig. 6 corresponds to the area of the (gµ−τ, mZ′) parameter where the +conditions τtrans < 1 s and ℓZ′ < 20 km are simultaneously satisfied. From our discussion above, +the shortening of the neutrino burst duration is so severe in this area that it can be already ruled +out by the SN 1987A measurement. +6 Although we should also expect a reduction in the neutrino diffusion time when ℓZ′ ∼ 1 km, the precise calculation of +this reduction is out of the scope of this work and is left for the future. +14 + +100 +101 +102 +103 +mZ′ [MeV] +10 +10 +10 +9 +10 +8 +10 +7 +10 +6 +g +Neff +Z′ = Rns +Z′ = 0.1 Rns +trans = 1 s +trans = 10 s +L +L +k=1 +k=2 +k=3 +FIG. 6. From top to bottom, values of the coupling as a function of the mediator mass for which the mediator +decay length is equal to the radius of the neutrinosphere (in solid lines) and the ones for which τtrans is equal +to 1 s (solid) and 10 s (dashed dotted) for the U(1)Lµ−Lτ model. We show this for the three inner shells of the +star. The coloured areas indicate the regions of the parameter space where the decay length is larger than the +neutrino sphere radius and τtrans is smaller than 1 s for each shell. In thicker lines we can see the lower limit +for which this happens in every shell. In grey we show cosmological bounds from Big Bang nucleosynthesis [13]. +10 +7 +10 +5 +10 +3 +10 +1 +101 +103 +Z′ [s] +mZ′ = 10 MeV +B +L +10 +10 +10 +9 +10 +8 +10 +7 +gB +L +10 +3 +10 +1 +101 +103 +Z′ [km] +k=1 +k=2 +k=3 +k=4 +k=5 +k=6 +10 +7 +10 +5 +10 +3 +10 +1 +101 +103 +Z′ [s] +mZ′ = 50 MeV +10 +10 +10 +9 +10 +8 +10 +7 +gB +L +10 +3 +10 +1 +101 +103 +Z′ [km] +10 +7 +10 +5 +10 +3 +10 +1 +101 +103 +Z′ [s] +mZ′ = 100 MeV +10 +10 +10 +9 +10 +8 +10 +7 +gB +L +10 +3 +10 +1 +101 +103 +Z′ [km] +FIG. 7. +The same as in Fig. 5 for the U(1)B−L model and only considering muon and tau neutrinos. +We also represent with dashed-dotted lines and dotted lines the values of the coupling for which +τtrans < 10 s and ℓZ′ < 2 km, respectively. We can expect that in the region between these lines +the duration and spectrum of the neutrino burst will also be modified with respect to the SM +prediction, although in a more subtle way. This region is susceptible to be tested in future SN +15 + +10 +3 +100 +103 +106 +109 +Z′ [s] +mZ′ = 10 MeV +B +L, +e +10 +10 +10 +9 +10 +8 +10 +7 +gB +L +10 +2 +100 +102 +104 +Z′ [km] +k=1 +k=2 +k=3 +k=4 +k=5 +k=6 +10 +3 +100 +103 +106 +109 +Z′ [s] +mZ′ = 50 MeV +10 +10 +10 +9 +10 +8 +10 +7 +gB +L +10 +2 +100 +102 +104 +Z′ [km] +10 +3 +100 +103 +106 +109 +Z′ [s] +mZ′ = 100 MeV +10 +10 +10 +9 +10 +8 +10 +7 +gB +L +10 +2 +100 +102 +104 +Z′ [km] +FIG. 8. The same as in Fig. 7 but for electron neutrinos. For reference, the dotted lines correspond to the +cases for which the timescale of the interaction is 0.1 (which corresponds to τtrans = 1 s), and where the decay +length of the mediator is equal to the neutrinosphere radius. +100 +101 +102 +103 +mZ′ [MeV] +10 +10 +10 +9 +10 +8 +10 +7 +10 +6 +gB +L +Neff +U70 +E137 +E137 +Z′ = Rns +Z′ = 0.1 Rns +trans = 1 s +trans = 10 s +B +L, +, +k=1 +k=2 +k=3 +100 +101 +102 +103 +mZ′ [MeV] +10 +10 +10 +9 +10 +8 +10 +7 +10 +6 +gB +L +Neff +U70 +E137 +E137 +Z′ = Rns +Z′ = 0.1 Rns +trans = 1 s +trans = 10 s +B +L, +e +k=1 +k=2 +k=3 +FIG. 9. The same as in Fig. 6, but for the U(1)B−L model. The plot on the left only considers muon and tau +neutrinos and the plot on the right only considers electron neutrinos. In grey we show complementary bounds +from different experiments (see Ref. [57] for more details). +data with more statistics. +• U(1)B−L +In this case, the Z′ can also decay into νe¯νe and e−e+. For 1 MeV ≪ mZ′ < 200 MeV, Br(Z′ → +νe¯νe) = 1/5 and Br(Z′ → e−e+) = 2/5. +Figs. 7 and 8 show the neutrino-antineutrino interaction timescale for νµ,τ and νe, respectively, +along with the corresponding values of ℓZ′. In the upper panels the dotted lines correspond to +τν¯ν→Z′ = 0.02 s for muon and tau neutrinos and τν¯ν→Z′ = 0.1 s for electron neutrinos, which is +equivalent in both cases to τtrans = 1 s. In the lower panels, the dotted lines indicate ℓZ′ > 20 km. +For νe, the conditions τν¯ν→Z′ < 0.1 s and ℓZ′ > 20 km are not simultaneously satisfied. The +16 + +reason is that the ¯νe density in the inner core is very suppressed. Thus, in the U(1)B−L model, +the shortening of the burst takes place due to νµ¯νµ → Z′ and ντ ¯ντ → Z′. +The resulting bounds on the (gµ−τ, mZ′) parameter space are shown in Fig. 9, for muon and tau +neutrinos (left panel) and electron neutrinos (right panel). The shaded area corresponds to the +are where τtrans < 1 s and ℓZ′ < 20 km, which rules out a new region of the U(1)B−L parameter +space. On the right panel, we can see that no region of the U(1)B−L model can be excluded by +only considering electron neutrinos. As we did for the U(1)Lµ−Lτ case, we also show the lines +where τtrans < 10 s and ℓZ′ < 2 km, which might also lead to an observable modification of the +neutrino signal in future SN data. +Finally, even if testing this is out of the scope of this work, it must be noticed that the decay of +Z′ into e−e+ outside the neutrinosphere can warm up the outer shells and create shock waves +with drastic consequences for the SN explosion and the photon flux reaching the Earth. +Within the U(1)Lµ−Lτ and the U(1)B−L models for mZ′ ∼ few × 10 MeV and gµ−τ ∼ 10−9 − 10−8, +muon and tau neutrino-antineutrino pairs from the hotter inner core can be transferred outside the +proto-NS so the spectrum of neutrinos reaching the Earth will be harder. +Such prediction can in +principle be tested by observation. This also increases the prospect of detection of diffuse SN neutrino +flux as for this detection, the energy threshold is a limiting factor. Besides, within the U(1)B−L model, +the possibility of Z′ decay to e−e+ pair right outside proto-NS can have drastic consequences for the +thermodynamics of the star beyond the neutrinosphere and subsequently for shock revival. These effects +merit a dedicated analysis which is left for future work. +V. +CONCLUSIONS +In this article, we have investigated the resonant production of low-mass vector mediators from +neutrino-antineutrino interactions in the core of proto-NS, determining the conditions under which +these processes can significantly alter neutrino diffusion in SN explosions. +We point out that the +resonant production of a long-lived Z′ and its subsequent decay into neutrinos can significantly shorten +the duration of the observed neutrino burst, providing a new way to test neutrino self-interactions. +As a concrete example, we have computed the rate of neutrino self-interaction in two well-motivated +new physics scenarios that feature a new vector mediator, namely U(1)B−L and U(1)Lµ−Lτ . In order to +calculate the rate of ν¯ν → (Z′)∗ → ν¯ν in the nuclear medium, we have considered the radial dependence +of the density, temperature and chemical potentials of the particles inside the proto-NS. We have found +that thanks to their vanishing chemical potential, the rate of muon or tau neutrino self-interaction can +exceed that of the Standard Model scattering off nucleons when the resonant condition for Z′ is fulfilled, +even for couplings as small as gµ−τ ∼ 10−6 −10−5. Within the U(1)B−L model, electron neutrinos also +receive new contributions to their scatterings but the effect is relevant only in outer shells, where the +chemical potential of νe is lower and the ¯νe density is therefore higher. +We have first discussed that, contrary to previous claims, in the range of parameter space where +neutrino-antineutrino interactions dominate over the SM neutrino-nucleon scattering, the diffusion +time does not significantly change, according to the argument of Ref. [19]. This is because the velocity +distributions of neutrinos inside the core are isotropic. In particular, the U(1)Lµ−Lτ solution for the +muon magnetic moment anomaly is not ruled out for this reason. We have then examined whether +frequent self-interaction can redistribute energy of neutrinos and antineutrinos, leading to a change in +the burst duration. However, we found out that the effects of energy redistribution are too small to +lead to a discernible impact in the SN 1987A data. More work is needed to determine whether the +subtle changes in the neutrino spectrum can be observable with future SN data. +Finally, for the first time, we have pointed out that, for small values of the couplings where the decay +length of Z′ is comparable or larger than the size of the proto-NS, the neutrino burst duration can be +significantly reduced. Neutrinos and antineutrinos from the inner core can produce on-shell Z′ bosons, +which leave the core unimpeded and then decay back to neutrino-antineutrino pairs. Since the total +binding energy of the star (∼ 1053 erg) is still transferred outside the star in the form of neutrinos, the +canonical cooling bound does not apply. However, three observable features of the neutrino burst can +be significantly altered: its duration, the energy spectrum, and the flavour composition. +17 + +We have focused on the first effect, showing that for 5 MeV < mZ′ < 200 MeV, there is an area of the +parameter space with couplings of the order of 10−10 − 10−8 where the duration of the neutrino burst +is shorter than 10 s and the decay length of the Z′ exceeds the proto-NS radius. We have determined +the region of parameter space where the predicted neutrino burst duration is less than 1 s, so that +the deviation from the observed duration of SN 1987A burst is significant enough. This rules out new +regions of the parameter space of both the U(1)Lµ−Lτ and U(1)B−L models. Future SN data, with +more statistics and a better measurement of the neutrino spectrum, together with improvements on +the SN models, might allow to test a larger part of parameter space predicting time duration shorter +than 10 s. +These results are relevant for any other model that features new MeV scale mediators that couple to +the neutrino sector such as light mediator models with NSI. +ACKNOWLEDGEMENTS +We would like to thank Juan Antonio Aguilar-Saavedra, Rafael Aoude, Ivan Esteban, Patrick Fold- +enauer, Luca Mantani, and M. ´Angeles P´erez-Garc´ıa, for useful discussions and comments. +DGC +acknowledges support from the Spanish Ministerio de Universidades under grant SI2/PBG/2020- +00005. +The work of M.C. was partially funded by the F.R.S.-FNRS through the MISU conven- +tion F.6001.19. +YF would like to acknowledge support from the ICTP through the Associates +Programme and from the Simons Foundation through grant number 284558FY19 and from Sara- +madan under contract No. ISEF/M/401439. 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D +45 (1992) 1557–1568. +20 + diff --git a/fdAyT4oBgHgl3EQfxPkj/content/tmp_files/load_file.txt b/fdAyT4oBgHgl3EQfxPkj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..87dc7f72ae213c0cd05c77e0407a3a2163fc84a2 --- /dev/null +++ b/fdAyT4oBgHgl3EQfxPkj/content/tmp_files/load_file.txt @@ -0,0 +1,1025 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf,len=1024 +page_content='IFT-UAM/CSIC-22-130 FTUAM-22-2 Constraints from the duration of supernova neutrino burst on resonant light gauge boson production by neutrinos David Cerde˜no,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' ∗ Marina Cerme˜no,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' † and Yasaman Farzan4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' ‡ 1Instituto de F´ısica Te´orica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' IFT-UAM/CSIC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 28049 Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Spain 2Departamento de F´ısica Te´orica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Universidad Aut´onoma de Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 28049 Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Spain 3Centre for Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Particle Physics and Phenomenology (CP3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Universit´e catholique de Louvain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Chemin du Cyclotron 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='B-1348 Louvain-la-Neuve,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Belgium 4School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Institute for Research in Fundamental Sciences (IPM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Box 19395-5531, Tehran, Iran In this article, we study the resonant production of low-mass vector mediators from neutrino- antineutrino interactions in the core of proto-neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We show that this process can signif- icantly alter neutrino diffusion in the first seconds after the supernova explosion, providing a new way to test neutrino self-interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Taking into account the radial dependence of the density, energy, and temperature inside the proto-neutron star, we compute the neutrino-antineutrino inter- action rate in the star interior in two well-motivated new physics scenarios, namely U(1)B−L and U(1)Lµ−Lτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' First, we determine the values of the coupling above which the neutrino self-interaction dominates over the Standard Model neutrino-nucleon scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Due to their vanishing chemical potentials, this effect is more important for muon and tau neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We argue that, although in this regime a redistribution of the neutrino energies might take place, this only affects a small part of the neutrino population and it cannot be constrained with the SN 1987A data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Thus, contrary to previous claims, the region of the parameter space where U(1)Lµ−Lτ explains the discrepancy in the muon anomalous magnetic moment is not ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We then focus on small gauge couplings, for which the decay length of the new gauge boson is comparable to (or larger than) the size of proto-neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We show that in this regime, the resonant production of a long-lived Z′ and its subsequent decay into neutrinos outside the proto-neutron star can significantly reduce the duration of the neutrino burst, and values of the coupling as small as O(10−9) can be probed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This rules out new areas of the parameter space of the U(1)Lµ−Lτ and U(1)B−L models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' These results are relevant for any other model that features new MeV-scale mediators that couple to the neutrino sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' ∗ davidg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='cerdeno@uam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='es † marina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='cermeno@ift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='es ‡ yasaman@theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='ipm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='ir arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='00661v1 [hep-ph] 2 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' INTRODUCTION Core-collapse supernovae (SN) are the violent explosions that result from the rapid collapse of gi- ant stars at the end of their thermonuclear evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' These very energetic phenomena provide a unique window to test new physics beyond the Standard Model (SM), especially in the neutrino sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Neutrinos are copiously produced during the SN collapse and the subsequent cooling of the resulting proto-neutron star (proto-NS), as confirmed by the observation of the burst from SN 1987A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The production of new exotic particles inside SN cores is severely constrained by the measured neutrino flux from SN 1987A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For example, if such particles are long-lived and very weakly-interacting, they may efficiently transfer energy from the core and thus induce energy loss that exceeds the measured flux of 1053 erg [1–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Moreover, if this particle mediates new neutrino interactions with matter fields, the neutrino mean free path can be reduced, modifying the observed ∼ 10 second duration of the neutrino burst from SN 1987A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This has been used to derive bounds on the neutrino scattering cross section with stellar matter, which ultimately can be translated into constraints on the parameters of the underlying theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' If the new physics is due to a light mediator between neutrinos and SM particles, the constraints on the mediator mass and couplings are complementary to those obtained from other experimental limits in the range of masses between 1 MeV and 1 GeV, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [12, 14, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' If the mass of the mediator (for example, a new vector Z′) is comparable to the temperature inside the proto-NS (MeV scale), the process ν¯ν → Z′∗ → ν¯ν can be resonant when the mediator is produced on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' These interactions are especially important for muon and tau neutrinos, for which scatterings are more frequent due to their vanishing chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [18], it was argued that the resonant Z′ production from neutrino-antineutrino pair in the SN core can have an important impact on the neutrino diffusion time in the U(1)Lµ−Lτ model, which poses a challenge to the solution to the muon anomalous magnetic moment, (g −2)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' It should be noted that although the SN 1987A neutrino signal was only detected through the positrons produced by electron antineutrinos, thanks to the oscillation phenomenon, neutrinos or antineutrinos of any flavour generated in the SN have a O(1) probability to be detected as νe or ¯νe after they exit the star core and propagate to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Thus, if muon and tau neutrinos have a diffusion time that exceeds ∼ 1 second (a factor 10 enhancement is expected for the signal time since the stored thermal energy, Etot th , in matter continues to be emitted in neutrinos even after the first neutrinos escape), this should have been observed as well in electron neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, if neutrino self-interactions are strong enough to make them behave as a tightly coupled per- fect fluid with only neutrino self-interactions, the neutrino gas expands regardless of the self-interaction strength, not affecting the neutrino diffusion time [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In essence, as long as the neutrino velocity dis- tribution is isotropic, neutrino self-interactions cannot change this distribution and therefore they do not affect their diffusion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Due to the frequent scattering on nucleons inside the proto-NS, the neutrino-antineutrino velocity distribution is already isotropic, so this argument applies to the generic case of new low-mass vector mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This prevented early bounds on the neutrino-neutrino scat- tering cross section, which applied to very large values σνν ∼ 10−35 cm2 [20] as well as the range of couplings in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' It would therefore seem that the solution to the (g − 2)µ is safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In this article, we provide a way to circumvent the argument of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [19] and allow to set constraints on neutrino self-interactions based on the duration and spectrum of the resulting neutrino signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We first point out that the resonant enhancement of neutrino-antineutrino interactions via MeV scale mediators can lead to a re-distribution of the neutrino energies in the SN interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Although this can make low-energy neutrinos more trapped, this effect depends crucially on the fraction of low-energy neutrinos, which we show is rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Then we show that, if the mediator decay length is of the order of (or larger than) the proto-NS radius, the properties of the observed neutrino signal (duration and spectrum) can be altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Neutrinos and antineutrinos deep inside the proto-NS core can convert to a Z′ within time scales much shorter than the diffusion time and they can be transferred outside the neutrinosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Since the Z′ boson decays back to ν¯ν outside the core, the total energy emitted in the form of neutrinos would match the binding energy of the star, and cooling bounds would not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, these processes can leave its imprint in three observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' First, since the Z′ can directly come out of the inner hot core (without being cooled in the outer shells), the energy spectrum of neutrinos leaving the star will be hotter than the SM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This consideration has been used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [21] to constrain the parameter space of models featuring a light scalar boson coupled to neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Second, since the Z′ diffuses much faster than neutrinos, this process would considerably shorten the duration 2 of the neutrino burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For the first time, we point this potential effect in the present paper and derive bounds on the coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Finally, the distortion in the flavor composition of the neutrino flux is another potential effect which depends on the flavor structure of the Z′ couplings to the neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Based on these arguments, we re-evaluate previous constraints for MeV scale mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' As concrete examples, we derive bounds for the two well-motivated gauged U(1)B−L and U(1)Lµ−Lτ models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' These constructions are simple anomaly-free extensions of the SM which feature a Z′ boson that mediates new interactions in the neutrino sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, our results can readily be applied also for other models with light mediators, such as the models for large Non-Standard neutrino Interaction (NSI) with matter fields [22–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Recent works have also considered some neutrino self-interaction effects on SN dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For exam- ple, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [29] strong neutrino self-interactions outside the proto-NS have been used to extract bounds on light scalars based on the SN diffusion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Likewise, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [30] self-interactions were considered between SN neutrinos and the cosmic neutrino background in order to obtain constraints based on the delay of SN neutrinos to reach the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Although 2 to 4 processes were shown to affect the shock revival in the SN neutrino core, ν¯ν → ν¯ν resonant processes were not included, since they do not change the number of neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In addition, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [21, 31] derived constraints based on the modification of the neutrino flux due to the production of massive bosons in the SN core, coming from neutrino-neutrino interactions, and their subsequent decays into neutrinos outside the proto-NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, these bounds only apply for very small values of the neutrino mediator couplings, when the product of the coupling and the mediator mass is below ∼ 10−7 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This article is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In Section II we introduce the U(1)Lµ−Lτ and U(1)B−L models, which incorporate a new vector coupling to the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In Section III we summarise the neutrino diffusion process in a proto-NS when only SM interactions are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We then discuss the uncertainties in the determination of the neutrino burst duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In the next two sections, we consider new-physics contributions to neutrino-antineutrino scattering, including the resonant production of the mediator on- shell, as well as the radial dependence of the density and temperature inside the proto-NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In Section IV, we study the effects in two complementary regions of values for the gauge coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1 addresses the large coupling regime, for which we compute the regions of the parameter space where the neutrino-antineutrino annihilation dominates over the SM contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We estimate the relevance of the potential effect and discuss that it is not discernible with the SN 1987A data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='2, we study the small coupling regime, where the gauge mediator can decay in a distance comparable to the size of the proto-NS, leading to a shorter neutrino burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Based on the observed duration of the SN 1987A neutrino signal, we exclude new regions of the parameter space in the U(1)B−L and U(1)Lµ−Lτ models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Our conclusions are presented in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' MODELS WITH LIGHT VECTOR MEDIATORS The recent measurement of the muon anomalous magnetic moment, (g−2)µ, by the E989 experiment [32] has confirmed a previously observed [33–35] deviation with respect to the SM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This can be interpreted as a hint of new physics in the leptonic sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Among the different beyond the Standard Model (BSM) realisations that could account for such an excess, the U(1)Lµ−Lτ gauge anomaly-free model is the simplest extension of the SM that would explain this observation introducing a light vector boson [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Moreover, such a model can offer an explanation for the nearly maximum mixing angle between the second and third generations of neutrinos [37], alleviate the tension between late and early time determinations of the Hubble constant [13] as well as accommodate dark matter (DM) with the correct relic abundance [38–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Due to the extremely small coupling to electrons or quarks, the U(1)Lµ−Lτ vector boson is difficult to test in collider or fixed target experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, experiments looking for new physics in the neutrino sector together with SN and NS signals offer an opportunity to search for this new vector boson exploiting its tree-level couplings to the muon and tau neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Recent works [13, 15, 18, 43, 44] have provided constraints over the model by studying its cosmological and astrophysical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Bounds on the effective number of relativistic degrees of freedom from big bang nucleosynthesis [18, 44] and stellar cooling [43] strongly constrain new vector bosons with masses below 1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, there is still room for heavier mediators which can explain the (g − 2)µ measurement as well as be produced 3 in the interior of the proto-NS core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The Lagrangian of this model can be expressed as LLµ−Lτ = LSM − 1 4Z′αβZ′ αβ + m2 Z′ 2 Z′ αZ′α + Z′ αJα µ−τ, (1) where mZ′ is the mass of the gauge boson, and Z′ αβ ≡ ∂αZ′ β − ∂βZ′ α is the field strength tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The µ − τ current is Jα µ−τ = gµ−τ (¯µγαµ + ¯νµγαPLνµ − ¯τγατ − ¯ντγαPLντ) , (2) where PL = 1 2 (1 − γ5) is the left chirality projector and gµ−τ is the gauge coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The rest frame partial widths for Z′ decays to charged leptons β = µ, τ and neutrinos νβ = νµ,τ can be written as ΓZ′→β+β− = g2 µ−τmZ′ 12π � 1 + 2m2 β m2 Z′ � � 1 − 4m2 β m2 Z′ , (3) ΓZ′→¯νβνβ = g2 µ−τmZ′ 24π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (4) Note that for mZ′ < 2mµ ∼ 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='3 MeV the only possible decay of the mediator is into µ and τ neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In our analysis we will consider 1 MeV ≤ mZ′ ≤ 1 GeV and therefore the decay into muons will only take place in a particular region of our parameter space (for mZ′ ∼ 1 GeV the branching ratio for the Z′ decay into muons is ΓZ′→µ+µ−/ � ΓZ′→¯νµνµ + ΓZ′→¯ντ ντ + ΓZ′→µ+µ−� ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Another well-motivated model that we study is the U(1)B−L gauge model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In this case the Z′ vector mediator couples at tree level to quarks and all leptons and neutrino flavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The Lagrangian can be written as LB−L = LSM − gB−LZ′ µ � 1 3 � q ¯qγµq − � α ¯lαγµlα − � α ¯ναγµνα � , (5) where all quarks have the same charge, gB−L/3, and all charged and neutral leptons have a charge of −gB−L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The Z′ may also interact with SM fermions via kinetic mixing with either the SM Z boson or the photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Here we assume that the couplings to SM fermions induced through kinetic mixing are subdominant to the direct couplings and neglect the kinetic mixing effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Regarding the decay of this mediator, we have to add two decay channels to those of U(1)Lµ−Lτ because the Z′ boson can now also decay into electrons and electron neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Thus, the decay widths are those of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (4) with β = e, µ, τ and substituting gµ−τ by gB−L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' STANDARD PICTURE OF NEUTRINO DIFFUSION IN A PROTO-NS The diffusion of neutrinos from the core of proto-NS is a complicated process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Let us first focus on the diffusion of νµ, ντ, ¯νµ and ¯ντ (collectively denoted as νx) within the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' During the short νe-burst and accretion periods (t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='7 s), the νe and ¯νe fluxes are much larger than the νx flux but, in the proto-NS cooling period, the luminosities in the form of all three neutrino flavors and their antiparticles become equal up to a precision of 10% [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The total binding energy of the proto-NS (from the binding energy of the collapsing star) is of the order of 1053 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This energy will be depleted with a gray body radiation of neutrinos and antineutrinos of all three flavors from the neutrinosphere (at a radius of ∼ 15 km for νe and ∼ 20 km for νx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' At the onset of the cooling phase (∼ 1 s after the bounce), the luminosity is of the order of ∼ 2 × 1052 erg s−1 for each neutrino and antineutrino species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, the outer shells cool down fast, the neutrinosphere recedes to smaller radii and the luminosity quickly drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The neutrino emission is backed up with the diffusion of neutrinos from the inner shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Due to 4 R (km) T (MeV) nB(fm−3) Ye µ∗ n (MeV) µ∗ p (MeV) µ∗ νe (MeV) m⋆ N (MeV) k = 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='0 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='3 496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='6 405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='4 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='6 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='6 k = 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='5 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='28 530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='0 458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='3 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='7 384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='9 k = 3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='0 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='25 656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='5 601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='9 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='9 599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='4 k = 4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='0 33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='2 779.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='8 723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='0 786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='0 k = 5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='5 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1 858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='7 813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='4 857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='0 k = 6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='0 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='05 917.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='2 893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='5 915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='9 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Values of neutron effective chemical potential, µ∗ n, proton effective chemical potential, µ∗ p, electron neutrino effective chemical potential, µ∗ ν, and nucleon effective mass, m⋆ N, for the spherical shells (labeled by the index k and defined by an outer radius R) that we consider at 1 s after bounce, with a baryonic density, nB, temperature, T and electron fraction, Ye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Temperatures, densities and electron fraction are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' multiple scattering, these neutrinos take a sizable time to reach the outer shells of the proto-NS, from where they are radiated out with a time scale of [46] tE ∼ 3 π2 Etot th 2Eν th R2 ns � 1 λν � , (6) where Etot th and Eν th are respectively the total baryon and neutrino thermal energies, ⟨1/λν⟩ is the average of the inverse of the neutrino mean free path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Remember that R2 ns⟨1/λν⟩ (where Rns is the radius of the neutrinosphere) gives the time scale of the diffusion of a single particle with a velocity of light and with random walk steps of λν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Within the SM, νx scattering is dominated by neutral current scattering on nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The cross section is proportional to ⟨E2 ν⟩, which varies across the core radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The neutrino interaction rate with other particles in the interior of the proto-NS, as well as its mean free path and diffusion time, depend on the nuclear medium properties, such as the baryonic density, nB, temperature, T, the effective nucleon masses, m⋆ N, and the neutrino and nucleon effective chemical potentials, µ∗ νe, µ∗ n, µ∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Note that the effective masses and chemical potentials of the particles in the nuclear medium are different from the naked values in vacuum by the presence of meson fields [47]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In Table I, the temperatures, densities, and lepton fractions for the different shells of the star for a time of 1 second after bounce, obtained from the simulation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [45], are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The effective electron neutrino and nucleon chemical potentials and the effective nucleon masses were derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [17] invoking the TM1 model, widely used in current numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This was done for a 18 M⊙ progenitor in a relativistic mean field approach, after solving the equations of motion and imposing the equilibrium conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Note that since the masses of muons and tau leptons are considerably larger than the temperature in the core, charged-current reactions for the µ and τ neutrinos cannot take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Thus, unless weak magnetism effects [48] are considered, the processes creating νµ and ντ produce the same amount of ¯νµ and ¯ντ [46] which implies µνµ = µ¯νµ = µντ = µ¯ντ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Note that in this case the mean neutrino energy is ⟨Eν⟩ ∼ πT while for electron neutrinos ⟨Eν⟩ ∼ µ∗ ν + πT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The neutrino distribution function is given by f(Eνβ, µ∗ νβ, T) = 1 1 + e (Eνβ −µ∗νβ )/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (7) The same expression holds for antineutrinos with µ∗ ¯νβ = −µ∗ νβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The neutrino density is computed as nνβ(µ∗ νβ, T) = 1 (2π)3 � ∞ 0 4πE2 νβf(Eνβ, µ∗ νβ, T)dEνβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (8) 1 In order to obtain the electron effective chemical potential, the equilibrium equation that must be solved involves effective meson fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In particular, µn + µνe = µp + µe + 2gρ < ρ >, where ρ is an effective field responsible of the strong interaction in the model used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 5 0 100 200 300 400 500 E [MeV] 10 12 10 10 10 8 10 6 10 4 10 2 100 102 104 E2f (E )[MeV2] k = 1 e e , , , 0 100 200 300 400 500 E [MeV] 10 12 10 10 10 8 10 6 10 4 10 2 100 102 104 E2f (E )[MeV2] k = 6 e e , , , FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Energy distribution of electron (green) and muon/tau (black) neutrinos (solid) and antineutrinos (dashed) as a function of the neutrino energy for shells k = 1 (left) and k = 6 (right) of Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 1, we have represented the energy distribution, E2 νβf(Eνβ, µ∗ νβ, T), for muon and tau neutrinos (black) and for electron neutrinos (green solid) and antineutrinos (green dashed), as a function of the neutrino energy for the first and sixth shells of the star (see Table I), on the left and right plots respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Since the chemical potential for muon/tau neutrinos and antineutrinos is zero, their distribution functions coincide, fνµ,τ = f¯νµ,τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, for electron antineutrinos µ∗ ¯νe = −µ∗ νe and, therefore, in the inner shells, where the electron neutrino chemical potential reaches the highest values, their distribution functions substantially differ, fνe ≫ f¯νe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This, in turn, leads to a negligible density of electron antineutrinos in the inner shells, which suppresses the νe¯νe interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In the outer shells, the electron neutrino and antineutrino chemical potentials significantly decrease in absolute value and therefore fνe ∼ f¯νe ∼ fνµ,τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Due to the high matter density and frequent collisions with the SN medium inside the proto-NS, flavour conversions are expected to be suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Neutrino oscillation becomes possible only after neutrinos exit the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' It should be pointed out that there are uncertainties in the duration of the neutrino signal from SN 1987A, mainly associated with the low statistics of the observed neutrinos and the unknown relative time offsets of the three detectors (Kamiokande II, IMB, and Baksan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Likewise, the SM prediction is also subject to uncertainties, for example in the choice of the equation of state for nuclear matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The duration of the neutrino signal in the SM can be computed assuming that neutrino-nucleon interactions are the leading contribution to the neutrino mean free path, λνβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The neutrino diffusion time can be computed as c∆tνβ = n � k=1 � R2 k − R2 k−1 � � 1/λνβ � k , (9) where k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 6 are the different shells of Table I for a time after bounce of 1 second,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Rk are the radius of each shell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' and � 1/λνβ � k is the average of the inverse of the mean free path in each shell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' ⟨1/λνβ⟩ = � dEνβf(Eνβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' µ∗ νβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' T)E2 νβλ−1 νβ (Eνβ) � dEνβf(Eνβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' µ∗νβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' T)E2νβ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (10) where Eνβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' EN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' ⃗pνβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' ⃗pN are the energies and momenta of the incoming neutrino and nucleon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' E′ νβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' E′ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' ⃗p′νβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' ⃗p′N are those of the outgoing states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' and pνβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' pN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' p′ νβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' p′ N are the corresponding four-momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In this expression, β = e, µ, τ indicates the neutrino flavour, N = n, p refers to the nucleon states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=', neutrons and protons, and λ−1 νβ (Eνβ) = � N � 2 d3 ⃗ pN (2π)3 f(EN, µ∗ N, T)| ⃗ vνβ − ⃗vN|σνβ,NF(E′ ν, E′ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (11) 6 νβ ¯νβ να ¯να Z′ νβ ¯να νβ ¯να Z′ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' New physics contribution to neutrino-antineutrino scattering through a low-mass vector boson, Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The diagram on the left represents the s-channel that is the leading contribution for the relevant range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For the U(1)Lµ−Lτ model α, β = µ, τ while for U(1)B−L, α, β = e, µ, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In this expression, | ⃗ vνβ − ⃗vN| is the relative velocity between the neutrino and the target, σνβ,N is the neutrino-nucleon scattering cross section, f(EN, µ∗ N, T)) is the Fermi Dirac distribution function for the incoming nucleon and F(E′ ν, E′ N) = (1 − f(E′ ν, µ∗ ν, T))(1 − f(E′ N, µ∗ N, T)) accounts for the Pauli blocking of the outgoing states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' If we consider only SM interactions, we obtain ∆tνµ,τ SM ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='3 s for muon and tau neutrinos and ∆tνe SM ∼ 3 s for electron neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This leads to tE ∼ 10 s in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (6), due to the fact that the stored thermal energy in matter continues to be emitted in neutrinos even after the first neutrinos escape, which is compatible with the observed duration of the neutrino signal from SN 1987A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' IMPACT OF RESONANT PRODUCTION OF NEW LIGHT MEDIATORS ON NEUTRINO DIFFUSION New physics in the neutrino sector can alter the neutrino-nucleus scattering cross section, thereby leading to a longer neutrino burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Compatibility with SN 1987A observation leads to upper bounds on the relevant new physics couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, as we explained in the Introduction, according to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [19] this does not apply to neutrino-neutrino or neutrino-antineutrino interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This argument is valid as long as neutrinos follow an equilibrium distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, this condition might not hold if a new mediator is produced on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In this section, we analyse two complementary regimes: when the new gauge coupling is large (so that neutrino self-interactions are comparable or exceed the SM contribution) and when it is small (so that the new mediator can travel a long distance inside the proto-NS star or even exit before decaying back to neutrinos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Large coupling regime In this section, we discuss how on-shell Z′ production can affect the neutrino burst duration in the relatively large coupling regime where the rate of neutrino-antineutrino interaction is comparable or larger than the neutrino-nucleon interaction rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 2, we show the neutrino-antineutrino scattering diagrams via the new vector mediator, Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The cross section of this process can be greatly enhanced through s-channel resonance when the mediator is produced on-shell2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This is very sensitive to the energy distribution of neutrinos in the SN core of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The t-channel contribution would be leading at very large values of the coupling (since it scales with the fourth power of the coupling) and small mediator masses (due to collinear enhancement), 2 If the new gauge coupling is so large that Z′ reaches thermal equilibrium with the neutrino gas and the plasma, it can in principle receive a share of the entropy content of the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We may therefore wonder whether this significantly reduces the temperature relative to the SM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Since neutrinos below the so-called energy neutrinosphere are in thermal equilibrium with the plasma, which act as a huge thermal energy source (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=', Etot th /Eν th ≫ 1), the addition of the three bosonic degrees of freedom associated with the three polarizations of the Z′ boson can only change the temperature by a negligible amount suppressed by Eν th/Etot th ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We therefore assume that around 1 s after the bounce, the temperature profile of the core is similar to what is predicted within the SM even in the limit of large coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 7 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=', mZ′ ∼ 1 MeV for gµ−τ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1 or for gB−L ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1, however, these regions of the parameter space are generally ruled out by existing experimental and observational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We have explicitly checked that, when the resonant condition is met, the main contribution to the neutrino-antineutrino cross section is given by the s-channel diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Likewise, other processes such as neutrino-neutrino scattering and neutrino-lepton scattering, both of which would proceed through t−channel, are sub- leading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The s-channel resonance in neutrino-antineutrino scattering is not relevant in the SM, since the typical neutrino energies inside the proto-NS are not high enough to produce the Z boson on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Because of this, in the previous works where the impact of new neutrino interactions in SN dynamics via light mediators was studied, such as Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [12, 14, 17], neutrino-antineutrino scattering was not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, in models with new gauge bosons in the MeV range, the resonance can significantly increase the neutrino-antineutrino scattering cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The models that we consider are lepton flavor conserving so νβ will fuse only with ¯νβ to produce an on-shell Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The impact of scatterings of neutrinos off nucleons on the average diffusion time is quite different from the effects of neutrino scattering off the background antineutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Nucleons are heavy particles with mass much larger than the temperature and therefore the energies of neutrinos in the SN core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' As a result, the scattering of a single neutrino (or antineutrino) off a nucleon changes its direction and therefore prolongs its diffusion time as formulated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, when pairs of neutrino and an- tineutrinos in a neutrino-antineutrino gas with isotropic velocity distribution scatter off each other, the angular distribution of the velocities does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [19], this means that the diffu- sion time will not be proportional to 1/λνβ ¯νβ→Z′→νβ ¯νβ ∼ Rνβ ¯νβ→Z′→νβ ¯νβ, where Rνβ ¯νβ→Z′→νβ ¯νβ is the neutrino-antineutrino scattering rate, even when 1/λνβ ¯νβ→Z′→νβ ¯νβ ≫ 1/λνβ in which λνβ ¯νβ→Z′→νβ ¯νβ and λνβ are respectively the mean free path of neutrinos due to scattering off antineutrinos and off nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, in this limit, the neutrino self-scattering can in principle redistribute the energies of neutrinos with a rate larger than that of scattering off the background electrons which brings the distribution back to the thermal Fermi-Dirac distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Let us consider a neutrino with energy E1 ∼ T scattering off an antineutrino whose momentum makes an angle of θ with the direction of the initial neutrino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In order for such a pair to produce an on-shell Z′, the antineutrino energy should be E2 = m2 Z′ 2E1(1 − cos θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (12) Simple kinematics3 show that the final neutrino and antineutrino from the Z′ decay will have flat energy distribution in the range [(E1 + E2)(1 − vZ′)/2, (E1 + E2)(1 + vZ′)/2 in which vZ′ = (1 − m2 Z′/(E1 + E2)2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For E2 ∼ E1 ∼ T, the energies of the final particles will be of the same order as those of the initial neutrinos so their diffusion time cannot be significantly altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Let us consider the limit that E2 ≫ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Considering that the scattering cross section of neutrinos off nuclei is proportional to the square of the neutrino energy, this can in principle change the diffusion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' If the rate of neutrino self-scattering is small, the impact will of course be tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' On the other hand, in case the rate of scattering off antineutrinos (or neutrinos) with energy E2 such that E2 > Elim ≫ T 4 is larger than the rate of scattering off the electrons (which thermalizes back the energy distribution), the higher energy tail of the neutrino distribution (with energies exceeding Elim) will be used up, without having time to be replaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Thus, independently of the value of the coupling, the fraction of neutrinos or antineutrinos that come out of the thermal Fermi-Dirac distribution will be very tiny and given by the ratio of the number density of antineutrinos with energy larger than Elim (nlim) to the total number density of neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This can be understood with the following argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The dynamics of nlim can be described as ˙nlim = −ΓSM(nlim−neq lim)−ΓNEW nlim where ΓSM is determined by the rate of scattering off the electrons and neq lim is the value of nlim computed with the Fermi- Dirac distribution neq lim = � ∞ Elim f(Eν, −µ∗ ν, T)E2 νdEν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The asymptotic (stable) value of nlim will be given by nasym lim = neq limΓSM/ΓNEW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Let us denote the number density of neutrinos scattered up to higher energies out of the Fermi-Dirac distribution by nout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The evolution of nout can be written as 3 If Z′ scatters multiple times before decay, it will reach thermal equilibrium with the neutrinos and this argument does not therefore apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Thermalization of Z′ however requires σ(Z′ν → Z′ν)nν/ΓZ′ > 1 which can be achieved only for couplings larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 4 In this discussion, Elim is taken an arbitrary value which Elim ≫ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 8 ˙nout = −Γ′ SMnout + ΓNEW nasym lim .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Thus, the asymptotic solution which shows the density of neutrinos kicked out of equilibrium is nout = (ΓSM/Γ′ SM)neq lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For Elim ≫ T this quantity is suppressed by a factor � ∞ Elim f(Eν, µ∗ ν, T)E2 νdEν � ∞ 0 f(Eν, −µ∗ν, T)E2νdEν ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (13) A less extreme scenario is the one where E1 = πT −∆ and E2 as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (12), where 0 < ∆ < πT is an arbitrary energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For example, let us consider muon and tau neutrinos with energies of E1 = 15 MeV, interacting with antineutrinos with energies E2 = m2 Z′/(30 MeV(1 − cos θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' If the mediator mass is mZ′ = 50 MeV, the antineutrino energies must be E2 ≳ 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='7 MeV in order to produce the mediator on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' From the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 1, we can see that, for these neutrino and antineutrino energies, E2 νfν(Eν) can be of order 1 and 100, respectively, in the first shell of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' After the interaction, ν1 will gain energy and therefore its interactions with nucleons will be stronger, since σν,N ∝ E2 ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' As a result, ν1 will be more bounded to stellar matter and stay for longer in the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This picture is consistent with what is shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [49] for a scenario of relatively strong DM-nucleon and DM-neutrino interactions inside SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In this case, DM particles are thermalised with the stellar material (as our neutrinos and antineutrinos are) and neutrino-DM interactions bound neutrinos to the star out to larger radii and lower temperatures, increasing the neutrino diffusion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Note that in page 20 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [50], they comment about these results and argue that what Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [49] obtained is similar to what would be expected from simply increasing the strength of neutrino interactions with regular matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Besides, they compare this scenario with the one where DM is not thermalised with the stellar material and they state that in the latter case strong DM-neutrino interactions are similar to strong neutrino self-interactions in SN since they both involve the emission of a strongly-coupled gas, and hence strong DM-neutrino interactions do not significantly affect the cooling time for SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The lower energy tail of the spectrum can therefore be scattered up to higher energies via Z′ resonance interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We should however remember that the fraction of muon or tau neutrinos with Eν < T (Eν < T/3) is only 8 % (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='48%) of the whole number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Thus, even if all of neutrinos in the low energy tail scatter up, the impact on the diffusion time is not observable in SN 1987A data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' A more detailed analysis is needed to determine whether this can be a noticeable feature in the event of future SN detection with a greatly improved statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In the following, we would like to study the regions of the parameter space for the two well-motivated U(1)Lµ−Lτ and U(1)B−L models where this effect could take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Implications for the parameter space of the U(1)Lµ−Lτ and U(1)B−L models To find the regions where neutrino-antineutrino interactions are as frequent or more than neutrino- nucleon interactions, we need to calculate the neutrino-antineutrino scattering rate via the new Z′ mediator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The average of the neutrino-antineutrino scattering rate can be calculated by integrating the scattering cross section as follows, ⟨R⟩νβ ¯νβ→Z′→να¯να = � dEνβf(Eνβ, µ∗ νβ, T)E2 νβRνβ ¯νβ→Z′→να¯να(Eνβ) � dEνβf(Eνβ, µ∗νβ, T)E2νβ , (14) where β = e, µ, τ indicates the neutrino flavour and Rνβ ¯νβ→Z′→να¯να(Eνβ) = � d3⃗p¯νβ (2π)3 f(E¯νβ, µ∗ ¯νβ, T)|⃗vνβ − ⃗v¯νβ|σνβ,¯νβ (15) is the neutrino-antineutrino scattering rate via Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Note that we are considering massless neutrinos and therefore |⃗pνβ| = Eνβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For the νβ¯νβ → Z′ → να¯να process, the factor |⃗vνβ − ⃗v¯νβ|σνβ,¯νβ can be written as |⃗vνβ − ⃗v¯νβ|σνβ,¯νβ = � d3⃗p′να (2π)32E′να � d3⃗p′¯να (2π)32E′ ¯να (2π)4δ(4)(pνβ + p¯νβ − p′ να + p′ ¯να) |M|2 νβ,¯νβ 4EνβE¯νβ F(E′ να, E′ ¯να), (16) 9 where Eνβ, E¯νβ, ⃗pνβ, ⃗p¯νβ are the energies and momenta of the incoming particles, E′ να, E′ ¯να, ⃗p′να, ⃗p′¯να are those of the outgoing states, and pνβ, p¯νβ, p′ να, p′ ¯να are the corresponding four-momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The factor F(E′ να, E′ ¯να) = (1 − f(E′ να, µ∗ να, T))(1 − f(E′ ¯να, µ∗ ¯να, T)) accounts for Pauli blocking in the outgoing states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Since, in the parameter space under analysis, the resonant production of the Z′ boson is the leading new physics process and ΓZ′→¯νβνβ/mZ′ ≪ 1 is fulfilled we can use the narrow width approximation (NWA) in order to perform the calculation of the neutrino-antineutrino interaction rate for these new physics interactions, Rνβ ¯νβ→Z′→να¯να(Eνβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In this limit, it can be shown that Rνβ ¯νβ→Z′→να¯να(Eνβ) = 1 32π � ∞ Emin ¯νβ dE¯νβ f(E¯νβ, µ∗ ¯νβ, T)E¯νβ E¯νβ + Eνβ � mZ′ E¯νβEνβ �2 |M|2 νβ ¯νβ→Z′ ΓZ′→να¯να Γtot Z′ , (17) with Emin ¯νβ = m2 Z′/(4Eνβ) and Γtot Z′ = � α ΓZ′→¯νανα + ΓZ′→β+β−, where β = µ, τ for the U(1)Lµ−Lτ model and β = e, µ, τ for the U(1)B−L model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Note that the decay into taus and mesons will not be open for the values of the mediator masses considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The squared amplitudes can be written as |M|2 Z′→νβ ¯νβ = g2 µ−τm2 Z′/2 for the U(1)Lµ−Lτ model and |M|2 Z′→νβ ¯νβ = g2 B−Lm2 Z′/2 for the U(1)B−L model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Here we are neglecting Pauli blocking for neutrinos and antineutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For muon and tau neutrinos, since their chemical potential vanishes, it is clear that (1 − f(E′ ¯ν, µ∗ ¯ν, T)) ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In the case of electron neutrinos, for the relevant range of outgoing energies in the integration, Pauli blocking does not have an important impact in our final results,5 except for the case νµ¯νµ → νe¯νe which is totally Fermi blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For this reason, in our calculations, we do not take into account νµ¯νµ → νe¯νe interactions in the first forth shells of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In principle, one can think that the annihilation of neutrino-antineutrino into muons via s-channel Z′ diagram may have an impact on the physics of the proto-NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, these processes generate the same amount of muons and antimuons, not changing their chemical potential, and therefore we do not expect the equation of state of the nuclear matter to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Besides, even though muon neutrinos (antineutrinos) can scatter on antimuons (muons), as the muon density and its scattering cross section with muon neutrinos and antineutrinos are much smaller than that of nucleons, the mean free path of scattering on muons (antimuons) is negligible compared to the scattering on nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The only effect we could expect comes from the decay of muons into muon neutrinos close to the neutrinosphere, which can change equality between the muon neutrino and tau neutrino fluxes that come out of the neutrinosphere, having consequences for collective neutrino oscillation [51–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For completeness, we have also explicitly checked that processes at one loop, such as neutrino- antineutrino annihilation into µ+µ− and the subsequent µ+µ− → ν¯ν are sub-leading for the calculation of the neutrino-antineutrino scattering rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In this subsection, we show the parameter space from which the rate of neutrino-antineutrino inter- action can exceed that of the neutrino scattering off nucleons for the two specific models, U(1)Lµ−Lτ and U(1)B−L, one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' As we mentioned above, these regions cannot be ruled out using SN 1987A data, but they are indicative of the regions that might be explored when a better measurement of the SN neutrino flux (perhaps with better statistics from future detectors) is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We shall compare them with the range ruled out with different observables and considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' U(1)Lµ−Lτ In this scenario, the new neutrino couplings to electrons and nucleons are extremely suppressed, and only the contributions to νµ,τ scattering have to be included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The mean free path of electron neutrinos is not modified, and therefore charged-current β-processes do not need to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We have represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 3 the values of gµ−τ as a function of mZ′ for which the new physics contribution to the neutrino scattering becomes as important as the SM one for each of the six 5 This is equivalent to what we observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [17] when comparing SN bounds for the lepton number violating (where the chemical potential vanishes and there is no Pauli blocking) and lepton number conserving models with scalar mediators for the same incoming neutrino energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 10 100 101 102 103 mZ′ [MeV] 10 7 10 6 10 5 10 4 10 3 10 2 g Neff White Dwarfs (g 2) CHARM-II COHERENT CsI COHERENT Ar BaBar 4 L L k=1 k=2 k=3 k=4 k=5 k=6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Values of (gµ−τ, mZ′) for which the inverse neutrino mean-free path of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (17) equals the SM value in each of the proto-NS shells of Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The band in red shows the region in the parameter space consistent with the (g − 2)µ measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Gray areas show different complementary bounds from Neff [13], Borexino [56], white dwarf cooling [57], COHERENT CsI [58] and LAr [56], neutrino tridents (Charm-II) [29, 59] and BaBar [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' shells described in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We should emphasize that, contrary to what might be inferred from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [41] the region compatible with (g − 2)µ is not yet ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The inner shells of the proto-NS (k = 1 − 4) are characterised by high temperatures and nucleon densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' These regions give the largest contribution to the neutrino scattering cross section both in the SM and when new physics is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Due to the larger available neutrino energies, in these inner regions the resonant condition for the light mediator holds for larger masses (the tail of the neutrino distribution function - albeit very small - is still considerably larger than in the outer regions as we showed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' From these lines, we see that a coupling of the order of gµ−τ ∼ 10−4 − 10−5 is sufficient for the new physics diagrams to be comparable to the SM contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' A small kink at 210 MeV= 2mµ can be observed in all the lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This is due to the opening of the Z′ → µ+µ− decay channel, which slightly increases the total Z′ decay width in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (17), reducing the cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In the outer shells (k = 5, 6) the nucleon density is considerably smaller (the neutrinosphere is located at ∼ 20 km for muon and tau neutrinos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This leads to a large suppression of SM processes (neutrino-neutron scattering) and it makes it easier for the new physics to dominate (considerably reducing the values of gµ−τ at which this occurs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Also, the temperature is much smaller in these regions, which means that the resonant condition can only be fulfilled for small masses of mZ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For this reason, the lines at which the new physics equals the SM contributions move to lower values of gµ−τ and mZ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, the time scale of neutrino diffusion from these two last layers up to the neutrinosphere is much smaller than one second, and therefore they are not relevant for the diffusion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' What sets the time scale of cooling of the outer shells is the graybody ”surface” cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Since the new interaction does not absorb νx, it cannot affect the grayness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For comparison, we show the existing bounds on the parameter space from various observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 11 100 101 102 103 mZ′ [MeV] 10 7 10 6 10 5 10 4 10 3 10 2 gB L NA64 E141 CHARM II Orsay BaBar B L, , k=1 k=2 k=3 k=4 k=5 k=6 100 101 102 103 mZ′ [MeV] 10 7 10 6 10 5 10 4 10 3 10 2 gB L NA64 E141 CHARM II Orsay BaBar B L, e k=1 k=2 k=3 k=4 k=5 k=6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 3, but for the U(1)B−L model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The plot on the left only considers muon and tau neutrinos and the plot on the right only considers electron neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In grey we show complementary bounds from different experiments [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' U(1)B−L In this scenario, electron neutrinos can also interact via the new force and the Z′ also mediates neutrino-nucleon and neutrino-electron interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, since the muon and tau neutrino chemical potentials are zero, neutrino-antineutrino scattering is more efficient for muon/tau neu- trinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In particular, in the inner shells of the star the number density of electron antineutrinos is negligible and therefore these interactions are significantly suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Since the electron neutrino chemical potential decreases with the radial distance to the centre of the star, electron neutrino- antineutrino scatterings become more frequent in the outer shells of the star, where the rate of νe¯νe scattering becomes comparable to the one for muon/tau neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Besides, for ∼ 1 after the bounce, neutrino-electron interactions are smaller than the neutrino- nucleon scatterings, both in the SM [45] and for our new physics interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Note that ne ≪ nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 4, we show the values of gB−L as a function of the mediator mass for which the new physics contribution to the neutrino scattering becomes as important as the SM one for each of the six shells of the proto-NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The plot on the left corresponds to muon and tau neutrinos and it is analogous to the U(1)Lµ−Lτ case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The plot on the right represents the results for electron neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Since the electron antineutrino abundance in the inner shells of the proto-NS is largely suppressed by its chemical potential, the new physics contribution is small in these regions and a very large coupling is needed to match the SM value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Small coupling regime In this section, we focus on the range of parameter space with very small couplings, where the decay length of Z′ can be comparable to or larger than the size of the proto-NS core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' If Z′ decays back only into ν¯ν, this would lead to neutrinos effectively escaping earlier, shortening the length of the SN neutrino burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Similarly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (17), the rate of the scattering of a single ν off any ¯ν in the medium can be computed via the following relation, Rνβ ¯νβ→Z′ = 1 32π � ∞ Emin ¯νβ dE¯νβ f(E¯νβ, µ∗ ¯νβ, T)E¯νβ E¯νβ + Eνβ � mZ′ E¯νβEνβ �2 |M|2 νβ ¯νβ→Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (18) The time scale of the interaction can be defined as τνβ ¯νβ→Z′ = c/Rνβ ¯νβ→Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 12 The decay length of Z′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' ℓZ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' is given by lifetime multiplied by the relativistic boost factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' γZ′ = EZ′/mZ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' averaged over the energy of Z′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' which is set by the energies of the neutrino and antineutrino producing the mediator on-shell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' ℓZ′ = � d3⃗pνβ (2π)3 f(Eνβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' µ∗ νβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' T) � d3⃗p¯νβ (2π)3 f(E¯νβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' µ∗ ¯νβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' T)σνβ ¯νβ→Z′| ⃗ vνβ − ⃗v¯νβ|γZ′vZ′ ℏ Γtot Z′ � d3⃗pνβ (2π)3 f(Eνβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' µ∗νβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' T) � d3⃗p¯νβ (2π)3 f(E¯νβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' µ∗ ¯νβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' T)σνβ ¯νβ→Z′| ⃗ vνβ − ⃗v¯νβ| ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (19) where vZ′ is the velocity of Z′ and σνβ ¯νβ→Z′ is the cross section for the νβ¯νβ → Z′ interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Since the temperature and therefore the energy distribution of neutrinos and antineutrinos producing Z′ depend on the distance from the center, ℓZ′ also varies with the distance from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We have computed ℓZ′ for each shell described in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' As we shall see, there is a range of couplings at which τν¯ν→Z′ for muon and tau neutrinos is much shorter than the typical time scale of neutrino diffusion time while ℓZ′ is larger than (or of the same order as) the radius of the neutrinosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Thus, neutrinos and antineutrinos can coalesce into an on-shell Z′ as they diffuse out of the proto-NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The Z′ travels unhindered outside the neutrinosphere and decays back into a ν¯ν pair outside (or close to) the neutrinosphere, where ν and ¯ν can straightly come out of the star without scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The outcome is that the average diffusion time for νµ and ντ and their antiparticles deep inside the core can be reduced to τνµ,τ ¯νµ,τ →Z′, which is much smaller than the typical diffusion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' At a snapshot of ∼ 1 s after the bounce (at the onset of the cooling phase), the energy of the νµ, ¯νµ, ντ and ¯ντ gases altogether, Eνµ,τ th account for about 2% of the total thermal energy Etot th of the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Likewise, the energy stored in the νe gas in the core, Eνe th is about 10 % of the total thermal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Considering that neutrinos are in thermal equilibrium with each other as well as with matter inside the core, the time scale over which the total binding energy can be transferred outside via Z′ can be estimated as τ µ,τ trans = τνµ,τ ¯νµ,τ →Z′ Etot th /Eνµ,τ th ∼ 50 τνµ,τ ¯νµ,τ →Z′ , (20) τ e trans = τνe¯νe→Z′ Etot th /Eνe th ∼ 10 τνe¯νe→Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' (21) If τtrans is of the order of 10 s (the SM prediction of the neutrino burst), the time duration can be significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, the change may be hidden within the uncertainties in the duration measurement and prediction as already discussed in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For τtrans ≪ 10 s, the predicted duration will be so short that it can already be ruled out via the measurement of the SN 1987A neutrino burst duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This has two important consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' First, the energy spectrum of neutrinos reaching the Earth will be harder than what expected in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This feature has been discussed within the Majoron model in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [21, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, the total energy carried away via neutrinos will be the same as what predicted in the SM which amounts to the binding energy of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Thus, the cooling argument (that is setting luminosity in form of Z′ equal to neutrino luminosities in the SM) is therefore not applicable, although it points towards the same part of parameter space as our burst duration argument does [11, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The cooling argument is only valid for massless mediators which cannot decay back into neutrinos [9, 10, 62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Second, the flavor composition of neutrinos transferred by Z′ can be different from the SM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For example, in the U(1)Lµ−Lτ model, neutrinos reaching out of the neutrinosphere will be composed of νµ, ντ, ¯νµ and ¯ντ with equal spectra and vanishing νe and ¯νe components, up to the corrections from the contributions from the standard neutrino diffusion from the neutrinosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Of course, these neutrinos will then go through flavor conversion traversing the outer shells of the star (and if ℓZ′ is larger than the star size, in the vacuum between the star and the Earth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Fluxes of νe and ¯νe will be produced via oscillation but the ratio between electron neutrino and muon and tau neutrino fluxes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=', Fνe/Fνµ,τ , as well as the equivalent for electron antineutrinos, F¯νe/Fµ,τ, at Earth will be different from the SM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Within the U(1)B−L model, Z′ can decay into νe¯νe, νµ¯νµ and ντ ¯ντ in equal amounts so we expect Fνe = F¯νe = Fνµ = F¯νµ = Fντ = F¯ντ both at a distance of ℓZ′ from the core center and at the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Notice that even oscillation (or matter effects in the star) cannot alter this flavor democracy because of the unitarity of the PMNS matrix and the fact that � β P(να → νβ) = � β P(¯να → ¯νβ) = 1, where P(να → νβ) and P(¯να → ¯νβ) are the conversion probabilities of neutrinos and antineutrinos of flavour α to neutrinos and antineutrinos of flavour β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 13 10 7 10 5 10 3 10 1 101 103 Z′ [s] mZ′ = 10 MeV L L 10 10 10 9 10 8 10 7 g 10 3 10 1 101 103 Z′ [km] k=1 k=2 k=3 k=4 k=5 k=6 10 7 10 5 10 3 10 1 101 103 Z′ [s] mZ′ = 50 MeV 10 10 10 9 10 8 10 7 g 10 3 10 1 101 103 Z′ [km] 10 7 10 5 10 3 10 1 101 103 Z′ [s] mZ′ = 100 MeV 10 10 10 9 10 8 10 7 g 10 3 10 1 101 103 Z′ [km] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The time scale of the neutrino-antineutrino interaction producing the Z′ vector mediator on-shell and the decay length of Z′ as a function of the coupling for the U(1)Lµ−Lτ model in the upper and lower panels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We show in different colours the corresponding values for each shell of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Different masses are depicted in the three columns showed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In particular, mZ′ = 10, 50, 100 MeV for the left, central and right columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For reference, the dotted lines correspond to the cases for which the timescale of the interaction is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='02 s, and where the decay length of the mediator is equal to the neutrinosphere radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Implications for the parameter space of the U(1)Lµ−Lτ and U(1)B−L models U(1)Lµ−Lτ In the upper panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 5, we represent the time scale of νµ¯νµ → Z′ or ντ ¯ντ → Z′ inside different shells of the proto-NS as a function of the gauge coupling, for various values of the mediator mass (mZ′ = 10, 50, 100 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The horizontal dotted line corresponds to τν¯ν→Z′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='02 s, which is equivalent to τtrans = 1 s for muon and tau neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The lower panels show ℓZ′ for the different shells and the same choice of mediator masses, and the horizontal dotted line highlights the value of the neutrinosphere at 20 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' From these plots we can infer the range of values of the gauge coupling for which τν¯ν→Z′ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='02 s and ℓZ′ is equal or larger to the neutrinosphere radius6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Notice that ℓZ′ is shorter in the outer shells because the Z′ produced with smaller temperatures have a smaller boost factor, EZ′/mZ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Also, the ν¯ν → Z′ interaction rate is maximised when the temperature of the layer is similar to ∼ mZ′/2π (since the average of neutrino and antineutrino energies for a given temperature of the star is ∼ πT) and consequently, τν¯ν→Z′ becomes smaller in those shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This behaviour can be observed in the three columns of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In particular, the value of τν¯ν→Z′ in the outer layer increases with mZ′, since the temperature is lower and Z′ production is suppressed by the Boltzmann factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The shaded region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 6 corresponds to the area of the (gµ−τ, mZ′) parameter where the conditions τtrans < 1 s and ℓZ′ < 20 km are simultaneously satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' From our discussion above, the shortening of the neutrino burst duration is so severe in this area that it can be already ruled out by the SN 1987A measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 6 Although we should also expect a reduction in the neutrino diffusion time when ℓZ′ ∼ 1 km, the precise calculation of this reduction is out of the scope of this work and is left for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 14 100 101 102 103 mZ′ [MeV] 10 10 10 9 10 8 10 7 10 6 g Neff Z′ = Rns Z′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1 Rns trans = 1 s trans = 10 s L L k=1 k=2 k=3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' From top to bottom, values of the coupling as a function of the mediator mass for which the mediator decay length is equal to the radius of the neutrinosphere (in solid lines) and the ones for which τtrans is equal to 1 s (solid) and 10 s (dashed dotted) for the U(1)Lµ−Lτ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We show this for the three inner shells of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The coloured areas indicate the regions of the parameter space where the decay length is larger than the neutrino sphere radius and τtrans is smaller than 1 s for each shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In thicker lines we can see the lower limit for which this happens in every shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In grey we show cosmological bounds from Big Bang nucleosynthesis [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 10 7 10 5 10 3 10 1 101 103 Z′ [s] mZ′ = 10 MeV B L 10 10 10 9 10 8 10 7 gB L 10 3 10 1 101 103 Z′ [km] k=1 k=2 k=3 k=4 k=5 k=6 10 7 10 5 10 3 10 1 101 103 Z′ [s] mZ′ = 50 MeV 10 10 10 9 10 8 10 7 gB L 10 3 10 1 101 103 Z′ [km] 10 7 10 5 10 3 10 1 101 103 Z′ [s] mZ′ = 100 MeV 10 10 10 9 10 8 10 7 gB L 10 3 10 1 101 103 Z′ [km] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 5 for the U(1)B−L model and only considering muon and tau neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We also represent with dashed-dotted lines and dotted lines the values of the coupling for which τtrans < 10 s and ℓZ′ < 2 km, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We can expect that in the region between these lines the duration and spectrum of the neutrino burst will also be modified with respect to the SM prediction, although in a more subtle way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This region is susceptible to be tested in future SN 15 10 3 100 103 106 109 Z′ [s] mZ′ = 10 MeV B L, e 10 10 10 9 10 8 10 7 gB L 10 2 100 102 104 Z′ [km] k=1 k=2 k=3 k=4 k=5 k=6 10 3 100 103 106 109 Z′ [s] mZ′ = 50 MeV 10 10 10 9 10 8 10 7 gB L 10 2 100 102 104 Z′ [km] 10 3 100 103 106 109 Z′ [s] mZ′ = 100 MeV 10 10 10 9 10 8 10 7 gB L 10 2 100 102 104 Z′ [km] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 7 but for electron neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For reference, the dotted lines correspond to the cases for which the timescale of the interaction is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1 (which corresponds to τtrans = 1 s), and where the decay length of the mediator is equal to the neutrinosphere radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 100 101 102 103 mZ′ [MeV] 10 10 10 9 10 8 10 7 10 6 gB L Neff U70 E137 E137 Z′ = Rns Z′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1 Rns trans = 1 s trans = 10 s B L, , k=1 k=2 k=3 100 101 102 103 mZ′ [MeV] 10 10 10 9 10 8 10 7 10 6 gB L Neff U70 E137 E137 Z′ = Rns Z′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1 Rns trans = 1 s trans = 10 s B L, e k=1 k=2 k=3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 6, but for the U(1)B−L model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The plot on the left only considers muon and tau neutrinos and the plot on the right only considers electron neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In grey we show complementary bounds from different experiments (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [57] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' data with more statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' U(1)B−L In this case, the Z′ can also decay into νe¯νe and e−e+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For 1 MeV ≪ mZ′ < 200 MeV, Br(Z′ → νe¯νe) = 1/5 and Br(Z′ → e−e+) = 2/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 7 and 8 show the neutrino-antineutrino interaction timescale for νµ,τ and νe, respectively, along with the corresponding values of ℓZ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In the upper panels the dotted lines correspond to τν¯ν→Z′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='02 s for muon and tau neutrinos and τν¯ν→Z′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1 s for electron neutrinos, which is equivalent in both cases to τtrans = 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In the lower panels, the dotted lines indicate ℓZ′ > 20 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' For νe, the conditions τν¯ν→Z′ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='1 s and ℓZ′ > 20 km are not simultaneously satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The 16 reason is that the ¯νe density in the inner core is very suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Thus, in the U(1)B−L model, the shortening of the burst takes place due to νµ¯νµ → Z′ and ντ ¯ντ → Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The resulting bounds on the (gµ−τ, mZ′) parameter space are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 9, for muon and tau neutrinos (left panel) and electron neutrinos (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The shaded area corresponds to the are where τtrans < 1 s and ℓZ′ < 20 km, which rules out a new region of the U(1)B−L parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' On the right panel, we can see that no region of the U(1)B−L model can be excluded by only considering electron neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' As we did for the U(1)Lµ−Lτ case, we also show the lines where τtrans < 10 s and ℓZ′ < 2 km, which might also lead to an observable modification of the neutrino signal in future SN data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Finally, even if testing this is out of the scope of this work, it must be noticed that the decay of Z′ into e−e+ outside the neutrinosphere can warm up the outer shells and create shock waves with drastic consequences for the SN explosion and the photon flux reaching the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Within the U(1)Lµ−Lτ and the U(1)B−L models for mZ′ ∼ few × 10 MeV and gµ−τ ∼ 10−9 − 10−8, muon and tau neutrino-antineutrino pairs from the hotter inner core can be transferred outside the proto-NS so the spectrum of neutrinos reaching the Earth will be harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Such prediction can in principle be tested by observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This also increases the prospect of detection of diffuse SN neutrino flux as for this detection, the energy threshold is a limiting factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Besides, within the U(1)B−L model, the possibility of Z′ decay to e−e+ pair right outside proto-NS can have drastic consequences for the thermodynamics of the star beyond the neutrinosphere and subsequently for shock revival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' These effects merit a dedicated analysis which is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' CONCLUSIONS In this article, we have investigated the resonant production of low-mass vector mediators from neutrino-antineutrino interactions in the core of proto-NS, determining the conditions under which these processes can significantly alter neutrino diffusion in SN explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We point out that the resonant production of a long-lived Z′ and its subsequent decay into neutrinos can significantly shorten the duration of the observed neutrino burst, providing a new way to test neutrino self-interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' As a concrete example, we have computed the rate of neutrino self-interaction in two well-motivated new physics scenarios that feature a new vector mediator, namely U(1)B−L and U(1)Lµ−Lτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In order to calculate the rate of ν¯ν → (Z′)∗ → ν¯ν in the nuclear medium, we have considered the radial dependence of the density, temperature and chemical potentials of the particles inside the proto-NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We have found that thanks to their vanishing chemical potential, the rate of muon or tau neutrino self-interaction can exceed that of the Standard Model scattering off nucleons when the resonant condition for Z′ is fulfilled, even for couplings as small as gµ−τ ∼ 10−6 −10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Within the U(1)B−L model, electron neutrinos also receive new contributions to their scatterings but the effect is relevant only in outer shells, where the chemical potential of νe is lower and the ¯νe density is therefore higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We have first discussed that, contrary to previous claims, in the range of parameter space where neutrino-antineutrino interactions dominate over the SM neutrino-nucleon scattering, the diffusion time does not significantly change, according to the argument of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This is because the velocity distributions of neutrinos inside the core are isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' In particular, the U(1)Lµ−Lτ solution for the muon magnetic moment anomaly is not ruled out for this reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We have then examined whether frequent self-interaction can redistribute energy of neutrinos and antineutrinos, leading to a change in the burst duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, we found out that the effects of energy redistribution are too small to lead to a discernible impact in the SN 1987A data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' More work is needed to determine whether the subtle changes in the neutrino spectrum can be observable with future SN data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Finally, for the first time, we have pointed out that, for small values of the couplings where the decay length of Z′ is comparable or larger than the size of the proto-NS, the neutrino burst duration can be significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Neutrinos and antineutrinos from the inner core can produce on-shell Z′ bosons, which leave the core unimpeded and then decay back to neutrino-antineutrino pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Since the total binding energy of the star (∼ 1053 erg) is still transferred outside the star in the form of neutrinos, the canonical cooling bound does not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' However, three observable features of the neutrino burst can be significantly altered: its duration, the energy spectrum, and the flavour composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 17 We have focused on the first effect, showing that for 5 MeV < mZ′ < 200 MeV, there is an area of the parameter space with couplings of the order of 10−10 − 10−8 where the duration of the neutrino burst is shorter than 10 s and the decay length of the Z′ exceeds the proto-NS radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' We have determined the region of parameter space where the predicted neutrino burst duration is less than 1 s, so that the deviation from the observed duration of SN 1987A burst is significant enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' This rules out new regions of the parameter space of both the U(1)Lµ−Lτ and U(1)B−L models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' Future SN data, with more statistics and a better measurement of the neutrino spectrum, together with improvements on the SN models, might allow to test a larger part of parameter space predicting time duration shorter than 10 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' These results are relevant for any other model that features new MeV scale mediators that couple to the neutrino sector such as light mediator models with NSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We would like to thank Juan Antonio Aguilar-Saavedra, Rafael Aoude, Ivan Esteban, Patrick Fold- enauer, Luca Mantani, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' ´Angeles P´erez-Garc´ıa, for useful discussions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' DGC acknowledges support from the Spanish Ministerio de Universidades under grant SI2/PBG/2020- 00005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' The work of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' was partially funded by the F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='-FNRS through the MISU conven- tion F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='6001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' YF would like to acknowledge support from the ICTP through the Associates Programme and from the Simons Foundation through grant number 284558FY19 and from Sara- madan under contract No.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} +page_content=' 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAyT4oBgHgl3EQfxPkj/content/2301.00661v1.pdf'} diff --git a/fdFAT4oBgHgl3EQf7x7c/content/tmp_files/2301.08747v1.pdf.txt b/fdFAT4oBgHgl3EQf7x7c/content/tmp_files/2301.08747v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..84fa349f9ac449a3eb885512361929d9c61886af --- /dev/null +++ b/fdFAT4oBgHgl3EQf7x7c/content/tmp_files/2301.08747v1.pdf.txt @@ -0,0 +1,597 @@ +Drawing Diestel-Leader graphs in 3D +Amandine Escalier +January 24, 2023 +Abstract +In this short note we give some code to represent Diestel-Leader graphs in 3D. The +code is written in TikZ. +0 +2 +4 +6 +0 +10 +20 +0 +1 +2 +3 +x +y +z +0 +2 +4 +6 +0 +10 +20 +0 +1 +2 +3 +x +y +z +Figure 1: Two views of the Diestel-Leader graph DL(3, 2) +The history of Diestel-Leader graphs takes its root in the following question asked by +Woess [Woe91]: is every connected locally finite vertex-transitive graph quasi-isometric +to some Cayley graph? In the hope of answering no to this question, Diestel and Leader +[DL01] defined what we now call Diestel-Leader graphs. However it was only later, in the +famous papers of Eskin, Fisher an Whyte [EFW07, EFW12] that it was showed that some +of the aforementioned graphs are not quasi-isometric to any Cayley graph. +In this note we give a code to draw these graphs in 3D, using TikZ. Readers only +interested by producing an illustration of some DL(p, q) can jump to the last pages of this +article (or the end of the .tex file) and copy-paste the code given in Section 2.2, write the +wanted values of p and q (line 29) and then compile. Readers wishing to change the code +can rely on the description made in Section 2.1. We start this note with a short reminder +of the definition of Diestel-Leader graphs. +1 +arXiv:2301.08747v1 [math.GM] 20 Jan 2023 + +1 +Diestel-Leader graphs +We recall here the definition of Diestel-Leader graphs. We refer to [Woe91, Section 2] for +more details. +1.1 +Tree and horocycles +Let q ⩾ 2 and denote by T = Tq the homogenous tree of degree q + 1. Denote by d the +usual graph distance on T fixing to 1 the length of an edge. +A geodesic ray is an infinite sequence (vn)n∈N of vertices of T such that d(vi, vj) = +|i − j|, for all i, j ∈ N. We say that two rays are equivalent if ther symmetric difference1 is +finite. We call end of Tq an equivalence class of rays in T and denote by ∂T the space of +ends of T. +Let ˆT := ∂T ∪ T. For any elements x, y ∈ ˆT there is a unique geodesic in ˆT, denoted +by xy, that connects x and y. +Now fix an end w ∈ ∂T, the confluent of two elements x, y ∈ ˆT\w with respect to w, +denoted by x ⋏ y, is defined as the element c = x ⋏ y such that xw ∩ yw = cw, that is to +say: the confluent is the point where the two geodesics xw and yw towards w meet (see +Figure 2). +w +c +x +y +yc +cw +xc +Figure 2: Example of the confluent of two points in T2 +Now, fix a root vertex o ∈ T. We define below a Busemann function, which will allow +us to endow our tree with some height notion. +Definition 1.1. Let w ∈ ∂Tq. The Busemann function with respect to w is the map +h : T → Z defined by +h(x) = d(x, x ⋏ o) − d(o, x ⋏ o). +Example 1.2. Let’s turn back to Figure 2 and let o = y be the root. +Then for x +represented in the figure, we have x ⋏ o = c and thus d(x, x ⋏ o) = 2 and d(o, x ⋏ o) = 1. +Therefore h(x) = 1. +Definition 1.3. Let w ∈ ∂Tq and k ∈ N. The horocycle with respect to w, denoted by +Hk, is the set Hk = {x ∈ T : h(x) = k}. +1Recall that the symmetric difference of two sets A and B is defined by |A△B| = (A\B) ∪ (B\A) +2 + +We refer to Figure 3 for an illustration. Note that every horocycle in ˆT is infinite. Every +vertex x in a horocycle Hk has exactly one neighbour x− (called the predecessor) in Hk−1 +and exactly q neighbours (called successors) in Hk+1 (see Figure 3 for an illustration). +w +x− +x +x+ +2 +x+ +1 +Hk +Hk−1 +Hk+1 +Hk+2 +Hk−2 +Figure 3: Horocycles, predecessor, successors +1.2 +Diestel-Leader graphs +Now fix p, q ⩾ 2 and consider Tp and Tq with respective roots op and oq, and respective +reference ends wp and wq. +Definition 1.4. The Diestel-Leader graph DL(p, q) is the graph whith set of vertices +V (DL(p, q)) = +� +(x, y) ∈ Tp × Tq : h(x) = −h(y) +� +. +and where there is an edge between two elements (x1, y1) and (x2, y2), if and only if (x1, x2) +is an edge in Tp and (y1, y2) is an edge in Tq. +If there is an edge between two vertices (x1, y1) and (x2, y2) in DL(p, q), then remark +that either +• x2 is one of the p childs of x1 in Tp and in this case y2 is the only predecessor of y1 +in Tq, namely y2 = y− +1 (see Figure 4a); +• or x2 = x− +1 is the unique predecessor of x1 in Tp and in this case y2 is one of the q +childs of y1 in Tq (see Figure 4b). +We represent these two cases in Figure 4. The edge is drawn in blue, the p-regular tree +in orange and the q-regular tree in brown. In light blue is represented the Diestel-Leader +graph, we refer to Figure 5 for a drawing of DL(p, q) itself. Note that in Figure 5, the +corresponding Tp and Tq are drawn on the planes x = 0 and y = 0 respectively. +Remark 1.5. When p = q the graph DL(p, q) is a Cayley graph of the Lamplighter +group Z/pZ ≀ Z. +2 +The code +The complete code is given in page 6, and written in TikZ. We start by some comments +on how the coordinates were computed and how the loops work. +3 + +0 +2 +4 +6 +0 +10 +20 +0 +1 +2 +3 +(x2, y2) +(x1, y1) +y1 +x1 +x2 +y2 = y− +1 +x +y +z +(a) When x2 is a child of x1 and y2 = y− +1 +0 +2 +4 +6 +0 +10 +20 +0 +1 +2 +3 +(x1, y1) +(x2, y2) +y2 +x2 = x− +1 +x1 +y1 +x +y +z +(b) When y2 is a child of y1 and x2 = x− +1 +Figure 4: Drawing an edge in DL(p, q): two cases +0 +1 +2 +3 +0 +5 +0 +1 +2 +x +y +z +(a) DL(3, 2) +0 +1 +2 +3 +0 +10 +0 +1 +2 +x +y +z +(b) DL(4, 2) +Figure 5: Two different Diestel-Leader graphs, represented at the same scale +4 + +2.1 +Comments on the code +Variables +The main variables are \p, \q (line 29) and \layers (line 35). The first two +variables correspond to the number of childs in the first and the second tree respectively, +that is to say the p and q in DL(p, q). Finally, the height of the graph is parametrized by +\layers: there are \layers+1 different heights of vertices appearing on the graph. +With the values given in the example in Section 2.2, we produce the left image in +Figure 1. +Finally changing the values in view={165}{10} (line 17) will change the point of view +(see for example Figure 1). For more details on how to parametrize the view, see the +pgfplots manual [Feu21, Section 4.11.1, page 311]. +Structure of the code +The code is composed of one main loop cut in two parts: the first +one that draws the tree on the plane y = 0, and the second one that draws simultaneously +the tree located on the plane x = 0 and the Diestel-Leader graph. +In this description “height” will stand for the z-coordinates in the picture. +Lines 58 to 72 We start by drawing the regular tree of degree p represented on the plane +y = 0, in orange in the pictures. This tree grows from the bottom to the top. +At height h there are ph vertices to draw and the space between two consecutive +nodes at this height is pspace = players-h (in particular at the top, when h = layers, +the space between two nodes is equal to 1). +Moreover the first horizontal node at height h has for horizontal coordinates the +middle of [0, players−h − 1] that is to say +players−h/2 − 0.5 = pspace/2 − 0.5, +and therefore, if h ̸= layers, the first of its child (at height n = h+1) has coordinates +the middle of [0, players−n − 1], namely players−h/(2p) − 0.5 = pspace/(2p) − 0.5. +1 +2 +3 +4 +n +layers= +ph vertices at height h +h = n − 1 +p childs +p childs +p childs +p childs +0 +1 +2 +ph − 1 +pspace = players-h +Figure 6: Ranges of the different loop variables for the first tree +Now fix n ∈ {1, . . . , layers}, the loop starting at line 58 goes over all the nodes at +height h = n − 1 and draws the edges between a node at height n − 1 and its q +childs at height n. To do so, it fixes the horizontal position k ∈ {1, . . . , pn−1 − 1} of +the chosen node at height n − 1. This node thus has to be shifted to the right by +pshift = k ∗ pspace and hence drawn at the coordinates +5 + +(pspace − 0.5 + k ∗ pspace, 0, n − 1) = (pspace − 0.5 + pshift, 0, n − 1). +The childs are then located at height n. Similarly as above, the first child to be drawn +has for horizontal coordinates the middle of [0, qlayers−n−1 − 1] and the “\child-th” +is shifted to right by child ∗ pspace. Hence the coordinates of the “\child-th” child +is given by +(pspace/q − 0.5 + pshift + child ∗ pspace, 0, n − 1). +Lines 79 to 116 This loop draws the brown tree on the plane x = 0 and the Diestel- +Leader graph at the same time. +Lines 79 to 95 The variable k at line 79 goes through the qrev = qlayers−n ver- +tices at height n. +Denote by v the k-th vertex at this height. +The loop +for \child in {0,...,\q-1} goes through the q childs of v and draw the +edge between v and the \child-th child at height n − 1. The coordinates of the +nodes are computed in the same way as in the previous loop. (Remark that +the heights in the current loop are swapped: the parent is at height n and the +childs are at height n−1 whereas in the previous one, the childs were at height +n and the parent at height n − 1.) +Lines 99 to 114 Inside the loop for \child in {0,...,\q-1} (starting line 81) +lies also the part that draws the Diestel-Leader. Remember that v is the current +vertex at height n in the q-regular tree Tq. +The loop for \kk in {0,...,\pnm-1} goes through all the vertices in the +orange tree, drawn at height n. It will correspond to all the couple (uv) such +that u ∈ Tp is drawn at the same height (the z-coordinates) than v. To a given +child c of v in Tp, corresponds then the edge in the Diestel-Leader linking (u, v) +to (u−, c), where we recall that u− is the unique predecesor of u in Tp (see +page 3). Hence the point (u, v) will be drawn at height n and will have the +same x-coordinates than u and same y-coordinates than v, namely +� +� +� +� +� +� +� +x += pspace/(2p) − 0.5 + pshiftprime + childprime ∗ pspace/p, +y += qn/2 − 0.5 + qshift, +z += n +Similarly, the node (u−, c) will be drawn at height n−1 and will have the same +x-coordinates than u− and same y-coordinates than c, namely +� +� +� +� +� +� +� +x += pspace/2 − 0.5 + pshiftprime, +y += qn/2 − 0.5 + qshift + child ∗ qn/q, +z += n − 1 +2.2 +The code +1 +\documentclass[border=0mm]{standalone} +2 +\usepackage[x11names]{xcolor} +3 +\usepackage{tikz} +6 + +4 +\usetikzlibrary{calc,math,backgrounds} +5 +\usepackage{ifthen} +6 +\usepackage{pgfplots} +7 +\pgfplotsset{compat=1.18} +8 +% ----------------------------------------------------------------- +9 +% ---------------------------- Colors (Colorblind friendly) +10 +% ----------------------------------------------------------------- +11 +\definecolor{MFCB}{cmyk}{0,0.06,0.20,0.6} +12 +\colorlet{Orange}{DarkOrange3!85} +13 +% + DeepSkyBlue4 +14 +% ----------------------------------------------------------------- +15 +\begin{document} +16 +\begin{tikzpicture}%[scale=0.5] %Uncomment to change the scale +17 +\begin{axis}[ +18 +view={165}{10}, % Change the point of view. +19 +% view={150}{10}, % Change the point of view. +20 +xlabel=$x$, +21 +zlabel=$z$, +22 +ylabel=$y$, +23 +] +24 +\tikzmath{ +25 +% ------------------------------------ +26 +% ------------------------------------ +27 +% ---- PARAMETERS p AND q OF DL(p,q) +28 +% ------------------------------------ +29 +% ------------------------------------ +30 +\p=2; \q=3; +31 +% ------------------------------------ +32 +% ------------------------------------ +33 +% ----- HEIGHT PARAMETER +34 +% ------------------------------------ +35 +% ------------------------------------ +36 +\layers=3;% If layer=3 then there are +37 +%4 heights appearing, numbered 0 1 2 3 +38 +% ------------------------------------ +39 +% ------------------------------------ +40 +for \n in {1,...,\layers}{% Vertical, n stands for the height +41 +% ----------------------------------------- +42 +% Stored variables +43 +% ----------------------------------------- +44 +% % For the q regular tree +45 +\qrev=pow(\q,\layers-\n);% Stores the value 2^(L-n) +46 +\qn=pow(\q,\n);%Stores the value q^n +47 +% For the p regular tree drawn from bottom to top +48 +\pspace=pow(\p,\layers+1-\n); % (space between two nodes +7 + +49 +% at height layers-(n-1)) +50 +\pnm=pow(\p,\n-1);% Stores p^(layers-nprime)=p^(n-1) +51 +% -------------------------------------------------------- +52 +% Regular tree of degree p drawn on the plane y=0 +53 +% Drawn starting from the bottom to the top +54 +% -------------------------------------------------------- +55 +for \k in {0,...,\pnm-1}{%Horizontal +56 +\pshift=\k*\pspace;% Horizontal shift: k*p^n +57 +for \child in {0,...,\p-1}{% Child of the considered node +58 +{ +59 +% draw an edge +60 +\addplot3[Orange!20,thick] coordinates% +61 +{%The vertex at the top (ie. the child) at height n +62 +(\pspace/(2*\p)-0.5+\pshift+\child*\pspace/\p,0,\n) +63 +% The vertex below (ie. the parent) at height n-1 +64 +(\pspace/2-0.5+\pshift,0,\n-1)}; +65 +}; % End of edges drawing +66 +}; % End of the loop “for \child in” +67 +};% End of the loop “for \k in ” +68 +% ========================================================= +69 +% Regular tree of degree q drawn on the plane x=0 +70 +% ---------------- AND ----------------------------------- +71 +% The Diestel-Leader graph +72 +% ========================================================= +73 +for \k in {0,...,\qrev-1}{%Horizontal +74 +\qshift=\k*\qn;% Horizontal shift: k*q^n +75 +for \child in {0,...,\q-1}{%Goes through the q childs +76 +%in the q-regular tree +77 +% +78 +% The Tree drawn on the plane x=0 (the light brown one) +79 +% +80 +{ %Drawing the edge +81 +\begin{scope}[on background layer] +82 +\addplot3[MFCB!20,thick] coordinates% +83 +% vertex at height n +84 +{(0,\qn/2-0.5+\qshift,\n) +85 +% child at height n-1 +86 +(0,\qn/(2*\q)-0.5+\qshift+\child*\qn/\q,\n-1)}; +87 +\end{scope} +88 +};%End of drawing for the edge of the tree +89 +% +90 +% The Diestel-Leader +91 +% +92 +for \kk in {0,...,\pnm-1}{ +93 +\pshiftprime=\kk*\pspace;% horizontal shift +8 + +94 +for \childprime in {0,...,\p-1}{%Goes through the p +95 +% childs in the p-regular tree +96 +{% +97 +% draw a blue edge of the Diest-Leader +98 +\addplot3[DeepSkyBlue4,thick] coordinates% +99 +{%The vertex at the top +100 +(\pspace/(2*\p)-0.5+\pshiftprime% +101 ++\childprime*\pspace/\p, +102 +\qn/2-0.5+\qshift,\n) +103 +% The vertex at height n-1 +104 +(\pspace/2-0.5+\pshiftprime,\qn/(2*\q)% +105 +-0.5+\qshift+\child*\qn/\q,\n-1)}; +106 +};%End drawing +107 +};% End if the loop “for childprime in” +108 +};% End if the loop “for kk in” +109 +}; % End of the loop “for \child in” +110 +};% End of the loop “for k in” +111 +};% End of the loop “for n in” +112 +}%End Tikzmath +113 +\end{axis} +114 +\end{tikzpicture} +115 +\end{document} +9 + +References +[DL01] +R. Diestel and I. Leader. A conjecture concerning a limit of non-cayley graphs. +Journal of Algebraic Combinatorics, 14:17–25, 2001. +[EFW07] A Eskin, D Fisher, and K. Whyte. Quasi-isometries and rigidity of solvable +groups. Pure and Applied Mathematics Quarterly, 3(4):927–947, 2007. +[EFW12] A Eskin, D Fisher, and K. Whyte. Coarse differentiation of quasi-isometries i: +Spaces not quasi-isometric to cayley graphs. Annals of Mathematics, 176(1):221– +260, 2012. +[Feu21] +C. +Feuersänger. +Manual +for +package +pgfplots, +2021. +https://ctan.org/pkg/pgfplots. +[Woe91] +Wolfgang Woess. Topological groups and infinite graphs. Discrete Mathematics, +95(1):373–384, 1991. +Acknowledgments We thank Tom Ferragut, Giles Gardam and Théo Laurent for their +useful remarks and comments. +Fundings The athor is funded by the Deutsche Forschungsgemeinschaft (DFG, German +Research Foundation) – Project-ID 427320536 – SFB 1442, as well as under Germany’s +Excellence Strategy EXC 2044 –390685587, Mathematics Münster: Dynamics–Geometry– +Structure. +Amandine Escalier +Mathematisches Institut, +Fachbereich Mathematik und Informatik der Universität Münster, +Orléans-Ring 12, +48149 Münster, +Germany +10 + diff --git a/fdFAT4oBgHgl3EQf7x7c/content/tmp_files/load_file.txt b/fdFAT4oBgHgl3EQf7x7c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d75bcd4718c46848f62e6bb85bc1131c229e73c3 --- /dev/null +++ b/fdFAT4oBgHgl3EQf7x7c/content/tmp_files/load_file.txt @@ -0,0 +1,207 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf,len=206 +page_content='Drawing Diestel-Leader graphs in 3D Amandine Escalier January 24, 2023 Abstract In this short note we give some code to represent Diestel-Leader graphs in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' The code is written in TikZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' 0 2 4 6 0 10 20 0 1 2 3 x y z 0 2 4 6 0 10 20 0 1 2 3 x y z Figure 1: Two views of the Diestel-Leader graph DL(3, 2) The history of Diestel-Leader graphs takes its root in the following question asked by Woess [Woe91]: is every connected locally finite vertex-transitive graph quasi-isometric to some Cayley graph?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' In the hope of answering no to this question, Diestel and Leader [DL01] defined what we now call Diestel-Leader graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' However it was only later, in the famous papers of Eskin, Fisher an Whyte [EFW07, EFW12] that it was showed that some of the aforementioned graphs are not quasi-isometric to any Cayley graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' In this note we give a code to draw these graphs in 3D, using TikZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Readers only interested by producing an illustration of some DL(p, q) can jump to the last pages of this article (or the end of the .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='tex file) and copy-paste the code given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='2, write the wanted values of p and q (line 29) and then compile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Readers wishing to change the code can rely on the description made in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' We start this note with a short reminder of the definition of Diestel-Leader graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='08747v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='GM] 20 Jan 2023 1 Diestel-Leader graphs We recall here the definition of Diestel-Leader graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' We refer to [Woe91, Section 2] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='1 Tree and horocycles Let q ⩾ 2 and denote by T = Tq the homogenous tree of degree q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Denote by d the usual graph distance on T fixing to 1 the length of an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' A geodesic ray is an infinite sequence (vn)n∈N of vertices of T such that d(vi, vj) = |i − j|, for all i, j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' We say that two rays are equivalent if ther symmetric difference1 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' We call end of Tq an equivalence class of rays in T and denote by ∂T the space of ends of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Let ˆT := ∂T ∪ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' For any elements x, y ∈ ˆT there is a unique geodesic in ˆT, denoted by xy, that connects x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Now fix an end w ∈ ∂T, the confluent of two elements x, y ∈ ˆT\\w with respect to w, denoted by x ⋏ y, is defined as the element c = x ⋏ y such that xw ∩ yw = cw, that is to say: the confluent is the point where the two geodesics xw and yw towards w meet (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' w c x y yc cw xc Figure 2: Example of the confluent of two points in T2 Now, fix a root vertex o ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' We define below a Busemann function, which will allow us to endow our tree with some height notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Let w ∈ ∂Tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' The Busemann function with respect to w is the map h : T → Z defined by h(x) = d(x, x ⋏ o) − d(o, x ⋏ o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Let’s turn back to Figure 2 and let o = y be the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Then for x represented in the figure, we have x ⋏ o = c and thus d(x, x ⋏ o) = 2 and d(o, x ⋏ o) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Therefore h(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Let w ∈ ∂Tq and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' The horocycle with respect to w, denoted by Hk, is the set Hk = {x ∈ T : h(x) = k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' 1Recall that the symmetric difference of two sets A and B is defined by |A△B| = (A\\B) ∪ (B\\A) 2 We refer to Figure 3 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Note that every horocycle in ˆT is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Every vertex x in a horocycle Hk has exactly one neighbour x− (called the predecessor) in Hk−1 and exactly q neighbours (called successors) in Hk+1 (see Figure 3 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' w x− x x+ 2 x+ 1 Hk Hk−1 Hk+1 Hk+2 Hk−2 Figure 3: Horocycles, predecessor, successors 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='2 Diestel-Leader graphs Now fix p, q ⩾ 2 and consider Tp and Tq with respective roots op and oq, and respective reference ends wp and wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' The Diestel-Leader graph DL(p, q) is the graph whith set of vertices V (DL(p, q)) = � (x, y) ∈ Tp × Tq : h(x) = −h(y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' and where there is an edge between two elements (x1, y1) and (x2, y2), if and only if (x1, x2) is an edge in Tp and (y1, y2) is an edge in Tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' If there is an edge between two vertices (x1, y1) and (x2, y2) in DL(p, q), then remark that either x2 is one of the p childs of x1 in Tp and in this case y2 is the only predecessor of y1 in Tq, namely y2 = y− 1 (see Figure 4a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' or x2 = x− 1 is the unique predecessor of x1 in Tp and in this case y2 is one of the q childs of y1 in Tq (see Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' We represent these two cases in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' The edge is drawn in blue, the p-regular tree in orange and the q-regular tree in brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' In light blue is represented the Diestel-Leader graph, we refer to Figure 5 for a drawing of DL(p, q) itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Note that in Figure 5, the corresponding Tp and Tq are drawn on the planes x = 0 and y = 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' When p = q the graph DL(p, q) is a Cayley graph of the Lamplighter group Z/pZ ≀ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' 2 The code The complete code is given in page 6, and written in TikZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' We start by some comments on how the coordinates were computed and how the loops work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' 3 0 2 4 6 0 10 20 0 1 2 3 (x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' y2) (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' y1) y1 x1 x2 y2 = y− 1 x y z (a) When x2 is a child of x1 and y2 = y− 1 0 2 4 6 0 10 20 0 1 2 3 (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' y1) (x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' y2) y2 x2 = x− 1 x1 y1 x y z (b) When y2 is a child of y1 and x2 = x− 1 Figure 4: Drawing an edge in DL(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' q): two cases 0 1 2 3 0 5 0 1 2 x y z (a) DL(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' 2) 0 1 2 3 0 10 0 1 2 x y z (b) DL(4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' 2) Figure 5: Two different Diestel-Leader graphs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' represented at the same scale 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='1 Comments on the code Variables The main variables are \\p, \\q (line 29) and \\layers (line 35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' The first two variables correspond to the number of childs in the first and the second tree respectively, that is to say the p and q in DL(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Finally, the height of the graph is parametrized by \\layers: there are \\layers+1 different heights of vertices appearing on the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' With the values given in the example in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='2, we produce the left image in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Finally changing the values in view={165}{10} (line 17) will change the point of view (see for example Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' For more details on how to parametrize the view, see the pgfplots manual [Feu21, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='1, page 311].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Structure of the code The code is composed of one main loop cut in two parts: the first one that draws the tree on the plane y = 0, and the second one that draws simultaneously the tree located on the plane x = 0 and the Diestel-Leader graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' In this description “height” will stand for the z-coordinates in the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Lines 58 to 72 We start by drawing the regular tree of degree p represented on the plane y = 0, in orange in the pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' This tree grows from the bottom to the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' At height h there are ph vertices to draw and the space between two consecutive nodes at this height is pspace = players-h (in particular at the top, when h = layers, the space between two nodes is equal to 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Moreover the first horizontal node at height h has for horizontal coordinates the middle of [0, players−h − 1] that is to say players−h/2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5 = pspace/2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5, and therefore, if h ̸= layers, the first of its child (at height n = h+1) has coordinates the middle of [0, players−n − 1], namely players−h/(2p) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5 = pspace/(2p) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' 1 2 3 4 n layers= ph vertices at height h h = n − 1 p childs p childs p childs p childs 0 1 2 ph − 1 pspace = players-h Figure 6: Ranges of the different loop variables for the first tree Now fix n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' , layers}, the loop starting at line 58 goes over all the nodes at height h = n − 1 and draws the edges between a node at height n − 1 and its q childs at height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' To do so, it fixes the horizontal position k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' , pn−1 − 1} of the chosen node at height n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' This node thus has to be shifted to the right by pshift = k ∗ pspace and hence drawn at the coordinates 5 (pspace − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5 + k ∗ pspace, 0, n − 1) = (pspace − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5 + pshift, 0, n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' The childs are then located at height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Similarly as above, the first child to be drawn has for horizontal coordinates the middle of [0, qlayers−n−1 − 1] and the “\\child-th” is shifted to right by child ∗ pspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Hence the coordinates of the “\\child-th” child is given by (pspace/q − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5 + pshift + child ∗ pspace, 0, n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Lines 79 to 116 This loop draws the brown tree on the plane x = 0 and the Diestel- Leader graph at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Lines 79 to 95 The variable k at line 79 goes through the qrev = qlayers−n ver- tices at height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Denote by v the k-th vertex at this height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' The loop for \\child in {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=',\\q-1} goes through the q childs of v and draw the edge between v and the \\child-th child at height n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' The coordinates of the nodes are computed in the same way as in the previous loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' (Remark that the heights in the current loop are swapped: the parent is at height n and the childs are at height n−1 whereas in the previous one, the childs were at height n and the parent at height n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=') Lines 99 to 114 Inside the loop for \\child in {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=',\\q-1} (starting line 81) lies also the part that draws the Diestel-Leader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Remember that v is the current vertex at height n in the q-regular tree Tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' The loop for \\kk in {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=',\\pnm-1} goes through all the vertices in the orange tree, drawn at height n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' It will correspond to all the couple (uv) such that u ∈ Tp is drawn at the same height (the z-coordinates) than v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' To a given child c of v in Tp, corresponds then the edge in the Diestel-Leader linking (u, v) to (u−, c), where we recall that u− is the unique predecesor of u in Tp (see page 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Hence the point (u, v) will be drawn at height n and will have the same x-coordinates than u and same y-coordinates than v, namely � � � � � � � x = pspace/(2p) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5 + pshiftprime + childprime ∗ pspace/p, y = qn/2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5 + qshift, z = n Similarly, the node (u−, c) will be drawn at height n−1 and will have the same x-coordinates than u− and same y-coordinates than c, namely � � � � � � � x = pspace/2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5 + pshiftprime, y = qn/2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5 + qshift + child ∗ qn/q, z = n − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='2 The code 1 \\documentclass[border=0mm]{standalone} 2 \\usepackage[x11names]{xcolor} 3 \\usepackage{tikz} 6 4 \\usetikzlibrary{calc,math,backgrounds} 5 \\usepackage{ifthen} 6 \\usepackage{pgfplots} 7 \\pgfplotsset{compat=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='18} 8 % ----------------------------------------------------------------- 9 % ---------------------------- Colors (Colorblind friendly) 10 % ----------------------------------------------------------------- 11 \\definecolor{MFCB}{cmyk}{0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='06,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='20,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='6} 12 \\colorlet{Orange}{DarkOrange3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='85} 13 % + DeepSkyBlue4 14 % ----------------------------------------------------------------- 15 \\begin{document} 16 \\begin{tikzpicture}%[scale=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5] %Uncomment to change the scale 17 \\begin{axis}[ 18 view={165}{10}, % Change the point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' 19 % view={150}{10}, % Change the point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' 20 xlabel=$x$, 21 zlabel=$z$, 22 ylabel=$y$, 23 ] 24 \\tikzmath{ 25 % ------------------------------------ 26 % ------------------------------------ 27 % ---- PARAMETERS p AND q OF DL(p,q) 28 % ------------------------------------ 29 % ------------------------------------ 30 \\p=2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' \\q=3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' 31 % ------------------------------------ 32 % ------------------------------------ 33 % ----- HEIGHT PARAMETER 34 % ------------------------------------ 35 % ------------------------------------ 36 \\layers=3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='% If layer=3 then there are 37 %4 heights appearing, numbered 0 1 2 3 38 % ------------------------------------ 39 % ------------------------------------ 40 for \\n in {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=',\\layers}{% Vertical, n stands for the height 41 % ----------------------------------------- 42 % Stored variables 43 % ----------------------------------------- 44 % % For the q regular tree 45 \\qrev=pow(\\q,\\layers-\\n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='% Stores the value 2^(L-n) 46 \\qn=pow(\\q,\\n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='%Stores the value q^n 47 % For the p regular tree drawn from bottom to top 48 \\pspace=pow(\\p,\\layers+1-\\n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' % (space between two nodes 7 49 % at height layers-(n-1)) 50 \\pnm=pow(\\p,\\n-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='% Stores p^(layers-nprime)=p^(n-1) 51 % -------------------------------------------------------- 52 % Regular tree of degree p drawn on the plane y=0 53 % Drawn starting from the bottom to the top 54 % -------------------------------------------------------- 55 for \\k in {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=',\\pnm-1}{%Horizontal 56 \\pshift=\\k*\\pspace;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='% Horizontal shift: k*p^n 57 for \\child in {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=',\\p-1}{% Child of the considered node 58 { 59 % draw an edge 60 \\addplot3[Orange!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='20,thick] coordinates% 61 {%The vertex at the top (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' the child) at height n 62 (\\pspace/(2*\\p)-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5+\\pshift+\\child*\\pspace/\\p,0,\\n) 63 % The vertex below (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' the parent) at height n-1 64 (\\pspace/2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5+\\pshift,0,\\n-1)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' 65 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' % End of edges drawing 66 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' % End of the loop “for \\child in” 67 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='% End of the loop “for \\k in ” 68 % ========================================================= 69 % Regular tree of degree q drawn on the plane x=0 70 % ---------------- AND ----------------------------------- 71 % The Diestel-Leader graph 72 % ========================================================= 73 for \\k in {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='..' metadata={'source': 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\\qn/2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5+\\qshift,\\n) 103 % The vertex at height n-1 104 (\\pspace/2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5+\\pshiftprime,\\qn/(2*\\q)% 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='5+\\qshift+\\child*\\qn/\\q,\\n-1)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' 106 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='%End drawing 107 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='% End if the loop “for childprime in” 108 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='% End if the loop “for kk in” 109 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' % End of the loop “for \\child in” 110 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='% End of the loop “for k in” 111 };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='% End of the loop “for n in” 112 }%End Tikzmath 113 \\end{axis} 114 \\end{tikzpicture} 115 \\end{document} 9 References [DL01] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Diestel and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Leader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' A conjecture concerning a limit of non-cayley graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Journal of Algebraic Combinatorics, 14:17–25, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' [EFW07] A Eskin, D Fisher, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Whyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Quasi-isometries and rigidity of solvable groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Pure and Applied Mathematics Quarterly, 3(4):927–947, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' [EFW12] A Eskin, D Fisher, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Whyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Coarse differentiation of quasi-isometries i: Spaces not quasi-isometric to cayley graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Annals of Mathematics, 176(1):221– 260, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' [Feu21] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Feuersänger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Manual for package pgfplots, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' https://ctan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content='org/pkg/pgfplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' [Woe91] Wolfgang Woess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Topological groups and infinite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Discrete Mathematics, 95(1):373–384, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Acknowledgments We thank Tom Ferragut, Giles Gardam and Théo Laurent for their useful remarks and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Fundings The athor is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 427320536 – SFB 1442, as well as under Germany’s Excellence Strategy EXC 2044 –390685587, Mathematics Münster: Dynamics–Geometry– Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} +page_content=' Amandine Escalier Mathematisches Institut, Fachbereich Mathematik und Informatik der Universität Münster, Orléans-Ring 12, 48149 Münster, Germany 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdFAT4oBgHgl3EQf7x7c/content/2301.08747v1.pdf'} diff --git a/hdE3T4oBgHgl3EQfIQm9/vector_store/index.pkl b/hdE3T4oBgHgl3EQfIQm9/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..febe197ac998efd79d0a5363a84a74e2e3bc3987 --- /dev/null +++ b/hdE3T4oBgHgl3EQfIQm9/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:169705eb8bdcc4a1bb623ff800c52cbe128a6eceb924d913e50b5a3a9c63d8d2 +size 204334 diff --git a/iNE1T4oBgHgl3EQffgTu/vector_store/index.faiss b/iNE1T4oBgHgl3EQffgTu/vector_store/index.faiss new file mode 100644 index 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Baltaa, Yannick Nagelc, Hang Yind, Alisa Rupenyana,b,∗, John Lygerosa +aAutomatic Control Laboratory, ETH Zurich, Physikstrasse 3, 8092, Zurich, Switzerland +bInspire AG, Technoparkstrasse 1, 8005, Zurich, Switzerland +cNematX AG, Vladimir-Prelog-Weg 5, 8093, Zurich, Switzerland +dDepartment of Mechanical and Process Engineering, ETH Zurich, Leonhardstrasse 21, 8092, Zurich, Switzerland +Abstract +When manufacturing parts using fused filament fabrication and anisotropic polymers, the mechanical properties of a +manufactured component are strongly dependent on the print trajectory orientation. We conduct non-planar slicing and +optimize the print trajectories to maximize the alignment between the material deposition direction and the stress flow +induced by a predefined load case. The trajectory optimization framework considers manufacturability constraints +in the form of uniform layer height and line spacing. We demonstrate the method by manufacturing a load bearing +mechanical bracket using a 5-axis 3D printer and a liquid crystal polymer material. The failure strength and stiffness +of the optimized bracket are improved by a factor of 44 and 6 respectively when compared with conventional printing. +Keywords: Fused Filament Fabrication, Trajectory Optimization, Non-Planar Printing, Stress Alignment, Anisotropy +1. Introduction +In most 3D-printing applications, the material is formed in 2D layers via the printing technology of interest (e.g. +material extrusion, laser power, or lithography). By forming subsequent layers on top of the previous ones, a 3D +object is obtained. Fused filament fabrication (FFF), also known as fused deposition modeling, is one of the most +common 3D printing processes [1]. Traditionally, in FFF, a thermoplastic material is deposited in predefined paths via +a numerically controlled heated extruder to create a part in a layer-wise fashion. The design geometry is sliced into +planar layers that are parallel to the print bed (in the X-Y plane, Fig. 1a), and a machine instruction file (e.g. G-Code) +is generated for the printing process. As the final 3D part is obtained by superposition of 2D layers in the Z-axis, the +conventional FFF approach is also referred to as 2.5D printing. +Based on the printing material, design geometry, printing machine, and other parameters, the design geometry is +often oriented to maximize mechanical properties of interest, which may be dimensional requirements, or mechanical +∗Corresponding author +Email addresses: xaguidetti@control.ee.ethz.ch (Xavier Guidetti), ebalta@control.ee.ethz.ch (Efe C. Balta), +yannick.nagel@nematx.com (Yannick Nagel), hanyin@student.ethz.ch (Hang Yin), ralisa@control.ee.ethz.ch (Alisa Rupenyan), +lygeros@control.ee.ethz.ch (John Lygeros) +Preprint submitted to Additive Manufacturing +January 13, 2023 +arXiv:2301.04999v1 [math.OC] 12 Jan 2023 + +x +y +z +Print bed +Planar +layers +Heated extruder +Thermoplastic filament +(a) 2.5D process +x +y +z +Print bed +Non-planar +layers +Heated extruder +Thermoplastic filament +(b) Non-planar process +Figure 1: Simplified representation of the 2.5D and non-planar fused filament fabrication processes +2 + +strength conditions [2, 3, 4, 5]. While 2.5D FFF printing provides material and geometry flexibility as a manufacturing +process, the layer-wise processing of the parts in a fixed horizontal plane may limit the mechanical performance of the +printed parts [6, 7], as the inter-layer interfaces are prone to failures [8, 9]. For example, if there is oblique loading, or +if the printed part does not follow a simple planar surface, the stress flow often crosses through the inter-layer bonding +locations creating premature failures under loading. While preliminary work exists in the literature for characterizing +the part failure under loading [10], the planar layer constraint in 2.5D FFF limits the applicability of the technology +in mechanically demanding use cases. Similar considerations are also relevant at the intra-layer scale, especially with +anisotropic polymers, for which stress-alignment dramatically improves mechanical properties [11]. +Here instead, we study a non-planar FFF process where layers are defined by arbitrary curves and surfaces in a 3D +space. Non-planar layers provide flexibility in aligning the stress flow with the deposited beads within a layer [12]. +Non-planar FFF printing approaches commonly use a 5-axis printer to deposit material along non-planar trajectories, +such that the stress flow on the designed geometry aligns with the layers deposition direction, improving mechanical +properties [13, 14]. While there are methods in the literature to optimize print trajectories for non-planar layers, ensur- +ing manufacturability constraints under various geometrical features (e.g. holes) is often a challenge [15]. Moreover, +constraining the print paths to meet conditions required by the material (e.g. uniform line spacing, or constant layer +height) has not been addressed. The printability constraints are particularly tight for anisotropic materials. Anisotropy +is however needed to exploit the potential of optimized non-planar printing [16]. +In this work, we utilize novel anisotropic polymer materials and design and manufacture non-planar print trajecto- +ries that are aligned with the stress flow induced by the loading conditions. The resulting design framework considers +the loading conditions and the stress flow on the given design geometry, as well as the material properties of the +printed polymer to optimize the mechanical strength through a novel approach. Section 2 discusses the existing liter- +ature and the shortcomings that are addressed by our method. In Section 3, we introduce the anisotropic polymer we +utilized and the details of our print optimization method. In Section 4, the method is applied to several demonstrator +and benchmarking geometries. Section 5 details the experimental results produced by manufacturing and testing one +of the geometries. Finally, in Section 6, we discuss the results and compare them to the 2.5D approach. +2. Background +Multiple works have investigated the field of non-planar printing [17]. In conformal printing, extrusion is con- +ducted along non-planar trajectories to produce a thin shell over a non-planar substrate. This technique has been used +in [18] to finish a piece by removing the stair artifacts produced by conventional 2.5D printing. Further extending into +non-planar complexity, [12] presents algorithms for printing complex geometries with a 5-axis printer, but the results +are not demonstrated on a real system. Similarly, print trajectory planning for non-planar robotic deposition is studied +in [19], ignoring, however, stress flow alignment and further manufacturability constraints for the printing; the same +holds for [12]. Full end-to-end implementation for non-planar FFF optimization requires knowledge of the extru- +3 + +sion dynamics [20, 21], machine kinematics [14, 22], and efficient trajectory optimization formulations [23, 22] that +consider process constraints [14], material, printed geometry [24, 25, 26], and stress flow field [13]. Consequently, +improving the non-planar printing process performance is not trivial and requires developing advanced methods to op- +timize print path trajectories for arbitrary geometries under a given stress flow field and material extrusion constraints. +Recent work has explored the deposition of fiber-reinforced filaments in the stress flow directions [27, 28]. Such +applications are limited due to the continuity constraints on the deposited fibers and result in restricted flexibility +as fibers need to be cut at the end of each individual deposition path. Other works on fiber-reinforced materials +avoid the cutting problem by producing continuous trajectories [29], however without optimizing for the stress flow. +Stress-orientation in 2.5D has been studied in [30], but only for planar stress. +In recent research, a Liquid Crystal Polymer (LCP) material for 3D printing has been developed, patented, and +adopted by NematX AG1. The material alignment in the direction of extrusion greatly improves the mechanical prop- +erties in the given direction, resulting in anisotropic material properties [11]. By leveraging such materials, it is +possible to improve the mechanical properties of the FFF printed parts, using non-planar printing methods. While +this possibility has been demonstrated in customized applications, currently there exists no rigorous framework that +exploits the anisotropic material properties to improve mechanical strength of FFF through non-planar printing. The +quality of LCP printed parts largely depends on a) the quality of slicing in the desired layer thickness across all layers +and how each layer supports a subsequent one (see [26] for 2.5D printing), and b) the quality of trajectory generation +in each layer and the homogeneous spacing of the deposited lines (see [31] for 2.5D printing). Previous approaches +[13] have successfully tackled the optimization problem for isotropic materials such as polylactic acid (PLA), neglect- +ing the practical constraints of high-quality FFF printing. This resulted, for example, in toolpaths containing large +variations in line spacing or layer height. With anisotropic polymers like LCPs that require small nozzles, low layer +heights and minimal line spacing changes, these variations are too large to print a part successfully. Additionally, the +application of the method from [13] to common load bearing brackets often resulted in print paths that are physically +not printable due to their orientation. +Our novel approach aims to improve the mechanical properties of non-planar FFF printed parts by utilizing +anisotropic materials such as LCPs [11] or fiber-reinforced filaments [32, 16]. We formulate an optimization frame- +work that includes the part properties (load case, material anisotropy) and the manufacturability constraints (extrusion +and machine dynamics). The main contributions of this work are +• A novel stress-aligned non-planar print trajectory generation method that considers layer thickness and line +spacing constraints; +• A non-planar FFF printing framework suitable for mechanical design geometries with turbulent stress flows +generated by holes or complex mechanical features; +1https://nematx.com/ +4 + +• The experimental application of stress-aligned non-planar FFF printing with anisotropic polymers to achieve +44× improvement in mechanical strength over baseline approaches. +To demonstrate the slicing and print trajectory optimization methods that we propose, we will use a load bearing +mechanical piece, that we refer to as the fork for simplicity. Figure 2 shows the fork together with its load case. +The loads create a non-obvious stress flow through the main body and the two arms, which makes stress-aligned +filament deposition challenging. The fork is particularly challenging for the classical 2.5D approach. Irrespective of +the orientation of the piece on the print bed (with or without support), slicing in planar parallel layers will not yield +unbroken print lines traveling through the entire part, with an adverse impact mechanical properties. Moreover, as the +main body splits in two arms, and as the different sections of the fork have different widths, it is difficult to find print +trajectories leading to constantly-spaced, long, unbroken lines. +Load +Load +y +x +z +Fixed +Figure 2: Fork demonstrator geometry and loads. The part is fixed in position via screws in the holes. +3. Material and methods +3.1. Material +The print trajectory optimization method we propose has been developed to exploit the properties of anisotropic +materials for FFF. A notable example are LCPs [11], developed and currently used for high-end applications by +NematX AG. LCPs are composed of aromatic thermotropic polyesters that self-assemble into nematic domains (i.e. +the long axes of the molecules are arranged in parallel, but not in well defined planes) when heated above their melting +temperature. Alignment of different domains is achieved by extrusion through the heated nozzle of a FFF machine: +the elongation and shear forces produced when the material traverses the nozzle reorient the nematic domains in +5 + +the extrusion direction. Studies conducted on the mechanical properties of LCP parts printed with unidirectional +deposition of the filaments have shown that properties are strongly dependent on the printing orientation. For example, +comparison between tensile samples printed 90° (i.e. load perpendicular to the planar print layers surfaces) and 0° (i.e. +filaments aligned to the load) to the loading direction has shown a 7.0× and 3.6× increase in the Young’s modulus and +ultimate tensile strength respectively (see Fig. 3 of [11]). These results clearly show how non-planar stress-aligned +deposition is required to fully exploit the properties of LCPs. The material is very sensitive to the parameters of the +FFF deposition process [31]. The best mechanical properties can be achieved when the layer height and line spacing +are kept as constant as possible, which in turn allows to print with constant flow rate and temperature and produces +the highest print quality. Additionally, during the deposition of tightly packed lines (i.e. whose spacing is significantly +smaller than the nozzle diameter), the drag created by the nozzle printing the next line causes misalignment in the +previously deposited monomers, further reducing performance [33]. The variations in the layer height (±30%) or line +spacing (±50%) that are achieved in the literature of stress-aligned FFF [13] can be very detrimental to the mechanical +properties of LCP printed parts. +3.2. Methods +Offset Slicing +Print Trajectory Opt +• +Stress projection and preprocessing (pic of stress and crit region) +• +Computation of the trajectory-generating scalar field +• +Extraction of the print trajectories and post-processing +Manufacturing +1. Finite Element Analysis +2. Offset Slicing +3. Print Trajectory Optimization +4. Manufacturing +a. +Stress projection and +pre-processing +c. +Extraction of the print +trajectories and post-processing +b. +Computation of the trajectory- +generating scalar field +Figure 3: Graphical flowchart of the stress-aligned trajectory optimization method, demonstrated on the fork geometry +Figure 3 provides a graphical overview of the proposed method for stress-aligned trajectory optimization. For a +given geometry and load case, the first step is to compute the stress in the part using Finite Element Analysis (FEA) +(Section 3.2.1). We then divide the geometry in constantly spaced non-planar slices. The orientation of the slices is +6 + +selected to reduce inter-layer stress, allow long and unbroken fibers across the entire geometry and maximize print +quality through uniform layer height (Section 3.2.2). The third step is to fill each non-planar layer with optimized print +trajectories. We first project the stress computed in FEA on the slice surface, and pre-process it so that it matches the +structure required by the optimization problem. Solving the problem corresponds to computing a trajectory-generating +scalar field in each layer. The isostatic lines (isolines) of this field – which we post-process for manufacturability – are +the stress-aligned and homogeneously spaced print trajectories that form the layer (Section 3.2.3). Once each layer +has been optimized, we join all the trajectories in an ordered sequence of points that is sent to a 5-axis FFF machine +to manufacture the part. In the next sections we explain each step of the method in detail. +3.2.1. Finite Element Analysis +To generate stress-aligned print trajectories, we must first compute the stress in the parts we intend to manufacture. +This is achieved using FEA to simulate the specific load case for which a piece is going to be optimized [34]. We begin +by subdividing the 3D representation of a part in a tetrahedral mesh. After having applied the boundary conditions +corresponding to the forces on the part and having assigned to the part isotropic properties (i.e. an arbitrary Young’s +modulus and a Poisson’s ratio ν = 0.3), we solve the resulting discretized problem leading to the Cauchy stress tensor +σ for each node in the mesh. A general stress tensor is typically decomposed using eigenvalue decomposition to +obtain the principal stresses and principal directions [35]. In a reference frame oriented along the principal directions, +the stress tensor can be represented as a diagonal matrix where the diagonal entries σ1, σ2 and σ3 are ordered so +that |σ1| ≤ |σ2| ≤ |σ3|. We call σ1 the minimum principal stress and σ3 the maximum principal stress and the +corresponding normalized eigenvectors minimum principal stress direction and maximum principal stress direction. +They represent the direction in which the principal stresses act. The maximum and minimum principal stress directions +are orthogonal. We will call principal stress vectors the vectors obtained by multiplying a principal stress with +the corresponding principal stress direction and stress flow the vector field that regroups the principal stress vectors +computed at all the nodes. +3.2.2. Offset Slicing +Conventionally, in 2.5D printing the desired component is sliced in evenly spaced layers that are parallel to the +print bed. The homogeneity of the layers allows one to obtain high-quality prints with excellent layer adhesion, +producing mechanically strong pieces. We adapt this slicing scheme to the non-planar context, where high-degrees- +of-freedom machines can extrude along curved layers. As in 2.5D printing, we choose as our first slice the contact +surface between the part and its (curved) support. Then, we generate the following slices so that each one has a +constant distance from the one below (Fig. 6a). Given a desired geometry and having chosen which one of its faces +will be resting on a support piece, we select this entire face as our first slice. Then, we follow the geodesic heat +method (Alg. 1) proposed in [36] to compute the geodesic distance of a domain (the desired geometry) from a subset +(the first slice). +7 + +Algorithm 1: The Geodesic Heat Method [36] +1 Compute the flow of heat u from a contact surface, by integrating the heat flow ˙u = ∆u for a fixed small time t; +2 From the temperature gradient ∇u, compute the normalized and negated vector field X = −∇u/|∇u|; +3 Recover the final distance φ, whose gradient follows X, by solving the Poisson equation ∆φ = ∇ · X. +The practitioner must first select the orientation of the piece on the print bed or support. Clearly, when working +with the objective to maximize the alignment between the deposited fibers and the maximum principal stress flow, +we would like the normals to the print layers to be aligned with the minimum principal stress flow. Existing methods +[13] that generate slices maximizing the alignment between layer normals and minimum principal stress vectors +simply push the orientation problem downstream, forcing the practitioner to look for ways to manufacture irregularly +oriented layers whose complexity leads to poor prints. Often, this results in twisted or intertwined layers which cannot +be physically manufactured. As our method focuses on manufacturability, the initial choice of orientation is actually +an advantage, as it allows one to select the orientation producing the best print quality. Additionally, when printing +mechanical pieces designed for a well defined load case, we noticed it is fairly easy and intuitive to find an initial +orientation generating a satisfactory layer alignment. We support this claim empirically in Section 4.1. +3.2.3. Print Trajectory Optimization +The starting point of our approach to print trajectory optimization is the work of Fang et al. [13], that we embellish +with numerous refinement steps which we discuss in detail. We first note that irrespective of the slicing method, +there will be regions where the maximum principal stress vector is not tangent to the surface of the slice. Thus, +we begin by projecting the maximum principal stress flow field onto the slice. Because of the structure of the print +path optimization problem, we then need to rotate the projected vectors around each surface normal to obtain their +orthogonal vectors f⊥. For a vector f (belonging to the stress flow F) and the corresponding surface normal n, we +compute f⊥ as +u = f − f · n +n · nn , +f⊥ = u × n +∥u × n∥ . +(1) +We regroup all calculated f⊥ in the projected, orthogonal and normalized maximum principal stress flow F⊥. +The heterogeneous stress orientations produced by the FEA, coupled with the fact that surface normals can lie on +both side of the slice, can create ambiguities in the stress flow F⊥. This undesired effect occurs in regions where the +vectors composing the stress flow are aligned (approximately) in the same direction but have opposite orientations. As +the toolpath optimization problem that we pose is sensitive to vector orientation ambiguity, we propose to pre-process +the stress flow. We first analyze all the vectors in F⊥ to determine their main Cartesian component. We compute +id = arg max +i +∥F⊥ · ei∥ , +(2) +where id represents the Cartesian component to which the vectors in F⊥ have the best alignment and ei, i = 1, 2, 3 are +8 + +the unit axes of a Cartesian reference frame in R3. Now, we can rectify the stress flow by calculating +fr = +������������� +f⊥ +if f⊥ · eid ≥ 0 +−f⊥ +otherwise +(3) +for all vectors in F⊥. We then regroup the resulting fr in the rectified stress flow Fr. Figure 5 includes an example +of stress flow rectification. Intuitively, the rectification consists in flipping the vectors that do not lie in main stress +flow orientation. The rectification ensures that the print trajectories that we then generate are homogeneous and easily +manufacturable. We point out that our method requires the stress flow to be rectified, but is not sensitive to the rectified +vectors orientation. This is because the extruded material should simply align with the main traction/compression +direction, making the problem invariant to 180° rotations of the stress flow. +Next we distinguish critical stress regions from uncritical ones. We consider a node as critical if the stress is both +sufficiently anisotropic and sufficiently large. The former condition translates to +|σ3| +|σ1| > θa , +(4) +where θa is the anisotropy threshold; the latter to +|σ3| +maxnodes |σ3| > θs , +(5) +where θs is the stress significance threshold. The idea behind this classification is that some regions of the desired +geometry undergo large and quasi unidirectional stresses: in these regions the alignment between deposited material +and the stress flow is crucial. Other regions are subject to low or isotropic stresses and the material alignment is not +important. In uncritical regions, we can print in the most convenient direction, which is typically the continuation of +the print direction used in the critical regions. This allows us to deposit long unbroken fibers in our manufactured +piece, fully exploiting LCPs properties. Figure 4 illustrates on an example geometry the critical regions corresponding +to different choices of the stress anisotropy and significance parameters. +It is now possible to extrapolate the stress flow fr of the nodes at the critical region boundary to the neighboring +uncritical nodes. We propagate the critical stress flow, until all the uncritical nodes have their original stress vector +replaced by an extrapolated (and normalized) critical stress vector. The extrapolation is conducted by solving an +optimization problem that minimizes the Dirichlet energy of the stress flow in the uncritical region, constrained by the +boundary conditions with the critical region. The optimal extrapolated stress flow is the minimizer of the constrained +problem +ˆFuc = arg minFuc +∥GFuc∥2 +s.t. +fuc = fr +for each node at the boundary +fuc · n = 0 +for each node in the uncritical region , +(6) +where Fuc regroups all the stress flow vectors fuc of the uncritical regions and G is the numerical gradient over the +triangular mesh of the uncritical region. Solving (6) corresponds to finding the nodal vectors that minimize the +9 + +(a) θa = 3 , θs = 0.1 +(b) θa = 15 , θs = 0.1 +(c) θa = 3 , θs = 0.2 +(d) θa = 15 , θs = 0.2 +Figure 4: Critical region shape (in red) as a function of the stress anisotropy and significance parameters θa and θs, shown on a slice of the fork +demonstrator geometry for the load case that will be discussed in Sec. 4. The same example slice will be used in all other figures. Experimentation +suggests that the method produces better trajectories when selecting parameters that produce a simply connected critical region, which in this +scenario corresponds to case (a). +10 + +orientation variability in the entire vector field. The two constraints enforce the requirement that the vectors at the +critical-uncritical boundary are fixed and that the solutions found in the uncritical region are tangent to the surface. +The solution ˆFuc replaces the stress flow of the uncritical nodes. Figure 5 includes an example of the propagation +process. +(a) Before stress flow rectification +(b) After stress flow rectification +(c) After stress flow propagation +Figure 5: Rectification and propagation of the stress flow, shown on a slice of the fork demonstrator geometry for the load case that will be +discussed in Sec. 4. Note the agreement in the vectors orientation after rectification and the change of stress flow direction in the uncritical region +after rectification. +Having applied all the previous steps to the results of the FEA simulation, to each node in the slice corresponds a +pre-processed orthogonal stress vector fp. We regroup all fp in Fp. We are ultimately interested in solving the problem +ˆφ = arg min +φ +� +S +∥∇φ − Fp∥2ds , +(7) +to find a scalar field ˆφ whose gradient matches as closely as possible the orthogonal stress flow Fp on the slice S . +Given the discretized nature of our problem, we reformulate the optimization problem (7) as the following regularized +optimization problem +ˆφ = arg min +φ +∥Gφ − Fp∥2 + ϵ∥φ∥2, +(8) +where G is the numerical gradient over the triangular mesh of the slice and ϵ is the coefficient of the regularizer. There +exists a closed form solution of (8), +ˆφ = (G⊤G + ϵI)−1G⊤Fp . +(9) +We can now produce the optimized print trajectories, which are isolines of the scalar field ˆφ that we just obtained. +Calculating the isolines is a straightforward operation that just requires extracting equally spaced isolines on the slice +mesh, based on ˆφ . The isolines spacing will correspond to the distance between two print trajectories. We conduct a +11 + +hfinal post-processing on the print trajectories by cubic spline re-sampling [37]. We first fit a cubic smoothing spline +interpolation, which produces a quasi-identical trajectory, but reduces major roughness. The spline is then sampled at +even distances to produce a sequence of points that represent the print trajectory. This final smoothing step improves +manufacturability and ensures that the mechanical properties of the part will not be limited by a poor quality print. +4. Implementation +In the following sections, we apply the slicing and trajectory optimization methods that we have just described +to the fork demonstrator geometry. The entire implementation is conducted using MATLAB. We first mesh the part, +apply the loads shown in Fig. 2 and conduct FEA to compute the Cauchy stress tensors that will be used in slicing and +trajectory optimization. +4.1. Offset Slicing +For each node of the FEA mesh, we compare the the maximum principal stress vectors with the slicing surfaces +normals. Following Sec. 3.2.3, the ideal slicing is obtained when each maximum principal stress is perpendicular to +the corresponding slice normal. Given the maximum principal stress vector f and the slicing surface normal n (both +assumed to be normalized) at the node j, we compute the nodal stress-alignment as +γj = ∥f × n∥ . +(10) +We calculate the average slicing stress-alignment over the entire piece by considering the critical regions only. For a +meshed piece containing J critical nodes, we obtain +¯γ = 1 +J +� +critical j +γ j . +(11) +The proposed metric will produce ¯γ ∈ [0, 1], where ¯γ = 1 corresponds to a perfectly aligned slicing and ¯γ = 0 to ta +case where the maximum principal stress vector is perpendicular to the slices. A comparison of our method (Fig. 6a) +with standard planar slicing along the X, Y, and Z axes of the demonstrator bracket reference frame is quantified in +Table 1. Both the offset slicing method we propose and the planar slicing along the Y-axis have very high alignment +Table 1: Average slicing stress-alignment for different slicing methods (evaluated in the critical regions only), with identical layers distance. +Offset +Planar X +Planar Y +Planar Z +¯γ +0.96 +0.38 +0.98 +0.87 +scores ¯γ. The Y-axis planar slicing, however, produces a large fraction of slices that belong exclusively to the fork +arms. As a consequence, the manufactured part would be more prone to inter-layer fractures in the arms. Conversely, +12 + +(a) Offset slicing produced by our method +(b) Example of one slice obtained via offset slicing (light blue) with the critical +maximum principal stress directions (dark blue) belonging to the slice. Note how +most of the vectors are approximately tangent to the slice surface, producing an +average slicing stress-alignment ¯γ ≈ 1. +(c) Critical region (in red) on one slice of the part. The critical region forms a sim- +ply connected surface and excludes the areas where the stress is low or isotropic. +(d) Trajectory-generating scalar field on one slice of the part. The color map is +associated to the value of the scalars associated to each node, showing a smooth +gradient. +(e) Isolines extracted from the trajectory-generating scalar field on one slice of the part. Note how the lines are very uniformly distributed, ensuring that the trajectories +can produce a high quality print. Additionally, the trajectories follow the arms of the bracket closely and connect the extremities of the parts with unbroken print lines. +Figure 6: Main steps of the implementation on the fork geometry +13 + +Z +XZ +XZ +XZ +Xthe proposed offset slicing allows to work on layers that span the entire piece, significantly increasing the strength of +the printed part and making offset slicing superior to all planar slicing methods. Figure 6b shows one of the slices +obtained using offset slicing, together with the critical maximum principal stress directions that were used to compute +γ on the depicted slice. +4.2. Print Trajectory Optimization +Next, we generate the print trajectories for each slice individually. We first project and rectify the stress flow on a +slice, and then determine the critical regions. We set the parameters of Equations (4) and (5) to θa = 3 and θs = 0.1, +to produce a simply connected critical region that covers most of each slice. The following step consists in solving +Eq. (9) to compute the values of the trajectory-generating scalar field over the nodes of each meshed slice. We show +the critical region and the scalar field corresponding to the example slice in Fig. 6c and 6d. Finally, it is possible +to extract the isolines of the scalar field that was just calculated. The isolines spacing corresponds to the distance +between two neighboring print trajectories, and can be easily selected to match the material and printer requirements. +Figure 6e shows the isolines that were obtained from the scalar field of Fig. 6d. In order to maximize the quality of +the print, we post-process the isolines. These are first interpolated using a cubic spline with a smoothing parameter +p = 0.95, to smoothen the small-scale wiggles that can be caused by meshing and numerical approximation. Then +we add contour lines to each slice and trim the smoothed isolines to avoid overlap with the contours during printing. +We finally connect sequentially the adjacent print lines and generate travel moves where required. Figure 7 shows the +final print trajectory that will be sent to the printer to manufacture one layer. +Figures 8 and 9 illustrate the application of our method to two benchmarking geometries taken from the literature, +which are commonly known as bunny head and topology optimization [13]. In table 2, we show the average trajectory +stress-alignment ¯β for the three geometries that we have demonstrated. Similarly to the metric ¯γ that we have defined +previously, we first compute the stress-alignment +βk = ∥f · d∥ , +(12) +where k indicates a point on the sampled print trajectory, and f and d are the maximum principal stress and print +direction at that location. We then average βk across all K critical points on the print trajectory to produce +¯β = 1 +K +� +critical k +βk . +(13) +The values of β range between 0 (no alignment) to 1 (perfect alignment). The results in Table 2 show that the +trajectories produced with our method excellent alignment with the maximum principal stress flow. +Finally, we show the distribution of distances between print trajectories in Fig. 10. A tight distribution corresponds +to excellent manufacturability, as the lines are very uniformly spaced. Other methods in the literature produce distri- +butions that appear to be sums of Gaussians, with significant tails extending away from the nominal spacing (Fig. 13 +of [13]). In Fig. 10, this behavior is completely absent: the distributions are extremely tight and centered exactly on +14 + +Finish +Start +Figure 7: Post-processed print trajectories on one layer of the part. Travel moves are depicted in red. +15 + +(a) Offset slicing +(b) Isolines and scalar field on one layer of the part +Figure 8: Application of our method to the benchmarking piece known as bunny head +(a) Offset slicing +(b) Isolines and scalar field on one layer of the part +Figure 9: Application of our method to the benchmarking piece known as topology optimization +Table 2: Average trajectory stress-alignment for different geometries (evaluated in the critical regions only) +Fork +Bunny Head +Topology +Optimization +¯β +0.94 +0.74 +0.88 +16 + +Z +XZ +Y +Xthe nominal line spacing distance. Table 3 compares the mean and variance of the distribution of the distances pro- +duced by our method and by the method by Fang et al. [13]. While the means are well centered around the nominal +line spacing for both methods (i.e. mean normalized distance ≈ 1), the variance produced by [13] is worse due to the +tails of the distribution. Our method, which targets manufacturability via homogeneous line spacing, exhibits instead +very low distances variance. This – paired with the homogeneous layer height produced by offset slicing – ensures +that very high-quality prints can be achieved with all materials, and particularly with sensitive ones such as LCPs. +0 +0.5 +1 +1.5 +2 +Normalized distance [0.4 mm] +0 +10 +20 +30 +40 +Density [%] +Fork +Bunny +Head +Topology +Optimization +Figure 10: Distribution of distances between print trajectories for the fork, bunny head and topology optimization geometries. The distance has +been normalized using the nominal line spacing of 0.4 mm that was selected when generating the trajectories. +Table 3: Mean and variance of the distribution of distances using the proposed method (see Fig. 10) and the method by Fang et al [13]. The data +from Fang et al. have been extracted from Fig. 13 of [13]. In both methods, the data have been normalized against the nominal line spacing and are +thus adimensional. +Fork +Bunny Head +Topology Optimization +Mean +Variance +Mean +Variance +Mean +Variance +Proposed method +1.01 +5.0 × 10−3 +1.01 +4.4 × 10−3 +1.03 +13.2 × 10−3 +Fang et al. [13] +N/A +N/A +0.97 +31.4 × 10−3 +0.98 +31.3 × 10−3 +5. Experimental Results +We have used the fork demonstrator geometry to validate experimentally the approach we have proposed and +quantify the improvements achieved through stress-aligned print trajectory optimization. The parts were manufactured +using a modified 5-axis computer numerical control (CNC) machine (5AxisMaker). The CNC machine was modified +by replacing the milling head with a commercial direct drive filament extruder (E3D Hemera) and was controlled +17 + +using Mach32 on a Windows PC. Figure 11 shows the custom printer and highlights its main components and axes. +y +z +x +r1 +r2 +Printed part +Extruder +Controller PC +Axes drives +Figure 11: Custom 5-axis FFF machine with highlighted principal components. +(a) Manufactured part attached to the fixture +(b) Part and fixture mounted on the testing rig +Figure 12: Installation of a fork manufactured using stress-aligned trajectories and LCPs on the testing rig +We tested the printed samples according to the load case for which the optimization was conducted (see Fig. 2). To +do so, we used a Zwick Z020 universal testing machine with a 20 kN capacity load cell. Each fork was attached to +the machine bottom sample holder with two screws passing through their holes. Then, the machine testing head was +lowered at a speed of 2 mm/ min. The tests were stopped when the applied force dropped under a predefined threshold +value, which is typically reached when the specimen is completely broken. During the test, the displacement and +applied force were recorded to trace a force-displacement curve and to compute the failure force and stiffness of each +printed specimen. +2https://www.machsupport.com/software/mach3/ +18 + +5.1. Experimental Printer Impact +Table 4: Failure force and stiffness of the isotropic PLA samples, used to evaluate the impact of the 5-axis experimental printer on mechanical +properties +PLA +2.5D +Prusa +Non-planar +Experimental +Failure force [N] +80.0 +52.9 +Stiffness [N/mm] +6.0 +4.6 +As the non-planar parts have been manufactured on an experimental machine, we expected the quality of our +non-planar prints to be inferior to what commercial printers achieve. Low print quality negatively affects the me- +chanical properties of printed components. Thus, we first evaluated the impact of our experimental 5-axis machine +on part properties by printing and comparing two forks with an isotropic material (PLA). The 2.5D fork was sliced +with a commercial planar slicer (Prusa Slicer) and printed on a Prusa i3 MK3s; the non-planar one was sliced with +our proposed method and printed on the experimental 5-axis machine. Both trajectory generation methods were set to +produce a layer thickness of 0.1 mm and a line spacing (corresponding to the extrusion width) of 0.4 mm. We used two +contour lines and printed at the same feed rate in both cases. The 2.5D slicer was set to produce an aligned rectilinear +infill with 100% density and to slice the part along the Z axis of the demonstrator bracket reference frame. Theoret- +ically, given the isotropic properties of PLA, the mechanical properties of the conventional and optimized samples +should be similar. However, the parts manufactured with the commercial printer rarely contain defects; conversely, +we expected the complexity and the experimental nature of our printer to increase the number of imperfections in the +part. The results in Table 4 – that we discuss comprehensively in Sec. 6 – show that the print imperfections produced +by the experimental machine during non-planar printing worsen the mechanical properties of the fork. Pictures of the +PLA samples after failure are shown in Fig. 14a and 14b. +5.2. LCPs results +After assessing the impact of the experimental 5-axis printer using isotropic polymers, we finally evaluated the +performance of our method on anisotropic materials. Following the same specifications discussed above, we printed +two samples using LCPs. The 2.5D sample was produced by planar slicing on a Prusa i3 MK3s, while the non-planar +sample was sliced with our method and printed on the 5-axis machine. The results are contained in Table 5 and +discussed in Sec. 6. Figure 13 depicts the force-displacement plots of the two samples. Pictures of the LCP samples +after failure are shown in Fig. 14c and 14d. +19 + +Table 5: Failure force and stiffness of the anisotropic LCP samples. The non-planar samples were sliced with the proposed optimization method +and considerably outperform conventional 2.5D slicing. +LCP +2.5D +Conventional +Non-planar +Optimized +Failure force [N] +2.6 +114.7 +Stiffness [N/mm] +2.4 +14.8 +0 +5 +10 +15 +20 +25 +Displacement [mm] +0 +20 +40 +60 +80 +100 +120 +Force [N] +1 +2 +3 +(1) Non-planar LCP +(2) 2.5D LCP +(3) 2.5D PLA +Figure 13: Force-displacement plot of the tested samples. To facilitate the comparison, the plot includes the curve of the 2.5D PLA sample produced +with a conventional material, machine and slicing method. +20 + +6. Discussion +The experiments demonstrate how print trajectory optimization vastly improves the mechanical properties of the +parts manufactured with anisotropic materials. The failure force of the LCP samples has increased by a factor of 44, +and the stiffness by a factor of 6. In light of the experimental nature of the equipment used for non-planar prints (whose +impact with respect to commercial-grade 2.5D prints has been quantified to -34% failure force and -23% stiffness in +Sec. 5.1), the improvements obtained with LCP samples are even more remarkable. +(a) 2.5D PLA +(b) Non-planar PLA +(c) 2.5D LCP +(d) Non-planar LCP +Figure 14: Failure during testing of the four printed combinations of material and trajectory generation method +Figure 14 offers important insights to better understand the quantitative results of the experiments. Looking first at +the 2.5D prints (Fig. 14a and 14c), both pieces break at similar locations, in the inclined part of the bracket, where the +part cross section is minimal and horizontal layers have a very small surface. Particularly for LCPs, it is evident how +the material has poor properties when the load is perpendicular to the layers’ surface, producing inter-layer fractures. +Thanks to its better layer adhesion properties, PLA breaks at a similar location but without a clear inter-layer failure. +Moving to the case of non-planar prints, we observe how PLA (Fig. 14b) breaks at the lower curvature region. As +discussed previously, PLA is quite insensitive to print orientation, and the fracture should ideally happen as in Fig. +14a. However, the lower curvature has been found to be the most complex region to print with the 5-axis equipment. +The machine kinematics make high-quality material deposition in convex curvatures difficult due to large accelerations +(and consequently vibrations) in the axes. It appeared upon observation that at the lower curvature the material was +partially over-extruded. We believe that this defect is the cause of the mechanical properties worsening found in the +non-planar PLA part, as well as for the failure location shown in Fig. 14b. Finally, Fig. 14d suggests that, despite the +printing defects, the non-planar LCP fork fails at the location where FEA predicted the largest stress. Failure takes +21 + +place when the deposited LCP fibers – which are stressed in the deposition direction – break under tension. Thanks +to the optimized layers and print trajectories orientation, the part does not suffer from inter-layer or inter-trajectory +fractures, and the anisotropic nature of LCPs is fully exploited to produce excellent mechanical properties. +7. Conclusions +In multi-axis 3D printing, it is possible to produce components using filaments that are deposited in almost any +desired spatial orientation. This freedom can be exploited to improve the mechanical properties of the printed parts, +particularly when using anisotropic polymers as feed stock. By aligning the material deposition direction with that +of the stress flow induced by a component’s load case, it is possible to dramatically improve the failure force and +stiffness of the part. +In this work, we have proposed a method for non-planar slicing and print trajectory optimization that consid- +ers manufacturing constraints such as constant layer thickness and line spacing. These constraints are particularly +important to obtain high-quality prints and avoid early failure of the parts. Using computations on benchmarking +geometries, we showed that our method produces extremely homogeneous and thus well manufacturable trajecto- +ries when compared to existing approaches. Our framework is suitable for complex geometries and stress flows +frequently encountered in common mechanical components. We have conducted experiments to manufacture load- +bearing brackets using LCP material from NematX AG on a 5-axis 3D printer. The results have demonstrated that, +despite the introduction of deposition defects linked with the complexity of 5-axis printing, the method fully exploits +the strongly anisotropic nature of LCPs. The optimized prints achieve a 44× improvement in failure strength and a 6× +improvement in stiffness, compared with conventional planar printing. +Funding +This work was supported by the Swiss Innovation Agency (Innosuisse, grant №102.617) and by the Swiss National +Science Foundation under NCCR Automation (grant №180545). +Acknowledgments +We acknowledge the support of NematX AG that provided the LCP material and experimental setup for this work. +References +[1] The state of 3d printing report: 2021 by sculpteo, https://www.sculpteo.com/en/ebooks/state-of-3d-printing-report-2021/, +(Accessed on 07/05/2022) (2021). +[2] M. J. Hooshmand, S. Mansour, A. 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Wolberg, Cubic spline interpolation: A review, Columbia University Computer Science Technical Reports (1988). +doi:10.7916/ +D82Z1DMQ. +24 + diff --git a/iNE4T4oBgHgl3EQfSgxY/content/tmp_files/load_file.txt b/iNE4T4oBgHgl3EQfSgxY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9d5ecfa69adda73fbf3fe76918595149bfaddd6 --- /dev/null +++ b/iNE4T4oBgHgl3EQfSgxY/content/tmp_files/load_file.txt @@ -0,0 +1,736 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf,len=735 +page_content='Stress Flow Guided Non-Planar Print Trajectory Optimization for Fused Filament Fabrication of Anisotropic Polymers Xavier Guidettia,b, Efe C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Baltaa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Physikstrasse 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 8092,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Switzerland bInspire AG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Technoparkstrasse 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 8005,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Switzerland cNematX AG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Vladimir-Prelog-Weg 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 8093,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Switzerland dDepartment of Mechanical and Process Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' ETH Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Leonhardstrasse 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 8092,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Switzerland Abstract When manufacturing parts using fused filament fabrication and anisotropic polymers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' the mechanical properties of a manufactured component are strongly dependent on the print trajectory orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We conduct non-planar slicing and optimize the print trajectories to maximize the alignment between the material deposition direction and the stress flow induced by a predefined load case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The trajectory optimization framework considers manufacturability constraints in the form of uniform layer height and line spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We demonstrate the method by manufacturing a load bearing mechanical bracket using a 5-axis 3D printer and a liquid crystal polymer material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The failure strength and stiffness of the optimized bracket are improved by a factor of 44 and 6 respectively when compared with conventional printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Keywords: Fused Filament Fabrication, Trajectory Optimization, Non-Planar Printing, Stress Alignment, Anisotropy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Introduction In most 3D-printing applications, the material is formed in 2D layers via the printing technology of interest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' material extrusion, laser power, or lithography).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' By forming subsequent layers on top of the previous ones, a 3D object is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Fused filament fabrication (FFF), also known as fused deposition modeling, is one of the most common 3D printing processes [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Traditionally, in FFF, a thermoplastic material is deposited in predefined paths via a numerically controlled heated extruder to create a part in a layer-wise fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The design geometry is sliced into planar layers that are parallel to the print bed (in the X-Y plane, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 1a), and a machine instruction file (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' G-Code) is generated for the printing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' As the final 3D part is obtained by superposition of 2D layers in the Z-axis, the conventional FFF approach is also referred to as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Based on the printing material, design geometry, printing machine, and other parameters, the design geometry is often oriented to maximize mechanical properties of interest, which may be dimensional requirements, or mechanical ∗Corresponding author Email addresses: xaguidetti@control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='ch (Xavier Guidetti), ebalta@control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='ch (Efe C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Balta), yannick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='nagel@nematx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='com (Yannick Nagel), hanyin@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='ch (Hang Yin), ralisa@control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='ch (Alisa Rupenyan), lygeros@control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='ch (John Lygeros) Preprint submitted to Additive Manufacturing January 13, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='04999v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='OC] 12 Jan 2023 x y z Print bed Planar layers Heated extruder Thermoplastic filament (a) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D process x y z Print bed Non-planar layers Heated extruder Thermoplastic filament (b) Non-planar process Figure 1: Simplified representation of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D and non-planar fused filament fabrication processes 2 strength conditions [2, 3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' While 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D FFF printing provides material and geometry flexibility as a manufacturing process, the layer-wise processing of the parts in a fixed horizontal plane may limit the mechanical performance of the printed parts [6, 7], as the inter-layer interfaces are prone to failures [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' For example, if there is oblique loading, or if the printed part does not follow a simple planar surface, the stress flow often crosses through the inter-layer bonding locations creating premature failures under loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' While preliminary work exists in the literature for characterizing the part failure under loading [10], the planar layer constraint in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D FFF limits the applicability of the technology in mechanically demanding use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Similar considerations are also relevant at the intra-layer scale, especially with anisotropic polymers, for which stress-alignment dramatically improves mechanical properties [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Here instead, we study a non-planar FFF process where layers are defined by arbitrary curves and surfaces in a 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Non-planar layers provide flexibility in aligning the stress flow with the deposited beads within a layer [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Non-planar FFF printing approaches commonly use a 5-axis printer to deposit material along non-planar trajectories, such that the stress flow on the designed geometry aligns with the layers deposition direction, improving mechanical properties [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' While there are methods in the literature to optimize print trajectories for non-planar layers, ensur- ing manufacturability constraints under various geometrical features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' holes) is often a challenge [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Moreover, constraining the print paths to meet conditions required by the material (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' uniform line spacing, or constant layer height) has not been addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The printability constraints are particularly tight for anisotropic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Anisotropy is however needed to exploit the potential of optimized non-planar printing [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' In this work, we utilize novel anisotropic polymer materials and design and manufacture non-planar print trajecto- ries that are aligned with the stress flow induced by the loading conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The resulting design framework considers the loading conditions and the stress flow on the given design geometry, as well as the material properties of the printed polymer to optimize the mechanical strength through a novel approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Section 2 discusses the existing liter- ature and the shortcomings that are addressed by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' In Section 3, we introduce the anisotropic polymer we utilized and the details of our print optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' In Section 4, the method is applied to several demonstrator and benchmarking geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Section 5 details the experimental results produced by manufacturing and testing one of the geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Finally, in Section 6, we discuss the results and compare them to the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Background Multiple works have investigated the field of non-planar printing [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' In conformal printing, extrusion is con- ducted along non-planar trajectories to produce a thin shell over a non-planar substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' This technique has been used in [18] to finish a piece by removing the stair artifacts produced by conventional 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Further extending into non-planar complexity, [12] presents algorithms for printing complex geometries with a 5-axis printer, but the results are not demonstrated on a real system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Similarly, print trajectory planning for non-planar robotic deposition is studied in [19], ignoring, however, stress flow alignment and further manufacturability constraints for the printing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' the same holds for [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Full end-to-end implementation for non-planar FFF optimization requires knowledge of the extru- 3 sion dynamics [20, 21], machine kinematics [14, 22], and efficient trajectory optimization formulations [23, 22] that consider process constraints [14], material, printed geometry [24, 25, 26], and stress flow field [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Consequently, improving the non-planar printing process performance is not trivial and requires developing advanced methods to op- timize print path trajectories for arbitrary geometries under a given stress flow field and material extrusion constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Recent work has explored the deposition of fiber-reinforced filaments in the stress flow directions [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Such applications are limited due to the continuity constraints on the deposited fibers and result in restricted flexibility as fibers need to be cut at the end of each individual deposition path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Other works on fiber-reinforced materials avoid the cutting problem by producing continuous trajectories [29], however without optimizing for the stress flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Stress-orientation in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D has been studied in [30], but only for planar stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' In recent research, a Liquid Crystal Polymer (LCP) material for 3D printing has been developed, patented, and adopted by NematX AG1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The material alignment in the direction of extrusion greatly improves the mechanical prop- erties in the given direction, resulting in anisotropic material properties [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' By leveraging such materials, it is possible to improve the mechanical properties of the FFF printed parts, using non-planar printing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' While this possibility has been demonstrated in customized applications, currently there exists no rigorous framework that exploits the anisotropic material properties to improve mechanical strength of FFF through non-planar printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The quality of LCP printed parts largely depends on a) the quality of slicing in the desired layer thickness across all layers and how each layer supports a subsequent one (see [26] for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D printing), and b) the quality of trajectory generation in each layer and the homogeneous spacing of the deposited lines (see [31] for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D printing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Previous approaches [13] have successfully tackled the optimization problem for isotropic materials such as polylactic acid (PLA), neglect- ing the practical constraints of high-quality FFF printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' This resulted, for example, in toolpaths containing large variations in line spacing or layer height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' With anisotropic polymers like LCPs that require small nozzles, low layer heights and minimal line spacing changes, these variations are too large to print a part successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Additionally, the application of the method from [13] to common load bearing brackets often resulted in print paths that are physically not printable due to their orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Our novel approach aims to improve the mechanical properties of non-planar FFF printed parts by utilizing anisotropic materials such as LCPs [11] or fiber-reinforced filaments [32, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We formulate an optimization frame- work that includes the part properties (load case, material anisotropy) and the manufacturability constraints (extrusion and machine dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The main contributions of this work are A novel stress-aligned non-planar print trajectory generation method that considers layer thickness and line spacing constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' A non-planar FFF printing framework suitable for mechanical design geometries with turbulent stress flows generated by holes or complex mechanical features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 1https://nematx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='com/ 4 The experimental application of stress-aligned non-planar FFF printing with anisotropic polymers to achieve 44× improvement in mechanical strength over baseline approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' To demonstrate the slicing and print trajectory optimization methods that we propose, we will use a load bearing mechanical piece, that we refer to as the fork for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Figure 2 shows the fork together with its load case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The loads create a non-obvious stress flow through the main body and the two arms, which makes stress-aligned filament deposition challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The fork is particularly challenging for the classical 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Irrespective of the orientation of the piece on the print bed (with or without support), slicing in planar parallel layers will not yield unbroken print lines traveling through the entire part, with an adverse impact mechanical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Moreover, as the main body splits in two arms, and as the different sections of the fork have different widths, it is difficult to find print trajectories leading to constantly-spaced, long, unbroken lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Load Load y x z Fixed Figure 2: Fork demonstrator geometry and loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The part is fixed in position via screws in the holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Material and methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Material The print trajectory optimization method we propose has been developed to exploit the properties of anisotropic materials for FFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' A notable example are LCPs [11], developed and currently used for high-end applications by NematX AG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' LCPs are composed of aromatic thermotropic polyesters that self-assemble into nematic domains (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' the long axes of the molecules are arranged in parallel, but not in well defined planes) when heated above their melting temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Alignment of different domains is achieved by extrusion through the heated nozzle of a FFF machine: the elongation and shear forces produced when the material traverses the nozzle reorient the nematic domains in 5 the extrusion direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Studies conducted on the mechanical properties of LCP parts printed with unidirectional deposition of the filaments have shown that properties are strongly dependent on the printing orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' For example, comparison between tensile samples printed 90° (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' load perpendicular to the planar print layers surfaces) and 0° (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' filaments aligned to the load) to the loading direction has shown a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='0× and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='6× increase in the Young’s modulus and ultimate tensile strength respectively (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 3 of [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' These results clearly show how non-planar stress-aligned deposition is required to fully exploit the properties of LCPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The material is very sensitive to the parameters of the FFF deposition process [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The best mechanical properties can be achieved when the layer height and line spacing are kept as constant as possible, which in turn allows to print with constant flow rate and temperature and produces the highest print quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Additionally, during the deposition of tightly packed lines (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' whose spacing is significantly smaller than the nozzle diameter), the drag created by the nozzle printing the next line causes misalignment in the previously deposited monomers, further reducing performance [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The variations in the layer height (±30%) or line spacing (±50%) that are achieved in the literature of stress-aligned FFF [13] can be very detrimental to the mechanical properties of LCP printed parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Methods Offset Slicing Print Trajectory Opt Stress projection and preprocessing (pic of stress and crit region) Computation of the trajectory-generating scalar field Extraction of the print trajectories and post-processing Manufacturing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Finite Element Analysis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Offset Slicing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Print Trajectory Optimization 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Manufacturing a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Stress projection and pre-processing c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Extraction of the print trajectories and post-processing b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Computation of the trajectory- generating scalar field Figure 3: Graphical flowchart of the stress-aligned trajectory optimization method, demonstrated on the fork geometry Figure 3 provides a graphical overview of the proposed method for stress-aligned trajectory optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' For a given geometry and load case, the first step is to compute the stress in the part using Finite Element Analysis (FEA) (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We then divide the geometry in constantly spaced non-planar slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The orientation of the slices is 6 selected to reduce inter-layer stress, allow long and unbroken fibers across the entire geometry and maximize print quality through uniform layer height (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The third step is to fill each non-planar layer with optimized print trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We first project the stress computed in FEA on the slice surface, and pre-process it so that it matches the structure required by the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Solving the problem corresponds to computing a trajectory-generating scalar field in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The isostatic lines (isolines) of this field – which we post-process for manufacturability – are the stress-aligned and homogeneously spaced print trajectories that form the layer (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Once each layer has been optimized, we join all the trajectories in an ordered sequence of points that is sent to a 5-axis FFF machine to manufacture the part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' In the next sections we explain each step of the method in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Finite Element Analysis To generate stress-aligned print trajectories, we must first compute the stress in the parts we intend to manufacture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' This is achieved using FEA to simulate the specific load case for which a piece is going to be optimized [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We begin by subdividing the 3D representation of a part in a tetrahedral mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' After having applied the boundary conditions corresponding to the forces on the part and having assigned to the part isotropic properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' an arbitrary Young’s modulus and a Poisson’s ratio ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='3), we solve the resulting discretized problem leading to the Cauchy stress tensor σ for each node in the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' A general stress tensor is typically decomposed using eigenvalue decomposition to obtain the principal stresses and principal directions [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' In a reference frame oriented along the principal directions, the stress tensor can be represented as a diagonal matrix where the diagonal entries σ1, σ2 and σ3 are ordered so that |σ1| ≤ |σ2| ≤ |σ3|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We call σ1 the minimum principal stress and σ3 the maximum principal stress and the corresponding normalized eigenvectors minimum principal stress direction and maximum principal stress direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' They represent the direction in which the principal stresses act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The maximum and minimum principal stress directions are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We will call principal stress vectors the vectors obtained by multiplying a principal stress with the corresponding principal stress direction and stress flow the vector field that regroups the principal stress vectors computed at all the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Offset Slicing Conventionally, in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D printing the desired component is sliced in evenly spaced layers that are parallel to the print bed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The homogeneity of the layers allows one to obtain high-quality prints with excellent layer adhesion, producing mechanically strong pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We adapt this slicing scheme to the non-planar context, where high-degrees- of-freedom machines can extrude along curved layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' As in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D printing, we choose as our first slice the contact surface between the part and its (curved) support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Then, we generate the following slices so that each one has a constant distance from the one below (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Given a desired geometry and having chosen which one of its faces will be resting on a support piece, we select this entire face as our first slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Then, we follow the geodesic heat method (Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 1) proposed in [36] to compute the geodesic distance of a domain (the desired geometry) from a subset (the first slice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 7 Algorithm 1: The Geodesic Heat Method [36] 1 Compute the flow of heat u from a contact surface, by integrating the heat flow ˙u = ∆u for a fixed small time t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 2 From the temperature gradient ∇u, compute the normalized and negated vector field X = −∇u/|∇u|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 3 Recover the final distance φ, whose gradient follows X, by solving the Poisson equation ∆φ = ∇ · X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The practitioner must first select the orientation of the piece on the print bed or support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Clearly, when working with the objective to maximize the alignment between the deposited fibers and the maximum principal stress flow, we would like the normals to the print layers to be aligned with the minimum principal stress flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Existing methods [13] that generate slices maximizing the alignment between layer normals and minimum principal stress vectors simply push the orientation problem downstream, forcing the practitioner to look for ways to manufacture irregularly oriented layers whose complexity leads to poor prints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Often, this results in twisted or intertwined layers which cannot be physically manufactured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' As our method focuses on manufacturability, the initial choice of orientation is actually an advantage, as it allows one to select the orientation producing the best print quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Additionally, when printing mechanical pieces designed for a well defined load case, we noticed it is fairly easy and intuitive to find an initial orientation generating a satisfactory layer alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We support this claim empirically in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Print Trajectory Optimization The starting point of our approach to print trajectory optimization is the work of Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' [13], that we embellish with numerous refinement steps which we discuss in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We first note that irrespective of the slicing method, there will be regions where the maximum principal stress vector is not tangent to the surface of the slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Thus, we begin by projecting the maximum principal stress flow field onto the slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Because of the structure of the print path optimization problem, we then need to rotate the projected vectors around each surface normal to obtain their orthogonal vectors f⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' For a vector f (belonging to the stress flow F) and the corresponding surface normal n, we compute f⊥ as u = f − f · n n · nn , f⊥ = u × n ∥u × n∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' (1) We regroup all calculated f⊥ in the projected, orthogonal and normalized maximum principal stress flow F⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The heterogeneous stress orientations produced by the FEA, coupled with the fact that surface normals can lie on both side of the slice, can create ambiguities in the stress flow F⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' This undesired effect occurs in regions where the vectors composing the stress flow are aligned (approximately) in the same direction but have opposite orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' As the toolpath optimization problem that we pose is sensitive to vector orientation ambiguity, we propose to pre-process the stress flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We first analyze all the vectors in F⊥ to determine their main Cartesian component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We compute id = arg max i ∥F⊥ · ei∥ , (2) where id represents the Cartesian component to which the vectors in F⊥ have the best alignment and ei, i = 1, 2, 3 are 8 the unit axes of a Cartesian reference frame in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Now, we can rectify the stress flow by calculating fr = ������������� f⊥ if f⊥ · eid ≥ 0 −f⊥ otherwise (3) for all vectors in F⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We then regroup the resulting fr in the rectified stress flow Fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Figure 5 includes an example of stress flow rectification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Intuitively, the rectification consists in flipping the vectors that do not lie in main stress flow orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The rectification ensures that the print trajectories that we then generate are homogeneous and easily manufacturable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We point out that our method requires the stress flow to be rectified, but is not sensitive to the rectified vectors orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' This is because the extruded material should simply align with the main traction/compression direction, making the problem invariant to 180° rotations of the stress flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Next we distinguish critical stress regions from uncritical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We consider a node as critical if the stress is both sufficiently anisotropic and sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The former condition translates to |σ3| |σ1| > θa , (4) where θa is the anisotropy threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' the latter to |σ3| maxnodes |σ3| > θs , (5) where θs is the stress significance threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The idea behind this classification is that some regions of the desired geometry undergo large and quasi unidirectional stresses: in these regions the alignment between deposited material and the stress flow is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Other regions are subject to low or isotropic stresses and the material alignment is not important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' In uncritical regions, we can print in the most convenient direction, which is typically the continuation of the print direction used in the critical regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' This allows us to deposit long unbroken fibers in our manufactured piece, fully exploiting LCPs properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Figure 4 illustrates on an example geometry the critical regions corresponding to different choices of the stress anisotropy and significance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' It is now possible to extrapolate the stress flow fr of the nodes at the critical region boundary to the neighboring uncritical nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We propagate the critical stress flow, until all the uncritical nodes have their original stress vector replaced by an extrapolated (and normalized) critical stress vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The extrapolation is conducted by solving an optimization problem that minimizes the Dirichlet energy of the stress flow in the uncritical region, constrained by the boundary conditions with the critical region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The optimal extrapolated stress flow is the minimizer of the constrained problem ˆFuc = arg minFuc ∥GFuc∥2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' fuc = fr for each node at the boundary fuc · n = 0 for each node in the uncritical region , (6) where Fuc regroups all the stress flow vectors fuc of the uncritical regions and G is the numerical gradient over the triangular mesh of the uncritical region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Solving (6) corresponds to finding the nodal vectors that minimize the 9 (a) θa = 3 , θs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='1 (b) θa = 15 , θs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='1 (c) θa = 3 , θs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2 (d) θa = 15 , θs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2 Figure 4: Critical region shape (in red) as a function of the stress anisotropy and significance parameters θa and θs, shown on a slice of the fork demonstrator geometry for the load case that will be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The same example slice will be used in all other figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Experimentation suggests that the method produces better trajectories when selecting parameters that produce a simply connected critical region, which in this scenario corresponds to case (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 10 orientation variability in the entire vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The two constraints enforce the requirement that the vectors at the critical-uncritical boundary are fixed and that the solutions found in the uncritical region are tangent to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The solution ˆFuc replaces the stress flow of the uncritical nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Figure 5 includes an example of the propagation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' (a) Before stress flow rectification (b) After stress flow rectification (c) After stress flow propagation Figure 5: Rectification and propagation of the stress flow, shown on a slice of the fork demonstrator geometry for the load case that will be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Note the agreement in the vectors orientation after rectification and the change of stress flow direction in the uncritical region after rectification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Having applied all the previous steps to the results of the FEA simulation, to each node in the slice corresponds a pre-processed orthogonal stress vector fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We regroup all fp in Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We are ultimately interested in solving the problem ˆφ = arg min φ � S ∥∇φ − Fp∥2ds , (7) to find a scalar field ˆφ whose gradient matches as closely as possible the orthogonal stress flow Fp on the slice S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Given the discretized nature of our problem, we reformulate the optimization problem (7) as the following regularized optimization problem ˆφ = arg min φ ∥Gφ − Fp∥2 + ϵ∥φ∥2, (8) where G is the numerical gradient over the triangular mesh of the slice and ϵ is the coefficient of the regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' There exists a closed form solution of (8), ˆφ = (G⊤G + ϵI)−1G⊤Fp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' (9) We can now produce the optimized print trajectories, which are isolines of the scalar field ˆφ that we just obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Calculating the isolines is a straightforward operation that just requires extracting equally spaced isolines on the slice mesh, based on ˆφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The isolines spacing will correspond to the distance between two print trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We conduct a 11 hfinal post-processing on the print trajectories by cubic spline re-sampling [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We first fit a cubic smoothing spline interpolation, which produces a quasi-identical trajectory, but reduces major roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The spline is then sampled at even distances to produce a sequence of points that represent the print trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' This final smoothing step improves manufacturability and ensures that the mechanical properties of the part will not be limited by a poor quality print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Implementation In the following sections, we apply the slicing and trajectory optimization methods that we have just described to the fork demonstrator geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The entire implementation is conducted using MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We first mesh the part, apply the loads shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 2 and conduct FEA to compute the Cauchy stress tensors that will be used in slicing and trajectory optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Offset Slicing For each node of the FEA mesh, we compare the the maximum principal stress vectors with the slicing surfaces normals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Following Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='3, the ideal slicing is obtained when each maximum principal stress is perpendicular to the corresponding slice normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Given the maximum principal stress vector f and the slicing surface normal n (both assumed to be normalized) at the node j, we compute the nodal stress-alignment as γj = ∥f × n∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' (10) We calculate the average slicing stress-alignment over the entire piece by considering the critical regions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' For a meshed piece containing J critical nodes, we obtain ¯γ = 1 J � critical j γ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' (11) The proposed metric will produce ¯γ ∈ [0, 1], where ¯γ = 1 corresponds to a perfectly aligned slicing and ¯γ = 0 to ta case where the maximum principal stress vector is perpendicular to the slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' A comparison of our method (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 6a) with standard planar slicing along the X, Y, and Z axes of the demonstrator bracket reference frame is quantified in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Both the offset slicing method we propose and the planar slicing along the Y-axis have very high alignment Table 1: Average slicing stress-alignment for different slicing methods (evaluated in the critical regions only), with identical layers distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Offset Planar X Planar Y Planar Z ¯γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='87 scores ¯γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The Y-axis planar slicing, however, produces a large fraction of slices that belong exclusively to the fork arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' As a consequence, the manufactured part would be more prone to inter-layer fractures in the arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Conversely, 12 (a) Offset slicing produced by our method (b) Example of one slice obtained via offset slicing (light blue) with the critical maximum principal stress directions (dark blue) belonging to the slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Note how most of the vectors are approximately tangent to the slice surface, producing an average slicing stress-alignment ¯γ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' (c) Critical region (in red) on one slice of the part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The critical region forms a sim- ply connected surface and excludes the areas where the stress is low or isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' (d) Trajectory-generating scalar field on one slice of the part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The color map is associated to the value of the scalars associated to each node, showing a smooth gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' (e) Isolines extracted from the trajectory-generating scalar field on one slice of the part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Note how the lines are very uniformly distributed, ensuring that the trajectories can produce a high quality print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Additionally, the trajectories follow the arms of the bracket closely and connect the extremities of the parts with unbroken print lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Figure 6: Main steps of the implementation on the fork geometry 13 Z XZ XZ XZ Xthe proposed offset slicing allows to work on layers that span the entire piece, significantly increasing the strength of the printed part and making offset slicing superior to all planar slicing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Figure 6b shows one of the slices obtained using offset slicing, together with the critical maximum principal stress directions that were used to compute γ on the depicted slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Print Trajectory Optimization Next, we generate the print trajectories for each slice individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We first project and rectify the stress flow on a slice, and then determine the critical regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We set the parameters of Equations (4) and (5) to θa = 3 and θs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='1, to produce a simply connected critical region that covers most of each slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The following step consists in solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' (9) to compute the values of the trajectory-generating scalar field over the nodes of each meshed slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We show the critical region and the scalar field corresponding to the example slice in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 6c and 6d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Finally, it is possible to extract the isolines of the scalar field that was just calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The isolines spacing corresponds to the distance between two neighboring print trajectories, and can be easily selected to match the material and printer requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Figure 6e shows the isolines that were obtained from the scalar field of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 6d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' In order to maximize the quality of the print, we post-process the isolines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' These are first interpolated using a cubic spline with a smoothing parameter p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='95, to smoothen the small-scale wiggles that can be caused by meshing and numerical approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Then we add contour lines to each slice and trim the smoothed isolines to avoid overlap with the contours during printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We finally connect sequentially the adjacent print lines and generate travel moves where required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Figure 7 shows the final print trajectory that will be sent to the printer to manufacture one layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Figures 8 and 9 illustrate the application of our method to two benchmarking geometries taken from the literature, which are commonly known as bunny head and topology optimization [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' In table 2, we show the average trajectory stress-alignment ¯β for the three geometries that we have demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Similarly to the metric ¯γ that we have defined previously, we first compute the stress-alignment βk = ∥f · d∥ , (12) where k indicates a point on the sampled print trajectory, and f and d are the maximum principal stress and print direction at that location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We then average βk across all K critical points on the print trajectory to produce ¯β = 1 K � critical k βk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' (13) The values of β range between 0 (no alignment) to 1 (perfect alignment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The results in Table 2 show that the trajectories produced with our method excellent alignment with the maximum principal stress flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Finally, we show the distribution of distances between print trajectories in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' A tight distribution corresponds to excellent manufacturability, as the lines are very uniformly spaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Other methods in the literature produce distri- butions that appear to be sums of Gaussians, with significant tails extending away from the nominal spacing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 13 of [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 10, this behavior is completely absent: the distributions are extremely tight and centered exactly on 14 Finish Start Figure 7: Post-processed print trajectories on one layer of the part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Travel moves are depicted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 15 (a) Offset slicing (b) Isolines and scalar field on one layer of the part Figure 8: Application of our method to the benchmarking piece known as bunny head (a) Offset slicing (b) Isolines and scalar field on one layer of the part Figure 9: Application of our method to the benchmarking piece known as topology optimization Table 2: Average trajectory stress-alignment for different geometries (evaluated in the critical regions only) Fork Bunny Head Topology Optimization ¯β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='88 16 Z XZ Y Xthe nominal line spacing distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Table 3 compares the mean and variance of the distribution of the distances pro- duced by our method and by the method by Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' While the means are well centered around the nominal line spacing for both methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' mean normalized distance ≈ 1), the variance produced by [13] is worse due to the tails of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Our method, which targets manufacturability via homogeneous line spacing, exhibits instead very low distances variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' This – paired with the homogeneous layer height produced by offset slicing – ensures that very high-quality prints can be achieved with all materials, and particularly with sensitive ones such as LCPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5 2 Normalized distance [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='4 mm] 0 10 20 30 40 Density [%] Fork Bunny Head Topology Optimization Figure 10: Distribution of distances between print trajectories for the fork, bunny head and topology optimization geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The distance has been normalized using the nominal line spacing of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='4 mm that was selected when generating the trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Table 3: Mean and variance of the distribution of distances using the proposed method (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 10) and the method by Fang et al [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The data from Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' have been extracted from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 13 of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' In both methods, the data have been normalized against the nominal line spacing and are thus adimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Fork Bunny Head Topology Optimization Mean Variance Mean Variance Mean Variance Proposed method 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='0 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='4 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='03 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2 × 10−3 Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' [13] N/A N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='97 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='4 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='98 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='3 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Experimental Results We have used the fork demonstrator geometry to validate experimentally the approach we have proposed and quantify the improvements achieved through stress-aligned print trajectory optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The parts were manufactured using a modified 5-axis computer numerical control (CNC) machine (5AxisMaker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The CNC machine was modified by replacing the milling head with a commercial direct drive filament extruder (E3D Hemera) and was controlled 17 using Mach32 on a Windows PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Figure 11 shows the custom printer and highlights its main components and axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' y z x r1 r2 Printed part Extruder Controller PC Axes drives Figure 11: Custom 5-axis FFF machine with highlighted principal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' (a) Manufactured part attached to the fixture (b) Part and fixture mounted on the testing rig Figure 12: Installation of a fork manufactured using stress-aligned trajectories and LCPs on the testing rig We tested the printed samples according to the load case for which the optimization was conducted (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' To do so, we used a Zwick Z020 universal testing machine with a 20 kN capacity load cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Each fork was attached to the machine bottom sample holder with two screws passing through their holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Then, the machine testing head was lowered at a speed of 2 mm/ min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The tests were stopped when the applied force dropped under a predefined threshold value, which is typically reached when the specimen is completely broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' During the test, the displacement and applied force were recorded to trace a force-displacement curve and to compute the failure force and stiffness of each printed specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='machsupport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='com/software/mach3/ 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Experimental Printer Impact Table 4: Failure force and stiffness of the isotropic PLA samples, used to evaluate the impact of the 5-axis experimental printer on mechanical properties PLA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D Prusa Non-planar Experimental Failure force [N] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='9 Stiffness [N/mm] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='6 As the non-planar parts have been manufactured on an experimental machine, we expected the quality of our non-planar prints to be inferior to what commercial printers achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Low print quality negatively affects the me- chanical properties of printed components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Thus, we first evaluated the impact of our experimental 5-axis machine on part properties by printing and comparing two forks with an isotropic material (PLA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D fork was sliced with a commercial planar slicer (Prusa Slicer) and printed on a Prusa i3 MK3s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' the non-planar one was sliced with our proposed method and printed on the experimental 5-axis machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Both trajectory generation methods were set to produce a layer thickness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='1 mm and a line spacing (corresponding to the extrusion width) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='4 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We used two contour lines and printed at the same feed rate in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D slicer was set to produce an aligned rectilinear infill with 100% density and to slice the part along the Z axis of the demonstrator bracket reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Theoret- ically, given the isotropic properties of PLA, the mechanical properties of the conventional and optimized samples should be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' However, the parts manufactured with the commercial printer rarely contain defects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' conversely, we expected the complexity and the experimental nature of our printer to increase the number of imperfections in the part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The results in Table 4 – that we discuss comprehensively in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 6 – show that the print imperfections produced by the experimental machine during non-planar printing worsen the mechanical properties of the fork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Pictures of the PLA samples after failure are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 14a and 14b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' LCPs results After assessing the impact of the experimental 5-axis printer using isotropic polymers, we finally evaluated the performance of our method on anisotropic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Following the same specifications discussed above, we printed two samples using LCPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D sample was produced by planar slicing on a Prusa i3 MK3s, while the non-planar sample was sliced with our method and printed on the 5-axis machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The results are contained in Table 5 and discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Figure 13 depicts the force-displacement plots of the two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Pictures of the LCP samples after failure are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 14c and 14d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 19 Table 5: Failure force and stiffness of the anisotropic LCP samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The non-planar samples were sliced with the proposed optimization method and considerably outperform conventional 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' LCP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D Conventional Non-planar Optimized Failure force [N] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='6 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='7 Stiffness [N/mm] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='4 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='8 0 5 10 15 20 25 Displacement [mm] 0 20 40 60 80 100 120 Force [N] 1 2 3 (1) Non-planar LCP (2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D LCP (3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D PLA Figure 13: Force-displacement plot of the tested samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' To facilitate the comparison, the plot includes the curve of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D PLA sample produced with a conventional material, machine and slicing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Discussion The experiments demonstrate how print trajectory optimization vastly improves the mechanical properties of the parts manufactured with anisotropic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The failure force of the LCP samples has increased by a factor of 44, and the stiffness by a factor of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' In light of the experimental nature of the equipment used for non-planar prints (whose impact with respect to commercial-grade 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D prints has been quantified to -34% failure force and -23% stiffness in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='1), the improvements obtained with LCP samples are even more remarkable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' (a) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D PLA (b) Non-planar PLA (c) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D LCP (d) Non-planar LCP Figure 14: Failure during testing of the four printed combinations of material and trajectory generation method Figure 14 offers important insights to better understand the quantitative results of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Looking first at the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='5D prints (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 14a and 14c), both pieces break at similar locations, in the inclined part of the bracket, where the part cross section is minimal and horizontal layers have a very small surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Particularly for LCPs, it is evident how the material has poor properties when the load is perpendicular to the layers’ surface, producing inter-layer fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Thanks to its better layer adhesion properties, PLA breaks at a similar location but without a clear inter-layer failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Moving to the case of non-planar prints, we observe how PLA (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 14b) breaks at the lower curvature region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' As discussed previously, PLA is quite insensitive to print orientation, and the fracture should ideally happen as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 14a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' However, the lower curvature has been found to be the most complex region to print with the 5-axis equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The machine kinematics make high-quality material deposition in convex curvatures difficult due to large accelerations (and consequently vibrations) in the axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' It appeared upon observation that at the lower curvature the material was partially over-extruded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We believe that this defect is the cause of the mechanical properties worsening found in the non-planar PLA part, as well as for the failure location shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 14b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 14d suggests that, despite the printing defects, the non-planar LCP fork fails at the location where FEA predicted the largest stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Failure takes 21 place when the deposited LCP fibers – which are stressed in the deposition direction – break under tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Thanks to the optimized layers and print trajectories orientation, the part does not suffer from inter-layer or inter-trajectory fractures, and the anisotropic nature of LCPs is fully exploited to produce excellent mechanical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Conclusions In multi-axis 3D printing, it is possible to produce components using filaments that are deposited in almost any desired spatial orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' This freedom can be exploited to improve the mechanical properties of the printed parts, particularly when using anisotropic polymers as feed stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' By aligning the material deposition direction with that of the stress flow induced by a component’s load case, it is possible to dramatically improve the failure force and stiffness of the part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' In this work, we have proposed a method for non-planar slicing and print trajectory optimization that consid- ers manufacturing constraints such as constant layer thickness and line spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' These constraints are particularly important to obtain high-quality prints and avoid early failure of the parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Using computations on benchmarking geometries, we showed that our method produces extremely homogeneous and thus well manufacturable trajecto- ries when compared to existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Our framework is suitable for complex geometries and stress flows frequently encountered in common mechanical components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' We have conducted experiments to manufacture load- bearing brackets using LCP material from NematX AG on a 5-axis 3D printer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The results have demonstrated that, despite the introduction of deposition defects linked with the complexity of 5-axis printing, the method fully exploits the strongly anisotropic nature of LCPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' The optimized prints achieve a 44× improvement in failure strength and a 6× improvement in stiffness, compared with conventional planar printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Funding This work was supported by the Swiss Innovation Agency (Innosuisse, grant №102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content='617) and by the Swiss National Science Foundation under NCCR Automation (grant №180545).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' Acknowledgments We acknowledge the support of NematX AG that provided the LCP material and experimental setup for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE4T4oBgHgl3EQfSgxY/content/2301.04999v1.pdf'} +page_content=' References [1] The state of 3d 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0000000000000000000000000000000000000000..2f00d894dbc977552f5fd818d5ef69388124d9ef --- /dev/null +++ b/jdAyT4oBgHgl3EQfkfjo/content/tmp_files/2301.00436v1.pdf.txt @@ -0,0 +1,2350 @@ +Hierarchical Explanations for Video Action Recognition +Sadaf Gulshad, +Teng Long, +Nanne van Noord +University of Amsterdam +{s.gulshad, t.long, n.j.e.vannoord}@uva.nl +Abstract +We propose Hierarchical ProtoPNet: an interpretable +network that explains its reasoning process by consider- +ing the hierarchical relationship between classes. Different +from previous methods that explain their reasoning process +by dissecting the input image and finding the prototypical +parts responsible for the classification, we propose to ex- +plain the reasoning process for video action classification +by dissecting the input video frames on multiple levels of the +class hierarchy. The explanations leverage the hierarchy to +deal with uncertainty, akin to human reasoning: When we +observe water and human activity, but no definitive action it +can be recognized as the water sports parent class. Only af- +ter observing a person swimming can we definitively refine +it to the swimming action. Experiments on ActivityNet and +UCF-101 show performance improvements while providing +multi-level explanations. +1. Introduction +When describing the world around us we may do so at +different levels of granularity, depending on the information +available or the level of detail we intend to convey. For in- +stance, a video might open with a shot of a cheering crowd, +allowing us to recognize it as a a sports event, as the cam- +era then pans to the river we can deduce that it is a water +sports event. However, only when the raft comes into the +frame can we determine that it concerns rafting. Nonethe- +less, in our description of this video, we may still only refer +to it as a sports or water sports event. Our reasoning and +description processes build on the hierarchical relation be- +tween classes, allowing for navigation between generic and +specific. In this work, we implement this process for video +action recognition by learning hierarchical prototypes that +we leverage for improved classification performance and +explanations at multiple levels of granularity. +Despite the remarkable performance of neural networks +for video understanding tasks [4,6,14,17,34,37,38,50] it is +still hard to explain the decisions of these networks, which +is of utmost importance for practical application. This ne- +The evidence for this action to be Rafting: +it’s grandparent looks like +Rafting +Surfing +Frontcrawl +Breaststroke +Kayaking +This video action is predicted as Rafting because: +Explanation with a regular +ProtoPNet. +Sports +Water Sports +it looks like +it’s sibling looks like +it’s sibling looks like +it’s sibling looks like +it’s sibling looks like +it looks like +This video belongs to rafting +because +it looks like +it looks like +it looks like +This video belongs +to sports because +This video belongs +to water sports +because +This video belongs +to rafting because +Leftmost: Parts in the original video that are highly activated by the prototype. +Second column: Training videos where prototypes come from. Third column: +Prototypes. Rightmost: Saliency map in the original video that are highly +activated by the prototype. +Figure 1. Multi-level Explanations. Multi-level explanation for a +video of rafting with Hierarchical ProtoPNet, showing the expla- +nations at grandparent, parent, and class level. +cessity has led to a growing stream of research that focuses +on making models interpretable besides performing accu- +rately [13, 15, 19, 23, 45]. A promising line of explainabil- +ity methods are the case-based reasoning models [7, 9, 21], +which focus on learning prototypes during the training and +making predictions based on the learned prototypes while +inference. This enables this look like that type of expla- +nation. However, previous case-based reasoning works are +limited to 2D images and models. Moreover, they provide +a single level of explanations and in case of uncertainty, the +explanations can be as bad as arbitrary, as each explanation +is considered equally apart. In this work, we focus on cap- +turing the hierarchical relations between actions to provide +multi-level explanations for videos. A challenge for mod- +els with built-in explainability, such as case-based reason- +ing models, is that it introduces an accuracy-explainability +trade-off, where explainability comes at the cost of accu- +racy. With this paper, we aim to introduce a model with +built-in explainability, whilst being less affected by this +trade-off. To achieve this goal we are inspired by recent +works on learning hyperbolic embedding spaces rather than +arXiv:2301.00436v1 [cs.CV] 1 Jan 2023 + +euclidean for natural language processing [8, 44] and com- +puter vision tasks [1, 12, 26]. These works have demon- +strated that it is beneficial for performance to have the em- +bedding space be guided by hierarchical prior knowledge, +which we believe will also benefit explainability. This be- +lief is guided by the similarity between human intuition and +the representation of categories in the hyperbolic space. +The main contributions of our paper are: 1) We propose +Hierarchical ProtoPNet, a case-based reasoning model for +interpreting video action recognition. 2) We demonstrate +that while other interpretable models fail to provide expla- +nations in the presence of uncertainty or lack of informa- +tion our Hierarchical ProtoPNet can overcome these chal- +lenges by providing multi-level explanations i.e., at class, +parent, or grandparent level. 3) We perform a benchmark +and show that our Hierarchical ProtoPNet outperforms its +non-hierarchical counterpart. +2. Related Work +2.1. Interpretations for Videos +Interpretations for neural networks can be broadly clas- +sified into two categories: 1) fitting explanations to the +decisions of the network after it has been trained i.e. +posthoc [13, 15, 19, 27, 35], 2) building explanation mech- +anism inherent in the network i.e. +built-in explanations. +[23, 24, 45, 46]. In this work, we focus on learning seman- +tic representations which are used for classification during +training rather than explaining a black box network posthoc. +A great deal of previous work has focused on video ac- +tion recognition, detection, segmentation and more [4,6,14, +17, 34, 37, 38, 50], however, most of these works focus on +designing black box models for specific tasks. They do not +explain why a certain decision is made by the model. More- +over, most of the research in the visual explanations domain +focuses on images. Only a few works focus on the interpre- +tation of these networks for videos [2, 16, 23, 41, 42], and +it is not possible to directly apply image-based explanation +methods to videos due to an extra time dimension in videos. +[16] and [2] focus on visualizing spatio-temporal atten- +tion in RNNs, CNNs are used only to extract features. In- +spired by class activation maps (CAM) [54] for images [42] +extended it for videos by finding both regions and frames +responsible for classification. [23] utilized perturbations to +extract the most informative parts of the inputs responsible +for the outputs. Both [23, 42] are posthoc methods, which +means they do not use explanations during prediction there- +fore they might not be faithful to what the network com- +putes [9]. [41] introduced class feature pyramids, a method +that traverses through the whole network and searches for +the kernels at different depths of the network responsible +for classification, therefore this method is computationally +expensive. In contrast, we enable built-in multi-level expla- +nations that do not add any computational complexity. +In this paper, we enable multi-level explanations for +videos by learning hierarchical prototypes for each class +and tracing them back to input videos. +2.2. Prototype-based Interpretations +In machine learning the term “prototype” is used in var- +ious contexts, in zero-shot learning [22, 51–53] and few- +shot learning [32, 39] prototypes are points in the embed- +ding space representing a single class and the distance from +these prototypes is used during classification. However, in +our work prototypes are closer to the samples in the training +set, and multiple prototypes are used to represent each class. +They are optimized to resemble the training set in order to +provide visual explanations. +The idea to provide built-in explanations with prototypes +was first explored in [21], where the authors introduced a +prototype layer in the network with an encoder-decoder ar- +chitecture. The prototype layer stores weights which are +close to encoded training samples, and a decoder is used to +visualize them. However, their model fails to generate re- +alistic visualizations for natural images. Thus [7] proposed +to learn prototypes for each class and visualized them by +tracing them back to the input images without a decoder. +Our work is inspired by [7], but where their work is limited +to 2D images and provides only one-level explanations we +extend it to multi-level explanations for videos. +[36] focuses on reducing the number of prototypes for +each class by finding shared prototypes among classes. [28] +enhances the prototypical explanations by adding textual +explanations to explain prototypes. [49] introduced a dif- +ferent similarity metric for computing similarities between +prototypes and image patches. They also introduced a loss +function to enhance the diversity of prototypes within a +class. Deformable ProtoPNet [9] learns spatially flexible +prototypes to capture pose and context variations in the in- +put. All previous prototype-based explanation methods pro- +vide explanations without considering the hierarchical re- +lations between classes on well-defined image CUB birds +[48] and Stanford cars [20] datasets. In contrast inspired by +the human way of explanations we consider hierarchical re- +lations between classes while learning prototypes for each +class for video datasets. +Most closely related to our work are [29, 45]. [45] in- +troduced a dynamic prototype network (DPNet) for finding +temporal artifacts and unnatural movements in deep fake +videos. However, the goal of DPNet is different from ours +with only two target classes fake/real making the task eas- +ier. [29] introduced neural prototype trees by combining +CNN architecture with the soft binary trees for providing +local and global explanations. However, as the number of +prototypes depends upon the size of the tree learning a Pro- +toTree becomes computationally expensive. Our proposed + +multi-level explanations do not add any extra computational +complexity to the training. +2.3. Hyperbolic Embeddings +[30] demonstrated that hyperbolic embeddings can learn +hierarchical tree-like structures. +Later on, the effective- +ness of hyperbolic embeddings have been shown for tex- +tual [11,44,55] to visual data [1,12,18,26]. Hyperbolic em- +beddings have also been used for zero-shot learning [10,25] +and for video action recognition [26, 43]. The hierarchi- +cal relationship between videos and the hierarchical way of +explaining decisions for humans calls for the need of us- +ing hyperbolic spaces. In this work, we utilize hyperbolic +embeddings for learning hierarchical prototypes to provide +human-like explanations for video action recognition. +3. Hierarchical Action Embeddings +Incorporating the prior hierarchical knowledge about ac- +tions into the network requires that we represent them +as embeddings. +In this section we detail how to learn +those action embeddings in hyperbolic space, in the next +section, we explain how to learn hierarchical prototypes +that are optimized by aligning them to the action embed- +dings in hyperbolic space. Given the set of action classes +A = {1, 2, ..., |A|}, in hierarchical action recognition we +also consider their ancestor classes H = {|A| + 1, |A| + +2, ..., |A| + |H|}, which allows us to construct a hierarchi- +cal tree with three levels, i.e., grandparent, parent, and child +(see Figure 2 right). This process of embedding the hierar- +chies is performed once, offline, per dataset. However, this +process can easily be repeated for alternative hierarchies. +Learning Action Embeddings. We map the action hierar- +chy A ∪ H into the shared hyperbolic space Dn to obtain +hierarchical action embeddings, which are used as action +class templates in the next section. Let P = {(u, v)|u = +φ(v)} be the positive pair of v and its parent φ(v) and +N = {(u′, v′)|u′ ̸= φ(v′)} be the negative pairs. The dis- +criminative loss akin to [26]: +L(P, N, Φ) = LH(P, N) + λ · LS(Φ), +(1) +where Φ stands for class templates matrix and its c − th +column Φc is the template vector of class c in Dn. For the +loss function, LH encourages the preservation of parent- +child relations and LS enforces separation among different +sub-hierarchies. The first part LH is akin to [31], where the +wrongly positioned child-parent pairs will be penalized: +LH(P, N) = +� +(u,v)∈P +log +� +e−d(u,v) +� +(u,v′)∈N e−d(u,v′) +� +, (2) +where −d(u, v) is the hyperbolic distance between two ac- +tion embeddings u and v, which can be written in short- +hand notation: +d(v, u) := 2 arctanh (∥−v ⊕ u∥) , +(3) +where ⊕ indicates the M¨obius addition [47] in 1−curved +hyperbolic space Dn. +In the second part, we encourage the separation among +sibling relationships, where we update Φ with separation +loss: +LS(Φ) = − +� +i∈|A| +|| ˜ΦT +i ˜Φi||F + γ||( ˆ +Φi ˆ +Φi +T − I)||F , +(4) +where ˆΦ consists of the non-sibling vectors with respect to +action class i while ˜Φ consists of i’s sibling vectors. +After learning with the above objectives, we obtain Φ, a +matrix of action template vectors including both actions A +and ancestor (parent and grandparent) actions H. +4. Hierarchical ProtoPNet +Figure 2 gives an overview of our proposed Hierarchical +ProtoPNet for video action recognition. Our Hierarchical +ProtoPNet consists of a 3D-CNN backbone f for extracting +features from the video frames, and a prototype layer gp for +learning prototypes for each frame. The prototype layer is +followed by a fully connected layer h that combines the pro- +totype similarity scores maps them to the shared hyperbolic +space through exponential mapping. Prior knowledge about +the relations between actions, in the form of the action hier- +archy, are projected to the shared space through discrimina- +tive embeddings, as described in Section 3. Subsequently, +we use hyperbolic learning to obtain hierarchical prototypes +that enable multi-level explainability. +4.1. Feature Extraction +As the backbone architecture, we use the video action +classification network 3D-Resnet [14]. For each input video +v ∈ RW ×H×T ×3 with T frames it extracts video features +Z ∈ RW0×H0×T0×D with the spatial resolution W0 × H0, +frames T0 and channels D. A key aspect of this backbone +is that T0 < T due to temporal pooling, as such the features +Z are extracted for segments rather than individual frames. +Because of the temporal pooling, the prototypes learned by +our Hierarchical ProtoPNet are spatiotemporal thereby ex- +plaining which parts of the segment are indicative of the +action in the video. +4.2. Prototype Layer +Given the features extracted from the 3D-Resnet Z, two +layers of 1 × 1 × 1 convolutions with the LeakyReLU ac- +tivations are added for adjusting the number of channels +for the top layer. For each child A and its parent H ac- +tion, the network learns m and n prototypes respectively +P = {pj}m+n +j=1 , whose shape is W1 × H1 × T1 × D with + +RGB Frames +RGB Embedding +ResNet-3D +backbone in +Exponential map + to +Action Embedding +Hyperbolic Hierarchy on +3.25 +2.34 +1.76 +Prototype Layer +Fully connected +Layer +Similarity scores +1.02 +0.56 +Children Prototypes +Parent Prototypes +Hierarchical +Cross-entropy +Hyperbolic Matching +Figure 2. Overview of the Hierarchical ProtoPNet. The Resnet-3D backbone extracts video features and the prototype layer learns +prototypes for children and parents, these prototypes are then converted to a single similarity score through max pooling. Finally, scores are +converted from Rn to Dn through a fully connected layer followed by an exponential map, to the shared hyperbolic space for Hierarchical +learning. Actions are mapped onto the shared hyperbolic space by learning a discriminative embedding on Dn. +W1 ≤ W0, H1 ≤ H0 and T1 ≤ T0. As such each pro- +totype represents a spatiotemporal part of the video. Given +the convolutional output Z = f(v) and prototypes p a pro- +totype layer gp computes the distances between each pro- +totype pj and the patches from Z and converts them to the +similarity scores using +gp(pj, Z) = max +z∈Z log (||z − pj||2 +2 + 1) +(||z − pj||2 +2 + ϵ) , ϵ > 0 +(5) +The distances between each prototype and the patch deter- +mine the extent to which a prototype is present in the input. +In contrast with the prior ProtoPNet architectures, we mul- +tiply similarity scores with the weights of a fully connected +layer h to obtain embeddings to be projected in the hyper- +bolic joint space for learning hierarchical prototypes. +4.3. Hierarchical Video Embeddings. +The embeddings h = h(gp(p, f(v))) obtained from the +prototype layer are in the Euclidean space and can not be di- +rectly mapped into the hyperbolic embedding space, there- +fore, we use exponential mapping [11] to map video em- +beddings into the hyperbolic space. +expx(h) = x ⊕ +� +tanh +� +||h|| +1 − ∥x∥2 +� h +||h|| +� +(6) +where ⊕ indicates the 1−curved Mobius addition, x is the +tangent point connecting tangent space T0Dn to Dn. Differ- +ent values of x lead to different tangent spaces, to avoid any +ambiguities we set x = 0 and project the video embeddings +to the hyperbolic space for matching with the hierarchical +actions. +4.4. Training Hierarchical ProtoPNet +Our training process consists of a multi-step procedure: +initially epochs we perform warm-up of the newly added +layers. Following the warm-up, we train the entire network +end-to-end. Every 10 epochs we perform prototype projec- +tion, i.e., updating the prototype layer only, followed by a +phase of fine-tuning the layers after the prototype layer. +Video and Action Matching in the Hyperbolic Space +We aim to learn a latent space where patches important +for classification are clustered around similar prototypes. In +order to learn hierarchical prototypes we optimize the pro- +totypes P = {pj}m+n +j=1 +to match videos to hyperbolic ac- +tion embeddings, hence our optimization is supervised by +Φ ∈ Dn×(|A|). Let {(vi, yi)}N +i=1 be the training set, where +v ∈ RW ×H×T ×3 and yi ∈ A. Our goal is to solve: +Lcrs + λ1Lcls + λ2Lsep +(7) +Hierarchical Cross Entropy. The first term in our loss is +the hierarchical cross-entropy loss Lcrs which penalizes the +misclassification defined as: +Lcrs = 1 +N +N +� +i=1 +K +� +k=1 +yik log p(y = k|v) +(8) +The softmax in the cross entropy is defined as the negative +distance between video embeddings and the hierarchical ac- + +S +YoYo +: Bsby Cntuing +:lowing Cadon +:Bg +:Wehg wity +Fhisoa Ceken +Non strenuous +Ardhery: +ePuup +Sports +Bouing: +UCF-101 +Body-Motion +TargetSports +JaehThow +Strenuous +HnerThoy +Junping Jaek +Eroeut: +ts +SP +PoiagTxM +x +V +expx( +MLcRStion embeddings in the hyperbolic space: +p(y = k|v) = +exp(−d(he), Φk) +� +k′ exp(−d(he), Φk)), +(9) +where he = exp0(h) is applying exponential map to the +prototype output h. +Hierarchical Clustering. Our hierarchical clustering cost +encourages input images to have at least one patch from +features to be closer to a child, parent or grandparent class +prototype. +Lcls = 1 +N +N +� +i=1 +min +j:pj∈P|A|+|H| +min +z∈patches(f(vi)) ||z − pj||2 +2 +(10) +Hierarchical Separation. Our hierarchical separation cost +encourages the latent patches of the images to stay away +from the prototypes not belonging to the same child class or +parent class or grandparent class. +Lsep = − 1 +N +N +� +i=1 +min +j:pj /∈P|A|+|H| +min +z∈patches(f(vi)) ||z − pj||2 +2 +(11) +4.5. Prototype Projection +We project prototypes onto the closest video features +from the training videos. We do so for child, parent, and +grandparent action categories. Mathematically, for a proto- +type pj from child, parent or grandparent class i.e. pj ∈ +P|A|+|H|, we update the prototype layer as: +pj ← argmin +z∈Zj +||z − pj||2 +(12) +where Zj = {˜z : ˜z ∈ patches(f(vi)) ∀i s.t. yi = +|A| + |H|}. Our prototype layer is updated not only with +the prototypes belonging to the child class but also with the +parent and grandparent classes enabling the learning of hi- +erarchical relations between classes. +4.6. Prototype Visualization +To construct the visualizations the learned prototypes are +mapped to the spatio-temporal input space. We select the +patch which highly activates for the prototype pj by for- +warding the input v through the network and upsampling the +activation map generated by the prototype layer gp(pj, Z) +both spatially and temporally (for videos). We visualize pj +for child, parent, and grandparent classes providing expla- +nations at all levels of the hierarchy. +5. Experimental Setup +5.1. Datasets +To evaluate our Hierarchical ProtoPNet for videos we +conduct experiments on two video datasets: UCF-101 [40] +and Activity-Net1.3 [5]. For UCF-101 only one level ac- +tion hierarchy is available with the dataset, therefore we +define additional levels to complement the hierarchy. For +ActivityNet we used the hierarchy protocol provided with +the dataset. All the hierarchies are made available together +with our implementation. +Hierarchical UCF-101. +UCF-101 [40] contains 13,320 +videos belonging to 101 action categories with a total length +of 27 hours. We define two additional levels of hierarchy +with the number of classes at level one, two, and three being +5, 20, and 101 respectively. The classes at the third level of +the hierarchy are the 101 original classes of the dataset. The +full hierarchy is included in the supplementary material. +Hierarchical ActivityNet. ActivityNet [5] contains 14,950 +untrimmed videos with each video consisting of one or +more action segments belonging to 200 action classes with +a total length of approximately 648 hours. We use 10,024 +videos for training and 4,926 for validation. We follow [14] +and train and test our model on trimmed videos to determine +video-level accuracy. We follow [26] to define the class- +level action hierarchies using the hierarchies that come with +the dataset. It contains 200, 38, and 6 classes in level one, +two, and three respectively. +5.2. Implementation Details +The hierarchical action embeddings are generated by +training the model with Reimannian Adam optimizer [3], +implemented with geoopt and Pytorch [33]. Apart from the +one-time offline step of generating the hierarchical action +embedding our Hierarchical ProtoPNet is trained in an end- +to-end fashion. For feature extraction, we used Resnet-3D- +18 [14] pre-trained on Kinetics [6] and added two 1 × 1 +convolutional layers with the LeakyReLU, a prototype layer +and the final embedding layer. We perform prototype pro- +jection and visualization every 10 epochs. +We report results on two variations of our Hierarchi- +cal ProtoPNet: Hierarchical ProtoPNet with the hyperbolic +cross-entropy loss and Hierarchical ProtoPNet CPG (child, +parent and grandparent) with hyperbolic cross-entropy loss. +For the CPG variant, we compare between prototype pro- +jection with 5 prototypes per class and 10 prototypes per +class, to explore whether this additional supervision makes +it possible to use fewer prototypes. For comparison, we +adapt ProtoPNet [7] to videos by replacing the 2D ResNet +backbone with a 3D ResNet backbone. +Evaluation Metrics. We report the performance at both +clip level and video level. The clip level accuracy is the +rate of correct prediction of each clip, while the video level +accuracy is the majority vote of the predictions over all +the clips within the video. Additionally, to show the ben- +efit of using hierarchical learning, we report accuracy for +three metrics: the class accuracy is calculated as the rate + +Network +Accuracy +Sibling Accuracy +Cousin Accuracy +# of prototypes +per class +Non-Interpretable +Models +3D-Resnet [14] +83.34 +89.73 +93.62 +- +Resnet-Hyperbolic [26] +82.64 +89.99 +93.28 +- +Interpretable +Models +ProtoPNet [7] +78.30 +85.92 +90.98 +10 +Hierarchical ProtoPNet +79.49 +88.60 +92.79 +10 +Hierarchical ProtoPNet CPG +79.45 +88.88 +92.73 +5 +Hierarchical ProtoPNet CPG +80.40 +89.30 +93.02 +10 +Table 1. Clip level accuracy comparison for different models on UCF-101 videos. We observe that our hierarchical ProtoPNet CPG with +10 prototypes per class recovers the drop due to accuracy-explainability trade-off significantly while providing multi-level explanations. +of correct prediction in the hierarchical space 0-hop away +from the ground-truth, the sibling accuracy as the rate of +correct prediction 2-hops away from the ground-truth, and +the cousin accuracy as the 4-hops correct prediction rate. +Higher performance on the sibling and cousin metrics in- +dicates that misclassifications are to hierarchically nearby, +and therefore, semantically similar classes. +6. Experiments +6.1. Recognition Accuracy +Non-Interpretable Models. +The performance of non- +interpretable models on UCF-101 and ActivityNet are +shown in the top two rows of Tables 1, 2, and 3. +For +fair comparison both non-interpretable models, a regular +Resnet [14] and a hyperbolic Resnet [26], are trained end- +to-end with the same data augmentations and an equal num- +ber of epochs. The only difference between the two non- +interpretable models is that for the Resnet model the cat- +egories are separated through euclidean hyperplanes while +the Resnet-Hyperbolic utilizes hyperbolic embedding space +to separate categories. Our results for UCF-101 show that +both a regular Resnet and the hyperbolic Resnet perform +similarly at clip level (Table 1) and video level (Table 3). +For ActivityNet the clip-level class accuracy (Table 2) +is comparable across both networks, however, due to the +hierarchical learning of hyperbolic Resnet it shows better +sibling and cousin accuracy, additionally, it shows an im- +provement for class accuracy at the video level (see Table +3). Overall, we see comparable performance for the non- +interpretable networks on UCF-101 and improvements for +the Hyperbolic networks on ActivityNet. +Interpretable Models. The performance of interpretable +models on UCF-101 and ActivityNet are shown in the bot- +tom four rows of Table 1, 2, and 3. We report the results +for a regular ProtoPNet [7] adapted for videos and the vari- +ations of our Hierarchical ProtoPNet. +For UCF-101, with a regular ProtoPNet with 10 proto- +types per class, the accuracy drops considerably: the clip- +level class accuracy drops to 78.30 and the video-level accu- +racy to 84.48. This is because of the explainability-accuracy +trade-off common in explainable-AI, also reported in [7]. In +contrast, our hierarchical ProtoPNet is much less affected +and recovers the drop by 1.19 for class accuracy, 2.68, and +1.81 for sibling and cousin accuracies respectively. +Our +Hierarchical-ProtoPNet CPG with 5 prototypes per class +also shows similar improvement in performance, however, +increasing the number of prototypes to 10 per class im- +proves the performance by 2.10, 3.38, and 2.04 for class, +sibling, and cousin accuracies respectively. Moreover, the +performance at the video level reaches 86.22 (see Table 3). +Hence, all three variations of our proposed hierarchical Pro- +toPNet reduce the accuracy drop. +On ActivityNet (see Table 2) we observe a clear +accuracy-explainability trade-off for the regular ProtoPNet, +with drops in both the clip and video level accuracies. How- +ever, whilst our Hierarchical ProtoPNet shows a similar +drop in class accuracy we can observe that it partially re- +covers from this drop on the sibling and cousin metrics. +This behavior holds for both the Hierarchical ProtoPNet +CPG with 5 prototypes, and for CPG variant with 10 pro- +totypes per class we even see improvements for the sibling +and cousin metrics of 1.19 and 1.3 respectively. Whilst Ac- +tivityNet remains challenging, an improvement in sibling +and cousin accuracies is directly beneficial to the explain- +ability as demonstrated in Section 6.2. +Hence, we can observe that on both datasets our Hi- +erarchical ProtoPNet is less affected by the accuracy- +explainability trade-off whilst also providing multi-level ex- +planations. +6.2. Visual Explanations +Multi-level Explanations. Figure 3 shows an example of +multi-level explanations provided by our Hierarchical Pro- +toPNet. Our model learns to represent the video clip fea- +tures as hierarchical prototypes that belong to grandparent, +parent and child classes. +For example, in Figure 3 our +model has learned prototypes (only one out of ten proto- +types shown for better presentation) from the grandparent +class playing instruments, parent class percussion and the + +Network +Accuracy +Sibling Accuracy +Cousin Accuracy +# of prototypes +per class +Non-Interpretable +Models +3D-Resnet [14] +49.99 +51.59 +63.88 +- +Hyperbolic-Resnet [26] +49.95 +52.03 +65.44 +- +Interpretable +Models +ProtoPNet [7] +46.06 +47.74 +61.67 +10 +Hierarchical ProtoPNet +46.02 +48.76 +62.77 +10 +Hierarchical ProtoPNet CPG +45.67 +48.72 +62.84 +5 +Hierarchical ProtoPNet CPG +46.26 +48.93 +62.97 +10 +Table 2. Clip level accuracy comparison for different models on ActivityNet videos. We observe that our hierarchical ProtoPNet CPG +with 10 prototypes per class recovers the drop for siblings and cousins and shows comparable performance with regular ProtoPNet for class +accuracy while providing multi-level explanations. +Network +UCF-101 +Accuracy +ActivityNet +Accuracy +# of prototypes +per class +Non-Interpretable +Models +3D-Resnet [14] +87.92 +69.15 +- +Resnet-Hyperbolic [26] +87.15 +70.19 +- +Interpretable +Models +ProtoPNet [7] +84.48 +66.46 +10 +Hierarchical ProtoPNet +84.03 +64.66 +10 +Hierarchical ProtoPNet CPG +84.87 +63.82 +5 +Hierarchical ProtoPNet CPG +86.22 +66.48 +10 +Table 3. Video level accuracy comparison for different models on UCF-101 and ActivityNet videos. We observe that our hierarchical +ProtoPNet CPG with 10 prototypes per class recovers the drop at video level significantly for UCF-101 and shows comparable performance +with the regular ProtoPNet for ActivityNet while providing multi-level explanations. +The evidence for this action to be Rafting: +it’s grandparent looks like +it looks like +Rafting +Surfing +Frontcrawl +Breaststroke +Kayaking +This video action is predicted as Rafting because: +Explanation with a regular +ProtoPNet. +Sports +Water Sports +it looks like +it’s sibling looks like +it’s sibling looks like +it’s sibling looks like +it’s sibling looks like +it looks like +it looks like +This video belongs to playing +instruments because +This video belongs to +percussion because +This video belongs to +playing daf because +it looks like +This video belongs to rafting +because +Leftmost: Parts in the original video that are highly activated by the prototype. Second +column: Training videos where prototypes come from. Third column: Prototypes. +Rightmost: Saliency map in the original video that are highly activated by the prototype. +it also looks like +Figure 3. Multi-level Explanations. Schematic representation +of the hierarchical prototype-based reasoning process of our pro- +posed Hierarchical ProtoPNet. +action class playing daf. We see the similarity between the +grandparent and original video in the posture of players and +hand movement, for the parent we observe the similarity in +the hands movement, and finally the prototypes for the play- +ing daf class show greater similarity for both the instrument +and the hand movements. +Effectiveness of Multi-level Explanations. The learned +prototypes of the regular ProtoPNet with our Hierarchical +ProtoPNet are contrasted in Figure 4. In Figure 4 top, we +observe that the top prediction by both the networks regu- +lar ProtoPNet and Hierarchical-ProtoPNet CPG are correct +(i.e., biking), however the learned prototypes for ProtoPNet +focus only on the tyres or the background. While for our +Hierarchical ProtoPNet the prototypes are diverse, focus- +ing on the tyre and pedals. Moreover, the top 2 prediction +in the bottom half of Figure 4 for the ProtoPNet is Discus +Throw which is non-related to biking according to human +intuition, while for the hierarchical ProtoPNet it is Horse +Racing which relates to biking as it is a riding sport. We ob- +serve that the prototypes for Discus throw are random and +when we visualize the area highly activated by those pro- +totypes, it either focuses on the background or ground (see +Figure 4 last two rows). Instead for our Hierarchical ProtoP- +Net, the top 2 prediction is Horse Racing and the prototypes +predominantly focus on the riders. +In Figure 5 we show another scenario where the multi- +level explanations are useful. +We see that the original +video is misclassified intohorse riding class, however, for +the more abstract explanations we can observe that its par- + +Leftmost: Original Video Frame. Second column: Prototypes. Third column: Training videos where prototypes come from. Fourth column: Parts in the original video that +are highly activated by the prototype. Rightmost: Saliency map in the original video that are highly activated by the prototype. (*Repeat for hierarchical ProtoPNet). +ProtoPNet: Predicted Class: Biking +Hierarchical ProtoPNet: Predicted Class: Biking +Predicted Sibling Class: Discus Throw +Predicted Sibling Class: Horse Race +Top 1 Prediction +Top 2 Prediction +Figure 4. Effectiveness in contrast to regular ProtoPNet. The comparison between regular ProtoPNet and our Hierarchical ProtoPNet +shows that our model learns more diverse prototypes in a hierarchical way and the prototypes for the siblings are also intuitive for humans. +The evidence for this action to be Rafting: +it’s grandparent looks like +it looks like +Rafting +Surfing +Frontcrawl +Breaststroke +Kayaking +This video action is predicted as Rafting because: +Explanation with a regular +ProtoPNet. +Sports +Water Sports +it looks like +it’s sibling looks like +it’s sibling looks like +it’s sibling looks like +it’s sibling looks like +it looks like +it looks like +Prediction: +Sports +because +Prediction: +Riding sports +because +Prediction: +Horse riding +because +Leftmost: Original video. Second column: Three different training videos where prototypes come from. +Third column: Prototypes. +Figure 5. Effectiveness in case of failure. Our multi-level expla- +nations provide useful information even in the case of misclassifi- +cation through the prototypes learned for parent and grandparent +classes. +ent class riding sports and grandparent class sports are cor- +rectly recognized. Hence our hierarchical explanations give +us useful information even in the case of misclassification. +7. Conclusion +In this work, we proposed Hierarchical ProtoPNet for +video action recognition. By learning hierarchical proto- +types we are able to provide explanations at multiple lev- +els of granularity, not only explaining why it is classi- +fied as a certain class, but also what spatiotemporal parts +contribute to it belonging to parent categories. +Our re- +sults show that Hierarchical ProtoPNet outperforms a prior +non-hierarchical approach on UCF-101, whilst performing +equally well on ActivityNet. Additionally, we demonstrate +our multi-level explanations that make it possible to see +which spatiotemporal parts contribute to grandparent, par- +ent, and class-level classifications. +Our hierarchical ap- +proach thereby provides richer explanations whilst compro- +mising less performance to gain explainability. + +References +[1] Mina Ghadimi Atigh, Julian Schoep, Erman Acar, Nanne van +Noord, and Pascal Mettes. Hyperbolic image segmentation. +In Proceedings of the IEEE/CVF Conference on Computer +Vision and Pattern Recognition, pages 4453–4462, 2022. 2, +3 +[2] Sarah Adel Bargal, Andrea Zunino, Donghyun Kim, Jian- +ming Zhang, Vittorio Murino, and Stan Sclaroff. Excitation +backprop for rnns. 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We define three levels of +hierarchy with the number of classes at level one, two, and +three being 5, 20, and 101 respectively for UCF-101. For +example, one of the grand parent class is playing music, +parents are wind, string, percussion and children are playing +flute, playing guitar, drumming and more. The classes at +the third level (i.e., child level) of the hierarchy are the 101 +original classes of the dataset. +ActivityNet contains 200, 38, and 6 classes in level one, +two, and three respectively (see Figure 7). For instance, +one of the grand parent is personal care and the parents are +dress up, grooming, wash up and children are putting on +shoes, getting a haircut, shaving etc. +9. Qualitative Results +Figure 8, 9 and 10 show the quantitative results for our +multi-level explanations, while Figure 12 and 11 show the +effectiveness of our method in case of failure. + +   +Horse Riding : +   +Horse Race : +   +Biking : +   +Fencing : +   +Field Hockey Penalty : +   +Soccer Penalty : +   +Cricket Shot : +   +Cricket Bowling : +   +Billiards Shot : +   +Basketball Dunk : +   +Basketball Shooting : +   +Basketball Pitch : +   +Golf Swing : +  +Volleyball Spiking : +   +Tennis Swing : +   +Table Tennis Shot : +   +Clean and Jerk : +   +Bench Press : +   +Sumo Wrestling : +   +Punch : +   +Boxing Speed Bag : +   +Boxing Punching Bag : +   +Frisbee Catch : +   +Archery : +   +Bowling : +   +Javelin Throw : +   +Hammer Throw : +   +Throw Discus : +   +Shotput : +   +Surfing : +   +Rowing : +   +Rafting : +   +Breaststroke : +   +Front Crawl : +   +Kayaking : +   +Ice Dancing : +   +Skijet : +   +Skiing : +   +Sky Diving : +   +Diving : +   +Cliff Diving : +   +Still Rings : +   +Uneven Bars : +   +Pole Vault : +   +Long Jump : +   +Pommel Horse : +  +Parallel Bars : +   +High Jump : +   +Floor Gymnastics : +   +Balance Beam : +   +Playing Flute : +   +: Playing Sitar +   +: Playing Piano +  +: Playing Cello +   +: Playing Violin +   +: Playing Guitar +   +: Playing Dhol +   +: Playing Daf +   +: Playing Tabla +   +: Drumming +   +: Band Marching +   +: Military Parade +   +: Salsa Spins +   +: Head Massage +   +: Haircut +   +: Trampoline Jumping +   +: Tai Chi +   +: Rope Climbing +   +: Rock Climbing Indoor +   +: Push Ups +   +: Pull Ups +   +: Lunges +   +: Jumping Jack +   +: Handstand Walking +   +: Handstand Pushups +   +: Body Weight Squat +   +: Wall Pushups +   +: Walking with a Dog +   +: Swing +   +: Blowing Candles +   +: Baby Crawling +   +: Yo Yo +   +: Soccer Juggling +   +: Skate Boarding +   +: Nun Chucks +   +: Jump Rope +   +: Juggling Balls +   +: Hula Hoop +   +: Writing on Board +   +: Typing +   +: Pizza Tossing +   +: Mopping Floor +   +: Mixing Batter +   +: Knitting +   +: Hammering +   +: Cutting in Kitchen +   +: Shaving Beard +   +: Brushing Teeth +   +: Blow Dry Hair +   +: Apply Lipstick +   +: Apply Eye Makeu +Riding Sports +Combat Sports +Ball Sports +Racket Sports +Strength Sports +Target Sports +Water Sports +Ice or Snow Sports +Air sports +Gymnatics +Wind +String +Precussion +More than two persons interaction +Two persons interaction +Strenuous +Non strenuous +Exercising/playing games +Working +Self-grooming +Sports +Playing Music +Human-Human +Body-Motion +Human-Object +UCF-101 +UCF-101 +Human-Object +Body-Motion +Human-Human +Playing Music +Sports +Self-grooming +Working +Exercising/playing games +Non strenuous +Strenuous +Two persons interaction +More than two persons interaction +Precussion +String +Wind +Gymnatics +Air sports +Ice or Snow Sports +Water Sports +Target Sports +Strength Sports +Racket Sports +Ball Sports +Combat Sports +Riding Sports +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +  +   +   +   +   +   +   +  +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +  +   +   +   +   +   +   +   +   +   +   +   +   +   +Figure 6. Hierarchy for UCF-101. Schematic representation of the hierarchy defined for UCF-101 dataset. The three levels of hierarchy +are grand parent (playing music), parent (wind, string, percussion) and children (playing flute, playing guitar, drumming etc) classes. + +   +Sumo : +   +Arm Wrestling : +   +Doing a Powerbomb : +   +Doing Fencing : +   +Rope Skipping : +   +Kneeling : +   +Doing Crunches : +   +Snatch : +   +Clean and Jerk : +   +Playing Water Polo : +   +Windsurfing : +   +Plataform Diving : +   +Springboard Diving : +   +Waterskiing : +   +Wakeboarding : +   +Swimming : +   +Surfing : +   +Scuba Diving : +   +Spinning : +   +Using the Rowing Machine : +   +Elliptical Trainer : +   +Snowboarding : +   +Snow Tubing : +   +Skiing : +   +Curling : +   +Ice Fishing : +   +Skateboarding : +   +Rollerblading : +   +Longboarding : +   +Calf Roping : +   +Bullfighting : +   +Playing Badminton : +   +Playing Racquetball : +   +Playing Lacrosse : +   +Playing Squash : +   +Tennis Serve with Ball Bouncing : +   +Ping-Pong : +   +Volleyball : +   +Playing Beach Volleyball : +   +Futsal : +   +Beach Soccer : +   +Hurling : +   +Croquet : +   +Playing Field Hockey : +   +Playing Ice Hockey : +   +Dodgeball : +   +ayup Drill in Basketball : +   +Tai Chi : +   +Doing Karate : +   +Doing Kickboxing : +   +Capoeira : +   +Running a Marathon : +   +Pole Vault : +   +Archery : +   +Hammer Throw : +   +Shot Put : +   +Triple Jump : +   +Long Jump : +   +Javelin Throw : +   +Discus Throw : +   +High Jump : +   +Doing Motocross : +   +Bungee Jumping : +   +Paintball : +   +Powerbocking : +   +Rock Climbing : +   +Bmx : +   +Camel Ride : +   +Horseback Riding : +   +Playing Polo : +   +Baton Twirling : +   +Using Uneven Bars : +   +Using Parallel Bars : +   +Tumbling : +   +Using the Balance Beam : +   +Using the Pommel Horse : +   +Doing Step Aerobics : +   +Zumba : +   +Kayaking : +   +Sailing : +   +River Tubing : +   +Rafting : +   +Canoeing : +   +Playing Kickball : +   +Cricket : +   +Smoking a Cigarette : +   +Smoking Hookah : +   +Playing Ten Pins : +   +Hopscotch : +   +Tug of War : +   +Throwing Darts : +   +Table Soccer : +   +Slacklining : +   +Shuffleboard : +   +Rock-Paper-Scissors : +   +Riding Bumper Cars : +   +Playing Rubik Cube : +   +Playing Pool : +   +Kite Flying : +   +Hula Hoop : +   +: Hitting a Pinata +   +: Beer Pong +   +: Playing Blackjack +   +: Shaving +   +: Washing Hands +   +: Brushing Teeth +   +: Washing Face +   +: Gargling Mouthwash +   +: Shaving Legs +   +: Braiding Hair +   +: Blow-Drying Hair +   +: Getting a Haircut +   +: Getting a Piercing +   +: Getting a Tattoo +   +: Removing Curlers +   +: Applying Sunscreen +   +: Doing Nails +   +: Brushing Hair +   +: Putting on Makeup +   +: Putting on Shoes +   +: Putting in Contact Lenses +   +: Fixing Bicycle +   +: Removing Ice from Car +   +: Hand Car Wash +   +: Changing Car Wheel +   +: Assembling Bicycle +   +: Mowing the Lawn +   +: Trimming Branches or Hedges +   +: Spread Mulch +   +: Raking Leaves +   +: Cutting the Grass +   +: Blowing Leaves +   +: Painting +   +: Plastering +   +: Laying Tile +   +: Installing Carpet +   +: Hanging Wallpaper +   +: Decorating the Christmas Tree +   +: Cleaning Sink +   +: Carving Jack-O-Lanterns +   +: Chopping Wood +   +: Painting Furniture +   +: Wrapping Presents +   +: Painting Fence +   +: Roof Shingle Removal +   +: Fixing the Roof +   +: Shoveling Snow +   +: Polishing Shoes +   +: Cleaning Shoes +   +: Knitting +   +: Hand Washing Clothe +   +: Ironing Clothes +   +: Polishing Forniture +   +: Vacuuming Floor +   +: Cleaning Windows +   +: Mooping Floor +   +: Welding +   +: Waxing Skis +   +: Sharpening Knives +   +: Walking the Dog +   +: Grooming Horse +   +: Disc Dog +   +: Bathing Dog +   +: Grooming Dog +   +: Clipping Cat Claws +   +: Washing Dishes +   +: Mixing Drinks +   +: Making a Sandwich +   +: Preparing Salad +   +: Preparing Pasta +   +: Peeling Potatoes +   +: Making an Omelette +   +: Making a Lemonade +   +: Making a Cake +   +: Baking Cookies +   +: Drinking Beer +   +: Drinking Coffee +   +: Having an Ice Cream +   +: Playing Accordion +   +: Playing Violin +   +: Playing Piano +   +: Playing Flauta +   +: Playing Guitarra +   +: Playing Saxophone +   +: Playing Harmonica +   +: Playing Bagpipes +   +: Playing Drums +   +: Playing Congas +   +: Drum Corps +   +: Starting a Campfire +   +: Using the Monkey Bar +   +: Swinging at the Playgroun +   +: Fun Sliding Down +   +: Building Sandcastles +   +: Belly Dance +   +: Breakdancing +   +: Cumbia +   +: Cheerleading +   +: Tango +   +: Ballet +Wrestling +Working out +Weightlifting +Water sports +Using cardiovascular equipment +Skiing/ice skating/snowboarding +Roller sports +Rodeo competitions +Racquet sports +Playing volleyball +Playing soccer +Playing hockey +Playing basketball +Martial arts +Field sports +Extreme sports +Equestrian sports +Doing gymnastics +Doing aerobics +Boating +Bat-and-ball games +Tobacco and drug use +Playing games +Wash up +Grooming +Dress up +Vehicle repair and maintenance +Lawn/Garden/and Houseplants +Interior Maintenance/Repair/& Decoration +Exterior Maintenance/Repair/& Decoration +Cleaning and Laundry +Appliances/Tools/and Toys +Animals and Pets +Food and drink preparation +Eating and Drinking +Playing musical instruments +Park activities +Dancing +Sports/Exercise/and Recreation +Relaxing and Leisure +Personal Care +Household Activities +Eating and drinking Activities +Arts and Entertainment +ActivityNet +ActivityNet +Arts and Entertainment +Eating and drinking Activities +Household Activities +Personal Care +Relaxing and Leisure +Sports/Exercise/and Recreation +Dancing +Park activities +Playing musical instruments +Eating and Drinking +Food and drink preparation +Animals and Pets +Appliances/Tools/and Toys +Cleaning and Laundry +Exterior Maintenance/Repair/& Decoration +Interior Maintenance/Repair/& Decoration +Lawn/Garden/and Houseplants +Vehicle repair and maintenance +Dress up +Grooming +Wash up +Playing games +Tobacco and drug use +Bat-and-ball games +Boating +Doing aerobics +Doing gymnastics +Equestrian sports +Extreme sports +Field sports +Martial arts +Playing basketball +Playing hockey +Playing soccer +Playing volleyball +Racquet sports +Rodeo competitions +Roller sports +Skiing/ice skating/snowboarding +Using cardiovascular equipment +Water sports +Weightlifting +Working out +Wrestling +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +   +Figure 7. Hierarchy for ActivityNet. Schematic representation of the hierarchy defined for Activity dataset. The three levels of hierarchy +are grand parent (personal care), parent (dress up, grooming, wash up) and children (putting on shoes, getting a haircut, shaving etc) +classes. + +The evidence for this action to be Rafting: +it’s grandparent looks like +it looks like +Rafting +Surfing +Frontcrawl +Breaststroke +Kayaking +This video action is predicted as Rafting because: +Explanation with a regular +ProtoPNet. +Sports +Water Sports +it looks like +it’s sibling looks like +it’s sibling looks like +it’s sibling looks like +it’s sibling looks like +it looks like +it looks like +This video belongs to sports +because +This video belongs to +target sports because +This video belongs to +bowling because +it looks like +This video belongs to rafting +because +Leftmost: Parts in the original video that are highly activated by the prototype. Second +column: Training videos where prototypes come from. Third column: Prototypes. +Rightmost: Saliency map in the original video that are highly activated by the prototype. +it also looks like +Figure 8. Multi-level Explanations. Schematic representation of the hierarchical prototype-based reasoning process of our proposed +Hierarchical ProtoPNet. + +it looks like +it looks like +it looks like +This video belongs to +human-object interaction +because +This video belongs to +self grooming because +This video belongs to +blow dry because +Leftmost: Parts in the original video that are highly activated by the prototype. Second +column: Training videos where prototypes come from. Third column: Prototypes. +Rightmost: Saliency map in the original video that are highly activated by the prototype. +it also looks like +Figure 9. Multi-level Explanations. Schematic representation of the hierarchical prototype-based reasoning process of our proposed +Hierarchical ProtoPNet. + +it looks like +it looks like +it looks like +This video belongs to +sports because +This video belongs to +snow sports because +This video belongs to +skiing because +Leftmost: Parts in the original video that are highly activated by the prototype. Second +column: Training videos where prototypes come from. Third column: Prototypes. +Rightmost: Saliency map in the original video that are highly activated by the prototype. +it also looks like +Figure 10. Multi-level Explanations. Schematic representation of the hierarchical prototype-based reasoning process of our proposed +Hierarchical ProtoPNet. + +The evidence for this action to be Rafting: +it looks like +This video action is predicted as Rafting because: +Explanation with a regular +ProtoPNet. +it looks like +it looks like +Prediction: +Body-motion +because +Prediction: +Strenuous +because +Prediction: +Rock Climbing +Indoor +because +Leftmost: Original video. Second column: Three different training videos where prototypes come from. +Third column: Prototypes. +Figure 11. Effectiveness in case of failure. Our multi-level explanations provide useful information even in the case of misclassification +through the prototypes learned for parent and grandparent classes. +The evidence for this action to be Rafting: +it looks like +This video action is predicted as Rafting because: +Explanation with a regular +ProtoPNet. +it looks like +it looks like +Prediction: +Human-human +interaction +because +Prediction: +Two-persons +interaction +because +Prediction: +Head Massage +because +Leftmost: Original video. Second column: Three different training videos where prototypes come from. +Third column: Prototypes. +Figure 12. Effectiveness in case of failure. Our multi-level explanations provide useful information even in the case of misclassification +through the prototypes learned for parent and grandparent classes. + diff --git a/jdAyT4oBgHgl3EQfkfjo/content/tmp_files/load_file.txt b/jdAyT4oBgHgl3EQfkfjo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d279d866c10f5a53292a071d159b03cb89cc6cd2 --- /dev/null +++ b/jdAyT4oBgHgl3EQfkfjo/content/tmp_files/load_file.txt @@ -0,0 +1,1091 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf,len=1090 +page_content='Hierarchical Explanations for Video Action Recognition Sadaf Gulshad, Teng Long, Nanne van Noord University of Amsterdam {s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='gulshad, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='long, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='vannoord}@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='nl Abstract We propose Hierarchical ProtoPNet: an interpretable network that explains its reasoning process by consider- ing the hierarchical relationship between classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Different from previous methods that explain their reasoning process by dissecting the input image and finding the prototypical parts responsible for the classification, we propose to ex- plain the reasoning process for video action classification by dissecting the input video frames on multiple levels of the class hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The explanations leverage the hierarchy to deal with uncertainty, akin to human reasoning: When we observe water and human activity, but no definitive action it can be recognized as the water sports parent class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Only af- ter observing a person swimming can we definitively refine it to the swimming action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Experiments on ActivityNet and UCF-101 show performance improvements while providing multi-level explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Introduction When describing the world around us we may do so at different levels of granularity, depending on the information available or the level of detail we intend to convey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For in- stance, a video might open with a shot of a cheering crowd, allowing us to recognize it as a a sports event, as the cam- era then pans to the river we can deduce that it is a water sports event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' However, only when the raft comes into the frame can we determine that it concerns rafting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Nonethe- less, in our description of this video, we may still only refer to it as a sports or water sports event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our reasoning and description processes build on the hierarchical relation be- tween classes, allowing for navigation between generic and specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In this work, we implement this process for video action recognition by learning hierarchical prototypes that we leverage for improved classification performance and explanations at multiple levels of granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Despite the remarkable performance of neural networks for video understanding tasks [4,6,14,17,34,37,38,50] it is still hard to explain the decisions of these networks, which is of utmost importance for practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' This ne- The evidence for this action to be Rafting: it’s grandparent looks like Rafting Surfing Frontcrawl Breaststroke Kayaking This video action is predicted as Rafting because: Explanation with a regular ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Sports Water Sports it looks like it’s sibling looks like it’s sibling looks like it’s sibling looks like it’s sibling looks like it looks like This video belongs to rafting because it looks like it looks like it looks like This video belongs to sports because This video belongs to water sports because This video belongs to rafting because Leftmost: Parts in the original video that are highly activated by the prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Second column: Training videos where prototypes come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Third column: Prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Rightmost: Saliency map in the original video that are highly activated by the prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Multi-level Explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Multi-level explanation for a video of rafting with Hierarchical ProtoPNet, showing the expla- nations at grandparent, parent, and class level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' cessity has led to a growing stream of research that focuses on making models interpretable besides performing accu- rately [13, 15, 19, 23, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' A promising line of explainabil- ity methods are the case-based reasoning models [7, 9, 21], which focus on learning prototypes during the training and making predictions based on the learned prototypes while inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' This enables this look like that type of expla- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' However, previous case-based reasoning works are limited to 2D images and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Moreover, they provide a single level of explanations and in case of uncertainty, the explanations can be as bad as arbitrary, as each explanation is considered equally apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In this work, we focus on cap- turing the hierarchical relations between actions to provide multi-level explanations for videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' A challenge for mod- els with built-in explainability, such as case-based reason- ing models, is that it introduces an accuracy-explainability trade-off, where explainability comes at the cost of accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' With this paper, we aim to introduce a model with built-in explainability, whilst being less affected by this trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' To achieve this goal we are inspired by recent works on learning hyperbolic embedding spaces rather than arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='00436v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='CV] 1 Jan 2023 euclidean for natural language processing [8, 44] and com- puter vision tasks [1, 12, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' These works have demon- strated that it is beneficial for performance to have the em- bedding space be guided by hierarchical prior knowledge, which we believe will also benefit explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' This be- lief is guided by the similarity between human intuition and the representation of categories in the hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The main contributions of our paper are: 1) We propose Hierarchical ProtoPNet, a case-based reasoning model for interpreting video action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 2) We demonstrate that while other interpretable models fail to provide expla- nations in the presence of uncertainty or lack of informa- tion our Hierarchical ProtoPNet can overcome these chal- lenges by providing multi-level explanations i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=', at class, parent, or grandparent level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 3) We perform a benchmark and show that our Hierarchical ProtoPNet outperforms its non-hierarchical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Interpretations for Videos Interpretations for neural networks can be broadly clas- sified into two categories: 1) fitting explanations to the decisions of the network after it has been trained i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' posthoc [13, 15, 19, 27, 35], 2) building explanation mech- anism inherent in the network i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' built-in explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' [23, 24, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In this work, we focus on learning seman- tic representations which are used for classification during training rather than explaining a black box network posthoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' A great deal of previous work has focused on video ac- tion recognition, detection, segmentation and more [4,6,14, 17, 34, 37, 38, 50], however, most of these works focus on designing black box models for specific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' They do not explain why a certain decision is made by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' More- over, most of the research in the visual explanations domain focuses on images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Only a few works focus on the interpre- tation of these networks for videos [2, 16, 23, 41, 42], and it is not possible to directly apply image-based explanation methods to videos due to an extra time dimension in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' [16] and [2] focus on visualizing spatio-temporal atten- tion in RNNs, CNNs are used only to extract features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In- spired by class activation maps (CAM) [54] for images [42] extended it for videos by finding both regions and frames responsible for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' [23] utilized perturbations to extract the most informative parts of the inputs responsible for the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Both [23, 42] are posthoc methods, which means they do not use explanations during prediction there- fore they might not be faithful to what the network com- putes [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' [41] introduced class feature pyramids, a method that traverses through the whole network and searches for the kernels at different depths of the network responsible for classification, therefore this method is computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In contrast, we enable built-in multi-level expla- nations that do not add any computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In this paper, we enable multi-level explanations for videos by learning hierarchical prototypes for each class and tracing them back to input videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Prototype-based Interpretations In machine learning the term “prototype” is used in var- ious contexts, in zero-shot learning [22, 51–53] and few- shot learning [32, 39] prototypes are points in the embed- ding space representing a single class and the distance from these prototypes is used during classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' However, in our work prototypes are closer to the samples in the training set, and multiple prototypes are used to represent each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' They are optimized to resemble the training set in order to provide visual explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The idea to provide built-in explanations with prototypes was first explored in [21], where the authors introduced a prototype layer in the network with an encoder-decoder ar- chitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The prototype layer stores weights which are close to encoded training samples, and a decoder is used to visualize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' However, their model fails to generate re- alistic visualizations for natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Thus [7] proposed to learn prototypes for each class and visualized them by tracing them back to the input images without a decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our work is inspired by [7], but where their work is limited to 2D images and provides only one-level explanations we extend it to multi-level explanations for videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' [36] focuses on reducing the number of prototypes for each class by finding shared prototypes among classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' [28] enhances the prototypical explanations by adding textual explanations to explain prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' [49] introduced a dif- ferent similarity metric for computing similarities between prototypes and image patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' They also introduced a loss function to enhance the diversity of prototypes within a class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Deformable ProtoPNet [9] learns spatially flexible prototypes to capture pose and context variations in the in- put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' All previous prototype-based explanation methods pro- vide explanations without considering the hierarchical re- lations between classes on well-defined image CUB birds [48] and Stanford cars [20] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In contrast inspired by the human way of explanations we consider hierarchical re- lations between classes while learning prototypes for each class for video datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Most closely related to our work are [29, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' [45] in- troduced a dynamic prototype network (DPNet) for finding temporal artifacts and unnatural movements in deep fake videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' However, the goal of DPNet is different from ours with only two target classes fake/real making the task eas- ier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' [29] introduced neural prototype trees by combining CNN architecture with the soft binary trees for providing local and global explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' However, as the number of prototypes depends upon the size of the tree learning a Pro- toTree becomes computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our proposed multi-level explanations do not add any extra computational complexity to the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hyperbolic Embeddings [30] demonstrated that hyperbolic embeddings can learn hierarchical tree-like structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Later on, the effective- ness of hyperbolic embeddings have been shown for tex- tual [11,44,55] to visual data [1,12,18,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hyperbolic em- beddings have also been used for zero-shot learning [10,25] and for video action recognition [26, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The hierarchi- cal relationship between videos and the hierarchical way of explaining decisions for humans calls for the need of us- ing hyperbolic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In this work, we utilize hyperbolic embeddings for learning hierarchical prototypes to provide human-like explanations for video action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hierarchical Action Embeddings Incorporating the prior hierarchical knowledge about ac- tions into the network requires that we represent them as embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In this section we detail how to learn those action embeddings in hyperbolic space, in the next section, we explain how to learn hierarchical prototypes that are optimized by aligning them to the action embed- dings in hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Given the set of action classes A = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=', |A|}, in hierarchical action recognition we also consider their ancestor classes H = {|A| + 1, |A| + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=', |A| + |H|}, which allows us to construct a hierarchi- cal tree with three levels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=', grandparent, parent, and child (see Figure 2 right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' This process of embedding the hierar- chies is performed once, offline, per dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' However, this process can easily be repeated for alternative hierarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Learning Action Embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We map the action hierar- chy A ∪ H into the shared hyperbolic space Dn to obtain hierarchical action embeddings, which are used as action class templates in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Let P = {(u, v)|u = φ(v)} be the positive pair of v and its parent φ(v) and N = {(u′, v′)|u′ ̸= φ(v′)} be the negative pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The dis- criminative loss akin to [26]: L(P, N, Φ) = LH(P, N) + λ · LS(Φ), (1) where Φ stands for class templates matrix and its c − th column Φc is the template vector of class c in Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For the loss function, LH encourages the preservation of parent- child relations and LS enforces separation among different sub-hierarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The first part LH is akin to [31], where the wrongly positioned child-parent pairs will be penalized: LH(P, N) = � (u,v)∈P log � e−d(u,v) � (u,v′)∈N e−d(u,v′) � , (2) where −d(u, v) is the hyperbolic distance between two ac- tion embeddings u and v, which can be written in short- hand notation: d(v, u) := 2 arctanh (∥−v ⊕ u∥) , (3) where ⊕ indicates the M¨obius addition [47] in 1−curved hyperbolic space Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In the second part, we encourage the separation among sibling relationships, where we update Φ with separation loss: LS(Φ) = − � i∈|A| || ˜ΦT i ˜Φi||F + γ||( ˆ Φi ˆ Φi T − I)||F , (4) where ˆΦ consists of the non-sibling vectors with respect to action class i while ˜Φ consists of i’s sibling vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' After learning with the above objectives, we obtain Φ, a matrix of action template vectors including both actions A and ancestor (parent and grandparent) actions H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hierarchical ProtoPNet Figure 2 gives an overview of our proposed Hierarchical ProtoPNet for video action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our Hierarchical ProtoPNet consists of a 3D-CNN backbone f for extracting features from the video frames, and a prototype layer gp for learning prototypes for each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The prototype layer is followed by a fully connected layer h that combines the pro- totype similarity scores maps them to the shared hyperbolic space through exponential mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Prior knowledge about the relations between actions, in the form of the action hier- archy, are projected to the shared space through discrimina- tive embeddings, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Subsequently, we use hyperbolic learning to obtain hierarchical prototypes that enable multi-level explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Feature Extraction As the backbone architecture, we use the video action classification network 3D-Resnet [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For each input video v ∈ RW ×H×T ×3 with T frames it extracts video features Z ∈ RW0×H0×T0×D with the spatial resolution W0 × H0, frames T0 and channels D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' A key aspect of this backbone is that T0 < T due to temporal pooling, as such the features Z are extracted for segments rather than individual frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Because of the temporal pooling, the prototypes learned by our Hierarchical ProtoPNet are spatiotemporal thereby ex- plaining which parts of the segment are indicative of the action in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Prototype Layer Given the features extracted from the 3D-Resnet Z, two layers of 1 × 1 × 1 convolutions with the LeakyReLU ac- tivations are added for adjusting the number of channels for the top layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For each child A and its parent H ac- tion, the network learns m and n prototypes respectively P = {pj}m+n j=1 , whose shape is W1 × H1 × T1 × D with RGB Frames RGB Embedding ResNet-3D backbone in Exponential map to Action Embedding Hyperbolic Hierarchy on 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='76 Prototype Layer Fully connected Layer Similarity scores 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='56 Children Prototypes Parent Prototypes Hierarchical Cross-entropy Hyperbolic Matching Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Overview of the Hierarchical ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The Resnet-3D backbone extracts video features and the prototype layer learns prototypes for children and parents, these prototypes are then converted to a single similarity score through max pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Finally, scores are converted from Rn to Dn through a fully connected layer followed by an exponential map, to the shared hyperbolic space for Hierarchical learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Actions are mapped onto the shared hyperbolic space by learning a discriminative embedding on Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' W1 ≤ W0, H1 ≤ H0 and T1 ≤ T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' As such each pro- totype represents a spatiotemporal part of the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Given the convolutional output Z = f(v) and prototypes p a pro- totype layer gp computes the distances between each pro- totype pj and the patches from Z and converts them to the similarity scores using gp(pj, Z) = max z∈Z log (||z − pj||2 2 + 1) (||z − pj||2 2 + ϵ) , ϵ > 0 (5) The distances between each prototype and the patch deter- mine the extent to which a prototype is present in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In contrast with the prior ProtoPNet architectures, we mul- tiply similarity scores with the weights of a fully connected layer h to obtain embeddings to be projected in the hyper- bolic joint space for learning hierarchical prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hierarchical Video Embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The embeddings h = h(gp(p, f(v))) obtained from the prototype layer are in the Euclidean space and can not be di- rectly mapped into the hyperbolic embedding space, there- fore, we use exponential mapping [11] to map video em- beddings into the hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' expx(h) = x ⊕ � tanh � ||h|| 1 − ∥x∥2 � h ||h|| � (6) where ⊕ indicates the 1−curved Mobius addition, x is the tangent point connecting tangent space T0Dn to Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Differ- ent values of x lead to different tangent spaces, to avoid any ambiguities we set x = 0 and project the video embeddings to the hyperbolic space for matching with the hierarchical actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Training Hierarchical ProtoPNet Our training process consists of a multi-step procedure: initially epochs we perform warm-up of the newly added layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Following the warm-up, we train the entire network end-to-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Every 10 epochs we perform prototype projec- tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=', updating the prototype layer only, followed by a phase of fine-tuning the layers after the prototype layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Video and Action Matching in the Hyperbolic Space We aim to learn a latent space where patches important for classification are clustered around similar prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In order to learn hierarchical prototypes we optimize the pro- totypes P = {pj}m+n j=1 to match videos to hyperbolic ac- tion embeddings, hence our optimization is supervised by Φ ∈ Dn×(|A|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Let {(vi, yi)}N i=1 be the training set, where v ∈ RW ×H×T ×3 and yi ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our goal is to solve: Lcrs + λ1Lcls + λ2Lsep (7) Hierarchical Cross Entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The first term in our loss is the hierarchical cross-entropy loss Lcrs which penalizes the misclassification defined as: Lcrs = 1 N N � i=1 K � k=1 yik log p(y = k|v) (8) The softmax in the cross entropy is defined as the negative distance between video embeddings and the hierarchical ac- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='S YoYo : Bsby Cntuing :lowing Cadon :Bg :Wehg wity Fhisoa Ceken Non strenuous Ardhery: ePuup Sports Bouing: UCF-101 Body-Motion TargetSports JaehThow Strenuous HnerThoy Junping Jaek Eroeut: ts SP PoiagTxM x V expx( MLcRStion embeddings in the hyperbolic space: p(y = k|v) = exp(−d(he),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Φk) � k′ exp(−d(he),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Φk)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' (9) where he = exp0(h) is applying exponential map to the prototype output h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hierarchical Clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our hierarchical clustering cost encourages input images to have at least one patch from features to be closer to a child, parent or grandparent class prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Lcls = 1 N N � i=1 min j:pj∈P|A|+|H| min z∈patches(f(vi)) ||z − pj||2 2 (10) Hierarchical Separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our hierarchical separation cost encourages the latent patches of the images to stay away from the prototypes not belonging to the same child class or parent class or grandparent class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Lsep = − 1 N N � i=1 min j:pj /∈P|A|+|H| min z∈patches(f(vi)) ||z − pj||2 2 (11) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Prototype Projection We project prototypes onto the closest video features from the training videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We do so for child, parent, and grandparent action categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Mathematically, for a proto- type pj from child, parent or grandparent class i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' pj ∈ P|A|+|H|, we update the prototype layer as: pj ← argmin z∈Zj ||z − pj||2 (12) where Zj = {˜z : ˜z ∈ patches(f(vi)) ∀i s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' yi = |A| + |H|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our prototype layer is updated not only with the prototypes belonging to the child class but also with the parent and grandparent classes enabling the learning of hi- erarchical relations between classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Prototype Visualization To construct the visualizations the learned prototypes are mapped to the spatio-temporal input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We select the patch which highly activates for the prototype pj by for- warding the input v through the network and upsampling the activation map generated by the prototype layer gp(pj, Z) both spatially and temporally (for videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We visualize pj for child, parent, and grandparent classes providing expla- nations at all levels of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Experimental Setup 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Datasets To evaluate our Hierarchical ProtoPNet for videos we conduct experiments on two video datasets: UCF-101 [40] and Activity-Net1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='3 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For UCF-101 only one level ac- tion hierarchy is available with the dataset, therefore we define additional levels to complement the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For ActivityNet we used the hierarchy protocol provided with the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' All the hierarchies are made available together with our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hierarchical UCF-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' UCF-101 [40] contains 13,320 videos belonging to 101 action categories with a total length of 27 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We define two additional levels of hierarchy with the number of classes at level one, two, and three being 5, 20, and 101 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The classes at the third level of the hierarchy are the 101 original classes of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The full hierarchy is included in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hierarchical ActivityNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' ActivityNet [5] contains 14,950 untrimmed videos with each video consisting of one or more action segments belonging to 200 action classes with a total length of approximately 648 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We use 10,024 videos for training and 4,926 for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We follow [14] and train and test our model on trimmed videos to determine video-level accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We follow [26] to define the class- level action hierarchies using the hierarchies that come with the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' It contains 200, 38, and 6 classes in level one, two, and three respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Implementation Details The hierarchical action embeddings are generated by training the model with Reimannian Adam optimizer [3], implemented with geoopt and Pytorch [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Apart from the one-time offline step of generating the hierarchical action embedding our Hierarchical ProtoPNet is trained in an end- to-end fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For feature extraction, we used Resnet-3D- 18 [14] pre-trained on Kinetics [6] and added two 1 × 1 convolutional layers with the LeakyReLU, a prototype layer and the final embedding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We perform prototype pro- jection and visualization every 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We report results on two variations of our Hierarchi- cal ProtoPNet: Hierarchical ProtoPNet with the hyperbolic cross-entropy loss and Hierarchical ProtoPNet CPG (child, parent and grandparent) with hyperbolic cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For the CPG variant, we compare between prototype pro- jection with 5 prototypes per class and 10 prototypes per class, to explore whether this additional supervision makes it possible to use fewer prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For comparison, we adapt ProtoPNet [7] to videos by replacing the 2D ResNet backbone with a 3D ResNet backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We report the performance at both clip level and video level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The clip level accuracy is the rate of correct prediction of each clip, while the video level accuracy is the majority vote of the predictions over all the clips within the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Additionally, to show the ben- efit of using hierarchical learning, we report accuracy for three metrics: the class accuracy is calculated as the rate Network Accuracy Sibling Accuracy Cousin Accuracy # of prototypes per class Non-Interpretable Models 3D-Resnet [14] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='34 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='73 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='62 - Resnet-Hyperbolic [26] 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='64 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='99 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='28 - Interpretable Models ProtoPNet [7] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='30 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='92 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='98 10 Hierarchical ProtoPNet 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='49 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='60 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='79 10 Hierarchical ProtoPNet CPG 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='45 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='88 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='73 5 Hierarchical ProtoPNet CPG 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='40 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='30 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='02 10 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Clip level accuracy comparison for different models on UCF-101 videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We observe that our hierarchical ProtoPNet CPG with 10 prototypes per class recovers the drop due to accuracy-explainability trade-off significantly while providing multi-level explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' of correct prediction in the hierarchical space 0-hop away from the ground-truth, the sibling accuracy as the rate of correct prediction 2-hops away from the ground-truth, and the cousin accuracy as the 4-hops correct prediction rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Higher performance on the sibling and cousin metrics in- dicates that misclassifications are to hierarchically nearby, and therefore, semantically similar classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Experiments 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Recognition Accuracy Non-Interpretable Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The performance of non- interpretable models on UCF-101 and ActivityNet are shown in the top two rows of Tables 1, 2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For fair comparison both non-interpretable models, a regular Resnet [14] and a hyperbolic Resnet [26], are trained end- to-end with the same data augmentations and an equal num- ber of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The only difference between the two non- interpretable models is that for the Resnet model the cat- egories are separated through euclidean hyperplanes while the Resnet-Hyperbolic utilizes hyperbolic embedding space to separate categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our results for UCF-101 show that both a regular Resnet and the hyperbolic Resnet perform similarly at clip level (Table 1) and video level (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For ActivityNet the clip-level class accuracy (Table 2) is comparable across both networks, however, due to the hierarchical learning of hyperbolic Resnet it shows better sibling and cousin accuracy, additionally, it shows an im- provement for class accuracy at the video level (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Overall, we see comparable performance for the non- interpretable networks on UCF-101 and improvements for the Hyperbolic networks on ActivityNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Interpretable Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The performance of interpretable models on UCF-101 and ActivityNet are shown in the bot- tom four rows of Table 1, 2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We report the results for a regular ProtoPNet [7] adapted for videos and the vari- ations of our Hierarchical ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For UCF-101, with a regular ProtoPNet with 10 proto- types per class, the accuracy drops considerably: the clip- level class accuracy drops to 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='30 and the video-level accu- racy to 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' This is because of the explainability-accuracy trade-off common in explainable-AI, also reported in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In contrast, our hierarchical ProtoPNet is much less affected and recovers the drop by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='19 for class accuracy, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='68, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='81 for sibling and cousin accuracies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our Hierarchical-ProtoPNet CPG with 5 prototypes per class also shows similar improvement in performance, however, increasing the number of prototypes to 10 per class im- proves the performance by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='10, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='38, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='04 for class, sibling, and cousin accuracies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Moreover, the performance at the video level reaches 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='22 (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hence, all three variations of our proposed hierarchical Pro- toPNet reduce the accuracy drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' On ActivityNet (see Table 2) we observe a clear accuracy-explainability trade-off for the regular ProtoPNet, with drops in both the clip and video level accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' How- ever, whilst our Hierarchical ProtoPNet shows a similar drop in class accuracy we can observe that it partially re- covers from this drop on the sibling and cousin metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' This behavior holds for both the Hierarchical ProtoPNet CPG with 5 prototypes, and for CPG variant with 10 pro- totypes per class we even see improvements for the sibling and cousin metrics of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='19 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Whilst Ac- tivityNet remains challenging, an improvement in sibling and cousin accuracies is directly beneficial to the explain- ability as demonstrated in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hence, we can observe that on both datasets our Hi- erarchical ProtoPNet is less affected by the accuracy- explainability trade-off whilst also providing multi-level ex- planations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Visual Explanations Multi-level Explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Figure 3 shows an example of multi-level explanations provided by our Hierarchical Pro- toPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our model learns to represent the video clip fea- tures as hierarchical prototypes that belong to grandparent, parent and child classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For example, in Figure 3 our model has learned prototypes (only one out of ten proto- types shown for better presentation) from the grandparent class playing instruments, parent class percussion and the Network Accuracy Sibling Accuracy Cousin Accuracy # of prototypes per class Non-Interpretable Models 3D-Resnet [14] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='99 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='59 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='88 - Hyperbolic-Resnet [26] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='95 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='03 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='44 - Interpretable Models ProtoPNet [7] 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='06 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='74 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='67 10 Hierarchical ProtoPNet 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='02 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='76 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='77 10 Hierarchical ProtoPNet CPG 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='67 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='72 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='84 5 Hierarchical ProtoPNet CPG 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='26 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='93 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='97 10 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Clip level accuracy comparison for different models on ActivityNet videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We observe that our hierarchical ProtoPNet CPG with 10 prototypes per class recovers the drop for siblings and cousins and shows comparable performance with regular ProtoPNet for class accuracy while providing multi-level explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Network UCF-101 Accuracy ActivityNet Accuracy # of prototypes per class Non-Interpretable Models 3D-Resnet [14] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='92 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='15 - Resnet-Hyperbolic [26] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='15 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='19 - Interpretable Models ProtoPNet [7] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='48 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='46 10 Hierarchical ProtoPNet 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='03 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='66 10 Hierarchical ProtoPNet CPG 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='87 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='82 5 Hierarchical ProtoPNet CPG 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='22 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='48 10 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Video level accuracy comparison for different models on UCF-101 and ActivityNet videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We observe that our hierarchical ProtoPNet CPG with 10 prototypes per class recovers the drop at video level significantly for UCF-101 and shows comparable performance with the regular ProtoPNet for ActivityNet while providing multi-level explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The evidence for this action to be Rafting: it’s grandparent looks like it looks like Rafting Surfing Frontcrawl Breaststroke Kayaking This video action is predicted as Rafting because: Explanation with a regular ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Sports Water Sports it looks like it’s sibling looks like it’s sibling looks like it’s sibling looks like it’s sibling looks like it looks like it looks like This video belongs to playing instruments because This video belongs to percussion because This video belongs to playing daf because it looks like This video belongs to rafting because Leftmost: Parts in the original video that are highly activated by the prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Second column: Training videos where prototypes come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Third column: Prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Rightmost: Saliency map in the original video that are highly activated by the prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' it also looks like Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Multi-level Explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Schematic representation of the hierarchical prototype-based reasoning process of our pro- posed Hierarchical ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' action class playing daf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We see the similarity between the grandparent and original video in the posture of players and hand movement, for the parent we observe the similarity in the hands movement, and finally the prototypes for the play- ing daf class show greater similarity for both the instrument and the hand movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Effectiveness of Multi-level Explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The learned prototypes of the regular ProtoPNet with our Hierarchical ProtoPNet are contrasted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In Figure 4 top, we observe that the top prediction by both the networks regu- lar ProtoPNet and Hierarchical-ProtoPNet CPG are correct (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=', biking), however the learned prototypes for ProtoPNet focus only on the tyres or the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' While for our Hierarchical ProtoPNet the prototypes are diverse, focus- ing on the tyre and pedals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Moreover, the top 2 prediction in the bottom half of Figure 4 for the ProtoPNet is Discus Throw which is non-related to biking according to human intuition, while for the hierarchical ProtoPNet it is Horse Racing which relates to biking as it is a riding sport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We ob- serve that the prototypes for Discus throw are random and when we visualize the area highly activated by those pro- totypes, it either focuses on the background or ground (see Figure 4 last two rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Instead for our Hierarchical ProtoP- Net, the top 2 prediction is Horse Racing and the prototypes predominantly focus on the riders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In Figure 5 we show another scenario where the multi- level explanations are useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We see that the original video is misclassified intohorse riding class, however, for the more abstract explanations we can observe that its par- Leftmost: Original Video Frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Second column: Prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Third column: Training videos where prototypes come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Fourth column: Parts in the original video that are highly activated by the prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Rightmost: Saliency map in the original video that are highly activated by the prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' (*Repeat for hierarchical ProtoPNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' ProtoPNet: Predicted Class: Biking Hierarchical ProtoPNet: Predicted Class: Biking Predicted Sibling Class: Discus Throw Predicted Sibling Class: Horse Race Top 1 Prediction Top 2 Prediction Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Effectiveness in contrast to regular ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The comparison between regular ProtoPNet and our Hierarchical ProtoPNet shows that our model learns more diverse prototypes in a hierarchical way and the prototypes for the siblings are also intuitive for humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The evidence for this action to be Rafting: it’s grandparent looks like it looks like Rafting Surfing Frontcrawl Breaststroke Kayaking This video action is predicted as Rafting because: Explanation with a regular ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Sports Water Sports it looks like it’s sibling looks like it’s sibling looks like it’s sibling looks like it’s sibling looks like it looks like it looks like Prediction: Sports because Prediction: Riding sports because Prediction: Horse riding because Leftmost: Original video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Second column: Three different training videos where prototypes come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Third column: Prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Effectiveness in case of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our multi-level expla- nations provide useful information even in the case of misclassifi- cation through the prototypes learned for parent and grandparent classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' ent class riding sports and grandparent class sports are cor- rectly recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hence our hierarchical explanations give us useful information even in the case of misclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Conclusion In this work, we proposed Hierarchical ProtoPNet for video action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' By learning hierarchical proto- types we are able to provide explanations at multiple lev- els of granularity, not only explaining why it is classi- fied as a certain class, but also what spatiotemporal parts contribute to it belonging to parent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our re- sults show that Hierarchical ProtoPNet outperforms a prior non-hierarchical approach on UCF-101, whilst performing equally well on ActivityNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Additionally, we demonstrate our multi-level explanations that make it possible to see which spatiotemporal parts contribute to grandparent, par- ent, and class-level classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our hierarchical ap- proach thereby provides richer explanations whilst compro- mising less performance to gain explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' References [1] Mina Ghadimi Atigh, Julian Schoep, Erman Acar, Nanne van Noord, and Pascal Mettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hyperbolic image segmentation.' 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Aditya Khosla, Agata Lapedriza, Aude Oliva, and Antonio Torralba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Learning deep features for discrimina- tive localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2921–2929, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 2 [55] Yudong Zhu, Di Zhou, Jinghui Xiao, Xin Jiang, Xiao Chen, and Qun Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hypertext: Endowing fasttext with hyperbolic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='16143, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hierarchies for UCF-101 and ActivityNet Figure 6 and 7 show the hierarchies for UCF-101 [40] and ActivityNet [5] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' We define three levels of hierarchy with the number of classes at level one, two, and three being 5, 20, and 101 respectively for UCF-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For example, one of the grand parent class is playing music, parents are wind, string, percussion and children are playing flute, playing guitar, drumming and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The classes at the third level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=', child level) of the hierarchy are the 101 original classes of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' ActivityNet contains 200, 38, and 6 classes in level one, two, and three respectively (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' For instance, one of the grand parent is personal care and the parents are dress up, grooming, wash up and children are putting on shoes, getting a haircut, shaving etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Qualitative Results Figure 8, 9 and 10 show the quantitative results for our multi-level explanations, while Figure 12 and 11 show the effectiveness of our method in case of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Sports ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Racket ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Sports ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Ball ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Sports ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Combat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Sports ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Riding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Sports ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hierarchy for UCF-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Schematic representation of the hierarchy defined for UCF-101 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The three levels of hierarchy are grand parent (playing music), parent (wind, string, percussion) and children (playing flute, playing guitar, drumming etc) classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Sumo : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Arm Wrestling : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Doing a Powerbomb : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Doing Fencing : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Rope Skipping : ' metadata={'source': 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+page_content='Plataform Diving : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Springboard Diving : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Waterskiing : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Wakeboarding : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Swimming : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Surfing : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Scuba Diving : ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Relaxing and Leisure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Sports/Exercise/and Recreation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Dancing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Park activities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Playing musical instruments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Eating and Drinking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Food and drink preparation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Animals and Pets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Appliances/Tools/and Toys ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Cleaning and Laundry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Exterior Maintenance/Repair/& Decoration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Interior Maintenance/Repair/& Decoration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Lawn/Garden/and Houseplants ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Vehicle repair and maintenance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Dress up ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Grooming ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Wash up ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Playing games ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Tobacco and drug use ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Bat and ball games ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Boating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Doing aerobics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Doing gymnastics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Equestrian sports ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Extreme sports ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Field sports ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Martial arts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Playing basketball ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Playing hockey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Playing soccer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Playing volleyball ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Racquet sports ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Rodeo competitions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Roller sports ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Skiing/ice skating/snowboarding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Using cardiovascular equipment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Water sports ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Weightlifting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Working out ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Wrestling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content='Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Hierarchy for ActivityNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Schematic representation of the hierarchy defined for Activity dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The three levels of hierarchy are grand parent (personal care), parent (dress up, grooming, wash up) and children (putting on shoes, getting a haircut, shaving etc) classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The evidence for this action to be Rafting: it’s grandparent looks like it looks like Rafting Surfing Frontcrawl Breaststroke Kayaking This video action is predicted as Rafting because: Explanation with a regular ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Sports Water Sports it looks like it’s sibling looks like it’s sibling looks like it’s sibling looks like it’s sibling looks like it looks like it looks like This video belongs to sports because This video belongs to target sports because This video belongs to bowling because it looks like This video belongs to rafting because Leftmost: Parts in the original video that are highly activated by the prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Second column: Training videos where prototypes come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Third column: Prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Rightmost: Saliency map in the original video that are highly activated by the prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' it also looks like Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Multi-level Explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Schematic representation of the hierarchical prototype-based reasoning process of our proposed Hierarchical ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' it looks like it looks like it looks like This video belongs to human-object interaction because This video belongs to self grooming because This video belongs to blow dry because Leftmost: Parts in the original video that are highly activated by the prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Second column: Training videos where prototypes come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Third column: Prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Rightmost: Saliency map in the original video that are highly activated by the prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' it also looks like Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Multi-level Explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Schematic representation of the hierarchical prototype-based reasoning process of our proposed Hierarchical ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' it looks like it looks like it looks like This video belongs to sports because This video belongs to snow sports because This video belongs to skiing because Leftmost: Parts in the original video that are highly activated by the prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Second column: Training videos where prototypes come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Third column: Prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Rightmost: Saliency map in the original video that are highly activated by the prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' it also looks like Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Multi-level Explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Schematic representation of the hierarchical prototype-based reasoning process of our proposed Hierarchical ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The evidence for this action to be Rafting: it looks like This video action is predicted as Rafting because: Explanation with a regular ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' it looks like it looks like Prediction: Body-motion because Prediction: Strenuous because Prediction: Rock Climbing Indoor because Leftmost: Original video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Second column: Three different training videos where prototypes come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Third column: Prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Effectiveness in case of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our multi-level explanations provide useful information even in the case of misclassification through the prototypes learned for parent and grandparent classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' The evidence for this action to be Rafting: it looks like This video action is predicted as Rafting because: Explanation with a regular ProtoPNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' it looks like it looks like Prediction: Human-human interaction because Prediction: Two-persons interaction because Prediction: Head Massage because Leftmost: Original video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Second column: Three different training videos where prototypes come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Third column: Prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Effectiveness in case of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} +page_content=' Our multi-level explanations provide useful information even in the case of misclassification through the prototypes learned for parent and grandparent classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAyT4oBgHgl3EQfkfjo/content/2301.00436v1.pdf'} diff --git a/ktE0T4oBgHgl3EQf7gLb/content/2301.02778v1.pdf b/ktE0T4oBgHgl3EQf7gLb/content/2301.02778v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0f0807065ab49eeb7c74dc0c0aca8ba6b8ba3bd3 --- /dev/null +++ b/ktE0T4oBgHgl3EQf7gLb/content/2301.02778v1.pdf @@ -0,0 +1,3 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diff --git a/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf b/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7e66796f2875be22b767e08080d257967f98f042 --- /dev/null +++ b/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ad8622f43d6240dcd95b0791ea8c200c5860948f9fe224599f6611d80f95dea0 +size 4887971 diff --git a/ltAyT4oBgHgl3EQfk_i4/content/tmp_files/2301.00445v1.pdf.txt b/ltAyT4oBgHgl3EQfk_i4/content/tmp_files/2301.00445v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..504f2370b61815af8a6d970a2443c1b365b4695c --- /dev/null +++ b/ltAyT4oBgHgl3EQfk_i4/content/tmp_files/2301.00445v1.pdf.txt @@ -0,0 +1,1851 @@ +arXiv:2301.00445v1 [math-ph] 1 Jan 2023 +Wigner equations for phonons transport and quantum heat flux +V.D. Camiolaa, V. Romanoa, G. Vitanzaa +, aDipartimento di Matematica e Informatica - Universit`a degli Studi di Catania +Abstract +Starting from the quantum Liouville equation for the density operator and applying the +Weyl quantization, Wigner equations for the longitudinal and transversal optical and acous- +tic phonons are deduced. The equations are valid for any solid, including 2D crystals like +graphene. With the use of Moyal’s calculus and its properties the pseudo-differential op- +erators are expanded up to the second order in ℏ. The phonon-phonon collision operators +are modelled in a BGK form and describe the relaxation of the Wigner functions to a +local equilibrium function, depending on a local equilibrium temperature which is definite +according to [1]. +An energy transport model is obtained by using the moment method with closures +based on a quantum version of the Maximum Entropy Principle. An explicit form of the +thermal conductivity with quantum correction is obtained under a suitable scaling. +1. Introduction +The use of the Wigner function is one of the most promising ways to study quantum +transport. Its main advantage is that a description similar to the classical or semiclassical +transport is obtained in a suitable phase-space. The mean values are expectation values +with respect to the Wigner function as if the latter were a probability density and the +semiclassical limit of the Wigner transport equation recovers, at least formally, the Boltz- +mann transport one. There is a huge body of literature regarding the Wigner equation and +the way to numerically solve it (see for example [2, 3, 4] and references therein). However, +the most of the works on the subject consider a quadratic dispersion relation for the en- +ergy. Instead, for several materials like semiconductors or semimetal, e.g. graphene, other +dispersion relations must be considered [5, 6, 7]. From the Wigner transport equation +quantum hydrodynamical models have been obtained in [8] for charge transport in silicon +in the case of parabolic bands, while in [9] the same has been devised for electrons moving +in graphene. +The enhanced miniaturization of electron and mechanical devices makes the thermal +effects increasingly relevant [10, 11] requiring the use of physically accurate models. At ki- +netic level a good description is that based on the semiclassical Peierls-Boltzmann equation +for each phonon branch. However, for typical lengths smaller than the phonon mean-free +path also quantum effects must be considered (see for example [11]). The Wigner equation +is a natural approach that better reveal the wave nature of phonons in such circumstances, +Preprint submitted to NLS +January 3, 2023 + +gives the Peierls-Boltzmann equation as semiclassical limit and still keeps the structure +of a kinetic formulation. In this work, the focus is on the acoustic and optical phonons +dynamics with a general dispersion relation. +In order to get insights into the quantum corrections, moment equations are deduced +from the corresponding Wigner equation. As in the classical case, one is led to a system +of balance equations that are not closed. So, the well-known problem of getting closure +relations arises, that is the issue to express the additional fields appearing in the moment +equations in terms of a set of fundamental variables, e.g. the phonon energy density and +energy flux. A sound way to accomplish this task is resorting to a quantum formulation +of the maximum entropy principle [12] (hereafter QMEP), formulated for the first time by +Jaynes [13]. Recently, a more formal theory has been developed in a series of papers [14, 15] +with several applications, for example for charge transport in semiconductors [8, 16, 17, 18]. +The interested reader is also referred to [19]. +We apply QMEP to the Wigner equations assuming the energy density and the energy +flux for each species of phonons as basic fields. By expanding up to the second order in +ℏ, quantum corrections to the semiclassical case [1] are deduced. In particular, in a long +time scaling an asymptotic expression for the heat flux is obtained. The latter consists +of a Fourier-like part with a highly nonlinear second order correction in the temperature +gradient. Explicit formulas for acoustic phonons in the Debye approximation are written. +The plan of the paper is as follows. In section 2, the semiclassical phonon transport is +summarized while in section 3 we write down the Wigner equations for acoustic and optical +phonons. Section 4 is dedicated to deducing the moment equations whose closure relations +are achieved by QMEP in section 5. In the last section a definition of local temperature +is introduced and an asymptotic expression of the quantum correction to the heat flux is +drawn. +2. Semiclassical phonon transport +In a crystal lattice the transport of energy is quantized in terms of quasi-particles named +phonons which are present with several branches and propagation modes. The latters vary +from a material to another but in any case they are grouped in acoustic and optical phonon +branches which, in turn, can oscillate in the longitudinal or transversal direction. The +complete dispersion relations can be usually obtained by a numerical approach in the first +Brillouin zone (FBZ) B. However, in the applications some standard approximations are +often adopted. +For the acoustic phonons, the Debye approximation for the dispersion relation εs(q) +is usually assumed, εs(q) = ℏωs(q) = ℏcs|q|, s = LA, TA. LA stands for longitudinal +acoustic while TA for transversal acoustic. cs is the sound speed of the s-branch and ℏ +denotes the reduced Planck constant. Consistently, the first Brillouin zone is extended to +Rd. Here d is the dimension of the space; d = 3 for bulk crystal while d = 2 for graphene +or similar 2D material like dichalcogenides. +For the longitudinal optical (LO) and the transversal optical (TO) phonon, the Einstein +dispersion relation, ℏωs ≈ const, with s = LO, TO, the phonon angular frequency, is usually +2 + +adopted. +Note that under such an assumption, the group velocity of the optical phonons is +negligible. +In some peculiar materials like graphene, it is customary to introduce also a fictitious +branch called K-phonons constituted by the phonons having wave vectors close to the +Dirac points, K or K′, in the first Brillouin zone (taking the origin in the center Γ of +FBZ). Also in this case the Einstein approximation is used on account of the limited +variability of the phonon energy near those points. Moreover, in graphene the phonons +are classified as in-plane, representing vibration parallel to the material, and out of plane, +representing vibrational mode orthogonal to the material. The LA, TA, LO, TO and K +phonons are in plane. The out of plane phonons belong to the acoustic branch and are +named ZA phonons. For them a quadratic dispersion relation is a good approximation +εZA(q) = ℏωZA(q) = ℏα|q|2, where α = 6.2 × 107m2/s (see [20]) +In the following, instead of the wave vector q we will use the phonon moment ℏq, but +we retain the same character for sake of simplifying the notation. So that εs(q) = cs|q|, +s = LA, TA and εZA(q) = α|q|2, where α = α/ℏ. +The thermal transport is usually described by macroscopic models, e.g. the Fourier one, +those based on the Maximum Entropy methods [19] or on phenomenological description +[10]. A more accurate way to tackle the question is to resort to semiclassical transport equa- +tions, the so-called Peierls-Boltzmann equations, for each phonon branch for the phonon +distributions fµ(t, x, q) +∂fµ +∂t + cµ · ∇xfµ = Cµ, +µ = LA, TA, . . . , +(1) +where cµ = ∇q (ℏωµ) is the group velocity of the µth phonon specie. +The phonon collision term Cµ splits into two terms +Cµ = Cµ +µ + +� +ν,ν̸=µ +Cν +µ, +ν = LA, TA, . . . . +(2) +Cµ +µ describes the phonon interaction within the same branch while Cν +µ describes the +phonon-phonon interaction between different species. To deal with the complete expres- +sions of the Cµ’s is a very complicated task even from a numerical point of view [21]. So, +they are usually simplified by the Bhatnagar-Gross-Krook (BGK) approximation +Cµ = −fµ − f LE +µ +τµ(q) +, +which mimics the relaxation of each phonon branch towards a common local equilibrium +condition, characterised by a local equilibrium temperature TL that is the same for each +phonon population. +The local equilibrium phonon distributions are given by the Bose-Einstein distributions +f LE +µ += +� +eℏωµ/kBTL − 1 +�−1 . +(3) +3 + +and the functions τµ are the phonon relaxation times. Additional BGK terms can be added +to include the interaction between pairs of different branches. +If we know the phonon distributions fµ’s, we can calculate the average phonon energy +densities +Wµ = +1 +(2π)2 +� +B +ℏωµ fµ dq, +(4) +and the expectation value of any function ψ(q) +Mψ = +1 +(2π)2 +� +B +ψ(q) fµ dq, +for example the energy flux if one takes ψ(q) = ℏωµvµ. +The modern devices, e.g. the electron ones like double gate MOSFETs (see [19]), are +undergoing more and more miniaturization. +This implies that the characteristic scales +are of the same order as the typical lengths where quantum effects become more and +more relevant. Therefore, quantum effects must be included and the semiclassical phonon +transport equations must be replaced by a more accurate model. +Among the possible +approaches, that one based on the Wigner equation has the advantage to be formulated +in a phase-space, allowing us to guess the feature of the solutions in analogy with the +semiclassical counterpart. +A huge literature has been devoted to the application of the Wigner equations to charge +transport (see [2, 3, 4]) but a limited use has been made for phonon transport. In the next +sections a transport model, based on the Wigner quasi distribution, will be devised for +phonon transport in nano-structures. +3. Phonon Wigner functions +The main point of our derivation is the kinetic description of a one-particle quantum +statistical state, given in terms of one-particle Wigner functions. Let us now briefly recall +the basic definitions and properties. A mixed (statistical) one-particle quantum state for +an ensemble of scalar particles in Rd is described by a density operator ˆρ, i.e. a bounded +non-negative operator with unit trace, acting on L2(Rd, C). Given the density operator +ˆρ on L2(Rd, C), the associated Wigner function, w = w(x, q), (x, q) ∈ R2d, is the inverse +Weyl quantization of ˆρ, +w = Op−1 +ℏ (ˆρ). +(5) +We recall that the Weyl quantization of a phase-space function (a symbol) a = a(x, q) is +the (Hermitian) operator Opℏ(a) formally defined by [22] +Opℏ(a)ψ(x) = +1 +(2πℏ)d +� +R2d a +�x + y +2 +, q +� +ψ(y)ei(x−y)·q/ℏdy dq +(6) +for any ψ ∈ L2(Rd, C). +The inverse quantization of ˆρ can be written as the Wigner +transform +w(x, q) = +� +Rd ρ(x + ξ/2, x − ξ/2)eiq·ξ/ℏdξ, +(7) +4 + +of the kernel ρ(x, y) of the density operator. +The dynamics of the time-dependent phonon Wigner functions gµ(x, q, t), µ = LA, TA, . . . +steams directly from the dynamics of the corresponding density operator ˆρµ(t), i.e. from +the Von Neumann or quantum Liouville equation +iℏ∂tˆρµ(t) = [ ˆHµ, ˆρµ(t)] := ˆHµˆρµ(t) − ˆρµ(t) ˆHµ, +(8) +where ˆHµ denotes the Hamiltonian operators of the µth phonons and [·, ·] the commutator. +If hµ = Op−1 +ℏ ( ˆHµ) is the symbol associated with ˆHµ, then, from Eq.s (8), we obtain the +Wigner equation for each phonon species +iℏ∂tgµ(x, q, t) = {hµ, gµ(x, q, t)}# := hµ#gµ(x, q, t) − gµ(x, q, t)#hµ. +(9) +With the symbol # we have denoted the Moyal (or twisted) product which translates the +product of operators at the level of symbols according to +a#b = Op−1 +ℏ (Opℏ(a)Opℏ(b)), +(10) +for any pair of symbols a and b. Here, we do not tackle the analytical issues which guarantee +the existence of the previous relations but limit ourselves to the remark that if two operators +are in the Hilbert-Schmidt class, that is the trace there exists and it is not negative and +bounded, then the product is still Hilbert-Schmidt and the Moyal calculus is well defined. +In the sequel, we will suppose that such conditions are valid. +The Moyal product, under suitable regularity assumptions (see [23]), possesses the +following formal semiclassical expansion +a#ℏb(x, q) = +� +α,β +�iℏ +2 +�|α|+|β| (−1)|β| +α!β! ∂α +x∂β +pa(x, q)∂β +x∂α +qb(x, q) +(11) +where α = (α1, ..., αd) ∈ Nd is a multi-index, |α| = � +i αi, α! = � +i αi!, ∂α +x = � +i ∂αi +xi and +similarly for β. +The expansion (11) can be rewritten as +a#ℏb(x, q) = +∞ +� +n=0 +ℏna#nb +(12) +where +a#nb(x, q) = +� +α,β,|α|+|β|=n +� i +2 +�n (−1)|β| +α!β! ∂α +x∂β +qa(x, q)∂β +x∂α +qb(x, q) +(13) +The first terms of (13) read +a#0b += +ab, +(14) +a#1b += +i +2(∇xa · ∇qb − ∇qa · ∇xb), +(15) +a#2b += +−1 +8(∇2 +xa : ∇2 +qb − 2∇x∇qa : ∇q∇xb + ∇2 +qa : ∇2 +xb). +(16) +5 + +where ∇2 denotes the Hessian matrix and : the contracted product of tensors. It is easy +to see that +a#nb(x, q) = (−1)nb#na(x, q), +that is the operation #n is commutative (respectively anticommutative) when n is even +(respectively odd). +If we neglect, for the moment, the phonon-phonon interactions, the Hamiltonian symbol +for each phonon branch is given by +hµ(q) = εµ(q) +µ = LA, TA, . . . . +(17) +By using the Moyal calculus, one can expand the second members of the previous +Wigner equations. Up to first order in ℏ2, we have +∂tgµ(t) + S[hµ]gµ(t) = 0, +µ = LA, TA, . . . , +(18) +where1 +S[hµ]gµ(x, q, t) := cµ · ∇xgµ(x, q, t) − ℏ2 +24 +∂3 +qhµ(q) +∂qi∂qj∂qk +∂3 +xgµ(x, q, t) +∂xi∂xj∂xk ++ O(ℏ4)) +µ = LA, TA, · · · . +(19) +The previous equations describe only ballistic transport and include only the harmonic +contribution to the Hamiltonian. In order to describe also intra and inter-branch phonon- +phonon interactions, an additional anharmonic term ˆHint encompassing the high order +correction to the Hamiltonian operator must be added. So doing, one gets the so-called +Wigner-Boltzmann equation +∂tgµ(x, q, t) + S[hµ]gµ(x, q, t) = Cµ(x, q, t), +µ = LA, TA, . . . , +(20) +In the quantum case the expression of Cµ is rather cumbersome. For electron transport in +semiconductors the interested reader can see [24]. In certain regimes it is justified to retain +the same form of the semiclassical collision operator as the semiclassical case [4]. Here, we +adopt a quantum BGK approach and model the collision terms as +Cµ = −(gµ − gLE +µ ) +τµ(q) +, +µ = LA, TA, . . . . +(21) +where gLE +µ +are now Wigner functions of local equilibrium which will be defined later. +The equation (20) along with the expression (21) for the collision operator represents +our starting point for the phonon transport. Note that for the optical phonons under the +Einstein approximation for the energy bands one has formally the same transport equation +as the semiclassical case because the group velocity vanishes. +1Summation over repeated indices is understood from 1 to d. +6 + +An alternative derivation of (20) can be obtained by explicitly writing the von Neumann +equation (see [18, 19] for the details). One obtains +S[hµ]gµ(t) = +i +ℏ(2π)d +� +Rd +x′×Rdν +� +ε +� +q + ℏ +2ν, t +� +− ε +� +q − ℏ +2ν, t +�� +gµ(x′, q, t)e−i(x′−x)·νdx′dν, +(22) +whose expansion is of course in agreement with the Moyal calculus. +4. Phonon Moment equations +Getting analytical solutions to equations (20)-(21) is a daunting task. Therefore, viable +approaches are numerical solutions based on finite differences or finite elements [2] or +stochastic solutions, e.g. those obtained with a suitable modification of the Monte Carlo +methods for the semiclassical Boltzmann equation [3]. +However, it is possible to have +simpler, even if approximate, models resorting to the moment method for the expectation +values of interest. In fact, it is well known that, although not positive definite, the Wigner +function is real and the expectation values of an operator can be formally obtained as an +average of the corresponding symbol with respect to gµ(x, q, t). So, for any regular enough +weight function ψ(q), let us introduce the short notation +< ψ > (x, t) := +1 +(2π)d +� +Rd ψ(q)gµ(x, q, t)dq, +(23) +which represents a partial average with respect to the phonon moment q. +More in general, if a = a(x, q) is a smooth symbol then it is possible to prove that the +expectation of the (hermitian) operator A = Opℏ(a) satisfies2 +E[A] = tr(ˆρA) = +� +R2d +ρ(x, y)kA(x, y)dxdy = +1 +(2π)d +� +R2d +a(x, q)gµ(x, q, t)dxdq += +� +Rd < a > (x, t)dx, +where kA(x, y) is the kernel of A. +We want to consider a minimum set of moments whose physical meaning is well clear. +In particular, we shall consider the phonon energy and energy flux of each branch +Wµ(x, t) =< hµ > (x, t), +Qµ(x, t) =< hµcµ > (x, t). +(24) +Note that the latter is directly related to the heat flux. +2Here we are considering a fixed instant of time. +7 + +The evolution equations for Wµ(x, t) and Qµ(x, t) are obtained by multiplying the +relative Wigner equation by hµ(q), and hµ(q)cµ and integrating with respect to q +∂tWµ(x, t) + +1 +(2π)d +� +Rdhµ(q)S[hµ]gµ dq = +1 +(2π)d +� +Rdhµ(q)Cµ dq, +∂tQµ(x, t) + +1 +(2π)d +� +Rdhµ(q)cµS[hµ]gµ dq = +1 +(2π)d +� +Rdhµ(q)cµCµ dq. +µ = LA, TA, . . . .(25) +We implicitly assume that the resulting integrals there exist, at least in the principal +value sense. In order to get some global insight from eq.s (25), we formally assume the +following expansions for each phonon branch3 +gµ(x, q, t) = g(0) +µ (x, q, t) + ℏ2g(2) +µ (x, q, t) + o(ℏ2). +(26) +It is possible to prove, at least formally [6], that the semiclassical Boltzmann equation +is recovered from the Wigner equation as ℏ �→ 0+. Therefore, g(0) +µ (x, q, t) can be considered +as the solution fµ of the semiclassical transport equation. Accordingly, we write +Wµ = W (0) +µ ++ ℏ2W (2) +µ ++ o(ℏ2), +Qµ = Q(0) +µ + ℏ2Q(2) +µ + o(ℏ2), +(27) +where +W (0) +µ += +1 +(2π)d +� +Rd hµg(0) +µ (x, q, t)dq, +W (2) +µ += +1 +(2π)d +� +Rd hµg(2) +µ (x, q, t)dq, +Q(0) +µ += +1 +(2π)d +� +Rd hµcµg(0) +µ (x, q, t)dq, +Q(2) +µ = +1 +(2π)d +� +Rd hµcµg(2) +µ (x, q, t)dq. +Regarding the moments of the collision terms, only with drastic simplifications analytical +expressions can be deduced. In analogy with the BGK approximation, if an average re- +laxation time independent on q is considered, one can expand the r.h.s. of eq.s (25) up to +first order in ℏ2 as a relaxation time terms +1 +(2π)d +� +Rdhµ(q)Cµ dq = −Wµ − W LE +µ +τ W +µ += −W (0) +µ +− W (0)LE +µ +τ W +µ +− ℏ2W (2) +µ +− W (2)LE +µ +τ W +µ ++ o(ℏ2), +1 +(2π)d +� +Rdhµ(q)cµCµ dq = −Qµ +τ Q +µ += −Q(0) +µ + ℏ2Q(2) +µ +τ Q +µ ++ o(ℏ2), +where +W LE +µ += +1 +(2π)d +� +Rdhµ(q)gLE +µ +dq. +Note that in the evaluation of the production term of the equations for the energy-fluxes the +isotropy of the equilibrium Wigner function has been invoked and therefore QLE +µ +vanishes. +3The coefficients of the odd powers in ℏ are assumed zero in according to the previous Moyal expansion. +8 + +The energy and energy-flux relaxation times, τ W +µ +and τ Q +µ +respectively, are assumed to +depends on the temperature, which will be definite in the next section, of the relative +branch. +Altogether, the resulting model is made of the following fluid-type equations + + + + + + + + + + + + + +∂tWµ + ∇x +� +Qµ − ℏ2 +24∂2Tµ +� += −W (0) +µ +− W (0)LE +µ +τ W +µ +− ℏ2W (2) +µ +− W (2)LE +µ +τ W +µ ++ o(ℏ2) +∂tQµ + ∇x +� +Jµ − ℏ2 +24∂2Uµ +� += −Q(0) +µ + ℏ2Q(2) +µ +τ Q +µ ++ o(ℏ2), +(28) +where Jµ = J(0) +µ + ℏ2J(2) +µ +with +J(0) +µ = +1 +(2π)d +� +Rd cµ ⊗ cµhµ(q)g(0) +µ (x, q, t)dq, +J(2) +µ = +1 +(2π)d +� +Rd cµ ⊗ cµhµ(q)g(2) +µ (x, q, t)dq, +and the complete symmetric tensors Tµ and Uµ have components +(Tijk)µ = +1 +(2π)d +� +Rd hµ +∂3hµ(q) +∂qi∂qj∂qk +g(0) +µ (x, q, t)dq, +(Uijkr)µ = +1 +(2π)d +� +Rd(cµ)rhµ(q) ∂3hµ(q) +∂qi∂qj∂qk +g(0) +µ (x, q, t)dq. +If we split into zero and first order in ℏ2, the evolution equations read +∂tW (0) +µ ++ ∇xQ(0) +µ = −W (0) +µ +− W (0)LE +µ +τ W +µ +(29) +∂tW (2) +µ ++ ∇xQ(2) +µ + +1 +(2π)d +∂3 +∂xi∂xj∂xk +� +Rd +hµ(q) +24 g(0) +µ +∂3 +∂qi∂qj∂qk +hµ(q)dq += −W (2) +µ +− W (2)LE +µ +τ W +µ +, +(30) +∂tQ(0) +µ + ∇xJ(0) +µ = −Q(0) +µ +τ Q +µ +, +(31) +∂tQ(2) +µ + ∇xJ(2) +µ + +1 +(2π)d +∂3 +∂xi∂xj∂xk +� +Rd cs +hµ(q) +24 +g(0) +µ +∂3 +∂qi∂qj∂qk +hµ(q)dq = −Q(2) +µ +τ Q +µ +. (32) +The zero order equations are the model already investigated in several papers [1, 7] where +is proved that it is a hyperbolic system of conservation law. So, finite propagation speed of +disturbances in energy is guaranteed to overcome the well-known paradox of the classical +9 + +Fourier law for the heat flux. However, the first order corrections in ℏ2 introduce dispersive +terms and it seems that in a quantum regime the requirement of finite propagation speed +of the thermal effects cannot be fulfilled. On the other hand, this is not surprising since +the nonlocal character of the quantum evolution equations can lead to energy propagation +without a bounded speed. +5. QMEP for the closure relations +The evolution equations (29)-(32) do not form a closed system of balance laws. If we +assume the energies Wµ and the energy-fluxes Qµ as the main fields, in order to get a set +of closed equations we need to express the additional fields Jµ, Tµ and Uµ as functions +of Wµ and Qµ. +A successful approach in a semiclassical setting is that based on the +Maximum Entropy Principle (MEP) (see also [19] for a complete review) which is based on +a pioneering paper of Jaynes [12, 13] who also proposed a way to extend the approach to +the quantum case. The MEP in a quantum setting has been the subject of several papers +[8, 14, 15, 16, 17] with several applications, e.g. to charge transport in graphene [1, 18]. +Here we will use such an approach for phonon transport. +The starting point is the entropy for the quantum system under consideration. +In +[18] the authors have employed the Von-Neumann entropy which, however, does not take +into account the statistical aspects. Therefore, we take as entropy a generalization of the +classical one for bosons. Let us introduce the operator +s(ˆρµ) = −kB[ˆρµ ln ˆρµ − (1 + ˆρµ) ln(1 + ˆρµ)], +(33) +which must be intended in the sense of the functional calculus. Here kB is the Boltzmann +constant. The entropy of the µ-th phonon branch reads +S(ˆρµ) = Tr{s(ˆρµ)} +which can be viewed as a quantum Bose-Einstein entropy. +According to MEP, we estimate ˆρµ with ˆρMEP +µ +which is obtained by maximizing S(ˆρµ) +under the constraints that some expectation values have to be preserved. In the semiclas- +sical point case, one maximizes the entropy preserving the values of the moments we have +taken as basic field variables +(Wµ(x, t), Qµ(x, t)) = +1 +(2π)d +� +Rd ψµ(q)gµ(x, q, t)dq = +1 +(2π)d +� +Rd ψµ(q)gMEP +µ +(x, q, t)dq, +(34) +where +ψµ(q) = (hµ(q), cµhµ(q)) +(35) +is the vector of the weight functions and gMEP +µ +is the Wigner function associated with +ˆρMEP +µ +. In the previous relations the time t and position x must be considered as fixed. +10 + +The quantum formulation of MEP is given in terms of expectation values +E1(t) = tr {ˆρµOpℏ(hµ(q))} (t), +E2(t) = tr {ˆρµOpℏ(cµhµ(q))} (t), +as follows: for fixed t +ˆρMEP +µ += argument max S(ˆρµ) +(36) +under the constraints +tr{ˆρMEP +µ +Opℏ(hµ(q))} = E1(t), +tr{ˆρMEP +µ +Opℏ(cµhµ(q))} = E2(t), +(37) +in the space of the Hilbert-Schmidt operators on L2(Rd, C) which are positive, with trace +one and such that the previous expectation values there exist. Note that we are applying +the maximization of the entropy for each phonon branch separately. In other words, we +are requiring the additivity of the entropy. +If we introduce the vector of the Lagrange multipliers +ηµ = (η0µ(x, t), η1µ(x, t)), +(38) +the vector of the moments +m[ρµ](x, t) := mµ(x, t) = +1 +(2π)d +� +Rd ψµ(q)gµ(x, q, t)dq, +(39) +and the vector of the moments which must be considered as known +Mµ(x, t) := (Wµ(x, t), Qµ(x, t)) , +(40) +the constrained optimization problem (36)-(37) can be rephrased as a saddle-point problem +for the Lagrangian +Lµ(ˆρµ, ηµ) += +S(ˆρµ) − +� +Rd ηµ · (mµ(x, t) − Mµ(x, t)) dx += +S(ˆρµ) − tr {ˆρµOpℏ(ηµ · hµ(q), cµhµ(q))} + +� +Rd ηµ · Mµ(x, t) dx +(41) +in the space of the admissible ˆρµ and smooth function ηµ. +If the Lagrangian Lµ(ˆρµ, ηµ) is Gˆateaux-differentiable with respect to ˆρµ, the first order +optimality conditions require +δLµ(ˆρµ, ηµ)(δˆρ) = 0 +for each Hilbert-Schmidt operators δˆρ on L2(Rd, C) which is positive, with trace one and +such that the previous expectation values there exist. +The existence of the first order Gˆateaux derivative is a consequence of the following +Lemma (for the proof see [25]; an elementary proof in the case of discrete spectrum is given +in [14]). +11 + +Lemma 1. If r(x) is a continuously differentiable increasing function on R+ then tr{r(ˆρ)} +is Gˆateaux-differentiable in the class of the Hermitian Hilbert-Schmidt positive operators +on L2(Rd, C). The Gˆateaux derivative along δρ is given by +δtr{r(ˆρ)}(δˆρ) = tr {r′(ˆρ)δˆρ} . +(42) +The extremality conditions for the unconstrained minimization problem (36)-(37) are +similar to that of the semiclassical case, as expressed by the following lemma (see [14]). +Lemma 2. The first order optimality condition for the minimization problem (36)-(37) is +equivalent to +ˆρµ = (s′)−1(Opℏ(ηµ · ψµ)) +(43) +where (s′)−1 is the inverse function of the first derivative of s. +Proof. By applying Lemma 1, the Gˆateaux derivative of the Lagrangian is given by +δLµ(ˆρµ, ηµ)(δˆρ) = tr {(s′(ˆρµ) − Opℏ(ηµ · ψµ)) δˆρ} +∀δˆρ perturbation in the class of the Hermitian Hilbert-Schmidt positive operators on +L2(Rd, C). This implies +s′(ˆρµ) = Opℏ(ηµ · ψµ). +□ +Since the function s(x) is concave, s′(x) is invertible. Explicitly we have +(s′)−1(z) = +1 +ez/kB − 1 +and the operator solving the first order optimality condition reads +ˆρ∗ +µ = (s′)−1(Opℏ(ηµ · ψµ)) = +1 +eOpℏ(ηµ·ψµ) − 1. +(44) +Moreover, such an operator is a point of maximum for the Lagrangian. +□ +Now, to complete the program we have to determine, among the smooth functions, the +Lagrange multipliers ηµ by solving the constraint +tr {ˆρµOpℏ(ηµ · (hµ(q), cµhµ(q))} − +� +Rd ηµ · Mµ(x, t) dx = 0. +(45) +If such an equation has a solution η∗ +µ, altogether the MEP density operator reads +ˆρMEP +µ += +1 +exp +� +Opℏ +� +η∗ +0µ(x, t)hµ(q) + η∗ +1µ(x, t) · cµhµ(q) +�� +− 1, +(46) +where we have rescaled the Lagrange multipliers including the factor 1/kB. +12 + +To determine conditions under which the equation (45) admits solutions is a very dif- +ficult task. Even in the semiclassical case there are examples (see [26]) of sets of moments +that cannot be moments of a MEP distribution. We will directly find out the solution at +least up to first order in ℏ2. +Once the MEP density function has been determined, the MEP Wigner function is +given by +gMEP +µ +(x, q, t) = Op−1 +ℏ (ˆρMEP +µ +) +which can be used to get the necessary closure relations by evaluating the additional fields +with gµ replaced by gMEP +µ +. +We remark that the constraints (45) can be more conveniently expressed as +1 +(2π)d +� +R2d +ηµ · ψµ(x, t)gMEP +µ +(x, q, t) dq dx − +� +Rd ηµ · Mµ(x, t) dx = 0 +and indeed we will require, in analogy with the semiclassical case, the stronger conditions +1 +(2π)d +� +Rd ψµ(x, t)gMEP +µ +(x, q, t) dq = Mµ(x, t), +where the Lagrange multipliers enter through gMEP +µ +(x, q, t). +5.1. Determination of the Lagrange Multipliers +For the sake of making lighter the notation, let us consider a single branch and drop the +index µ in the Wigner function in this section. We look formally for a solution in powers +of ℏ +gMEP = gMEP +0 ++ ℏgMEP +1 ++ ℏ2gMEP +2 ++ ... +(47) +firstly without taking into account the dependence of the Lagrange multipliers on ℏ. +Of course, on account of the properties of the Weyl quantization, gMEP +0 +is equal to the +semiclassical counterpart [22] +gMEP +0 += +1 +exp [η0(x, t)h(q) + η1(x, t) · ch(q)] − 1 +In order to determine the higher order terms gMEP +k +, k ≥ 1, given a symbol a(x, q) let us +introduce the so-called quantum exponential Exp(a) defined as +Exp(a) = Op−1 +ℏ [exp(Opℏ(a))] +which can be expanded as +Exp(a) = Exp0(a) + ℏExp1(a) + ℏ2Exp2(a) + ... +(48) +Proposition Let a(x, p) be a smooth symbol. Then the following expansion is valid +Exp(a) = exp(a) − ℏ2 +8 exp(a) +� +∂2a +∂xi∂xj +∂2a +∂pi∂pj +− +∂2a +∂xi∂pj +∂2a +∂pi∂xj ++ 1 +3 +∂2a +∂xi∂xj +∂a +∂pi +∂a +∂pj +−2 +3 +∂2a +∂xi∂pj +∂a +∂pi +∂a +∂xj ++ 1 +3 +∂2a +∂pi∂pj +∂a +∂xi +∂a +∂xj +� ++ O(ℏ4), (49) +13 + +where Einstein’s convention has been used. +□ +The proof can be found for example in [14]. +By using what is proved in [16], we have +gMEP +2n+1 = 0, +n ≥ 0, +(50a) +gMEP +2n += − +n−1 +� +m=0 +� +k+l+m=n +Exp2k(ξ)#2lgMEP +2m +eξ − 1 +, +n ≥ 1 +(50b) +where #2l are the even terms of the Moyal product expansion and +ξ = η0µ(x, t)h(q) + η1(x, t) · ch(q). +In particular +gMEP +1 += 0 +and +gMEP +2 += +−1 +8 +eξ +(eξ − 1)3 +� +(eξ + 1) +� +∂2ξ +∂xi∂xj +∂2ξ +∂qi∂qj +− +∂2ξ +∂xi∂qj +∂2ξ +∂qi∂xj +� +−(e2ξ + 4eξ + 1) +3(eξ − 1) +� +∂2ξ +∂xi∂xj +∂ξ +∂qi +∂ξ +∂qj +− 2 ∂2ξ +∂xi∂qj +∂ξ +∂qi +∂ξ +∂xj ++ +∂2ξ +∂qi∂qj +∂ξ +∂xi +∂ξ +∂xj +�� +Therefore, up to first order in ℏ2 we have +gMEP +µ += gMEP +0 ++ ℏ2gMEP +2 +. +and the constraints for each phonon branch read +W += +1 +(2π)d +� +Rd +h(q) +eξ − 1dq + ℏ2 +1 +(2π)d +� +Rd h(q)gMEP +2 +dq, +(51) +Q += +1 +(2π)d +� +Rd +ch(q) +eξ − 1dq + ℏ2 +1 +(2π)d +� +Rd ch(q)gMEP +2 +dq. +(52) +The previous equations form a nonlinear system of PDEs for the Lagrange multipliers +whose analytical solution seems very difficult to get. Indeed, the situation is even more +cumbersome because in a numerical scheme the inversion of the constraints should be +performed at each time step. +A viable strategy is to use the Lagrange multipliers as field variables by rewriting the +evolution equations (28) in the form +∇ηW ∂ +∂tηT + +d +� +i=1 +� +∇ηQi +∂ +∂xi +ηT − ℏ2 +24∇η +� +∇x∂2 +xT +� ∂ +∂xi +ηT +� += −W − W LE +τ W +, +(53) +∇ηQi +∂ +∂tηT + +d +� +j=1 +� +∇ηJ ∂ +∂xj +ηT − ℏ2 +24∇η +� +∂2 +xU +� ∂ +∂xj +ηT +� += − Qi +τ Q , +(54) +14 + +getting a highly nonlinear system of PDEs. Note that both ∇ηW and ∇ηQi contain space +derivatives of η. +A further simplification can be obtained by expanding the Lagrange multipliers as +η = η(0) + ℏ2η(2) + o(ℏ2). +Therefore, the basic fields are also expanded with respect to ℏ2 +W = W (0) + ℏ2W (2) + o(ℏ2), +Q = Q(0) + ℏ2Q(2) + o(ℏ2) +where +W (0) += +1 +(2π)d +� +Rd +h(q) +eξ(0) − 1dq, +W (2) += +− +1 +(2π)dη(2) · +� +Rd eξ(0) +h(q)ψ +� +eξ(0) − 1 +�2dq + +1 +(2π)d +� +Rd h(q)gMEP +2 +(η(0))dq, +Q(0) +i += +1 +(2π)d +� +Rd +cih(q) +eξ(0) − 1dq, +Q(2) +i += +− +1 +(2π)dη(0) · +� +Rd +ciψeξ(0)h(q) +(eξ(0) − 1)2 dq + +1 +(2π)d +� +Rd cih(q)gMEP +2 +(η(0))dq, +with ξ(0) = η(0) · ψ. +The balance equations become +∇η(0)W (0) ∂ +∂t(η(0))T + +d +� +i=1 +� +∇η(0)Q(0) ∂ +∂xi +(η(0))T +� += −W (0) − W (0)LE +τ W +(55) +∇η(0)Q(0) +i +∂ +∂t(η(0))T + +d +� +i=1 +� +∇η(0)J(0) ∂ +∂xj +(η(0))T +� += −Q(0) +i +τ Q , +(56) +∂tW (2) + ∇xQ(2) + +1 +(2π)d +∂3 +∂xi∂xj∂xk +� +Rd +h(q) +24 gMEP +0 +(η(0)) +∂3 +∂qi∂qj∂qk +h(q)dq += −W (2) − W (2)LE +τ W +, +(57) +∂tQ(2) + ∇xJ(2) + +1 +(2π)d +∂3 +∂xi∂xj∂xk +� +Rd ch(q) +24 gMEP +0 +(η(0)) +∂3 +∂qi∂qj∂qk +h(q)dq = −Q(2) +τ Q . (58) +We observe that the equations (55)-(56) decouple. Once they are solved, one can get the +second order term of the Lagrange multipliers from (57)-(58) which form a linear system +for η(2). This is rather beneficial from a computational point of view +Proposition 1. At zero order in ℏ2 the map η �→ M(η) is (locally) invertible. +Proposition 2. The equations (55)-(56) form a symmetric hyperbolic system of balance +laws. +The proofs can be found in [19]. +15 + +6. Local equilibrium temperature and heat conductivity +The concept of temperature out of equilibrium is a subtle topic and still a matter of +debate. In the case of charge transport in semiconductors often the phonons are considered +as a thermal bath and under some reasonable assumptions one can hypothesize that the +electrons are in thermal equilibrium with the bath. +In general if the dynamics of the +phonons must be included, a thermal bath for these does not exist, unless a thermostated +system is considered. Therefore, we need to introduce a local equilibrium temperature for +the overall phonon system. +In statistical mechanics, one of the most reasonable and adopted ways to generalize the +concept of temperature in a non-equilibrium state is that of relating it to the Lagrange +multipliers associated with the energy constraint. For the phonon transport in graphene, +an approach based on the Lagrange multipliers was followed in [1] (which the interested +reader is referred to for the details). Let us recall here the main features. At equilibrium, +the phonon temperatures and the corresponding Lagrange multipliers are related by +kB Tµ(x) = +1 +η0,µ(x) = +1 +η(0) +0,µ(x) +− ℏ2 η(2) +0,µ(x) +(η(0) +0,µ(x))2 + o(ℏ2). +If we assume that such relations hold, even out of equilibrium, the definition of the local +temperature can be given in terms of the Lagrangian multipliers as follows. +Definition 1. The local temperature of a system of two or more branches of phonons is +TLE := +1 +kBηLE +0 +(x), where ηLE +0 (x) is the common Lagrange multiplier that the occupation num- +bers of the branches, taken into account, would have if they were in the local thermodynamic +equilibrium corresponding to their total energy density, that is, the following: +W(ηLE +0 (x)) := +� +µ +Wµ(η0,µ(x)) = +� +µ +Wµ(ηLE +0 (x)), +(59) +where the sum runs over all the phonon branches. +At global equilibrium the temperature is constant T = ¯T and the Wigner function +reduced to the Bose -Einstein distribution +gµ = +� +ehµ(q)/kB ¯T − 1 +�−1 +. +(60) +with the same temperature for each phonon branch. +Let us consider a small perturbation δT(x) of the temperature in the sense that +δ(x)/ ¯T ≪ 1. We can expand gMEP +µ +in powers of δ(x)/ ¯T +gMEP +µ += +� +ehµ(q)/kB ¯T − 1 +�−1 ++ +� +ehµ(q)/kB ¯T − 1 +�−2 +ehµ(q)/kB ¯T hµ(q) +kB ¯T +δT(x) +¯T ++ℏ2 ¯T ∂gMEP +2,µ +( ¯T) +∂T +δT +¯T + o +� +ℏ2δT +¯T +� +. +16 + +We observe that typically the relaxation energy relaxation time is much longer than the +energy-flux relaxation times, that is τ Q ≪ τ W. In a long time scaling, much longer than +τ Q, we get +Qµ = −τ Q +� +∇xJµ + +ℏ2 +(2π)d +∂3 +∂xi∂xj∂xk +� +Rd ch(q) +24 gMEP +0,µ +(η(0)) +∂3 +∂qi∂qj∂qk +h(q)dq +� +. +(61) +The relation between the Lagrange multipliers and the basic fields, as seen, can hardly +be inverted analytically but a numerical procedure is necessary. However, if we consider +a situation where the system is not too far from the equilibrium an expansion of the +Lagrange multipliers around the equilibrium state can be performed. At equilibrium gMEP +is isotropic and therefore ηequil +1 += 0 and in a neighborhood of the equilibrium η1 remains +small. Therefore, for small deviations from the thermodynamic equilibrium the expansion +g(0)MEP +µ += +� +ehµ(q)/kBT − 1 +�−1 − +� +ehµ(q)/kBT − 1 +�−2 ehµ(q)/kBThµ(q)η1,µ · cµ + O(ℏ2). +is valid. +By substituting in (61) one gets up to first order in ℏ2 +Qµ = −τ Q∇xJµ +−τ Q +ℏ2 +(2π)d +∂3 +∂xi∂xj∂xk +� +Rd chµ(q) +24 +� +ehµ(q)/kBT − 1 +�−2 ehµ(q)/kBThµ(q)η1,µ · c +∂3 +∂qi∂qj∂qk +hµ(q)dq. +In particular, at the zero order we have +Q(0) +µ += −τ Q∇xJ(0) +µ = − +τ +(2π)d∇x +� +Rd cµ ⊗ cµhµ(q)g(0)MEP +µ +(x, q, t)dq. += − τ Q +(2π)d∇x +� +Rd cµ ⊗ cµhµ(q) +� +ehµ(q)/kBT − 1 +�−1 dq += − τ Q +(2π)d +� +Rd cµ ⊗ cµhµ(q) ∂ +∂T +� +ehµ(q)/kBT − 1 +�−1 dq ∇xTµ += − +τ Q +(2π)dkBT 2 +� +Rd cµ ⊗ cµh2 +µ(q) +ehµ(q)/kBT +(ehµ(q)/kBT − 1)2dq ∇xTµ +which can be written in the Fourier form +Q(0) +µ = −K(0) +µ ∇xTµ +with the thermal conductivity tensor given by +K(0) +µ = +τ Q +(2π)dkBT 2 +� +Rd cµ ⊗ cµh2 +µ(q) +ehµ(q)/kBT +(ehµ(q)/kBT − 1)2dq. +It is evident that Kµ is positive definite. +17 + +Observe that ∀n ∈ Sd +� +Sd ni1ni2 · · · nirdΩ = 0 +if r +odd, +Sd being the unit sphere in Rd. Therefore, if hµ(q) is isotropic Kµ is isotropic as well +K(0) +µ = 1 +dk(0)I, +with I identity matrix of order d and k(0) the zero order trace +k(0) = +τ Q +(2π)dkBT 2 +� +Rd |cµ|2 h2 +µ(q) +ehµ(q)/kBT +(ehµ(q)/kBT − 1)2dq. +The second order correction in ℏ2 reads +Q(2) +µ = − τ Q +(2π)d∇x +� +Rd cµ ⊗ cµhµ(q)g(2) +µ (η(0)(x, q, t))dq +− τ Q +(2π)d +∂3 +∂xi∂xj∂xk +� +Rd chµ(q) +24 +� +ehµ(q)/kBT − 1 +�−2 ehµ(q)/kBThµ(q)η1,µ · c +∂3 +∂qi∂qj∂qk +hµ(q)dq. +Indeed the last term in the previous relation is of order ℏ2δT +T +and can be considered +negligible for small deviations from local equilibrium. The remaining part gives a highly +nonlinear correction which cannot be put in a Fourier form. +As an example we consider the case of the longitudinal and transversal acoustic phonons +in the Debye approximation for a single branch. In such a case the corresponding symbol +of the phonon hamiltonian reads c|q| and therefore +k(0) +ac += +τ Q +(2π)dkBT 2ac +� +Rd c4|q|2 +ec|q|/kBTac +(ec|q|/kBTac − 1)2dq += +τ Qc4 +(2π)dkBT 2ac +mis(Sd) +� +∞ +0 +|q|d+1 +ec|q|/kBTac +(ec|q|/kBTac − 1)2d|q| += +kBτ Qc3−d +(2π)d +mis(Sd) (kBTac)d−1 +� +∞ +0 +zd+1 +ez +(ez − 1)2d z +(62) +where +mis(Sd) = 2πd/2 +Γ(d/2) +is the measure of Sd, Γ(x) being the Euler gamma function. The previous integral is con- +vergent for any d ∈ N. Observe that we get a dependence on the temperature proportional +18 + +to T d−1 +ac . Regarding the second order correction we observe that +gMEP +2 += −1 +8 +eξ +(eξ − 1)3 +� +c2(eξ + 1) +k2 +BT(x, t)4|q|2 +� +δij|q|2 +� +2 ∂T +∂xi +∂T +∂xj +− T +∂2T +∂xi∂xj +� ++qiqj +� +T +∂2T +∂xi∂xj +− 3 ∂T +∂xi +∂T +∂xj +�� +− +c3(e2ξ + 4eξ + 1) +3k3 +B|q|(eξ − 1)T(x, t)5 +� +(δij|q|2 − qiqj) ∂T +∂xi +∂T +∂xj +− qiqjT +∂2T +∂xi∂xj +�� += −1 +8 +c2eξ +(eξ − 1)3 +� (eξ + 1) +k2 +BT(x, t)4 +� +2|∇xT|2 − T∆xT + ninj +� +T +∂2T +∂xi∂xj +− 3 ∂T +∂xi +∂T +∂xj +�� +− c(e2ξ + 4eξ + 1)|q| +3k3 +B(eξ − 1)T(x, t)5 +� +(δij − ninj) ∂T +∂xi +∂T +∂xj +− ninjT +∂2T +∂xi∂xj +�� +with now ξ = c|q|/kBT. Therefore, the second order correction to the heat flux is given by +Q(2) +µ = −τ Q∇xJ(2) +µ +with +J(2) = +1 +(2π)d +� +Rd c ⊗ ch(q)gMEP +2 +dq = +c2 +(2π)d +� +Rd nhnkh(q)gMEP +2 +dq eh ⊗ ek := J(2) +hk eh ⊗ ek +(e1, e2, · · · , ed) being an orthonormal basis of Rd. +By taking into account the well-known formulas +� +Ω +nhnkdΩ = mis(Sd) +d +δij, +� +Ω +ninjnhnkdΩ = mis(Sd) +d(d + 2)(δijδhk + δihδjk + δikδjk) +and that +� +∞ +0 +h(q)eξ(eξ + 1) +(eξ − 1)3 +qd−1dq = c +�kBT +c +�d+1 � +∞ +0 +eξ(eξ + 1) +(eξ − 1)3 ξddξ := c +�kBT +c +�d+1 +I1(d), +� +∞ +0 +h(q)eξ(e2ξ + 4eξ + 1) +(eξ − 1)4 +qddq = c +�kBT +c +�d+2 � +∞ +0 +eξ(e2ξ + 4eξ + 1) +(eξ − 1)4 +ξd+1dξ +:= c +�kBT +c +�d+2 +I2(d), +the components of J(2) read +J(2) +hk = − +c3 +8(2π)d +mis(Sd) +d +1 +k2 +BT 4(x, t) +�kBT +c +�d+1 +�� +(2|∇xT|2 − T∆xT)I1(d) − 1 +3 +∂T +∂xi +∂T +∂xj +δijI2(d) +� +δhk + +�� +T +∂2T +∂xi∂xj +− 3 ∂T +∂xi +∂T +∂xj +� +I1(d) ++ 1 +3 +� ∂T +∂xi +∂T +∂xj ++ T +∂2T +∂xi∂xj +� +I2(d) +� +(δijδhk + δihδjk + δikδjk) +� +. +19 + +The integrals I1(d) and I2(d) are divergent in the cases d = 1 and d = 2. +As a +consequence, the quantum corrections are valid only in the bulk (d = 3) case where I1(3) = +π2, I2(3) = 4π2. This peculiarity is physically related to the density of states and the form +of the energy dispersion relations. +Conclusions and acknowledgements +The Wigner equation for phonons has been written in the case of a generic dispersion +relation. Moment equations have been deduced and closed by QMEP. Under a long-time +scaling an expression for the heat flux with a nonlinear quantum correction has been +obtained. The model is suited for the investigation in modern micro-devices where the +enhanced miniaturization makes thermal effects more and more relevant. +The authors acknowledge the support from INdAM (GNFM) and from Universit`a degli +Studi di Catania, Piano della Ricerca 2020/2022 Linea di intervento 2 ”QICT”, V. D. +Camiola acknowledges the financial support from the project AIM, Mobilit`a dei Ricercatori +Asse I del PON R & I 2014-2020, proposta AIM1893589. +Declarations +Conflicts of interest/Competing interests +The authors declare they have no financial interests. +Data availability +Data sharing is not applicable to this article as no new data were created or analyzed in +this study. +References +[1] Mascali, G., Romano, V.:“Charge Transport In Graphene Including Thermal Effects”. +SIAM J. APPL. MATH, Vol. 77, No. 2, pp. 593-613, (2017) Society for Industrial and +Applied Mathematics. doi: 10.1137/15M1052573. +[2] Morandi, O.; Sch¨urrer, F. “Wigner model for quantum transport in graphene”. J. +Phys. A Math. Theor. 2011, 44, 265301. +[3] Muscato, O.;Wagner,W. “A class of stochastic algorithms for theWigner equation”. +SIAM J. Sci. 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Journal of Statistical Physics, Vol. 112, Nos. 3/4, 587-628 (2003). +[15] Degond, P., M´ehats, F., Ringhofer, C.: +“Quantum Energy-Transport and Drift- +Diffusion Models”. Journal of Statistical Physics, Vol. 118, Nos. 3/4, 625-667 (2005). +[16] Barletti, L.: “Hydrodynamic equations for electrons in graphene obtained from the +maximum entropy principle”, J. Math. Phys., 55, 083303, 21 pp (2014). +[17] Barletti, L., Cintolesi, C.: +“Derivation of Isothermal Quantum Fluid Equations +with Fermi-Dirac and Bose-Einstein Statistics”. J Stat Phys 148,353–386 (2012). doi: +10.1007/s10955-012-0535-5. +[18] Luca, L., Romano, V.: +“Quantum corrected hydrodynamic models for charge +transport +in +graphene”. +Annals of +Physics, +Volume +406, +pp. +30-53 +(2019) +doi:10.1016/j.aop.2019.03.018. +[19] Camiola, V.D., Mascali, G., Romano, V.: “Charge Transport in Low Dimensional +Semiconductor Structures, The Maximum Entropy Approach”. Springer, (2020). +21 + +[20] Coco, M., Romano, V.: “Simulation of Electron-Phonon Coupling and Heating Dy- +namics in Suspended Monolayer Graphene Including All the Phonon Branches”, J. +Heat Transfer 140 092404-1 (2018). +[21] Srivastava, G. P.: “The Physics of Phonons”, Taylor and Francis (1990). +[22] Hall, B. C.: Quantum theory for Mathematicians, Springer (2013). +[23] Folland, G. B.: “Harmonic Analysis in Phase Space”, Princeton University Press +(1989). +[24] Frommlet, F., Markowich, P., Ringhofer, C.: “A Wignerfunction Approach to Phonon +Scattering”, VLSI Design 9 (4) 339-350 (1999). +[25] Nier, F.: “A variational formulation of Schr¨odinger-Poisson systems in dimension d ≤ +3”, Comm. in Partial Differential equations 18:7-8 1125-1147. +[26] Junk, M.: “Domain of definition of Levermore’s five moment system, J. Stat. Physics +93 1143-1167 (1998). +[27] Luca, L., Romano, V.: “Hydrodynamical models for charge transport in graphene +based on the Maximum Entropy Principle: the case of moments based on energy +powers”. Atti della Accademia Peloritana dei Pericolanti - Classe di Scienze Fisiche, +Matematiche e Naturali, [S.l.], p. A5, (2018) doi: 10.1478/AAPP.96S1A5. +22 + diff --git a/ltAyT4oBgHgl3EQfk_i4/content/tmp_files/load_file.txt b/ltAyT4oBgHgl3EQfk_i4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e3112ffe706fed450813906d1c19403d60e0d4a --- /dev/null +++ b/ltAyT4oBgHgl3EQfk_i4/content/tmp_files/load_file.txt @@ -0,0 +1,513 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf,len=512 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='00445v1 [math-ph] 1 Jan 2023 Wigner equations for phonons transport and quantum heat flux V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Camiolaa, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Romanoa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Vitanzaa , aDipartimento di Matematica e Informatica - Universit`a degli Studi di Catania Abstract Starting from the quantum Liouville equation for the density operator and applying the Weyl quantization, Wigner equations for the longitudinal and transversal optical and acous- tic phonons are deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The equations are valid for any solid, including 2D crystals like graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' With the use of Moyal’s calculus and its properties the pseudo-differential op- erators are expanded up to the second order in ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The phonon-phonon collision operators are modelled in a BGK form and describe the relaxation of the Wigner functions to a local equilibrium function, depending on a local equilibrium temperature which is definite according to [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' An energy transport model is obtained by using the moment method with closures based on a quantum version of the Maximum Entropy Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' An explicit form of the thermal conductivity with quantum correction is obtained under a suitable scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Introduction The use of the Wigner function is one of the most promising ways to study quantum transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Its main advantage is that a description similar to the classical or semiclassical transport is obtained in a suitable phase-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The mean values are expectation values with respect to the Wigner function as if the latter were a probability density and the semiclassical limit of the Wigner transport equation recovers, at least formally, the Boltz- mann transport one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' There is a huge body of literature regarding the Wigner equation and the way to numerically solve it (see for example [2, 3, 4] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' However, the most of the works on the subject consider a quadratic dispersion relation for the en- ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Instead, for several materials like semiconductors or semimetal, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' graphene, other dispersion relations must be considered [5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' From the Wigner transport equation quantum hydrodynamical models have been obtained in [8] for charge transport in silicon in the case of parabolic bands, while in [9] the same has been devised for electrons moving in graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The enhanced miniaturization of electron and mechanical devices makes the thermal effects increasingly relevant [10, 11] requiring the use of physically accurate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' At ki- netic level a good description is that based on the semiclassical Peierls-Boltzmann equation for each phonon branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' However, for typical lengths smaller than the phonon mean-free path also quantum effects must be considered (see for example [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The Wigner equation is a natural approach that better reveal the wave nature of phonons in such circumstances, Preprint submitted to NLS January 3, 2023 gives the Peierls-Boltzmann equation as semiclassical limit and still keeps the structure of a kinetic formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In this work, the focus is on the acoustic and optical phonons dynamics with a general dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In order to get insights into the quantum corrections, moment equations are deduced from the corresponding Wigner equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' As in the classical case, one is led to a system of balance equations that are not closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' So, the well-known problem of getting closure relations arises, that is the issue to express the additional fields appearing in the moment equations in terms of a set of fundamental variables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' the phonon energy density and energy flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' A sound way to accomplish this task is resorting to a quantum formulation of the maximum entropy principle [12] (hereafter QMEP), formulated for the first time by Jaynes [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Recently, a more formal theory has been developed in a series of papers [14, 15] with several applications, for example for charge transport in semiconductors [8, 16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The interested reader is also referred to [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' We apply QMEP to the Wigner equations assuming the energy density and the energy flux for each species of phonons as basic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' By expanding up to the second order in ℏ, quantum corrections to the semiclassical case [1] are deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In particular, in a long time scaling an asymptotic expression for the heat flux is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The latter consists of a Fourier-like part with a highly nonlinear second order correction in the temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Explicit formulas for acoustic phonons in the Debye approximation are written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The plan of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In section 2, the semiclassical phonon transport is summarized while in section 3 we write down the Wigner equations for acoustic and optical phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Section 4 is dedicated to deducing the moment equations whose closure relations are achieved by QMEP in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In the last section a definition of local temperature is introduced and an asymptotic expression of the quantum correction to the heat flux is drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Semiclassical phonon transport In a crystal lattice the transport of energy is quantized in terms of quasi-particles named phonons which are present with several branches and propagation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The latters vary from a material to another but in any case they are grouped in acoustic and optical phonon branches which, in turn, can oscillate in the longitudinal or transversal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The complete dispersion relations can be usually obtained by a numerical approach in the first Brillouin zone (FBZ) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' However, in the applications some standard approximations are often adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' For the acoustic phonons, the Debye approximation for the dispersion relation εs(q) is usually assumed, εs(q) = ℏωs(q) = ℏcs|q|, s = LA, TA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' LA stands for longitudinal acoustic while TA for transversal acoustic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' cs is the sound speed of the s-branch and ℏ denotes the reduced Planck constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Consistently, the first Brillouin zone is extended to Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Here d is the dimension of the space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' d = 3 for bulk crystal while d = 2 for graphene or similar 2D material like dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' For the longitudinal optical (LO) and the transversal optical (TO) phonon, the Einstein dispersion relation, ℏωs ≈ const, with s = LO, TO, the phonon angular frequency, is usually 2 adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Note that under such an assumption, the group velocity of the optical phonons is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In some peculiar materials like graphene, it is customary to introduce also a fictitious branch called K-phonons constituted by the phonons having wave vectors close to the Dirac points, K or K′, in the first Brillouin zone (taking the origin in the center Γ of FBZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Also in this case the Einstein approximation is used on account of the limited variability of the phonon energy near those points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Moreover, in graphene the phonons are classified as in-plane, representing vibration parallel to the material, and out of plane, representing vibrational mode orthogonal to the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The LA, TA, LO, TO and K phonons are in plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The out of plane phonons belong to the acoustic branch and are named ZA phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' For them a quadratic dispersion relation is a good approximation εZA(q) = ℏωZA(q) = ℏα|q|2, where α = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='2 × 107m2/s (see [20]) In the following, instead of the wave vector q we will use the phonon moment ℏq, but we retain the same character for sake of simplifying the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' So that εs(q) = cs|q|, s = LA, TA and εZA(q) = α|q|2, where α = α/ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The thermal transport is usually described by macroscopic models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' the Fourier one, those based on the Maximum Entropy methods [19] or on phenomenological description [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' A more accurate way to tackle the question is to resort to semiclassical transport equa- tions, the so-called Peierls-Boltzmann equations, for each phonon branch for the phonon distributions fµ(t, x, q) ∂fµ ∂t + cµ · ∇xfµ = Cµ, µ = LA, TA, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' , (1) where cµ = ∇q (ℏωµ) is the group velocity of the µth phonon specie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The phonon collision term Cµ splits into two terms Cµ = Cµ µ + � ν,ν̸=µ Cν µ, ν = LA, TA, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (2) Cµ µ describes the phonon interaction within the same branch while Cν µ describes the phonon-phonon interaction between different species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' To deal with the complete expres- sions of the Cµ’s is a very complicated task even from a numerical point of view [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' So, they are usually simplified by the Bhatnagar-Gross-Krook (BGK) approximation Cµ = −fµ − f LE µ τµ(q) , which mimics the relaxation of each phonon branch towards a common local equilibrium condition, characterised by a local equilibrium temperature TL that is the same for each phonon population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The local equilibrium phonon distributions are given by the Bose-Einstein distributions f LE µ = � eℏωµ/kBTL − 1 �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (3) 3 and the functions τµ are the phonon relaxation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Additional BGK terms can be added to include the interaction between pairs of different branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' If we know the phonon distributions fµ’s, we can calculate the average phonon energy densities Wµ = 1 (2π)2 � B ℏωµ fµ dq, (4) and the expectation value of any function ψ(q) Mψ = 1 (2π)2 � B ψ(q) fµ dq, for example the energy flux if one takes ψ(q) = ℏωµvµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The modern devices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' the electron ones like double gate MOSFETs (see [19]), are undergoing more and more miniaturization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' This implies that the characteristic scales are of the same order as the typical lengths where quantum effects become more and more relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Therefore, quantum effects must be included and the semiclassical phonon transport equations must be replaced by a more accurate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Among the possible approaches, that one based on the Wigner equation has the advantage to be formulated in a phase-space, allowing us to guess the feature of the solutions in analogy with the semiclassical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' A huge literature has been devoted to the application of the Wigner equations to charge transport (see [2, 3, 4]) but a limited use has been made for phonon transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In the next sections a transport model, based on the Wigner quasi distribution, will be devised for phonon transport in nano-structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Phonon Wigner functions The main point of our derivation is the kinetic description of a one-particle quantum statistical state, given in terms of one-particle Wigner functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Let us now briefly recall the basic definitions and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' A mixed (statistical) one-particle quantum state for an ensemble of scalar particles in Rd is described by a density operator ˆρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' a bounded non-negative operator with unit trace, acting on L2(Rd, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Given the density operator ˆρ on L2(Rd, C), the associated Wigner function, w = w(x, q), (x, q) ∈ R2d, is the inverse Weyl quantization of ˆρ, w = Op−1 ℏ (ˆρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (5) We recall that the Weyl quantization of a phase-space function (a symbol) a = a(x, q) is the (Hermitian) operator Opℏ(a) formally defined by [22] Opℏ(a)ψ(x) = 1 (2πℏ)d � R2d a �x + y 2 , q � ψ(y)ei(x−y)·q/ℏdy dq (6) for any ψ ∈ L2(Rd, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The inverse quantization of ˆρ can be written as the Wigner transform w(x, q) = � Rd ρ(x + ξ/2, x − ξ/2)eiq·ξ/ℏdξ, (7) 4 of the kernel ρ(x, y) of the density operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The dynamics of the time-dependent phonon Wigner functions gµ(x, q, t), µ = LA, TA, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' steams directly from the dynamics of the corresponding density operator ˆρµ(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' from the Von Neumann or quantum Liouville equation iℏ∂tˆρµ(t) = [ ˆHµ, ˆρµ(t)] := ˆHµˆρµ(t) − ˆρµ(t) ˆHµ, (8) where ˆHµ denotes the Hamiltonian operators of the µth phonons and [·, ·] the commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' If hµ = Op−1 ℏ ( ˆHµ) is the symbol associated with ˆHµ, then, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='s (8), we obtain the Wigner equation for each phonon species iℏ∂tgµ(x, q, t) = {hµ, gµ(x, q, t)}# := hµ#gµ(x, q, t) − gµ(x, q, t)#hµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (9) With the symbol # we have denoted the Moyal (or twisted) product which translates the product of operators at the level of symbols according to a#b = Op−1 ℏ (Opℏ(a)Opℏ(b)), (10) for any pair of symbols a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Here, we do not tackle the analytical issues which guarantee the existence of the previous relations but limit ourselves to the remark that if two operators are in the Hilbert-Schmidt class, that is the trace there exists and it is not negative and bounded, then the product is still Hilbert-Schmidt and the Moyal calculus is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In the sequel, we will suppose that such conditions are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The Moyal product, under suitable regularity assumptions (see [23]), possesses the following formal semiclassical expansion a#ℏb(x, q) = � α,β �iℏ 2 �|α|+|β| (−1)|β| α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' ∂α x∂β pa(x, q)∂β x∂α qb(x, q) (11) where α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=', αd) ∈ Nd is a multi-index, |α| = � i αi, α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' = � i αi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=', ∂α x = � i ∂αi xi and similarly for β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The expansion (11) can be rewritten as a#ℏb(x, q) = ∞ � n=0 ℏna#nb (12) where a#nb(x, q) = � α,β,|α|+|β|=n � i 2 �n (−1)|β| α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' ∂α x∂β qa(x, q)∂β x∂α qb(x, q) (13) The first terms of (13) read a#0b = ab, (14) a#1b = i 2(∇xa · ∇qb − ∇qa · ∇xb), (15) a#2b = −1 8(∇2 xa : ∇2 qb − 2∇x∇qa : ∇q∇xb + ∇2 qa : ∇2 xb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (16) 5 where ∇2 denotes the Hessian matrix and : the contracted product of tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' It is easy to see that a#nb(x, q) = (−1)nb#na(x, q), that is the operation #n is commutative (respectively anticommutative) when n is even (respectively odd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' If we neglect, for the moment, the phonon-phonon interactions, the Hamiltonian symbol for each phonon branch is given by hµ(q) = εµ(q) µ = LA, TA, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (17) By using the Moyal calculus, one can expand the second members of the previous Wigner equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Up to first order in ℏ2, we have ∂tgµ(t) + S[hµ]gµ(t) = 0, µ = LA, TA, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' , (18) where1 S[hµ]gµ(x, q, t) := cµ · ∇xgµ(x, q, t) − ℏ2 24 ∂3 qhµ(q) ∂qi∂qj∂qk ∂3 xgµ(x, q, t) ∂xi∂xj∂xk + O(ℏ4)) µ = LA, TA, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (19) The previous equations describe only ballistic transport and include only the harmonic contribution to the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In order to describe also intra and inter-branch phonon- phonon interactions, an additional anharmonic term ˆHint encompassing the high order correction to the Hamiltonian operator must be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' So doing, one gets the so-called Wigner-Boltzmann equation ∂tgµ(x, q, t) + S[hµ]gµ(x, q, t) = Cµ(x, q, t), µ = LA, TA, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' , (20) In the quantum case the expression of Cµ is rather cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' For electron transport in semiconductors the interested reader can see [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In certain regimes it is justified to retain the same form of the semiclassical collision operator as the semiclassical case [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Here, we adopt a quantum BGK approach and model the collision terms as Cµ = −(gµ − gLE µ ) τµ(q) , µ = LA, TA, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (21) where gLE µ are now Wigner functions of local equilibrium which will be defined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The equation (20) along with the expression (21) for the collision operator represents our starting point for the phonon transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Note that for the optical phonons under the Einstein approximation for the energy bands one has formally the same transport equation as the semiclassical case because the group velocity vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 1Summation over repeated indices is understood from 1 to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 6 An alternative derivation of (20) can be obtained by explicitly writing the von Neumann equation (see [18, 19] for the details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' One obtains S[hµ]gµ(t) = i ℏ(2π)d � Rd x′×Rdν � ε � q + ℏ 2ν, t � − ε � q − ℏ 2ν, t �� gµ(x′, q, t)e−i(x′−x)·νdx′dν, (22) whose expansion is of course in agreement with the Moyal calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Phonon Moment equations Getting analytical solutions to equations (20)-(21) is a daunting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Therefore, viable approaches are numerical solutions based on finite differences or finite elements [2] or stochastic solutions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' those obtained with a suitable modification of the Monte Carlo methods for the semiclassical Boltzmann equation [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' However, it is possible to have simpler, even if approximate, models resorting to the moment method for the expectation values of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In fact, it is well known that, although not positive definite, the Wigner function is real and the expectation values of an operator can be formally obtained as an average of the corresponding symbol with respect to gµ(x, q, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' So, for any regular enough weight function ψ(q), let us introduce the short notation < ψ > (x, t) := 1 (2π)d � Rd ψ(q)gµ(x, q, t)dq, (23) which represents a partial average with respect to the phonon moment q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' More in general, if a = a(x, q) is a smooth symbol then it is possible to prove that the expectation of the (hermitian) operator A = Opℏ(a) satisfies2 E[A] = tr(ˆρA) = � R2d ρ(x, y)kA(x, y)dxdy = 1 (2π)d � R2d a(x, q)gµ(x, q, t)dxdq = � Rd < a > (x, t)dx, where kA(x, y) is the kernel of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' We want to consider a minimum set of moments whose physical meaning is well clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In particular, we shall consider the phonon energy and energy flux of each branch Wµ(x, t) =< hµ > (x, t), Qµ(x, t) =< hµcµ > (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (24) Note that the latter is directly related to the heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 2Here we are considering a fixed instant of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 7 The evolution equations for Wµ(x, t) and Qµ(x, t) are obtained by multiplying the relative Wigner equation by hµ(q), and hµ(q)cµ and integrating with respect to q ∂tWµ(x, t) + 1 (2π)d � Rdhµ(q)S[hµ]gµ dq = 1 (2π)d � Rdhµ(q)Cµ dq, ∂tQµ(x, t) + 1 (2π)d � Rdhµ(q)cµS[hµ]gµ dq = 1 (2π)d � Rdhµ(q)cµCµ dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' µ = LA, TA, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (25) We implicitly assume that the resulting integrals there exist, at least in the principal value sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In order to get some global insight from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='s (25), we formally assume the following expansions for each phonon branch3 gµ(x, q, t) = g(0) µ (x, q, t) + ℏ2g(2) µ (x, q, t) + o(ℏ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (26) It is possible to prove, at least formally [6], that the semiclassical Boltzmann equation is recovered from the Wigner equation as ℏ �→ 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Therefore, g(0) µ (x, q, t) can be considered as the solution fµ of the semiclassical transport equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Accordingly, we write Wµ = W (0) µ + ℏ2W (2) µ + o(ℏ2), Qµ = Q(0) µ + ℏ2Q(2) µ + o(ℏ2), (27) where W (0) µ = 1 (2π)d � Rd hµg(0) µ (x, q, t)dq, W (2) µ = 1 (2π)d � Rd hµg(2) µ (x, q, t)dq, Q(0) µ = 1 (2π)d � Rd hµcµg(0) µ (x, q, t)dq, Q(2) µ = 1 (2π)d � Rd hµcµg(2) µ (x, q, t)dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Regarding the moments of the collision terms, only with drastic simplifications analytical expressions can be deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In analogy with the BGK approximation, if an average re- laxation time independent on q is considered, one can expand the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='s (25) up to first order in ℏ2 as a relaxation time terms 1 (2π)d � Rdhµ(q)Cµ dq = −Wµ − W LE µ τ W µ = −W (0) µ − W (0)LE µ τ W µ − ℏ2W (2) µ − W (2)LE µ τ W µ + o(ℏ2), 1 (2π)d � Rdhµ(q)cµCµ dq = −Qµ τ Q µ = −Q(0) µ + ℏ2Q(2) µ τ Q µ + o(ℏ2), where W LE µ = 1 (2π)d � Rdhµ(q)gLE µ dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Note that in the evaluation of the production term of the equations for the energy-fluxes the isotropy of the equilibrium Wigner function has been invoked and therefore QLE µ vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 3The coefficients of the odd powers in ℏ are assumed zero in according to the previous Moyal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 8 The energy and energy-flux relaxation times, τ W µ and τ Q µ respectively, are assumed to depends on the temperature, which will be definite in the next section, of the relative branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Altogether,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' the resulting model is made of the following fluid-type equations \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∂tWµ + ∇x � Qµ − ℏ2 24∂2Tµ � = −W (0) µ − W (0)LE µ τ W µ − ℏ2W (2) µ − W (2)LE µ τ W µ + o(ℏ2) ∂tQµ + ∇x � Jµ − ℏ2 24∂2Uµ � = −Q(0) µ + ℏ2Q(2) µ τ Q µ + o(ℏ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (28) where Jµ = J(0) µ + ℏ2J(2) µ with J(0) µ = 1 (2π)d � Rd cµ ⊗ cµhµ(q)g(0) µ (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t)dq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' J(2) µ = 1 (2π)d � Rd cµ ⊗ cµhµ(q)g(2) µ (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t)dq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' and the complete symmetric tensors Tµ and Uµ have components (Tijk)µ = 1 (2π)d � Rd hµ ∂3hµ(q) ∂qi∂qj∂qk g(0) µ (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t)dq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (Uijkr)µ = 1 (2π)d � Rd(cµ)rhµ(q) ∂3hµ(q) ∂qi∂qj∂qk g(0) µ (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t)dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' If we split into zero and first order in ℏ2, the evolution equations read ∂tW (0) µ + ∇xQ(0) µ = −W (0) µ − W (0)LE µ τ W µ (29) ∂tW (2) µ + ∇xQ(2) µ + 1 (2π)d ∂3 ∂xi∂xj∂xk � Rd hµ(q) 24 g(0) µ ∂3 ∂qi∂qj∂qk hµ(q)dq = −W (2) µ − W (2)LE µ τ W µ , (30) ∂tQ(0) µ + ∇xJ(0) µ = −Q(0) µ τ Q µ , (31) ∂tQ(2) µ + ∇xJ(2) µ + 1 (2π)d ∂3 ∂xi∂xj∂xk � Rd cs hµ(q) 24 g(0) µ ∂3 ∂qi∂qj∂qk hµ(q)dq = −Q(2) µ τ Q µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (32) The zero order equations are the model already investigated in several papers [1, 7] where is proved that it is a hyperbolic system of conservation law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' So, finite propagation speed of disturbances in energy is guaranteed to overcome the well-known paradox of the classical 9 Fourier law for the heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' However, the first order corrections in ℏ2 introduce dispersive terms and it seems that in a quantum regime the requirement of finite propagation speed of the thermal effects cannot be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' On the other hand, this is not surprising since the nonlocal character of the quantum evolution equations can lead to energy propagation without a bounded speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' QMEP for the closure relations The evolution equations (29)-(32) do not form a closed system of balance laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' If we assume the energies Wµ and the energy-fluxes Qµ as the main fields, in order to get a set of closed equations we need to express the additional fields Jµ, Tµ and Uµ as functions of Wµ and Qµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' A successful approach in a semiclassical setting is that based on the Maximum Entropy Principle (MEP) (see also [19] for a complete review) which is based on a pioneering paper of Jaynes [12, 13] who also proposed a way to extend the approach to the quantum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The MEP in a quantum setting has been the subject of several papers [8, 14, 15, 16, 17] with several applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' to charge transport in graphene [1, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Here we will use such an approach for phonon transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The starting point is the entropy for the quantum system under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In [18] the authors have employed the Von-Neumann entropy which, however, does not take into account the statistical aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Therefore, we take as entropy a generalization of the classical one for bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Let us introduce the operator s(ˆρµ) = −kB[ˆρµ ln ˆρµ − (1 + ˆρµ) ln(1 + ˆρµ)], (33) which must be intended in the sense of the functional calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Here kB is the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The entropy of the µ-th phonon branch reads S(ˆρµ) = Tr{s(ˆρµ)} which can be viewed as a quantum Bose-Einstein entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' According to MEP, we estimate ˆρµ with ˆρMEP µ which is obtained by maximizing S(ˆρµ) under the constraints that some expectation values have to be preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In the semiclas- sical point case, one maximizes the entropy preserving the values of the moments we have taken as basic field variables (Wµ(x, t), Qµ(x, t)) = 1 (2π)d � Rd ψµ(q)gµ(x, q, t)dq = 1 (2π)d � Rd ψµ(q)gMEP µ (x, q, t)dq, (34) where ψµ(q) = (hµ(q), cµhµ(q)) (35) is the vector of the weight functions and gMEP µ is the Wigner function associated with ˆρMEP µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In the previous relations the time t and position x must be considered as fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 10 The quantum formulation of MEP is given in terms of expectation values E1(t) = tr {ˆρµOpℏ(hµ(q))} (t), E2(t) = tr {ˆρµOpℏ(cµhµ(q))} (t), as follows: for fixed t ˆρMEP µ = argument max S(ˆρµ) (36) under the constraints tr{ˆρMEP µ Opℏ(hµ(q))} = E1(t), tr{ˆρMEP µ Opℏ(cµhµ(q))} = E2(t), (37) in the space of the Hilbert-Schmidt operators on L2(Rd, C) which are positive, with trace one and such that the previous expectation values there exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Note that we are applying the maximization of the entropy for each phonon branch separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In other words, we are requiring the additivity of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' If we introduce the vector of the Lagrange multipliers ηµ = (η0µ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' η1µ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (38) the vector of the moments m[ρµ](x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t) := mµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t) = 1 (2π)d � Rd ψµ(q)gµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t)dq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (39) and the vector of the moments which must be considered as known Mµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t) := (Wµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Qµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (40) the constrained optimization problem (36)-(37) can be rephrased as a saddle-point problem for the Lagrangian Lµ(ˆρµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' ηµ) = S(ˆρµ) − � Rd ηµ · (mµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t) − Mµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t)) dx = S(ˆρµ) − tr {ˆρµOpℏ(ηµ · hµ(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' cµhµ(q))} + � Rd ηµ · Mµ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t) dx (41) in the space of the admissible ˆρµ and smooth function ηµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' If the Lagrangian Lµ(ˆρµ, ηµ) is Gˆateaux-differentiable with respect to ˆρµ, the first order optimality conditions require δLµ(ˆρµ, ηµ)(δˆρ) = 0 for each Hilbert-Schmidt operators δˆρ on L2(Rd, C) which is positive, with trace one and such that the previous expectation values there exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The existence of the first order Gˆateaux derivative is a consequence of the following Lemma (for the proof see [25];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' an elementary proof in the case of discrete spectrum is given in [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 11 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' If r(x) is a continuously differentiable increasing function on R+ then tr{r(ˆρ)} is Gˆateaux-differentiable in the class of the Hermitian Hilbert-Schmidt positive operators on L2(Rd, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The Gˆateaux derivative along δρ is given by δtr{r(ˆρ)}(δˆρ) = tr {r′(ˆρ)δˆρ} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (42) The extremality conditions for the unconstrained minimization problem (36)-(37) are similar to that of the semiclassical case, as expressed by the following lemma (see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The first order optimality condition for the minimization problem (36)-(37) is equivalent to ˆρµ = (s′)−1(Opℏ(ηµ · ψµ)) (43) where (s′)−1 is the inverse function of the first derivative of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' By applying Lemma 1, the Gˆateaux derivative of the Lagrangian is given by δLµ(ˆρµ, ηµ)(δˆρ) = tr {(s′(ˆρµ) − Opℏ(ηµ · ψµ)) δˆρ} ∀δˆρ perturbation in the class of the Hermitian Hilbert-Schmidt positive operators on L2(Rd, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' This implies s′(ˆρµ) = Opℏ(ηµ · ψµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' □ Since the function s(x) is concave, s′(x) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Explicitly we have (s′)−1(z) = 1 ez/kB − 1 and the operator solving the first order optimality condition reads ˆρ∗ µ = (s′)−1(Opℏ(ηµ · ψµ)) = 1 eOpℏ(ηµ·ψµ) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (44) Moreover, such an operator is a point of maximum for the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' □ Now, to complete the program we have to determine, among the smooth functions, the Lagrange multipliers ηµ by solving the constraint tr {ˆρµOpℏ(ηµ · (hµ(q), cµhµ(q))} − � Rd ηµ · Mµ(x, t) dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (45) If such an equation has a solution η∗ µ, altogether the MEP density operator reads ˆρMEP µ = 1 exp � Opℏ � η∗ 0µ(x, t)hµ(q) + η∗ 1µ(x, t) · cµhµ(q) �� − 1, (46) where we have rescaled the Lagrange multipliers including the factor 1/kB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 12 To determine conditions under which the equation (45) admits solutions is a very dif- ficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Even in the semiclassical case there are examples (see [26]) of sets of moments that cannot be moments of a MEP distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' We will directly find out the solution at least up to first order in ℏ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Once the MEP density function has been determined, the MEP Wigner function is given by gMEP µ (x, q, t) = Op−1 ℏ (ˆρMEP µ ) which can be used to get the necessary closure relations by evaluating the additional fields with gµ replaced by gMEP µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' We remark that the constraints (45) can be more conveniently expressed as 1 (2π)d � R2d ηµ · ψµ(x, t)gMEP µ (x, q, t) dq dx − � Rd ηµ · Mµ(x, t) dx = 0 and indeed we will require, in analogy with the semiclassical case, the stronger conditions 1 (2π)d � Rd ψµ(x, t)gMEP µ (x, q, t) dq = Mµ(x, t), where the Lagrange multipliers enter through gMEP µ (x, q, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Determination of the Lagrange Multipliers For the sake of making lighter the notation, let us consider a single branch and drop the index µ in the Wigner function in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' We look formally for a solution in powers of ℏ gMEP = gMEP 0 + ℏgMEP 1 + ℏ2gMEP 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (47) firstly without taking into account the dependence of the Lagrange multipliers on ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Of course, on account of the properties of the Weyl quantization, gMEP 0 is equal to the semiclassical counterpart [22] gMEP 0 = 1 exp [η0(x, t)h(q) + η1(x, t) · ch(q)] − 1 In order to determine the higher order terms gMEP k , k ≥ 1, given a symbol a(x, q) let us introduce the so-called quantum exponential Exp(a) defined as Exp(a) = Op−1 ℏ [exp(Opℏ(a))] which can be expanded as Exp(a) = Exp0(a) + ℏExp1(a) + ℏ2Exp2(a) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (48) Proposition Let a(x, p) be a smooth symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Then the following expansion is valid Exp(a) = exp(a) − ℏ2 8 exp(a) � ∂2a ∂xi∂xj ∂2a ∂pi∂pj − ∂2a ∂xi∂pj ∂2a ∂pi∂xj + 1 3 ∂2a ∂xi∂xj ∂a ∂pi ∂a ∂pj −2 3 ∂2a ∂xi∂pj ∂a ∂pi ∂a ∂xj + 1 3 ∂2a ∂pi∂pj ∂a ∂xi ∂a ∂xj � + O(ℏ4), (49) 13 where Einstein’s convention has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' □ The proof can be found for example in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' By using what is proved in [16], we have gMEP 2n+1 = 0, n ≥ 0, (50a) gMEP 2n = − n−1 � m=0 � k+l+m=n Exp2k(ξ)#2lgMEP 2m eξ − 1 , n ≥ 1 (50b) where #2l are the even terms of the Moyal product expansion and ξ = η0µ(x, t)h(q) + η1(x, t) · ch(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In particular gMEP 1 = 0 and gMEP 2 = −1 8 eξ (eξ − 1)3 � (eξ + 1) � ∂2ξ ∂xi∂xj ∂2ξ ∂qi∂qj − ∂2ξ ∂xi∂qj ∂2ξ ∂qi∂xj � −(e2ξ + 4eξ + 1) 3(eξ − 1) � ∂2ξ ∂xi∂xj ∂ξ ∂qi ∂ξ ∂qj − 2 ∂2ξ ∂xi∂qj ∂ξ ∂qi ∂ξ ∂xj + ∂2ξ ∂qi∂qj ∂ξ ∂xi ∂ξ ∂xj �� Therefore, up to first order in ℏ2 we have gMEP µ = gMEP 0 + ℏ2gMEP 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' and the constraints for each phonon branch read W = 1 (2π)d � Rd h(q) eξ − 1dq + ℏ2 1 (2π)d � Rd h(q)gMEP 2 dq, (51) Q = 1 (2π)d � Rd ch(q) eξ − 1dq + ℏ2 1 (2π)d � Rd ch(q)gMEP 2 dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (52) The previous equations form a nonlinear system of PDEs for the Lagrange multipliers whose analytical solution seems very difficult to get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Indeed, the situation is even more cumbersome because in a numerical scheme the inversion of the constraints should be performed at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' A viable strategy is to use the Lagrange multipliers as field variables by rewriting the evolution equations (28) in the form ∇ηW ∂ ∂tηT + d � i=1 � ∇ηQi ∂ ∂xi ηT − ℏ2 24∇η � ∇x∂2 xT � ∂ ∂xi ηT � = −W − W LE τ W , (53) ∇ηQi ∂ ∂tηT + d � j=1 � ∇ηJ ∂ ∂xj ηT − ℏ2 24∇η � ∂2 xU � ∂ ∂xj ηT � = − Qi τ Q , (54) 14 getting a highly nonlinear system of PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Note that both ∇ηW and ∇ηQi contain space derivatives of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' A further simplification can be obtained by expanding the Lagrange multipliers as η = η(0) + ℏ2η(2) + o(ℏ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Therefore, the basic fields are also expanded with respect to ℏ2 W = W (0) + ℏ2W (2) + o(ℏ2), Q = Q(0) + ℏ2Q(2) + o(ℏ2) where W (0) = 1 (2π)d � Rd h(q) eξ(0) − 1dq, W (2) = − 1 (2π)dη(2) · � Rd eξ(0) h(q)ψ � eξ(0) − 1 �2dq + 1 (2π)d � Rd h(q)gMEP 2 (η(0))dq, Q(0) i = 1 (2π)d � Rd cih(q) eξ(0) − 1dq, Q(2) i = − 1 (2π)dη(0) · � Rd ciψeξ(0)h(q) (eξ(0) − 1)2 dq + 1 (2π)d � Rd cih(q)gMEP 2 (η(0))dq, with ξ(0) = η(0) · ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The balance equations become ∇η(0)W (0) ∂ ∂t(η(0))T + d � i=1 � ∇η(0)Q(0) ∂ ∂xi (η(0))T � = −W (0) − W (0)LE τ W (55) ∇η(0)Q(0) i ∂ ∂t(η(0))T + d � i=1 � ∇η(0)J(0) ∂ ∂xj (η(0))T � = −Q(0) i τ Q , (56) ∂tW (2) + ∇xQ(2) + 1 (2π)d ∂3 ∂xi∂xj∂xk � Rd h(q) 24 gMEP 0 (η(0)) ∂3 ∂qi∂qj∂qk h(q)dq = −W (2) − W (2)LE τ W , (57) ∂tQ(2) + ∇xJ(2) + 1 (2π)d ∂3 ∂xi∂xj∂xk � Rd ch(q) 24 gMEP 0 (η(0)) ∂3 ∂qi∂qj∂qk h(q)dq = −Q(2) τ Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (58) We observe that the equations (55)-(56) decouple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Once they are solved, one can get the second order term of the Lagrange multipliers from (57)-(58) which form a linear system for η(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' This is rather beneficial from a computational point of view Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' At zero order in ℏ2 the map η �→ M(η) is (locally) invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The equations (55)-(56) form a symmetric hyperbolic system of balance laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The proofs can be found in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Local equilibrium temperature and heat conductivity The concept of temperature out of equilibrium is a subtle topic and still a matter of debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In the case of charge transport in semiconductors often the phonons are considered as a thermal bath and under some reasonable assumptions one can hypothesize that the electrons are in thermal equilibrium with the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In general if the dynamics of the phonons must be included, a thermal bath for these does not exist, unless a thermostated system is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Therefore, we need to introduce a local equilibrium temperature for the overall phonon system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In statistical mechanics, one of the most reasonable and adopted ways to generalize the concept of temperature in a non-equilibrium state is that of relating it to the Lagrange multipliers associated with the energy constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' For the phonon transport in graphene, an approach based on the Lagrange multipliers was followed in [1] (which the interested reader is referred to for the details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Let us recall here the main features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' At equilibrium, the phonon temperatures and the corresponding Lagrange multipliers are related by kB Tµ(x) = 1 η0,µ(x) = 1 η(0) 0,µ(x) − ℏ2 η(2) 0,µ(x) (η(0) 0,µ(x))2 + o(ℏ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' If we assume that such relations hold, even out of equilibrium, the definition of the local temperature can be given in terms of the Lagrangian multipliers as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The local temperature of a system of two or more branches of phonons is TLE := 1 kBηLE 0 (x), where ηLE 0 (x) is the common Lagrange multiplier that the occupation num- bers of the branches, taken into account, would have if they were in the local thermodynamic equilibrium corresponding to their total energy density, that is, the following: W(ηLE 0 (x)) := � µ Wµ(η0,µ(x)) = � µ Wµ(ηLE 0 (x)), (59) where the sum runs over all the phonon branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' At global equilibrium the temperature is constant T = ¯T and the Wigner function reduced to the Bose -Einstein distribution gµ = � ehµ(q)/kB ¯T − 1 �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (60) with the same temperature for each phonon branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Let us consider a small perturbation δT(x) of the temperature in the sense that δ(x)/ ¯T ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' We can expand gMEP µ in powers of δ(x)/ ¯T gMEP µ = � ehµ(q)/kB ¯T − 1 �−1 + � ehµ(q)/kB ¯T − 1 �−2 ehµ(q)/kB ¯T hµ(q) kB ¯T δT(x) ¯T +ℏ2 ¯T ∂gMEP 2,µ ( ¯T) ∂T δT ¯T + o � ℏ2δT ¯T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 16 We observe that typically the relaxation energy relaxation time is much longer than the energy-flux relaxation times, that is τ Q ≪ τ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In a long time scaling, much longer than τ Q, we get Qµ = −τ Q � ∇xJµ + ℏ2 (2π)d ∂3 ∂xi∂xj∂xk � Rd ch(q) 24 gMEP 0,µ (η(0)) ∂3 ∂qi∂qj∂qk h(q)dq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' (61) The relation between the Lagrange multipliers and the basic fields, as seen, can hardly be inverted analytically but a numerical procedure is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' However, if we consider a situation where the system is not too far from the equilibrium an expansion of the Lagrange multipliers around the equilibrium state can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' At equilibrium gMEP is isotropic and therefore ηequil 1 = 0 and in a neighborhood of the equilibrium η1 remains small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Therefore, for small deviations from the thermodynamic equilibrium the expansion g(0)MEP µ = � ehµ(q)/kBT − 1 �−1 − � ehµ(q)/kBT − 1 �−2 ehµ(q)/kBThµ(q)η1,µ · cµ + O(ℏ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' By substituting in (61) one gets up to first order in ℏ2 Qµ = −τ Q∇xJµ −τ Q ℏ2 (2π)d ∂3 ∂xi∂xj∂xk � Rd chµ(q) 24 � ehµ(q)/kBT − 1 �−2 ehµ(q)/kBThµ(q)η1,µ · c ∂3 ∂qi∂qj∂qk hµ(q)dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In particular, at the zero order we have Q(0) µ = −τ Q∇xJ(0) µ = − τ (2π)d∇x � Rd cµ ⊗ cµhµ(q)g(0)MEP µ (x, q, t)dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' = − τ Q (2π)d∇x � Rd cµ ⊗ cµhµ(q) � ehµ(q)/kBT − 1 �−1 dq = − τ Q (2π)d � Rd cµ ⊗ cµhµ(q) ∂ ∂T � ehµ(q)/kBT − 1 �−1 dq ∇xTµ = − τ Q (2π)dkBT 2 � Rd cµ ⊗ cµh2 µ(q) ehµ(q)/kBT (ehµ(q)/kBT − 1)2dq ∇xTµ which can be written in the Fourier form Q(0) µ = −K(0) µ ∇xTµ with the thermal conductivity tensor given by K(0) µ = τ Q (2π)dkBT 2 � Rd cµ ⊗ cµh2 µ(q) ehµ(q)/kBT (ehµ(q)/kBT − 1)2dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' It is evident that Kµ is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 17 Observe that ∀n ∈ Sd � Sd ni1ni2 · · · nirdΩ = 0 if r odd, Sd being the unit sphere in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Therefore, if hµ(q) is isotropic Kµ is isotropic as well K(0) µ = 1 dk(0)I, with I identity matrix of order d and k(0) the zero order trace k(0) = τ Q (2π)dkBT 2 � Rd |cµ|2 h2 µ(q) ehµ(q)/kBT (ehµ(q)/kBT − 1)2dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The second order correction in ℏ2 reads Q(2) µ = − τ Q (2π)d∇x � Rd cµ ⊗ cµhµ(q)g(2) µ (η(0)(x, q, t))dq − τ Q (2π)d ∂3 ∂xi∂xj∂xk � Rd chµ(q) 24 � ehµ(q)/kBT − 1 �−2 ehµ(q)/kBThµ(q)η1,µ · c ∂3 ∂qi∂qj∂qk hµ(q)dq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Indeed the last term in the previous relation is of order ℏ2δT T and can be considered negligible for small deviations from local equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The remaining part gives a highly nonlinear correction which cannot be put in a Fourier form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' As an example we consider the case of the longitudinal and transversal acoustic phonons in the Debye approximation for a single branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' In such a case the corresponding symbol of the phonon hamiltonian reads c|q| and therefore k(0) ac = τ Q (2π)dkBT 2ac � Rd c4|q|2 ec|q|/kBTac (ec|q|/kBTac − 1)2dq = τ Qc4 (2π)dkBT 2ac mis(Sd) � +∞ 0 |q|d+1 ec|q|/kBTac (ec|q|/kBTac − 1)2d|q| = kBτ Qc3−d (2π)d mis(Sd) (kBTac)d−1 � +∞ 0 zd+1 ez (ez − 1)2d z (62) where mis(Sd) = 2πd/2 Γ(d/2) is the measure of Sd, Γ(x) being the Euler gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The previous integral is con- vergent for any d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Observe that we get a dependence on the temperature proportional 18 to T d−1 ac .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Regarding the second order correction we observe that gMEP 2 = −1 8 eξ (eξ − 1)3 � c2(eξ + 1) k2 BT(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t)4|q|2 � δij|q|2 � 2 ∂T ∂xi ∂T ∂xj − T ∂2T ∂xi∂xj � +qiqj � T ∂2T ∂xi∂xj − 3 ∂T ∂xi ∂T ∂xj �� − c3(e2ξ + 4eξ + 1) 3k3 B|q|(eξ − 1)T(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t)5 � (δij|q|2 − qiqj) ∂T ∂xi ∂T ∂xj − qiqjT ∂2T ∂xi∂xj �� = −1 8 c2eξ (eξ − 1)3 � (eξ + 1) k2 BT(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t)4 � 2|∇xT|2 − T∆xT + ninj � T ∂2T ∂xi∂xj − 3 ∂T ∂xi ∂T ∂xj �� − c(e2ξ + 4eξ + 1)|q| 3k3 B(eξ − 1)T(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t)5 � (δij − ninj) ∂T ∂xi ∂T ∂xj − ninjT ∂2T ∂xi∂xj �� with now ξ = c|q|/kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Therefore, the second order correction to the heat flux is given by Q(2) µ = −τ Q∇xJ(2) µ with J(2) = 1 (2π)d � Rd c ⊗ ch(q)gMEP 2 dq = c2 (2π)d � Rd nhnkh(q)gMEP 2 dq eh ⊗ ek := J(2) hk eh ⊗ ek (e1, e2, · · · , ed) being an orthonormal basis of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' By taking into account the well-known formulas � Ω nhnkdΩ = mis(Sd) d δij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' � Ω ninjnhnkdΩ = mis(Sd) d(d + 2)(δijδhk + δihδjk + δikδjk) and that � +∞ 0 h(q)eξ(eξ + 1) (eξ − 1)3 qd−1dq = c �kBT c �d+1 � +∞ 0 eξ(eξ + 1) (eξ − 1)3 ξddξ := c �kBT c �d+1 I1(d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' � +∞ 0 h(q)eξ(e2ξ + 4eξ + 1) (eξ − 1)4 qddq = c �kBT c �d+2 � +∞ 0 eξ(e2ξ + 4eξ + 1) (eξ − 1)4 ξd+1dξ := c �kBT c �d+2 I2(d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' the components of J(2) read J(2) hk = − c3 8(2π)d mis(Sd) d 1 k2 BT 4(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' t) �kBT c �d+1 �� (2|∇xT|2 − T∆xT)I1(d) − 1 3 ∂T ∂xi ∂T ∂xj δijI2(d) � δhk + �� T ∂2T ∂xi∂xj − 3 ∂T ∂xi ∂T ∂xj � I1(d) + 1 3 � ∂T ∂xi ∂T ∂xj + T ∂2T ∂xi∂xj � I2(d) � (δijδhk + δihδjk + δikδjk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' 19 The integrals I1(d) and I2(d) are divergent in the cases d = 1 and d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' As a consequence, the quantum corrections are valid only in the bulk (d = 3) case where I1(3) = π2, I2(3) = 4π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' This peculiarity is physically related to the density of states and the form of the energy dispersion relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Conclusions and acknowledgements The Wigner equation for phonons has been written in the case of a generic dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Moment equations have been deduced and closed by QMEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Under a long-time scaling an expression for the heat flux with a nonlinear quantum correction has been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The model is suited for the investigation in modern micro-devices where the enhanced miniaturization makes thermal effects more and more relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' The authors acknowledge the support from INdAM (GNFM) and from Universit`a degli Studi di Catania, Piano della Ricerca 2020/2022 Linea di intervento 2 ”QICT”, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Camiola acknowledges the financial support from the project AIM, Mobilit`a dei Ricercatori Asse I del PON R & I 2014-2020, proposta AIM1893589.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Declarations Conflicts of interest/Competing interests The authors declare they have no financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfk_i4/content/2301.00445v1.pdf'} +page_content=' Data availability Data sharing is not applicable to this article as no new data were created or analyzed in this study.' 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a/s9E5T4oBgHgl3EQfmQ_T/content/tmp_files/2301.05678v1.pdf.txt b/s9E5T4oBgHgl3EQfmQ_T/content/tmp_files/2301.05678v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..571f69f5c642c1b80b6a0e8518a0bad4bd18f524 --- /dev/null +++ b/s9E5T4oBgHgl3EQfmQ_T/content/tmp_files/2301.05678v1.pdf.txt @@ -0,0 +1,1938 @@ +arXiv:2301.05678v1 [math.CO] 13 Jan 2023 +A localized approach to generalized Tur´an problems +Rachel Kirsch +George Mason University +rkirsch4@gmu.edu +JD Nir +Toronto Metropolitan University +jd.nir@torontomu.ca +January 16, 2023 +Abstract +Generalized Tur´an problems ask for the maximum number of copies of a graph H in an +n-vertex, F-free graph, denoted by ex(n, H, F). We show how to extend the new, localized +approach of Bradaˇc, Malec, and Tompkins to generalized Tur´an problems. +We weight the +copies of H (typically taking H = Kt), instead of the edges, based on the size of the largest +clique, path, or star containing the vertices of the copy of H, and in each case prove a tight +upper bound on the sum of the weights. A consequence of our new localized theorems is an +asymptotic determination of ex(n, H, K1,r) for every H having at least one dominating vertex +and mex(m, H, K1,r) for every H having at least two dominating vertices. +1 +Introduction +Extremal graph theory is often considered the study of how easily measured global graph parameters, +such as the numbers of vertices and edges in a graph, influence its local substructures [7]. +An +archetypical result due to Tur´an describes which size cliques a graph is guaranteed to contain based +on its order and size: +Theorem (Tur´an [22]). Let n, r ≥ 1 be integers. If G is a Kr+1-free graph on n vertices (that is, no +subgraph of G is isomorphic to the complete graph Kr+1), then G contains at most n2 +2 (1 − 1 +r) edges. +This is denoted +ex(n, Kr+1) ≤ n2 +2 +� +1 − 1 +r +� +. +Furthermore, the Kr+1-free graph on n vertices with the greatest number of edges is the Tur´an +graph, Tr(n), in which the vertices of the graph are partitioned into r parts of sizes as close to equal +as possible, and vertices are adjacent if and only if they are in different parts. +Tur´an’s theorem has been generalized by many authors. +In 2015, Alon and Shikhelman [1], +expanding on several sporadic results (e.g. [3, 14, 15]), introduced the generalized Tur´an number +ex(n, H, F), which denotes the greatest number of subgraphs of an F-free graph on n vertices that +are isomorphic to H. +When maximizing the number of copies of H ̸= K2, it is possible to fix +the number of edges instead of the number of vertices. +Radcliffe and Uzzell [21] introduced the +generalized edge Tur´an number, mex(m, H, F), which denotes the greatest number of subgraphs that +are isomorphic to H in an F-free graph on m edges. +1 + +The quantities ex(n, H, F) and mex(m, H, F) have motivated many interesting results; see [5, +6, 11, 12, 13, 20] for an (incomplete) sample. However, these problems often stretch the notion of +“easily measurable” properties on which extremal graph theory is based. Though it is not known to +be NP-complete, counting the number of subgraphs of G that are isomorphic to H is considered a +challenging computation problem. Therefore, practically, it may be difficult to determine whether a +specific G contains a forbidden F even when the exact value of ex(n, H, F) is known. +Recently, Bradaˇc [4], based on a conjecture of Balogh and Lidick´y, gave a fundamentally different +generalization of Tur´an’s theorem: +Theorem (Bradaˇc [4]). Let G be a graph on n vertices. For each edge e ∈ E(G), define its weight +w(e) as +w(e) = +k +2(k − 1) +where k is the size of the largest clique in G containing e. Then +� +e∈E(G) +w(e) ≤ n2/4. +Bradaˇc’s result generalizes Tur´an’s theorem in the following sense: if it is known that G is Kr+1- +free, then, noting w(e) is decreasing in k, we see w(e) ≥ r/(2(r − 1)) for every edge e. Thus +|E(G)| · +r +(2(r − 1)) ≤ +� +e∈E(G) +w(e) ≤ n2 +4 +=⇒ |E(G)| ≤ n2 +2 +� +1 − 1 +r +� +. +The novelty of Bradaˇc’s result is in the local nature of the weight function. Rather than counting +the total number of edges in the entire graph, a global property, the weight we assign to each +edge depends only on the neighborhoods of the vertices of that edge which may be computed more +efficiently. +Inspired by Bradaˇc’s result, Malec and Tompkins [19] investigated other results which could be +“localized” in a similar fashion. In addition to giving a new proof of Bradaˇc’s result, they proved a +local version of another celebrated extremal graph theory result of Erd˝os and Gallai, as well as the +LYMB inequality (a generalization of Sperner’s theorem on boolean lattices), a generalization of the +Erd˝os-Ko-Rado theorem, and a theorem of Erd˝os and Szekeres on sequences. +Both Tur´an’s theorem and the theorem of Erd˝os and Gallai considered by Malec and Tompkins +tell us something about the graph based on the number of edges it contains. +Keeping in mind +that an edge is a clique containing two vertices, natural generalizations of both results to cliques of +larger size have been investigated. In this article, we show that these results, too, admit localized +generalizations. In fact, we prove several local results by weighting the cliques or other subgraphs of +G of a given size based on the maximum size of various substructures that contain them. In some +cases these “generalizations” actually establish new extremal results. +Our results follow a general framework. Each theorem concerns a target subgraph H, which in +many cases is a clique of some fixed size, and a family F = {F1, F2, . . .} of graphs in which Fi ⊆ Fi+1. +We first establish a size function which, given a copy of H in G, returns the largest Fi such that +some subgraph of G isomorphic to Fi contains H in some meaningful way. Then we define a weight +function which depends only on the size function of H and prove a bound on the sum of the weights +of every copy of H in G. In each case we show that the weight function is a decreasing function of +2 + +the size function, so a global upper bound on the size function implies a lower bound on the sum of +the weights, and we recover a “non-localized” theorem. +The rest of this paper is arranged as follows. We begin with some notational conventions and +preliminary results in Section 2. Then in Section 3, we weight t-cliques in G by the size of the largest +clique containing them to generalize Zykov’s theorem, itself a direct extension of Tur´an’s theorem. +We weight t-cliques by the longest path containing their vertices in Section 4, considering graphs of +fixed order in Section 4.1 and graphs of fixed size in Section 4.2. We weight a broad class of graphs, +including cliques, by the size of the largest star containing their vertices in Section 5.1, in which we +prove a family of novel generalized Tur´an and edge Tur´an results. We also give a hypergraph version +of one of these localized results in Section 5.2. We conclude with some open questions in Section 6. +2 +Preliminaries +2.1 +Notation +In addition to standard graph theoretic notation (see [2], for example), we establish the following +conventions. We define the path graph Pn to have n vertices and n − 1 edges and the star graph +Sr to have r leaves (and thus r + 1 total vertices). Given graphs G and H, we let N (H, G) denote +the number of subgraphs of G that are isomorphic to H. If N (H, G) = 0, we say G is H-free. As +mentioned in the introduction, if H and F are graphs, then +ex(n, H, F) = max{N (H, G) : |V (G)| = n and G is F-free} +and +mex(m, H, F) = max{N (H, G) : |E(G)| = m and G is F-free}. +We also have need to refer to the copies of H in G. When we do so for cliques, we refer to the +sets of vertices that span a complete subgraph and write +Kt(G) = {S ⊆ V (G) : G[S] ∼= Kt}. +For a more general graph H, we write H(G) to refer to the set of (not necessarily induced) subgraphs +of G that are isomorphic to H. +The size function and weight function(s) in each section are denoted by the notation shown below. +Cliques +Paths +Stars (graphs) +Stars (hypergraphs) +Section +3 +4 +5.1 +5.2 +Size function +αG +βG +θG +x +Weight function(s) +wG +pG, p′ +G +su +G +s +2.2 +Generalized binomial coefficients +We use generalized binomial coefficients when working with paths in Section 4 and hypergraphs in +Section 5. For a real number x ≥ k − 1 and a natural number k, the generalized binomial coefficient +�x +k +� +is defined as (x)(x − 1) · · ·(x − k + 1)/k!. When x < k − 1 we set +�x +k +� += 0. The function +�x +k +� +is +weakly increasing for all real numbers x and strictly increasing on x ≥ k − 1. +3 + +Observation 2.1. For all x ∈ R and n ∈ N, we have +2n +�x +n +� +≤ +�2x +n +� +, +with strict inequality when 2x + 1 > n ≥ 2. +Proof. The right side is always non-negative and is positive for 2x > n − 1. When x ≤ n − 1, the left +side is zero. When x > n − 1, the inequality is equivalent to +(2x)(2x − 2) · · · (2x − 2n + 2) ≤ (2x)(2x − 1) · · ·(2x − n + 1), +so is strict when n ≥ 2. +In Section 4 we also use the following observation and theorem. +Observation 2.2. Let G be a graph having at least one edge. Write |E(G)| in the form +�x +2 +� +, where +x ≥ 2 is real. Then |V (G)| ≥ x. +Proof. If |V (G)| < x then |V (G)| ≤ ⌈x⌉ − 1, so |E(G)| ≤ +�⌈x⌉−1 +2 +� +< +�x +2 +� +. Here we used that the +generalized binomial coefficient +�y +2 +� +is strictly increasing for all y ≥ 1. +Theorem 2.3 (Lov´asz [17]). Let t ≥ 2. Let G be a graph. Write the number of edges of G in the +form +�x +2 +� +, where x ≥ 1 is real. Then N (Kt, G) ≤ +�x +t +� +. +We also use generalized binomial coefficients in Section 5.2, where we introduce hypergraph +definitions and notation before stating Theorem 5.7, a bound on the number of t-cliques in a q- +uniform hypergraph based on the number of edges, which is a version of Lov´asz’s approximate form +of the Kruskal-Katona theorem. Theorem 2.3 is the special case of Theorem 5.7 corresponding to +graphs. +3 +Weighting by Maximum Clique Size +In this section we prove the following theorem which extends the localized version of Tur´an’s theorem +in [19] by assigning weights to cliques of any size, rather than just edges. Then we show that it +simultaneously extends a different extension of Tur´an’s theorem due to Zykov. +Theorem 3.1. Let t ≥ 2. For each T ∈ Kt(G), define +αG(T) = max{k : T ⊆ V (S) for some S ⊆ V (G) s.t. G[S] ∼= Kk} +and +wG(T) = αG(T)t +�αG(T) +t +�. +Then wG(T) is well-defined and decreasing in αG(T), and +w(G) = +� +T∈Kt(G) +wG(T) ≤ nt, +with equality if and only if G is a balanced multipartite graph with at least t parts. +4 + +Note that by setting t = 2, we recover the result which is Theorem 1 of both [4] and [19]. +� +T∈K2(G) +αG(e)2 +�αG(e) +2 +� = +� +e∈E(G) +2αG(e) +αG(e) − 1 ≤ n2 =⇒ +� +e∈E(G) +αG(e) +αG(e) − 1 ≤ n2 +2 . +Proof. First, αG(T) ≥ t because G[T] ∼= Kt, and therefore wG(T) is well-defined. We observe wG(T) +is decreasing in αG(T) as we can write +wG(T) = t! · +αG(T)t +�t−1 +i=0(αG(T) − i) += t! · +t−1 +� +i=1 +� +1 + +i +αG(T) − i +� +, +which is a non-empty product of functions decreasing in αG(T). +Let G be a graph on n ≥ 3 vertices. Note that if G contains an edge e that is not contained in any +t-clique, then e also is not contained in any larger clique, so Kt(G−e) = Kt(G), wG−e(T) = wG(T) for +every T ∈ Kt(G), and w(G) = w(G−e). Therefore we may assume that every edge in G is contained +in a t-clique. First assume there is r ≥ t such that G is complete r-partite with (non-empty) parts +A1, . . . , Ar. Then for each T ∈ Kt(G) we have αG(T) = r and wG(T) = rt/ +�r +t +� +. This gives +w(G) = rt +�r +t +�N (Kt, G) += rt +�r +t +� +� +S∈([r] +t ) +� +s∈S +|As| +To bound this sum, we relax the condition that the parts have integral sizes. By symmetry, the +real-valued polynomial function +f(x1, . . . , xr) = +� +S∈([r] +t ) +� +i∈S +xi +has a unique maximum when each xi = n/r, in which case each of the +�r +t +� +terms is (n/r)t, and thus +w(G) ≤ rt +�r +t +� · +�r +t +� +· +�n +r +�t += nt +with equality if and only if |Ai| = n/r. +Thus we may assume G is not complete multipartite. We use a technique introduced by Zykov [24] +sometimes called Zykov symmetrization. Suppose there are x, y, z ∈ V (G) such that x ∼ z but +x ̸∼ y ̸∼ z. We show that as long as such vertices exist, we can find G′ on n vertices such that +w(G′) > w(G). +If no such vertices exist, then x ̸∼ y and y ̸∼ z implies x ̸∼ z, which is to +say nonadjacent vertices can be partitioned into equivalence classes and therefore G is complete +multipartite. Thus producing such a G′ reduces this case to the complete multipartite case, and +furthermore proves any graph that meets the bound in the theorem must be complete multipartite. +For convenience, for v ∈ V (G), define +wG(v) = +� +T∈Kt(G) +s.t. v∈T +wG(T). +Assume without loss of generality that wG(x) ≥ wG(z) and consider two cases. +5 + +Case 3.1.1. wG(x) > wG(y). +Introduce a new vertex x′, add edges so that N(x′) = N(x), and let G′ = G − y + x′. Consider a +clique K′ (of any size) that is present in G′ but not in G. Such a clique must contain x′ and so must +be contained in NG′[x′] ∼= NG[x]. Therefore any T ∈ Kt(G) contained in K′ is also contained in some +clique K ⊆ V (G) of the same size. Thus every T ∈ Kt(G) ∩ Kt(G′) has αG(T) ≥ αG′(T). As wG(T) +is decreasing in αG(T), we have wG(T) ≤ wG′(T). Then +� +T∈Kt(G′) +wG′(T) = +� +T∈Kt(G′)\Kt(G) +wG′(T) + +� +T∈Kt(G′)∩Kt(G) +wG′(T) +≥ +� +T∈Kt(G′)\Kt(G) +wG′(T) + +� +T∈Kt(G−y) +wG(T) += wG′(x′) + +� � +T∈Kt(G) +wG(T) − wG(y) +� += wG(x) + +� +T∈Kt(G) +wG(T) − wG(y) +> +� +T∈Kt(G) +wG(T). +Case 3.1.2. wG(x) ≤ wG(y). +In this case we introduce two new vertices, y′ and y′′, add edges such that N(y′′) = N(y′) = N(y), +and define G′′ = G − x − z + y′ + y′′. As before, every clique K′ that is in G′′ but not in G must +contain y′ or y′′ (but not both as y′ ̸∼ y′′). Thus once again K′ must be contained in NG′′[y′] ∼= NG[y] +or NG′′[y′′] ∼= NG[y], so every T ∈ Kt(G) ∩ Kt(G′′) has αG(T) ≥ αG′′(T) and wG(T) ≤ wG′′(T). +Therefore +� +T∈Kt(G′′) +wG′′(T) = +� +T∈Kt(G′′)\Kt(G) +wG′′(T) + +� +T∈Kt(G′′)∩Kt(G) +wG′′(T) +≥ +� +T∈Kt(G′′)\Kt(G) +wG′′(T) + +� +T∈Kt(G−x−z) +wG(T) += wG′′(y′) + wG′′(y′′) + +� � +T∈Kt(G) +wG(T) − wG(x) − wG(z) + +� +T∈Kt(G) +s.t. x,z∈T +wG(T) +� += 2wG(y) + +� +T∈Kt(G) +wG(T) − wG(x) − wG(z) + +� +T∈Kt(G) +s.t. x,z∈T +wG(T) +≥ +� +T∈Kt(G) +wG(T) + +� +T∈Kt(G) +s.t. x,z∈T +wG(T) +> +� +T∈Kt(G) +wG(T), +where the last step holds because x ∼ z, and our initial assumption guarantees every edge is contained +in a t-clique. +6 + +One can also view Theorem 3.1 as a generalization of a theorem of Zykov [24] (and independently +Erd˝os [8]). In what is now considered the first generalized Tur´an result, Zykov proved that, among +Kr+1-free graphs, the Tur´an graph Tr(n) maximizes not only the number of edges but the number of +cliques of any size. +Theorem 3.2 (Zykov [24]). Let G be a Kr+1-free graph on n vertices. Then for any t ≥ 1, +N (Kt, G) ≤ +�r +t +� �n +r +�t +. +Note that Zykov actually proved the stronger result that ex(n, Kt, Kr+1) = N (Kt, Tr(n)), though +these results agree when r +�� n. +We can prove Theorem 3.2 as a consequence of Theorem 3.1: +Proof of Theorem 3.2. Let G be a an n-vertex, Kr+1-free graph. Then for each T ∈ Kt(G) we have +αG(T) ≤ r and, as wG(T) is decreasing in αG(T), wG(T) ≥ rt/ +�r +t +� +. Thus by Theorem 3.1, +N (Kt, G) · rt +�r +t +� = +� +T∈Kt(G) +rt +�r +t +� ≤ +� +T∈Kt(G) +wG(T) ≤ nt +and so +N (Kt, G) ≤ +�r +t +� +rt · nt = +�r +t +� �n +r +�t +. +4 +Weighting by Maximum Path Length +In 1959, Erd˝os and Gallai [9] proved that every graph with n vertices and m edges contains a path of +length at least 2m/n, and as a consequence ex(n, K2, Pr+1) ≤ (r−1)n +2 +, which, when r +�� n, is achieved by +a disjoint union of copies of Kr. Luo [18] determined ex(n, Kt, Pr+1) asymptotically, and Chakraborti +and Chen [5] then determined mex(m, Kt, Pr+1). Malec and Tompkins [19] gave a local version of the +Erd˝os-Gallai result. To extend their result to the t-clique versions, considering both graphs of fixed +order and graphs of fixed size, we define a more general size function for paths. +Definition 4.1. Let G be a graph on n vertices and t ≥ 2. For each T ∈ Kt(G), define +βG(T) = max{k : T ⊆ V (S) for some S ⊆ G s.t. S ∼= Pk+1}. +4.1 +Paths in graphs of fixed order +In this section we extend a result of Malec and Tompkins [19, Theorem 2] to cliques, which we will +demonstrate also generalizes a result of Luo [18]. +Theorem 4.2. Let G be a graph on n vertices and t ≥ 2. Define +pG(T) = +1 +�βG(T) +t−1 +�. +7 + +Then pG(T) is well-defined and decreasing in βG(T), and +� +T∈Kt(G) +pG(T) ≤ n +t , +with equality if and only if G is a disjoint union of complete graphs of order at least t. +We note that setting t = 2 does not quite recover [19, Theorem 2] as our weight function is not +equivalent at t = 2. Malec and Tompkins define βMT(e) to be the longest path containing the edge +e as a subgraph. This definition does not extend to larger cliques as paths do not contain them as +subgraphs. Instead, we merely require that all vertices of the clique appear in the path. Thus when +t = 2, unlike for Malec and Tompkins, the vertices of our edge may occur in the path without the +edge being part of the path. +Proof. First, we note any ordering of the vertices in any T ∈ Kt(G) is a path of length t − 1, so +βG(T) ≥ t − 1 and +�βG(T) +t−1 +� +> 0. Therefore pG(T) is well-defined and decreasing in βG(T). +We proceed by induction on n. When 1 ≤ n ≤ t − 1 we have +� +T∈Kt(G) +pG(T) = 0 < n +t . +If G is not connected, let C1, . . . , Cq be the components of G. Applying the inductive hypothesis +to each component, we have +� +T∈Kt(G) +pG(T) = +q +� +i=1 +� +T∈Kt(Ci) +pG(T) += +q +� +i=1 +� +T∈Kt(Ci) +pCi(T) +≤ +q +� +i=1 +|V (Ci)| +t +by induction += n +t . +Therefore we may assume G is connected and n ≥ t. Let r be the length of a longest path P ∼= Pr+1 +in G. +Case 4.2.1. There exists a cycle C containing the vertices of P. +Suppose for the sake of contradiction that there is a vertex u that is not on the cycle C. Since G +is connected, there is a path from u to C and then around C, which is longer than P, contradicting +that P is a longest path. Therefore every vertex of G is on C. Each T ∈ Kt(G) is contained in a +path of length n − 1 and has pG(T) = 1/ +�n−1 +t−1 +� +. +Hence, +� +T∈Kt(G) +pG(T) = +� +T∈Kt(G) +1 +�n−1 +t−1 +� = N (Kt, G) +�n−1 +t−1 +� +≤ +�n +t +� +�n−1 +t−1 +� = n +t , +and equality implies G ∼= Kn, with n ≥ t. +8 + +Case 4.2.2. There does not exist a cycle C containing the vertices of P. +Let v and w be the endpoints of P. Then {v, w} /∈ E(G). Label the vertices of P in order as +(v = u1, u2, . . . , ur+1 = w). Let V = {i : v ∼ ui} and W = {i : w ∼ ui−1}. If i ∈ V ∩ W then +(ui, u1, u2, . . . , ui−1, ur+1, ur, . . . , ui) is a cycle containing the vertices of P, so V ∩ W = ∅. Since +V, W ⊆ {2, . . . , r + 1}, we have d(v) + d(w) = |V | + |W| = |V ∪ W| ≤ r. Assume, without loss of +generality, that d(v) ≤ r/2. +Let R(v) be the set of t-cliques of G that contain v. Then |R(v)| ≤ +�d(v) +t−1 +� +. The fact that P is +a longest path implies both that βG(T) ≤ r and that every t-clique T in R(v) is contained in V (P) +(so βG(T) ≥ r), and therefore pG(T) = 1/ +� r +t−1 +� +for every T ∈ R(v). For any T ∈ Kt(G − v), we see +βG−v(T) ≤ βG(T) and therefore, as pG is decreasing in βG(T), pG−v(T) ≥ pG(T). +Applying the inductive hypothesis to G − v, we get +� +T∈Kt(G) +pG(T) = +� +T∈Kt(G−v) +pG(T) + +� +T∈R(v) +pG(T) +≤ +� +T∈Kt(G−v) +pG−v(T) + +� +T∈R(v) +pG(T) +≤ n − 1 +t ++ |R(v)| +� r +t−1 +� +≤ n − 1 +t ++ +�d(v) +t−1 +� +� r +t−1 +� +≤ n − 1 +t ++ +�r/2 +t−1 +� +� r +t−1 +� +≤ n − 1 +t ++ +(1/2)t−1� r +t−1 +� +� r +t−1 +� +by Observation 2.1 += n − 1 +t ++ +1 +2t−1 +≤ n − 1 +t ++ 1 +t += n +t +as 2t−1 ≥ t when t ≥ 2. When t ≥ 3, we see 2t−1 > t and thus if equality holds, no component is in +Case 4.2.2, so every component is in Case 4.2.1, and every component is a clique. +When t = 2, the claim that equality holds only for cliques follows from [19, Theorem 2]. Though, +as noted, our definition of βG(T) does not quite match their βMT(e), we have βMT(e) ≤ βG(e) as any +path containing an edge also contains the vertices of that edge. Because +pG(T) = +1 +�βG(T) +2−1 +� = +1 +βG(T) +is decreasing and matches Malec and Tompkins’ weight, we see +� +T∈K2(G) +pG(T) = +� +e∈E(G) +1 +βG(e) ≤ +� +e∈E(G) +1 +βMT(e). +9 + +Therefore if equality holds in our result, it also holds in Malec and Tompkins’ result, and by Theorem 2 +in [19], G is a disjoint union of complete graphs. +In 2018, Luo [18] extended Erd˝os and Gallai’s theorem by showing disjoint unions of cliques +maximize the number of cliques of any size. +Theorem 4.3 (Luo [18]). Let G be a Pr+1-free graph on n vertices. Then for any t ≤ r, +N (Kt, G) ≤ n +r +�r +t +� +. +Furthermore, equality holds if and only if r +�� n and G is a disjoint union of copies of Kr. +We note that Theorem 4.2 generalizes Theorem 4.3: +Proof. Let G be a Pr+1-free graph on n vertices. In such a graph G, every T ∈ Kt(G) has βG(T) ≤ +r − 1, so pG(T) ≥ +1 +(r−1 +t−1). By Theorem 4.2, +N (Kt, G) +�r−1 +t−1 +� += +� +T∈Kt(G) +1 +�r−1 +t−1 +� ≤ +� +T∈Kt(G) +pG(T) ≤ n +t , +so +N (Kt, G) ≤ n +t +�r − 1 +t − 1 +� += n +r +�r +t +� +. +By Theorem 4.2, equality implies that every component is a Pr+1-free clique, and that every t-clique +is contained in a Pr, so G is a disjoint union of copies of Kr. +4.2 +Paths in graphs of fixed size +Chakraborti and Chen [5] asked and answered the edge variant of this question by determining +mex(m, Kt, Pr+1) exactly for all values of the parameters m, t, and r. We prove a localized form of +their result, then show that it implies an asymptotic determination of mex(m, Kt, Pr+1). +Theorem 4.4. Let t ≥ 3. Let G be a graph having m edges. For each T ∈ Kt(G), define +p′ +G(T) = +1 +�βG(T)−1 +t−2 +�. +Then p′ +G(T) is well-defined and decreasing in βG(T), and +� +T∈Kt(G) +p′ +G(T) ≤ m +�t +2 +�, +with equality if and only if G is a disjoint union of complete graphs of order at least t and any number +of isolated vertices. +10 + +Proof. As before, any ordering of the vertices in any T ∈ Kt(G) is a path of length t − 1, so +βG(T) ≥ t − 1 and +�βG(T)−1 +t−2 +� +> 0. Therefore p′ +G(T) is well-defined and decreasing in βG(T). +We proceed by induction on m. When 1 ≤ m ≤ +�t +2 +� +− 1 we have +� +T∈Kt(G) +p′ +G(T) = 0 < m +�t +2 +�. +If G is not connected, let C1, . . . , Cq be the components of G. Applying the inductive hypothesis +to each component, we have +� +T∈Kt(G) +p′ +G(T) = +q +� +i=1 +� +T∈Kt(Ci) +p′ +G(T) += +q +� +i=1 +� +T∈Kt(Ci) +p′ +Ci(T) +≤ +q +� +i=1 +|E(Ci)| +�t +2 +� +by induction += m +�t +2 +�. +Therefore we may assume G is connected and m ≥ +�t +2 +� +. Let r be the length of a longest path +P ∼= Pr+1 in G. +Case 4.4.1. There exists a cycle C containing the vertices of P. +Suppose for the sake of contradiction that there is a vertex u that is not on the cycle C. Since G +is connected, there is a path from u to C and then around C, which is longer than P, contradicting +that P is a longest path. Therefore every vertex of G is on C. Each T ∈ Kt(G) is contained in a path +of length |V (G)| − 1, and has p′ +G(T) = 1/ +�|V (G)|−2 +t−2 +� +. Let x ≥ t be a real number satisfying m = +�x +2 +� +. +Then we have |V (G)| ≥ x by Observation 2.2 and, by Theorem 2.3, N (Kt, G) ≤ +�x +t +� +. Hence, +� +T∈Kt(G) +p′ +G(T) = +� +T∈Kt(G) +1 +�|V (G)|−2 +t−2 +� = N (Kt, G) +�|V (G)|−2 +t−2 +� ≤ +�x +t +� +�x−2 +t−2 +� = +�x +2 +� +�t +2 +� = m +�t +2 +�, +and equality implies +�|V (G)|−2 +t−2 +� += +�x−2 +t−2 +� +with x − 2 ≥ (t − 2) − 1, so x = |V (G)|. Then N (Kt, G) = +�|V (G)| +t +� +, so G ∼= Kx (and x ≥ t). +Case 4.4.2. There does not exist a cycle C containing the vertices of P. +Let v and w be the endpoints of P. Then {v, w} /∈ E(G). Label the vertices of P in order +as (v = u1, u2, . . . , ur+1 = w). Let V = {i : v ∼ ui} and W = {i : w ∼ ui−1}. If i ∈ V ∩ W +then (ui, u1, u2, . . . , ui−1, ur+1, ur, . . . , ui) is a cycle containing the vertices of P. Because V, W ⊆ +{2, . . . , r + 1}, we have d(v) + d(w) = |V | + |W| = |V ∪ W| ≤ r. Assume, without loss of generality, +that d(v) ≤ r/2. +Let R(v) be the set of t-cliques of G that contain v. Then |R(v)| ≤ +�d(v) +t−1 +� +. Since P is a longest +path, v and w have no neighbors outside of P, and so every t-clique in R(v) is contained in V (P). +11 + +Every T ∈ R(v) has p′ +G(T) = 1/ +�r−1 +t−2 +� +because it is contained in P (and there are no longer paths). +As noted above, we have βG−v(T) ≤ βG(T) for any T ∈ Kt(G − v) and as p′ +G is also decreasing in +βG(T), p′ +G−v(T) ≥ p′ +G(T). Applying the inductive hypothesis to G − v, we get +� +T∈Kt(G) +p′ +G(T) = +� +T∈Kt(G−v) +p′ +G(T) + +� +T∈R(v) +p′ +G(T) +≤ +� +T∈Kt(G−v) +p′ +G−v(T) + +� +T∈R(v) +p′ +G(T) +≤ m − d(v) +�t +2 +� ++ |R(v)| +�r−1 +t−2 +� +≤ m − d(v) +�t +2 +� ++ +�d(v) +t−1 +� +�r−1 +t−2 +� += m − d(v) +�t +2 +� ++ +�d(v)−1 +t−2 +� +�r−1 +t−2 +� · d(v) +t − 1 +≤ m − d(v) +�t +2 +� ++ +�r/2−1 +t−2 +� +�r−1 +t−2 +� · d(v) +t − 1 +≤ m − d(v) +�t +2 +� ++ +(1/2)t−2�r−2 +t−2 +� +�r−1 +t−2 +� +· d(v) +t − 1 +by Observation 2.1 += m − d(v) +�t +2 +� ++ r − t + 1 +r − 1 +· +1 +2t−1 · d(v) +t−1 +2 +< m − d(v) +�t +2 +� ++ d(v) +�t +2 +� += m +�t +2 +� +as 2t−1 > t and r−t+1 +r−1 +< 1 when t ≥ 3. Thus, if equality holds, no component is in Case 4.4.2, so +every component is in Case 4.4.1, and every component is a clique. +As mentioned, Chakraborti and Chen [5] determined mex(m, Kt, Pr+1) exactly, from which one +can derive the following weaker result: +Theorem 4.5 (Chakraborti and Chen [5]). For any 3 ≤ t ≤ r, if G is a Pr+1-free graph with m +edges, then +N (Kt, G) ≤ m +�r +2 +� · +�r +t +� +. +Furthermore, equality holds if and only if +�r +2 +� +divides m, and G is a disjoint union of copies of Kr. +We use Theorem 4.4 to prove Theorem 4.5: +Proof. Let G be a Pr+1-free graph. Then for each T ∈ Kt(G), we have βG(T) ≤ r − 1 and as p′ +G(T) +is decreasing in βG, +N (Kt, G) · +1 +�r−2 +t−2 +� ≤ +� +T∈Kt(G) +p′ +G(T) ≤ m +�t +2 +� +12 + +and therefore +N (Kt, G) ≤ m +�t +2 +� · +�r − 2 +t − 2 +� += m +�r +2 +� · +�r +t +� +. +Equality implies G is a disjoint union of Pr+1-free complete graphs, and every t-clique is contained +in a Pr, so G is a disjoint union of copies of Kr. +5 +Weighting by Maximum Star Size +5.1 +Graphs +In this section we consider generalized extremal problems of the form ex(n, H, Sr), forbidding the +star with r leaves. Unlike in previous sections where H was a clique, here we consider a broader +range of graphs H. We use H(G) to denote the number of (not necessarily induced) subgraphs of G +isomorphic to H. +Given a graph G and a set of vertices U ⊆ V (G), the common neighborhood of U in G is the set +of vertices of G adjacent to each vertex in U, or equivalently the intersection of the neighbor sets +of each vertex of U. The common degree of U, which we denote cdG(U), is the size of the common +neighborhood. Note that U is disjoint from its common neighborhood. +Denote the collection of dominating vertices of a graph G by Dom(G), and let dom(G) = +| Dom(G)|. Note that for any U ⊆ Dom(G), U is a clique, and the common neighborhood of U +is V (G) \ U. +The following theorem, which is the main result of this section, provides a template for localized +bounds on the number of copies of H in a graph based on the number of sets of dominating vertices +of a given size contained in H. We will focus the cases where these sets have size one or two, but we +provide the theorem in full generality. +Theorem 5.1. Let H be a graph on t vertices with dom(H) ≥ 1. For every graph G and each +T ∈ H(G), define +θG(T) = max{k : ∃v ∈ V (T) ⊆ V (S) s.t. V (S) ⊆ N[v] and S ∼= Sk} = max{dG(v) : v ∈ Dom(T)}, +and for each 1 ≤ u ≤ dom(H), define +su +G(T) = +1 +�θG(T)−u+1 +t−u +�. +Then for any 1 ≤ u ≤ dom(H) and any set of dominating vertices U of H with |U| = u, su +G(T) is +well-defined and decreasing in θG(T), and +� +T∈H(G) +su +G(T) ≤ N (H − U, Kt−u) +�dom(H) +u +� +· N (Ku, G). +Equality holds when G is a disjoint union of complete graphs of order at least t and any number of +components without u-cliques. +13 + +Proof. First, in any T ∈ H(G), there is a dominating vertex v of T which is the center of a star with +(at least) t − 1 leaves, so θG(T) ≥ t − 1 and +�θG(T)−u+1 +t−u +� +> 0. Therefore su +G(T) is well-defined and +decreasing on θG(T). +Let T ∈ H(G) and U ⊆ Dom(T) such that |U| = u. Recall U is a clique and the common +neighborhood of U in T is V (T) \ U. Thus for any v ∈ U, the vertices in the common neighborhood +of U in G together with the vertices U \{v} all are adjacent to v, forming a copy of ScdG(U)+u−1 whose +center is v ∈ T, and which contains all vertices of T. Therefore we have θG(T) ≥ cdG(U) + u − 1, or +θG(T) − u + 1 ≥ cdG(U) ≥ t − u because the common neighborhood of U in G contains at least the +t − u vertices of V (T) \ U. +For each T ∈ Kt(G), there are +�dom(H) +u +� +sets U ∈ Ku(G) such that each vertex of U is dominating +in T. For each U ∈ Ku(G), the number of copies T of H in G for which U ⊆ Dom(T) is at most +�cdG(U) +t−u +� +N (H − U, Kt−u): we choose t − u vertices from the common neighborhood of U in G to fill +out T and then choose how to embed the vertices of H − U, which can be done in at most as many +ways as embedding them into a clique of the same size. Therefore +�dom(H) +u +� � +T∈H(G) +su +G(T) = +� +U∈Ku(G) +� +T∈H(G) +U⊆Dom(T) +su +G(T) += +� +U∈Ku(G) +� +T∈H(G) +U⊆Dom(T) +1 +�θG(T)−u+1 +t−u +� +≤ +� +U∈Ku(G) +� +T∈H(G) +U⊆Dom(T) +1 +�cdG(U)+u−1−u+1 +t−u +� +≤ +� +U∈Ku(G) +N (H − U, Kt−u) +�cdG(U) +t−u +� +�cdG(U) +t−u +� += N (H − U, Kt−u)N (Ku, G). +When G is a disjoint union of complete graphs and any number of components without u-cliques, +let Kr be a component of G that contains a u-clique. Then θG(T) = r − 1 for every T ∈ H(G), and +su +G(T) = +1 +(r−u +t−u). The number of copies of H in this component is +�r +u +��r−u +t−u +� +N (H − U, Kt−u)/ +�dom H +u +� +, +which can be seen by first choosing a u-clique of the Kr to act as a selected set of u dominating +vertices of H, then choosing t − u of the r − u other vertices in the same component, then choosing +an embedding of H − U into those vertices (which is independent of the choice of U). Each copy of +H is counted this way once for each choice of selected set of u dominating vertices of H. Therefore +� +T∈H(Kr) +su +G(T) = +1 +�r−u +t−u +� +�r +u +��r − u +t − u +� +N (H − U, Kt−u)/ +�dom H +u +� += N (H − U, Kt−u) +�dom H +u +� +N (Ku, Kr). +If C1, . . . , Ck are the components of G then +� +T∈H(G) +su +G(T) = +k +� +i=1 +� +T∈H(Ci) +su +Ci(T) = N (H − U, Kt−u) +�dom H +u +� +k +� +i=1 +N (Ku, Ci) = N (H − U, Kt−u) +�dom H +u +� +N (Ku, G). +14 + +5.1.1 +Weighting t-cliques by maximum star size +By taking H = Kt and u ∈ {1, 2} in Theorem 5.1 we obtain the following corollaries. We note that +Proposition 1 in [19] is the case t = 2 of Corollary 5.2. +Corollary 5.2. For every n-vertex graph G and every clique size t ≥ 1, +� +T∈Kt(G) +s1 +G(T) = +� +T∈Kt(G) +1 +�θG(T) +t−1 +� ≤ n +t , +with equality if and only if G is a disjoint union of complete graphs of order at least t when t ≥ 3. +Proof. Let G be a graph on n vertices. We set H = Kt and u = 1 in Theorem 5.1, so s1 +G(T) = +1 +( +θG(T ) +t−1 ). +Any vertex v of Kt is a dominating vertex of Kt, so by Theorem 5.1 we have +� +T∈Kt(G) +1 +�θG(T) +t−1 +� ≤ N (Kt − {v}, Kt−1) +�dom(Kt) +1 +� +· N (K1, G) = n +t . +If equality holds, then, following the proof of Theorem 5.1, the number of t-cliques containing v is +exactly +�d(v) +t−1 +� +for every vertex v. For t ≥ 3 this implies every vertex has a complete neighborhood, so +every component is a complete graph. For any disjoint union of complete graphs G = Kn1 ∪· · ·∪Knk, +we have +� +T∈Kt(G) +1 +�θG(T) +t−1 +� = +k +� +i=1 +�ni +t +� +· +1 +�ni−1 +t−1 +� = +k +� +i=1 +ni +t = n +t . +Note that taking t = 1 in Corollary 5.2 gives the true, if trivial, statement that an n-vertex graph +G has at most n vertices. When t = 2, Malec and Tompkins note in [19, Proposition 1] that equality +holds if and only if every component of G is regular. +Corollary 5.3. For every m-edge graph G and every clique size t ≥ 2, +� +T∈Kt(G) +s2 +G(T) = +� +T∈Kt(G) +1 +�θG(T)−1 +t−2 +� ≤ m +�t +2 +�. +Equality holds when G is a disjoint union of complete graphs of order at least t and any number of +isolated vertices. +Proof. Let G be a graph on m edges. We set H = Kt and u = 2 in Theorem 5.1, so s2 +G(T) = +1 +( +θG(T )−1 +t−2 ). +Any two vertices v and w of Kt are dominating vertices of Kt, so by Theorem 5.1 we have +� +T∈Kt(G) +1 +�θG(T)−1 +t−2 +� ≤ N (Kt − {v, w}, Kt−2) +�dom(Kt) +2 +� +· N (K2, G) = m +�t +2 +�. +Corollary 5.2 and Corollary 5.3 in turn imply the following two known theorems on the maximum +number of t-cliques in bounded-degree graphs having a given number of vertices or a given number +of edges, respectively. These theorems have also been proved using essentially the same argument +as the one used in [23] to give an upper bound on the total number of cliques of all sizes. The first +theorem determines ex(n, Kt, Sr) asymptotically. +15 + +Corollary 5.4 (Wood [23]). Let t ≥ 1 and G be a graph on n vertices having ∆(G) ≤ r − 1. Then +N (Kt, G) ≤ n +r +�r +t +� +. +For t ≥ 3 equality holds if and only if r +�� n and G is a disjoint union of copies of Kr. +Proof. Let G be a graph on n vertices with maximum degree at most r − 1. Then θG(T) ≤ r − 1 for +every T ∈ Kt(G). By Corollary 5.2 we have +N (Kt, G) +�r−1 +t−1 +� += +� +T∈Kt(G) +1 +�r−1 +t−1 +� ≤ +� +T∈Kt(G) +1 +�θG(T) +t−1 +� ≤ n +t , +so N (Kt, G) ≤ n +t +�r−1 +t−1 +� += n +r +�r +t +� +. +If we instead fix the number of edges, we determine mex(m, Kt, Sr) asymptotically. +Corollary 5.5 (Wood [23]). Let t ≥ 2 and G be a graph on m edges with ∆(G) ≤ r − 1. Then +N (Kt, G) ≤ m +�r +2 +� +�r +t +� +. +Equality holds when +�r +2 +� �� m and G is a disjoint union of copies of Kr with any number of isolated +vertices. +Proof. Let G be a graph on m edges with maximum degree at most r − 1. Again θG(T) ≤ r − 1 for +every T ∈ Kt(G). By Corollary 5.3 we have +N (Kt, G) +�r−2 +t−2 +� += +� +T∈Kt(G) +1 +�r−2 +t−2 +� ≤ +� +T∈Kt(G) +1 +�θG(T)−1 +t−2 +� ≤ m +�t +2 +�, +so N (Kt, G) ≤ +m +(t +2) +�r−2 +t−2 +� += +m +(r +2) +�r +t +� +. +5.1.2 +Weighting copies of H by maximum star size +As mentioned, we can apply Theorem 5.1 to a broader class of graphs H than just cliques. This +allows us to prove novel asymptotic results on ex(n, H, Sr) for any H with at least one dominating +vertex and on mex(n, H, Sr) for H with at least two dominating vertices: +Theorem 5.6. Let H be a graph on t vertices. +(i) If H has at least one dominating vertex, then +ex(n, H, Sr) = (1 − o(1))N (H, ⌊n +r⌋Kr), +and +(ii) if H has at least two dominating vertices, then +mex(m, H, Sr) = (1 − o(1))N (H, +� +m +(r +2) +� +Kr). +16 + +Proof. Let G be a Sr-free graph on n vertices and m edges. Then for each T ∈ H(G), we have +θG(T) ≤ r − 1 and thus s1 +G(T) ≥ 1/ +�r−1 +t−1 +� +and s2 +G(T) ≥ 1/ +�r−2 +t−2 +� +. Therefore, as long as dom(H) ≥ 1, +let v be a dominating vertex and apply Theorem 5.1 with u = 1 to get +N (H, G) · +1 +�r−1 +t−1 +� ≤ +� +T∈H(G) +s1 +G(T) ≤ N (H − {v}, Kt−1) +dom(H) +· n +so that +N (H, G) ≤ n +�r − 1 +t − 1 +�N (H − {v}, Kt−1) +dom(H) += (1 − o(1))N (H, ⌊n +r ⌋Kr), +as, when r | n, we can count copies of H in ⌊n +r ⌋Kr by first choosing a vertex v of G to act as a selected +dominating vertex in H, then choosing t − 1 of the r − 1 other vertices in the same component, then +choosing an embedding of H − {v} into those vertices (which is independent of the choice of v). +Each copy of H is counted this way once for each choice of selected dominating vertex in H. The +asymptotic factor allows for n not divisible by r. Furthermore, as long as dom(H) ≥ 2, let v and w +both be dominating vertices and apply Theorem 5.1 with u = 2 to get +N (H, G) · +1 +�r−2 +t−2 +� ≤ +� +T∈H(G) +s2 +G(T) ≤ N (H − {v, w}, Kt−1) +dom(H) +· m +so that +N (H, G) ≤ m +�r − 2 +t − 2 +�N (H − {v, w}, Kt−1) +dom(H) += (1 − o(1))N (H, +� +m +(r +2) +� +Kr), +where similarly the asymptotic factor allows for m not divisible by +�r +2 +� +. +In both cases we achieve a matching lower bound by taking as many disjoint copies of Kr as +possible and making the remaining vertices independent or making the remaining edges a matching. +(For the values of n and m when we have some remaining vertices or edges, a better lower bound is +given by forming a clique with the remaining vertices or a colex graph with the remaining edges.) +Notice that cliques, stars, and all connected threshold graphs have at least one dominating vertex +so are included as possible graphs H in part (i) of Theorem 5.6. +5.2 +Hypergraphs +We now consider localized bounds for hypergraphs of bounded degree. Recall that a hypergraph is +q-uniform if every edge is a set of q vertices. The degree of a set of vertices I, denoted by d(I), is +the number of edges E that contain I. Letting i = |I|, the neighborhood of I is the (q − i)-uniform +hypergraph {E \ I : I ⊂ E ∈ E(H)}. For a q-uniform hypergraph H and 1 ≤ i < q, we write ∆i(H) +for the maximum degree d(I) over all sets I of i vertices. +For t ≥ q, we denote by K(q) +t +the complete q-uniform hypergraph on t vertices. We write Kt(H) +for the set of t-cliques in H, i.e., Kt(H) = {S ⊆ V (H) : H[S] ∼= K(q) +t }. We use the following upper +bound on N (K(q) +t , H), which is proved in [16, Theorem 32] as an immediate consequence of Lov´asz’ +approximate version of the Kruskal-Katona theorem. +Theorem 5.7 (Lov´asz [17]). Let q, t ∈ N with t ≥ q. Let H be a q-uniform hypergraph. Write the +number of edges of H in the form +�x +q +� +, where x ≥ q − 1 is real. Then N (K(q) +t , H) ≤ +�x +t +� +. +17 + +The following theorem generalizes Corollary 5.2 to q-uniform hypergraphs. Note that when q = 2, +we have i = 1 and x(I) = d(I) + i = d(v) + 1 for I = {v}, and so the function s(T) in the following +theorem can be thought of as an extension of the function s1 +G(T) of Theorem 5.1 to hypergraphs. +Theorem 5.8. Let t ≥ q > i ≥ 1 and suppose H is a q-uniform hypergraph on n vertices. For each +I ∈ +�V (H) +i +� +, define x(I) ≥ q − i − 1 by the equation d(I) = +�x(I)−i +q−i +� +, and, for each T ∈ Kt(H), define +x(T) = max +� +x(I) : I ∈ +�T +i +�� +and +s(T) = +1 +�x(T)−i +t−i +�. +Then s(T) is well-defined and decreasing as a function of x(T), +� +T∈Kt(H) +s(T) ≤ +�n +i +� +�t +i +� , +and there is an infinite family of hypergraphs that achieve the bound. +Proof. Let I ∈ +�V (H) +i +� +. For every T ∈ Kt(I), we have x(T) ≥ x(I) by definition. If Kt(I) is nonempty, +then d(I) ≥ +�t−i +q−i +� +and x(I) ≥ t. Therefore every T ∈ Kt(H) has x(T) ≥ t, so w(T) is a decreasing +function of x(T). Hence T ∈ Kt(I) implies w(T) ≤ 1/ +�x(I)−i +t−i +� +. Therefore +�t +i +� +� +T∈Kt(H) +s(T) = +� +I∈(V (H) +i ) +� +T∈Kt(I) +s(T) +≤ +� +I∈(V (H) +i ) +� +T∈Kt(I) +1 +�x(I)−i +t−i +� +≤ +� +I∈(V (H) +i ) +�x(I)−i +t−i +� +�x(I)−i +t−i +� += +�n +i +� +, +where the second inequality follows from applying Theorem 5.7 to the neighborhood of I. +Design theory provides an infinite family of graphs that meet this bound; we direct the reader +to [16] for more information on such hypergraphs. If H is a q-shadow of a Steiner system S(i, r, n) +for some r then by [16, Lemma 38(b)] we have x(I) = r for every I and x(T) = r for every T, so +s(T) = +1 +(r−i +t−i). By [16, Lemma 38(a)] we have N (K(q) +t , H) = +�r +t +�(n +i) +(r +i). Therefore +� +T∈Kt(H) +s(T) = +�r +t +��n +i +� +�r +i +��r−i +t−i +� = +�n +i +� +�t +i +� . +It seems interesting and challenging to characterize all of the extremal q-graphs in Theorem 5.8. +See [16, Theorem 43] for a related characterization of the extremal q-graphs in the non-localized +theorem. +As a corollary of Theorem 5.8 we obtain the following theorem of Radcliffe and the first author +on maximizing the number of t-cliques among bounded-degree q-uniform hypergraphs. +18 + +Theorem 5.9 (Kirsch and Radcliffe [16]). Let 1 ≤ i < q ≤ t and suppose H is an q-uniform +hypergraph on n vertices such that ∆i(H) ≤ +�x−i +q−i +� +for some real number x ≥ q. Then +N (K(q) +t , H) ≤ +�n +i +� +�x +i +� +�x +t +� +. +Proof using Theorem 5.8. The condition ∆i(H) ≤ +�x−i +q−i +� +implies that x(I) ≤ x for every I ∈ +�V (H) +i +� +, +so x(T) = max{x(I) : I ∈ +�T +i +� +} ≤ x for every T ∈ Kt(H). Theorem 5.8 gives +N (K(q) +t , H) +�x−i +t−i +� += +� +T∈Kt(H) +1 +�x−i +t−i +� ≤ +� +T∈Kt(H) +w(T) ≤ +�n +i +� +�t +i +� , +so N (K(q) +t , H) ≤ (n +i) +(t +i) +�x−i +t−i +� += (n +i) +(x +i) +�x +t +� +. +6 +Open Problems +We briefly mention a few additional instances of problems that we believe are amenable to localized +extensions. +The following conjecture is a localized form of a theorem of Frohmader [10], as phrased in [16, +Theorem 8], on maximizing the number of t-cliques among m-edge, Kr+1-free graphs. +Conjecture 6.1. Let t ≥ 2. For each T ∈ Kt(G), define +αG(T) = max{k : T ⊆ V (S) for some S ⊆ V (G) s.t. G[S] ∼= Kk} +and +w′ +G(T) = +�αG(T) +2 +�t/2 +�αG(T) +t +� . +For every m-edge graph G, +� +T∈Kt(G) +w′ +G(T) ≤ mt/2. +Many extremal results on paths, beginning with the results of Erd˝os and Gallai [9], are conse- +quences of extremal theorems regarding cycles. While the family of cycle graphs {C3, C4, . . .} does +not have the subgraph inclusion property shared by cliques, paths, and stars, these results consider +graphs of bounded circumference (that is, maximum cycle length). The techniques in this paper +often bounded a weight function by arguing a maximal structure could not be extended; cycles do +not allow such arguments, which could make proving localized results more difficult. Nevertheless, we +provide the following weight function and conjectures based on results of Luo [18] and Chakraborti +and Chen [6], respectively. +Definition 6.2. Let t ≥ 2. For each T ∈ Kt(G), define +γG(T) = max{k : T ⊆ V (S) for some S ⊆ G s.t. S ∼= Ck}. +19 + +Conjecture 6.3. Let t ≥ 2. For each T ∈ Kt(G), define +cG(T) = γG(T) − 1 +�γG(T) +t +� +. +Then cG(T) is well-defined and decreasing in γG(T), and +� +T∈Kt(G) +cG(T) ≤ n − 1, +with equality if and only if each 2-connected component of G is a complete graph of order at least t. +Conjecture 6.4. Let t ≥ 2. For each T ∈ Kt(G), define +c′ +G(T) = +�γG(T) +2 +� +�γG(T) +t +�. +Then c′ +G(T) is well-defined and decreasing in γG(T), and +� +T∈Kt(G) +c′ +G(T) ≤ m, +with equality if and only if each 2-connected component of G is a complete graph of order at least t +and any number of isolated vertices. +It may be possible to generalize Corollary 5.3 to hypergraphs. We make the following conjecture +as a localized version of Theorem 51 in [16], analogously to Theorem 5.8. +Conjecture 6.5. Let t ≥ q > i ≥ 1 and suppose H is a q-uniform hypergraph on m edges. For each +I ∈ +�V (H) +i +� +, define x(I) ≥ q − i − 1 by the equation d(I) = +�x(I)−i +q−i +� +, and, for each T ∈ Kt(H), define +x(T) = max +� +x(I) : I ∈ +�T +i +�� +and +s′(T) = +1 +�x(T)−q +t−q +�. +Then +� +T∈Kt(H) +s′(T) ≤ m +�t +q +�. +Finally, we used Theorem 5.8 to obtain new asymptotically tight bounds on ex(n, H, Sr) when +H has at least one dominating vertex and on mex(m, H, Sr) when H has at least two dominating +vertices. It may be possible to prove similar results for hypergraphs. +Question 6.6. Can Theorem 5.6 (or Theorem 5.1) be generalized to the setting of q-uniform hyper- +graphs with bounded maximum i-degree, perhaps with i = 1 or i = q − 1, in such a way as to obtain +new generalized Tur´an-type results for hypergraphs? +Acknowledgments +The authors thank Jamie Radcliffe for valuable discussions. +20 + +References +[1] Noga Alon and Clara Shikhelman, Many T copies in H-free graphs, J. Combin. Theory Ser. 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Lov´asz, Combinatorial problems and exercises, North-Holland Publishing Co., Amsterdam- +New York, 1979. MR 537284 +21 + +[18] Ruth Luo, The maximum number of cliques in graphs without long cycles, Journal of Combina- +torial Theory, Series B 128 (2018), 219–226. +[19] David Malec and Casey Tompkins, Localized versions of extremal problems, arXiv e-prints (2022), +arXiv:2205.12246. +[20] Natasha Morrison, JD Nir, Sergey Norin, Pawe�l Rza˙zewski, and Alexandra Wesolek, Every graph +is eventually Tur´an-good, arXiv e-prints (2022), arXiv:2208.08499. +[21] Jamie Radcliffe and Andrew Uzzell, Stability and Erd˝os–Stone type results for F-free graphs with +a fixed number of edges, arXiv e-prints (2018), arXiv:1810.04746. +[22] Paul Tur´an, Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapok 48 (1941), 436–452. +[23] David R Wood, On the maximum number of cliques in a graph, Graphs and Combinatorics 23 +(2007), no. 3, 337–352. +[24] Alexander Aleksandrovich Zykov, On some properties of linear complexes, Matematicheskii +sbornik 66 (1949), no. 2, 163–188. +22 + diff --git a/s9E5T4oBgHgl3EQfmQ_T/content/tmp_files/load_file.txt b/s9E5T4oBgHgl3EQfmQ_T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b6f2693bb50786d7a95745f1e673312dd425923d --- /dev/null +++ b/s9E5T4oBgHgl3EQfmQ_T/content/tmp_files/load_file.txt @@ -0,0 +1,649 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf,len=648 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='05678v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='CO] 13 Jan 2023 A localized approach to generalized Tur´an problems Rachel Kirsch George Mason University rkirsch4@gmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='edu JD Nir Toronto Metropolitan University jd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='nir@torontomu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='ca January 16, 2023 Abstract Generalized Tur´an problems ask for the maximum number of copies of a graph H in an n-vertex, F-free graph, denoted by ex(n, H, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We show how to extend the new, localized approach of Bradaˇc, Malec, and Tompkins to generalized Tur´an problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We weight the copies of H (typically taking H = Kt), instead of the edges, based on the size of the largest clique, path, or star containing the vertices of the copy of H, and in each case prove a tight upper bound on the sum of the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' A consequence of our new localized theorems is an asymptotic determination of ex(n, H, K1,r) for every H having at least one dominating vertex and mex(m, H, K1,r) for every H having at least two dominating vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 1 Introduction Extremal graph theory is often considered the study of how easily measured global graph parameters, such as the numbers of vertices and edges in a graph, influence its local substructures [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' An archetypical result due to Tur´an describes which size cliques a graph is guaranteed to contain based on its order and size: Theorem (Tur´an [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let n, r ≥ 1 be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' If G is a Kr+1-free graph on n vertices (that is, no subgraph of G is isomorphic to the complete graph Kr+1), then G contains at most n2 2 (1 − 1 r) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' This is denoted ex(n, Kr+1) ≤ n2 2 � 1 − 1 r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Furthermore, the Kr+1-free graph on n vertices with the greatest number of edges is the Tur´an graph, Tr(n), in which the vertices of the graph are partitioned into r parts of sizes as close to equal as possible, and vertices are adjacent if and only if they are in different parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Tur´an’s theorem has been generalized by many authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' In 2015, Alon and Shikhelman [1], expanding on several sporadic results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' [3, 14, 15]), introduced the generalized Tur´an number ex(n, H, F), which denotes the greatest number of subgraphs of an F-free graph on n vertices that are isomorphic to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' When maximizing the number of copies of H ̸= K2, it is possible to fix the number of edges instead of the number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Radcliffe and Uzzell [21] introduced the generalized edge Tur´an number, mex(m, H, F), which denotes the greatest number of subgraphs that are isomorphic to H in an F-free graph on m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 1 The quantities ex(n, H, F) and mex(m, H, F) have motivated many interesting results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' see [5, 6, 11, 12, 13, 20] for an (incomplete) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' However, these problems often stretch the notion of “easily measurable” properties on which extremal graph theory is based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Though it is not known to be NP-complete, counting the number of subgraphs of G that are isomorphic to H is considered a challenging computation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore, practically, it may be difficult to determine whether a specific G contains a forbidden F even when the exact value of ex(n, H, F) is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Recently, Bradaˇc [4], based on a conjecture of Balogh and Lidick´y, gave a fundamentally different generalization of Tur´an’s theorem: Theorem (Bradaˇc [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a graph on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For each edge e ∈ E(G), define its weight w(e) as w(e) = k 2(k − 1) where k is the size of the largest clique in G containing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then � e∈E(G) w(e) ≤ n2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Bradaˇc’s result generalizes Tur´an’s theorem in the following sense: if it is known that G is Kr+1- free, then, noting w(e) is decreasing in k, we see w(e) ≥ r/(2(r − 1)) for every edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Thus |E(G)| · r (2(r − 1)) ≤ � e∈E(G) w(e) ≤ n2 4 =⇒ |E(G)| ≤ n2 2 � 1 − 1 r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The novelty of Bradaˇc’s result is in the local nature of the weight function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Rather than counting the total number of edges in the entire graph, a global property, the weight we assign to each edge depends only on the neighborhoods of the vertices of that edge which may be computed more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Inspired by Bradaˇc’s result, Malec and Tompkins [19] investigated other results which could be “localized” in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' In addition to giving a new proof of Bradaˇc’s result, they proved a local version of another celebrated extremal graph theory result of Erd˝os and Gallai, as well as the LYMB inequality (a generalization of Sperner’s theorem on boolean lattices), a generalization of the Erd˝os-Ko-Rado theorem, and a theorem of Erd˝os and Szekeres on sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Both Tur´an’s theorem and the theorem of Erd˝os and Gallai considered by Malec and Tompkins tell us something about the graph based on the number of edges it contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Keeping in mind that an edge is a clique containing two vertices, natural generalizations of both results to cliques of larger size have been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' In this article, we show that these results, too, admit localized generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' In fact, we prove several local results by weighting the cliques or other subgraphs of G of a given size based on the maximum size of various substructures that contain them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' In some cases these “generalizations” actually establish new extremal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Our results follow a general framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Each theorem concerns a target subgraph H, which in many cases is a clique of some fixed size, and a family F = {F1, F2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='} of graphs in which Fi ⊆ Fi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We first establish a size function which, given a copy of H in G, returns the largest Fi such that some subgraph of G isomorphic to Fi contains H in some meaningful way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then we define a weight function which depends only on the size function of H and prove a bound on the sum of the weights of every copy of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' In each case we show that the weight function is a decreasing function of 2 the size function, so a global upper bound on the size function implies a lower bound on the sum of the weights, and we recover a “non-localized” theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The rest of this paper is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We begin with some notational conventions and preliminary results in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then in Section 3, we weight t-cliques in G by the size of the largest clique containing them to generalize Zykov’s theorem, itself a direct extension of Tur´an’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We weight t-cliques by the longest path containing their vertices in Section 4, considering graphs of fixed order in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 and graphs of fixed size in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We weight a broad class of graphs, including cliques, by the size of the largest star containing their vertices in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1, in which we prove a family of novel generalized Tur´an and edge Tur´an results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We also give a hypergraph version of one of these localized results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We conclude with some open questions in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 Notation In addition to standard graph theoretic notation (see [2], for example), we establish the following conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We define the path graph Pn to have n vertices and n − 1 edges and the star graph Sr to have r leaves (and thus r + 1 total vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Given graphs G and H, we let N (H, G) denote the number of subgraphs of G that are isomorphic to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' If N (H, G) = 0, we say G is H-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' As mentioned in the introduction, if H and F are graphs, then ex(n, H, F) = max{N (H, G) : |V (G)| = n and G is F-free} and mex(m, H, F) = max{N (H, G) : |E(G)| = m and G is F-free}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We also have need to refer to the copies of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' When we do so for cliques, we refer to the sets of vertices that span a complete subgraph and write Kt(G) = {S ⊆ V (G) : G[S] ∼= Kt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For a more general graph H, we write H(G) to refer to the set of (not necessarily induced) subgraphs of G that are isomorphic to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The size function and weight function(s) in each section are denoted by the notation shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Cliques Paths Stars (graphs) Stars (hypergraphs) Section 3 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2 Size function αG βG θG x Weight function(s) wG pG, p′ G su G s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2 Generalized binomial coefficients We use generalized binomial coefficients when working with paths in Section 4 and hypergraphs in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For a real number x ≥ k − 1 and a natural number k, the generalized binomial coefficient �x k � is defined as (x)(x − 1) · · ·(x − k + 1)/k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='. When x < k − 1 we set �x k � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The function �x k � is weakly increasing for all real numbers x and strictly increasing on x ≥ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 3 Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For all x ∈ R and n ∈ N, we have 2n �x n � ≤ �2x n � , with strict inequality when 2x + 1 > n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The right side is always non-negative and is positive for 2x > n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' When x ≤ n − 1, the left side is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' When x > n − 1, the inequality is equivalent to (2x)(2x − 2) · · · (2x − 2n + 2) ≤ (2x)(2x − 1) · · ·(2x − n + 1), so is strict when n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' In Section 4 we also use the following observation and theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a graph having at least one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Write |E(G)| in the form �x 2 � , where x ≥ 2 is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then |V (G)| ≥ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' If |V (G)| < x then |V (G)| ≤ ⌈x⌉ − 1, so |E(G)| ≤ �⌈x⌉−1 2 � < �x 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Here we used that the generalized binomial coefficient �y 2 � is strictly increasing for all y ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='3 (Lov´asz [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let t ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Write the number of edges of G in the form �x 2 � , where x ≥ 1 is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then N (Kt, G) ≤ �x t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We also use generalized binomial coefficients in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2, where we introduce hypergraph definitions and notation before stating Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='7, a bound on the number of t-cliques in a q- uniform hypergraph based on the number of edges, which is a version of Lov´asz’s approximate form of the Kruskal-Katona theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='3 is the special case of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='7 corresponding to graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 3 Weighting by Maximum Clique Size In this section we prove the following theorem which extends the localized version of Tur´an’s theorem in [19] by assigning weights to cliques of any size, rather than just edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then we show that it simultaneously extends a different extension of Tur´an’s theorem due to Zykov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let t ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For each T ∈ Kt(G), define αG(T) = max{k : T ⊆ V (S) for some S ⊆ V (G) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' G[S] ∼= Kk} and wG(T) = αG(T)t �αG(T) t �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then wG(T) is well-defined and decreasing in αG(T), and w(G) = � T∈Kt(G) wG(T) ≤ nt, with equality if and only if G is a balanced multipartite graph with at least t parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 4 Note that by setting t = 2, we recover the result which is Theorem 1 of both [4] and [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' � T∈K2(G) αG(e)2 �αG(e) 2 � = � e∈E(G) 2αG(e) αG(e) − 1 ≤ n2 =⇒ � e∈E(G) αG(e) αG(e) − 1 ≤ n2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' First, αG(T) ≥ t because G[T] ∼= Kt, and therefore wG(T) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We observe wG(T) is decreasing in αG(T) as we can write wG(T) = t!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' · αG(T)t �t−1 i=0(αG(T) − i) = t!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' · t−1 � i=1 � 1 + i αG(T) − i � , which is a non-empty product of functions decreasing in αG(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a graph on n ≥ 3 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Note that if G contains an edge e that is not contained in any t-clique, then e also is not contained in any larger clique, so Kt(G−e) = Kt(G), wG−e(T) = wG(T) for every T ∈ Kt(G), and w(G) = w(G−e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore we may assume that every edge in G is contained in a t-clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' First assume there is r ≥ t such that G is complete r-partite with (non-empty) parts A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' , Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then for each T ∈ Kt(G) we have αG(T) = r and wG(T) = rt/ �r t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' This gives w(G) = rt �r t �N (Kt, G) = rt �r t � � S∈([r] t ) � s∈S |As| To bound this sum, we relax the condition that the parts have integral sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' By symmetry, the real-valued polynomial function f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' , xr) = � S∈([r] t ) � i∈S xi has a unique maximum when each xi = n/r, in which case each of the �r t � terms is (n/r)t, and thus w(G) ≤ rt �r t � · �r t � �n r �t = nt with equality if and only if |Ai| = n/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Thus we may assume G is not complete multipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We use a technique introduced by Zykov [24] sometimes called Zykov symmetrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Suppose there are x, y, z ∈ V (G) such that x ∼ z but x ̸∼ y ̸∼ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We show that as long as such vertices exist, we can find G′ on n vertices such that w(G′) > w(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' If no such vertices exist, then x ̸∼ y and y ̸∼ z implies x ̸∼ z, which is to say nonadjacent vertices can be partitioned into equivalence classes and therefore G is complete multipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Thus producing such a G′ reduces this case to the complete multipartite case, and furthermore proves any graph that meets the bound in the theorem must be complete multipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For convenience, for v ∈ V (G), define wG(v) = � T∈Kt(G) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' v∈T wG(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Assume without loss of generality that wG(x) ≥ wG(z) and consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 5 Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' wG(x) > wG(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Introduce a new vertex x′, add edges so that N(x′) = N(x), and let G′ = G − y + x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Consider a clique K′ (of any size) that is present in G′ but not in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Such a clique must contain x′ and so must be contained in NG′[x′] ∼= NG[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore any T ∈ Kt(G) contained in K′ is also contained in some clique K ⊆ V (G) of the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Thus every T ∈ Kt(G) ∩ Kt(G′) has αG(T) ≥ αG′(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' As wG(T) is decreasing in αG(T), we have wG(T) ≤ wG′(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then � T∈Kt(G′) wG′(T) = � T∈Kt(G′)\\Kt(G) wG′(T) + � T∈Kt(G′)∩Kt(G) wG′(T) ≥ � T∈Kt(G′)\\Kt(G) wG′(T) + � T∈Kt(G−y) wG(T) = wG′(x′) + � � T∈Kt(G) wG(T) − wG(y) � = wG(x) + � T∈Kt(G) wG(T) − wG(y) > � T∈Kt(G) wG(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' wG(x) ≤ wG(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' In this case we introduce two new vertices, y′ and y′′, add edges such that N(y′′) = N(y′) = N(y), and define G′′ = G − x − z + y′ + y′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' As before, every clique K′ that is in G′′ but not in G must contain y′ or y′′ (but not both as y′ ̸∼ y′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Thus once again K′ must be contained in NG′′[y′] ∼= NG[y] or NG′′[y′′] ∼= NG[y], so every T ∈ Kt(G) ∩ Kt(G′′) has αG(T) ≥ αG′′(T) and wG(T) ≤ wG′′(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore � T∈Kt(G′′) wG′′(T) = � T∈Kt(G′′)\\Kt(G) wG′′(T) + � T∈Kt(G′′)∩Kt(G) wG′′(T) ≥ � T∈Kt(G′′)\\Kt(G) wG′′(T) + � T∈Kt(G−x−z) wG(T) = wG′′(y′) + wG′′(y′′) + � � T∈Kt(G) wG(T) − wG(x) − wG(z) + � T∈Kt(G) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' x,z∈T wG(T) � = 2wG(y) + � T∈Kt(G) wG(T) − wG(x) − wG(z) + � T∈Kt(G) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' x,z∈T wG(T) ≥ � T∈Kt(G) wG(T) + � T∈Kt(G) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' x,z∈T wG(T) > � T∈Kt(G) wG(T), where the last step holds because x ∼ z, and our initial assumption guarantees every edge is contained in a t-clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 6 One can also view Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 as a generalization of a theorem of Zykov [24] (and independently Erd˝os [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' In what is now considered the first generalized Tur´an result, Zykov proved that, among Kr+1-free graphs, the Tur´an graph Tr(n) maximizes not only the number of edges but the number of cliques of any size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2 (Zykov [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a Kr+1-free graph on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then for any t ≥ 1, N (Kt, G) ≤ �r t � �n r �t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Note that Zykov actually proved the stronger result that ex(n, Kt, Kr+1) = N (Kt, Tr(n)), though these results agree when r �� n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We can prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2 as a consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1: Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a an n-vertex, Kr+1-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then for each T ∈ Kt(G) we have αG(T) ≤ r and, as wG(T) is decreasing in αG(T), wG(T) ≥ rt/ �r t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Thus by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1, N (Kt, G) · rt �r t � = � T∈Kt(G) rt �r t � ≤ � T∈Kt(G) wG(T) ≤ nt and so N (Kt, G) ≤ �r t � rt · nt = �r t � �n r �t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 4 Weighting by Maximum Path Length In 1959, Erd˝os and Gallai [9] proved that every graph with n vertices and m edges contains a path of length at least 2m/n, and as a consequence ex(n, K2, Pr+1) ≤ (r−1)n 2 , which, when r �� n, is achieved by a disjoint union of copies of Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Luo [18] determined ex(n, Kt, Pr+1) asymptotically, and Chakraborti and Chen [5] then determined mex(m, Kt, Pr+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Malec and Tompkins [19] gave a local version of the Erd˝os-Gallai result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' To extend their result to the t-clique versions, considering both graphs of fixed order and graphs of fixed size, we define a more general size function for paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a graph on n vertices and t ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For each T ∈ Kt(G), define βG(T) = max{k : T ⊆ V (S) for some S ⊆ G s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' S ∼= Pk+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 Paths in graphs of fixed order In this section we extend a result of Malec and Tompkins [19, Theorem 2] to cliques, which we will demonstrate also generalizes a result of Luo [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a graph on n vertices and t ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Define pG(T) = 1 �βG(T) t−1 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 7 Then pG(T) is well-defined and decreasing in βG(T), and � T∈Kt(G) pG(T) ≤ n t , with equality if and only if G is a disjoint union of complete graphs of order at least t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We note that setting t = 2 does not quite recover [19, Theorem 2] as our weight function is not equivalent at t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Malec and Tompkins define βMT(e) to be the longest path containing the edge e as a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' This definition does not extend to larger cliques as paths do not contain them as subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Instead, we merely require that all vertices of the clique appear in the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Thus when t = 2, unlike for Malec and Tompkins, the vertices of our edge may occur in the path without the edge being part of the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' First, we note any ordering of the vertices in any T ∈ Kt(G) is a path of length t − 1, so βG(T) ≥ t − 1 and �βG(T) t−1 � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore pG(T) is well-defined and decreasing in βG(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We proceed by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' When 1 ≤ n ≤ t − 1 we have � T∈Kt(G) pG(T) = 0 < n t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' If G is not connected, let C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' , Cq be the components of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Applying the inductive hypothesis to each component, we have � T∈Kt(G) pG(T) = q � i=1 � T∈Kt(Ci) pG(T) = q � i=1 � T∈Kt(Ci) pCi(T) ≤ q � i=1 |V (Ci)| t by induction = n t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore we may assume G is connected and n ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let r be the length of a longest path P ∼= Pr+1 in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' There exists a cycle C containing the vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Suppose for the sake of contradiction that there is a vertex u that is not on the cycle C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Since G is connected, there is a path from u to C and then around C, which is longer than P, contradicting that P is a longest path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore every vertex of G is on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Each T ∈ Kt(G) is contained in a path of length n − 1 and has pG(T) = 1/ �n−1 t−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Hence, � T∈Kt(G) pG(T) = � T∈Kt(G) 1 �n−1 t−1 � = N (Kt, G) �n−1 t−1 � ≤ �n t � �n−1 t−1 � = n t , and equality implies G ∼= Kn, with n ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 8 Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' There does not exist a cycle C containing the vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let v and w be the endpoints of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then {v, w} /∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Label the vertices of P in order as (v = u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' , ur+1 = w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let V = {i : v ∼ ui} and W = {i : w ∼ ui−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' If i ∈ V ∩ W then (ui, u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' , ui−1, ur+1, ur, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' , ui) is a cycle containing the vertices of P, so V ∩ W = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Since V, W ⊆ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' , r + 1}, we have d(v) + d(w) = |V | + |W| = |V ∪ W| ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Assume, without loss of generality, that d(v) ≤ r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let R(v) be the set of t-cliques of G that contain v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then |R(v)| ≤ �d(v) t−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The fact that P is a longest path implies both that βG(T) ≤ r and that every t-clique T in R(v) is contained in V (P) (so βG(T) ≥ r), and therefore pG(T) = 1/ � r t−1 � for every T ∈ R(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For any T ∈ Kt(G − v), we see βG−v(T) ≤ βG(T) and therefore, as pG is decreasing in βG(T), pG−v(T) ≥ pG(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Applying the inductive hypothesis to G − v, we get � T∈Kt(G) pG(T) = � T∈Kt(G−v) pG(T) + � T∈R(v) pG(T) ≤ � T∈Kt(G−v) pG−v(T) + � T∈R(v) pG(T) ≤ n − 1 t + |R(v)| � r t−1 � ≤ n − 1 t + �d(v) t−1 � � r t−1 � ≤ n − 1 t + �r/2 t−1 � � r t−1 � ≤ n − 1 t + (1/2)t−1� r t−1 � � r t−1 � by Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 = n − 1 t + 1 2t−1 ≤ n − 1 t + 1 t = n t as 2t−1 ≥ t when t ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' When t ≥ 3, we see 2t−1 > t and thus if equality holds, no component is in Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2, so every component is in Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1, and every component is a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' When t = 2, the claim that equality holds only for cliques follows from [19, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Though, as noted, our definition of βG(T) does not quite match their βMT(e), we have βMT(e) ≤ βG(e) as any path containing an edge also contains the vertices of that edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Because pG(T) = 1 �βG(T) 2−1 � = 1 βG(T) is decreasing and matches Malec and Tompkins’ weight, we see � T∈K2(G) pG(T) = � e∈E(G) 1 βG(e) ≤ � e∈E(G) 1 βMT(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 9 Therefore if equality holds in our result, it also holds in Malec and Tompkins’ result, and by Theorem 2 in [19], G is a disjoint union of complete graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' In 2018, Luo [18] extended Erd˝os and Gallai’s theorem by showing disjoint unions of cliques maximize the number of cliques of any size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='3 (Luo [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a Pr+1-free graph on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then for any t ≤ r, N (Kt, G) ≤ n r �r t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Furthermore, equality holds if and only if r �� n and G is a disjoint union of copies of Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We note that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2 generalizes Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='3: Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a Pr+1-free graph on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' In such a graph G, every T ∈ Kt(G) has βG(T) ≤ r − 1, so pG(T) ≥ 1 (r−1 t−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2, N (Kt, G) �r−1 t−1 � = � T∈Kt(G) 1 �r−1 t−1 � ≤ � T∈Kt(G) pG(T) ≤ n t , so N (Kt, G) ≤ n t �r − 1 t − 1 � = n r �r t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2, equality implies that every component is a Pr+1-free clique, and that every t-clique is contained in a Pr, so G is a disjoint union of copies of Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2 Paths in graphs of fixed size Chakraborti and Chen [5] asked and answered the edge variant of this question by determining mex(m, Kt, Pr+1) exactly for all values of the parameters m, t, and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We prove a localized form of their result, then show that it implies an asymptotic determination of mex(m, Kt, Pr+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let t ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a graph having m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For each T ∈ Kt(G), define p′ G(T) = 1 �βG(T)−1 t−2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then p′ G(T) is well-defined and decreasing in βG(T), and � T∈Kt(G) p′ G(T) ≤ m �t 2 �, with equality if and only if G is a disjoint union of complete graphs of order at least t and any number of isolated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' As before, any ordering of the vertices in any T ∈ Kt(G) is a path of length t − 1, so βG(T) ≥ t − 1 and �βG(T)−1 t−2 � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore p′ G(T) is well-defined and decreasing in βG(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We proceed by induction on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' When 1 ≤ m ≤ �t 2 � − 1 we have � T∈Kt(G) p′ G(T) = 0 < m �t 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' If G is not connected, let C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' , Cq be the components of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Applying the inductive hypothesis to each component, we have � T∈Kt(G) p′ G(T) = q � i=1 � T∈Kt(Ci) p′ G(T) = q � i=1 � T∈Kt(Ci) p′ Ci(T) ≤ q � i=1 |E(Ci)| �t 2 � by induction = m �t 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore we may assume G is connected and m ≥ �t 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let r be the length of a longest path P ∼= Pr+1 in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' There exists a cycle C containing the vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Suppose for the sake of contradiction that there is a vertex u that is not on the cycle C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Since G is connected, there is a path from u to C and then around C, which is longer than P, contradicting that P is a longest path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore every vertex of G is on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Each T ∈ Kt(G) is contained in a path of length |V (G)| − 1, and has p′ G(T) = 1/ �|V (G)|−2 t−2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let x ≥ t be a real number satisfying m = �x 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then we have |V (G)| ≥ x by Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2 and, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='3, N (Kt, G) ≤ �x t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Hence, � T∈Kt(G) p′ G(T) = � T∈Kt(G) 1 �|V (G)|−2 t−2 � = N (Kt, G) �|V (G)|−2 t−2 � ≤ �x t � �x−2 t−2 � = �x 2 � �t 2 � = m �t 2 �, and equality implies �|V (G)|−2 t−2 � = �x−2 t−2 � with x − 2 ≥ (t − 2) − 1, so x = |V (G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then N (Kt, G) = �|V (G)| t � , so G ∼= Kx (and x ≥ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' There does not exist a cycle C containing the vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let v and w be the endpoints of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then {v, w} /∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Label the vertices of P in order as (v = u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' , ur+1 = w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let V = {i : v ∼ ui} and W = {i : w ∼ ui−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' If i ∈ V ∩ W then (ui, u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' , ui−1, ur+1, ur, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' , ui) is a cycle containing the vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Because V, W ⊆ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' , r + 1}, we have d(v) + d(w) = |V | + |W| = |V ∪ W| ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Assume, without loss of generality, that d(v) ≤ r/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let R(v) be the set of t-cliques of G that contain v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then |R(v)| ≤ �d(v) t−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Since P is a longest path, v and w have no neighbors outside of P, and so every t-clique in R(v) is contained in V (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 11 Every T ∈ R(v) has p′ G(T) = 1/ �r−1 t−2 � because it is contained in P (and there are no longer paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' As noted above, we have βG−v(T) ≤ βG(T) for any T ∈ Kt(G − v) and as p′ G is also decreasing in βG(T), p′ G−v(T) ≥ p′ G(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Applying the inductive hypothesis to G − v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' we get � T∈Kt(G) p′ G(T) = � T∈Kt(G−v) p′ G(T) + � T∈R(v) p′ G(T) ≤ � T∈Kt(G−v) p′ G−v(T) + � T∈R(v) p′ G(T) ≤ m − d(v) �t 2 � + |R(v)| �r−1 t−2 � ≤ m − d(v) �t 2 � + �d(v) t−1 � �r−1 t−2 � = m − d(v) �t 2 � + �d(v)−1 t−2 � �r−1 t−2 � · d(v) t − 1 ≤ m − d(v) �t 2 � + �r/2−1 t−2 � �r−1 t−2 � · d(v) t − 1 ≤ m − d(v) �t 2 � + (1/2)t−2�r−2 t−2 � �r−1 t−2 � d(v) t − 1 by Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 = m − d(v) �t 2 � + r − t + 1 r − 1 1 2t−1 · d(v) t−1 2 < m − d(v) �t 2 � + d(v) �t 2 � = m �t 2 � as 2t−1 > t and r−t+1 r−1 < 1 when t ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Thus, if equality holds, no component is in Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2, so every component is in Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1, and every component is a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' As mentioned, Chakraborti and Chen [5] determined mex(m, Kt, Pr+1) exactly, from which one can derive the following weaker result: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='5 (Chakraborti and Chen [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For any 3 ≤ t ≤ r, if G is a Pr+1-free graph with m edges, then N (Kt, G) ≤ m �r 2 � · �r t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Furthermore, equality holds if and only if �r 2 � divides m, and G is a disjoint union of copies of Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We use Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='4 to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='5: Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a Pr+1-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then for each T ∈ Kt(G), we have βG(T) ≤ r − 1 and as p′ G(T) is decreasing in βG, N (Kt, G) · 1 �r−2 t−2 � ≤ � T∈Kt(G) p′ G(T) ≤ m �t 2 � 12 and therefore N (Kt, G) ≤ m �t 2 � · �r − 2 t − 2 � = m �r 2 � · �r t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Equality implies G is a disjoint union of Pr+1-free complete graphs, and every t-clique is contained in a Pr, so G is a disjoint union of copies of Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 5 Weighting by Maximum Star Size 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 Graphs In this section we consider generalized extremal problems of the form ex(n, H, Sr), forbidding the star with r leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Unlike in previous sections where H was a clique, here we consider a broader range of graphs H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We use H(G) to denote the number of (not necessarily induced) subgraphs of G isomorphic to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Given a graph G and a set of vertices U ⊆ V (G), the common neighborhood of U in G is the set of vertices of G adjacent to each vertex in U, or equivalently the intersection of the neighbor sets of each vertex of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The common degree of U, which we denote cdG(U), is the size of the common neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Note that U is disjoint from its common neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Denote the collection of dominating vertices of a graph G by Dom(G), and let dom(G) = | Dom(G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Note that for any U ⊆ Dom(G), U is a clique, and the common neighborhood of U is V (G) \\ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The following theorem, which is the main result of this section, provides a template for localized bounds on the number of copies of H in a graph based on the number of sets of dominating vertices of a given size contained in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We will focus the cases where these sets have size one or two, but we provide the theorem in full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let H be a graph on t vertices with dom(H) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For every graph G and each T ∈ H(G), define θG(T) = max{k : ∃v ∈ V (T) ⊆ V (S) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' V (S) ⊆ N[v] and S ∼= Sk} = max{dG(v) : v ∈ Dom(T)}, and for each 1 ≤ u ≤ dom(H), define su G(T) = 1 �θG(T)−u+1 t−u �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then for any 1 ≤ u ≤ dom(H) and any set of dominating vertices U of H with |U| = u, su G(T) is well-defined and decreasing in θG(T), and � T∈H(G) su G(T) ≤ N (H − U, Kt−u) �dom(H) u � N (Ku, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Equality holds when G is a disjoint union of complete graphs of order at least t and any number of components without u-cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' First, in any T ∈ H(G), there is a dominating vertex v of T which is the center of a star with (at least) t − 1 leaves, so θG(T) ≥ t − 1 and �θG(T)−u+1 t−u � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore su G(T) is well-defined and decreasing on θG(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let T ∈ H(G) and U ⊆ Dom(T) such that |U| = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Recall U is a clique and the common neighborhood of U in T is V (T) \\ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Thus for any v ∈ U, the vertices in the common neighborhood of U in G together with the vertices U \\{v} all are adjacent to v, forming a copy of ScdG(U)+u−1 whose center is v ∈ T, and which contains all vertices of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore we have θG(T) ≥ cdG(U) + u − 1, or θG(T) − u + 1 ≥ cdG(U) ≥ t − u because the common neighborhood of U in G contains at least the t − u vertices of V (T) \\ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For each T ∈ Kt(G), there are �dom(H) u � sets U ∈ Ku(G) such that each vertex of U is dominating in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For each U ∈ Ku(G), the number of copies T of H in G for which U ⊆ Dom(T) is at most �cdG(U) t−u � N (H − U, Kt−u): we choose t − u vertices from the common neighborhood of U in G to fill out T and then choose how to embed the vertices of H − U, which can be done in at most as many ways as embedding them into a clique of the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore �dom(H) u � � T∈H(G) su G(T) = � U∈Ku(G) � T∈H(G) U⊆Dom(T) su G(T) = � U∈Ku(G) � T∈H(G) U⊆Dom(T) 1 �θG(T)−u+1 t−u � ≤ � U∈Ku(G) � T∈H(G) U⊆Dom(T) 1 �cdG(U)+u−1−u+1 t−u � ≤ � U∈Ku(G) N (H − U, Kt−u) �cdG(U) t−u � �cdG(U) t−u � = N (H − U, Kt−u)N (Ku, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' When G is a disjoint union of complete graphs and any number of components without u-cliques, let Kr be a component of G that contains a u-clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then θG(T) = r − 1 for every T ∈ H(G), and su G(T) = 1 (r−u t−u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The number of copies of H in this component is �r u ��r−u t−u � N (H − U, Kt−u)/ �dom H u � , which can be seen by first choosing a u-clique of the Kr to act as a selected set of u dominating vertices of H, then choosing t − u of the r − u other vertices in the same component, then choosing an embedding of H − U into those vertices (which is independent of the choice of U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Each copy of H is counted this way once for each choice of selected set of u dominating vertices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore � T∈H(Kr) su G(T) = 1 �r−u t−u � �r u ��r − u t − u � N (H − U, Kt−u)/ �dom H u � = N (H − U, Kt−u) �dom H u � N (Ku, Kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' If C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' , Ck are the components of G then � T∈H(G) su G(T) = k � i=1 � T∈H(Ci) su Ci(T) = N (H − U, Kt−u) �dom H u � k � i=1 N (Ku, Ci) = N (H − U, Kt−u) �dom H u � N (Ku, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 Weighting t-cliques by maximum star size By taking H = Kt and u ∈ {1, 2} in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 we obtain the following corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We note that Proposition 1 in [19] is the case t = 2 of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For every n-vertex graph G and every clique size t ≥ 1, � T∈Kt(G) s1 G(T) = � T∈Kt(G) 1 �θG(T) t−1 � ≤ n t , with equality if and only if G is a disjoint union of complete graphs of order at least t when t ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a graph on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We set H = Kt and u = 1 in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1, so s1 G(T) = 1 ( θG(T ) t−1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Any vertex v of Kt is a dominating vertex of Kt, so by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 we have � T∈Kt(G) 1 �θG(T) t−1 � ≤ N (Kt − {v}, Kt−1) �dom(Kt) 1 � N (K1, G) = n t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' If equality holds, then, following the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1, the number of t-cliques containing v is exactly �d(v) t−1 � for every vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For t ≥ 3 this implies every vertex has a complete neighborhood, so every component is a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For any disjoint union of complete graphs G = Kn1 ∪· · ·∪Knk, we have � T∈Kt(G) 1 �θG(T) t−1 � = k � i=1 �ni t � 1 �ni−1 t−1 � = k � i=1 ni t = n t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Note that taking t = 1 in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2 gives the true, if trivial, statement that an n-vertex graph G has at most n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' When t = 2, Malec and Tompkins note in [19, Proposition 1] that equality holds if and only if every component of G is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For every m-edge graph G and every clique size t ≥ 2, � T∈Kt(G) s2 G(T) = � T∈Kt(G) 1 �θG(T)−1 t−2 � ≤ m �t 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Equality holds when G is a disjoint union of complete graphs of order at least t and any number of isolated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a graph on m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We set H = Kt and u = 2 in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1, so s2 G(T) = 1 ( θG(T )−1 t−2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Any two vertices v and w of Kt are dominating vertices of Kt, so by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 we have � T∈Kt(G) 1 �θG(T)−1 t−2 � ≤ N (Kt − {v, w}, Kt−2) �dom(Kt) 2 � N (K2, G) = m �t 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='3 in turn imply the following two known theorems on the maximum number of t-cliques in bounded-degree graphs having a given number of vertices or a given number of edges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' These theorems have also been proved using essentially the same argument as the one used in [23] to give an upper bound on the total number of cliques of all sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The first theorem determines ex(n, Kt, Sr) asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 15 Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='4 (Wood [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let t ≥ 1 and G be a graph on n vertices having ∆(G) ≤ r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then N (Kt, G) ≤ n r �r t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For t ≥ 3 equality holds if and only if r �� n and G is a disjoint union of copies of Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a graph on n vertices with maximum degree at most r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then θG(T) ≤ r − 1 for every T ∈ Kt(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2 we have N (Kt, G) �r−1 t−1 � = � T∈Kt(G) 1 �r−1 t−1 � ≤ � T∈Kt(G) 1 �θG(T) t−1 � ≤ n t , so N (Kt, G) ≤ n t �r−1 t−1 � = n r �r t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' If we instead fix the number of edges, we determine mex(m, Kt, Sr) asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='5 (Wood [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let t ≥ 2 and G be a graph on m edges with ∆(G) ≤ r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then N (Kt, G) ≤ m �r 2 � �r t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Equality holds when �r 2 � �� m and G is a disjoint union of copies of Kr with any number of isolated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a graph on m edges with maximum degree at most r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Again θG(T) ≤ r − 1 for every T ∈ Kt(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='3 we have N (Kt, G) �r−2 t−2 � = � T∈Kt(G) 1 �r−2 t−2 � ≤ � T∈Kt(G) 1 �θG(T)−1 t−2 � ≤ m �t 2 �, so N (Kt, G) ≤ m (t 2) �r−2 t−2 � = m (r 2) �r t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2 Weighting copies of H by maximum star size As mentioned, we can apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 to a broader class of graphs H than just cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' This allows us to prove novel asymptotic results on ex(n, H, Sr) for any H with at least one dominating vertex and on mex(n, H, Sr) for H with at least two dominating vertices: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let H be a graph on t vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' (i) If H has at least one dominating vertex, then ex(n, H, Sr) = (1 − o(1))N (H, ⌊n r⌋Kr), and (ii) if H has at least two dominating vertices, then mex(m, H, Sr) = (1 − o(1))N (H, � m (r 2) � Kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let G be a Sr-free graph on n vertices and m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then for each T ∈ H(G), we have θG(T) ≤ r − 1 and thus s1 G(T) ≥ 1/ �r−1 t−1 � and s2 G(T) ≥ 1/ �r−2 t−2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore, as long as dom(H) ≥ 1, let v be a dominating vertex and apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 with u = 1 to get N (H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' G) · 1 �r−1 t−1 � ≤ � T∈H(G) s1 G(T) ≤ N (H − {v},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Kt−1) dom(H) n so that N (H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' G) ≤ n �r − 1 t − 1 �N (H − {v},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Kt−1) dom(H) = (1 − o(1))N (H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' ⌊n r ⌋Kr),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' when r | n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' we can count copies of H in ⌊n r ⌋Kr by first choosing a vertex v of G to act as a selected dominating vertex in H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' then choosing t − 1 of the r − 1 other vertices in the same component,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' then choosing an embedding of H − {v} into those vertices (which is independent of the choice of v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Each copy of H is counted this way once for each choice of selected dominating vertex in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The asymptotic factor allows for n not divisible by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Furthermore, as long as dom(H) ≥ 2, let v and w both be dominating vertices and apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 with u = 2 to get N (H, G) · 1 �r−2 t−2 � ≤ � T∈H(G) s2 G(T) ≤ N (H − {v, w}, Kt−1) dom(H) m so that N (H, G) ≤ m �r − 2 t − 2 �N (H − {v, w}, Kt−1) dom(H) = (1 − o(1))N (H, � m (r 2) � Kr), where similarly the asymptotic factor allows for m not divisible by �r 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' In both cases we achieve a matching lower bound by taking as many disjoint copies of Kr as possible and making the remaining vertices independent or making the remaining edges a matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' (For the values of n and m when we have some remaining vertices or edges, a better lower bound is given by forming a clique with the remaining vertices or a colex graph with the remaining edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=') Notice that cliques, stars, and all connected threshold graphs have at least one dominating vertex so are included as possible graphs H in part (i) of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2 Hypergraphs We now consider localized bounds for hypergraphs of bounded degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Recall that a hypergraph is q-uniform if every edge is a set of q vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The degree of a set of vertices I, denoted by d(I), is the number of edges E that contain I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Letting i = |I|, the neighborhood of I is the (q − i)-uniform hypergraph {E \\ I : I ⊂ E ∈ E(H)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For a q-uniform hypergraph H and 1 ≤ i < q, we write ∆i(H) for the maximum degree d(I) over all sets I of i vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For t ≥ q, we denote by K(q) t the complete q-uniform hypergraph on t vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We write Kt(H) for the set of t-cliques in H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=', Kt(H) = {S ⊆ V (H) : H[S] ∼= K(q) t }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We use the following upper bound on N (K(q) t , H), which is proved in [16, Theorem 32] as an immediate consequence of Lov´asz’ approximate version of the Kruskal-Katona theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='7 (Lov´asz [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let q, t ∈ N with t ≥ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let H be a q-uniform hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Write the number of edges of H in the form �x q � , where x ≥ q − 1 is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then N (K(q) t , H) ≤ �x t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 17 The following theorem generalizes Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2 to q-uniform hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Note that when q = 2, we have i = 1 and x(I) = d(I) + i = d(v) + 1 for I = {v}, and so the function s(T) in the following theorem can be thought of as an extension of the function s1 G(T) of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1 to hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let t ≥ q > i ≥ 1 and suppose H is a q-uniform hypergraph on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For each I ∈ �V (H) i � , define x(I) ≥ q − i − 1 by the equation d(I) = �x(I)−i q−i � , and, for each T ∈ Kt(H), define x(T) = max � x(I) : I ∈ �T i �� and s(T) = 1 �x(T)−i t−i �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then s(T) is well-defined and decreasing as a function of x(T), � T∈Kt(H) s(T) ≤ �n i � �t i � , and there is an infinite family of hypergraphs that achieve the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let I ∈ �V (H) i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For every T ∈ Kt(I), we have x(T) ≥ x(I) by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' If Kt(I) is nonempty, then d(I) ≥ �t−i q−i � and x(I) ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore every T ∈ Kt(H) has x(T) ≥ t, so w(T) is a decreasing function of x(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Hence T ∈ Kt(I) implies w(T) ≤ 1/ �x(I)−i t−i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore �t i � � T∈Kt(H) s(T) = � I∈(V (H) i ) � T∈Kt(I) s(T) ≤ � I∈(V (H) i ) � T∈Kt(I) 1 �x(I)−i t−i � ≤ � I∈(V (H) i ) �x(I)−i t−i � �x(I)−i t−i � = �n i � , where the second inequality follows from applying Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='7 to the neighborhood of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Design theory provides an infinite family of graphs that meet this bound;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' we direct the reader to [16] for more information on such hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' If H is a q-shadow of a Steiner system S(i, r, n) for some r then by [16, Lemma 38(b)] we have x(I) = r for every I and x(T) = r for every T, so s(T) = 1 (r−i t−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' By [16, Lemma 38(a)] we have N (K(q) t , H) = �r t �(n i) (r i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Therefore � T∈Kt(H) s(T) = �r t ��n i � �r i ��r−i t−i � = �n i � �t i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' It seems interesting and challenging to characterize all of the extremal q-graphs in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' See [16, Theorem 43] for a related characterization of the extremal q-graphs in the non-localized theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' As a corollary of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='8 we obtain the following theorem of Radcliffe and the first author on maximizing the number of t-cliques among bounded-degree q-uniform hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 18 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='9 (Kirsch and Radcliffe [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let 1 ≤ i < q ≤ t and suppose H is an q-uniform hypergraph on n vertices such that ∆i(H) ≤ �x−i q−i � for some real number x ≥ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then N (K(q) t , H) ≤ �n i � �x i � �x t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Proof using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The condition ∆i(H) ≤ �x−i q−i � implies that x(I) ≤ x for every I ∈ �V (H) i � , so x(T) = max{x(I) : I ∈ �T i � } ≤ x for every T ∈ Kt(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='8 gives N (K(q) t , H) �x−i t−i � = � T∈Kt(H) 1 �x−i t−i � ≤ � T∈Kt(H) w(T) ≤ �n i � �t i � , so N (K(q) t , H) ≤ (n i) (t i) �x−i t−i � = (n i) (x i) �x t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 6 Open Problems We briefly mention a few additional instances of problems that we believe are amenable to localized extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The following conjecture is a localized form of a theorem of Frohmader [10], as phrased in [16, Theorem 8], on maximizing the number of t-cliques among m-edge, Kr+1-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let t ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For each T ∈ Kt(G), define αG(T) = max{k : T ⊆ V (S) for some S ⊆ V (G) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' G[S] ∼= Kk} and w′ G(T) = �αG(T) 2 �t/2 �αG(T) t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For every m-edge graph G, � T∈Kt(G) w′ G(T) ≤ mt/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Many extremal results on paths, beginning with the results of Erd˝os and Gallai [9], are conse- quences of extremal theorems regarding cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' While the family of cycle graphs {C3, C4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='} does not have the subgraph inclusion property shared by cliques, paths, and stars, these results consider graphs of bounded circumference (that is, maximum cycle length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' The techniques in this paper often bounded a weight function by arguing a maximal structure could not be extended;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' cycles do not allow such arguments, which could make proving localized results more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Nevertheless, we provide the following weight function and conjectures based on results of Luo [18] and Chakraborti and Chen [6], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let t ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For each T ∈ Kt(G), define γG(T) = max{k : T ⊆ V (S) for some S ⊆ G s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' S ∼= Ck}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 19 Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let t ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For each T ∈ Kt(G), define cG(T) = γG(T) − 1 �γG(T) t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then cG(T) is well-defined and decreasing in γG(T), and � T∈Kt(G) cG(T) ≤ n − 1, with equality if and only if each 2-connected component of G is a complete graph of order at least t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let t ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For each T ∈ Kt(G), define c′ G(T) = �γG(T) 2 � �γG(T) t �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then c′ G(T) is well-defined and decreasing in γG(T), and � T∈Kt(G) c′ G(T) ≤ m, with equality if and only if each 2-connected component of G is a complete graph of order at least t and any number of isolated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' It may be possible to generalize Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='3 to hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' We make the following conjecture as a localized version of Theorem 51 in [16], analogously to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Let t ≥ q > i ≥ 1 and suppose H is a q-uniform hypergraph on m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' For each I ∈ �V (H) i � , define x(I) ≥ q − i − 1 by the equation d(I) = �x(I)−i q−i � , and, for each T ∈ Kt(H), define x(T) = max � x(I) : I ∈ �T i �� and s′(T) = 1 �x(T)−q t−q �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Then � T∈Kt(H) s′(T) ≤ m �t q �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Finally, we used Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='8 to obtain new asymptotically tight bounds on ex(n, H, Sr) when H has at least one dominating vertex and on mex(m, H, Sr) when H has at least two dominating vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' It may be possible to prove similar results for hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Can Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='6 (or Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content='1) be generalized to the setting of q-uniform hyper- graphs with bounded maximum i-degree, perhaps with i = 1 or i = q − 1, in such a way as to obtain new generalized Tur´an-type results for hypergraphs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' Acknowledgments The authors thank Jamie Radcliffe for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9E5T4oBgHgl3EQfmQ_T/content/2301.05678v1.pdf'} +page_content=' 20 References [1] Noga Alon and Clara Shikhelman, Many T copies in H-free graphs, J.' metadata={'source': 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@@ +arXiv:2301.06953v1 [astro-ph.GA] 17 Jan 2023 +Astronomy & Astrophysics manuscript no. FPaanda2 +©ESO 2023 +January 18, 2023 +A new framework for understanding the evolution +of early type galaxies +M. D’Onofrio,⋆1 and C. Chiosi1 +Department of Physics and Astronomy, University of Padua, Vicolo Osservatorio 3, I35122 Padova (Italy) +e-mail: mauro.donofrio@unipd.it +e-mail: cesare.chiosi@unipd.it +Received December, 2022; accepted January, 2023 +ABSTRACT +Context. We have recently suggested that the combination of the scalar virial theorem (Ms ∝ Reσ2) and the L = L′ +0σβ law, with +L′ +0 and β changing from galaxy to galaxy (and with time), can provide a new set of equations valid for investigating the evolution +of early-type galaxies (D’Onofrio & Chiosi 2022). These equations are able to account for the tilt of the Fundamental Plane and to +explain the observed distributions of early-type galaxies in all its projections. +Aims. In this paper we analyze the advantages offered by those equations, derive the β and L′ +0 parameters for real and simulated +galaxies, and demonstrate that, according to the value of β, galaxies can move only along some permitted directions in the fundamental +plane projections. Then, we show that simple galaxy models that grow in mass by infall of gas and form stars with a star formation +rate depending on the stellar velocity dispersion nicely reproduce the observed distributions of early-type galaxies in the Fundamental +Plane projections and yield βs that agree with the measured ones. +Methods. We derive the mutual relationships among the stellar mass, effective radius, velocity dispersion, and luminosity of early- +type galaxies as a function of β and calculate the coefficients of the Fundamental Plane. Then, using the simple infall models, we show +that the star formation history of early-type galaxies is compatible with the σ-dependent star formation rate, and that both positive +and negative values of β are possible in a standard theory of galaxy evolution. +Results. The parameter β(t) offers a new view of the evolution of early-type galaxies. In brief, i) it gives a coherent interpretation of +the Fundamental Plane and of the motions of galaxies in its projections; ii) it is the fingerprint of their evolution; iii) it measures the +degree of virialization of early-type galaxies; iv) and finally it allows us to infer their evolution in the near past. +Key words. galaxies: structure – galaxies: evolution – galaxies: ellipticals and lenticulars – galaxies: scaling relations +1. Introduction +This study is the latest of a series aimed at demonstrating that +the scaling relations (Sc-Rs) for early-type galaxies (ETGs), i.e. +the mutual correlations between the main structural parameters +of galaxies (e.g. the stellar mass Ms, the effective radius Re, the +effective surface intensity Ie, the luminosity L and the central +velocity dispersion σ)1, can be fully understood if we adopt a +new perspective in which the Virial Theorem (VT) of the stellar +systems is coupled to the galaxy luminosity taking into account +that this latter can randomly vary with time as a result of accre- +tion/depletion events associated to mergers/close encounters ex- +pected in the hierarchical galaxy formation scenario in addition +to the natural evolution of its stellar content. The new equation +governing the luminosity is expressed by +L(t) = L′ +0(t)σ(t)β(t). +(1) +in which L, L′ +0, σ and β are all functions of time and can vary +from galaxy to galaxy. This relation is formally equivalent to the +Faber & Jackson relation for ETGs (Faber & Jackson 1976), but +it has a profoundly different physical meaning. In this relation +⋆ Corresponding author: Mauro D’Onofrio +1 Therein after by structural parameters of a galaxy we mean those of +the above list, and leave aside the parameters that define the internal +structure such as the Sérsic index, the axial ratio, etc. Galaxies are con- +sidered point mass objects. +β and L′ +0 are free time-dependent parameters that can vary con- +siderably from galaxy to galaxy, according to the mass assembly +history and stellar evolution of each object. This relation empir- +ically encrypts the effects of all the above physical processes in +terms of luminosity and velocity dispersion variations, param- +eters that can both vary across time because galaxies evolve, +merge, and interact. +In our previous works we tried to highlight some of the +advantages offered by coupling the VT with the L = L′ +0σβ +law. The first efforts were dedicated to understand the origin +of the Fundamental Plane (FP) of ETGs and the distributions +observed in its 2D projections (D’Onofrio et al. 2017a, 2019, +2020; D’Onofrio & Chiosi 2021). While discussing this prob- +lems, D’Onofrio et al. (2017b) and D’Onofrio & Chiosi (2022) +advanced the idea that the explanation invoked for the origin of +the FP tilt (and its small scatter) should also account for the ob- +served distributions of galaxies in all the 2D projections of the +FP. The solution was found in the coupling of the VT with the +time-dependent L = L′ +0σβ relation. +The key idea behind this approach is that the luminosity of +galaxies is not simply related to the total stellar mass, but also to +random variations caused by mergers and interactions. This im- +plies that, accepting the L = L′ +0σβ law as an empirical descriptor +of the possible changes occurring in σ and L, one can describe +a galaxy with two different independent equations: the classical +scalar VT on the notion that galaxies are always very close to +Article number, page 1 of 25 + +A&A proofs: manuscript no. FPaanda2 +the mechanical equilibrium, and the L = L′ +0σβ law that fully ac- +counts for all possible processes taking place during the lifetime +of a galaxy. +Our previous studies have successfully shown that a β param- +eter changing with time and assuming either positive and nega- +tive values, can easily explain the movements and distribution in +the planes of the Sc-Rs. This approach is in fact able to explain +in a natural way the tilt of the FP, the existence of the Zone of +Exclusions (ZoE) observed in many Sc-Rs, and the direction of +motion derived from the changes in σ and L. +In this work we aim to provide evidences that such an ap- +proach gives a global interpretation of the Sc-Rs observed for +ETGs and that even the classical monolithic view of mass as- +sembly is in agreement with the idea of a variable β parameter +thus confirming the L = L′ +0(t)σβ(t) law. +The paper is organized as follows: Sec. 2 gives a short de- +scription of the samples of galaxies (both real and simulated) +used in this work; Sec. 3 is dedicated to the derivation of the +new equations of galaxy evolution and to the different relations +among the structural parameters in all FP projections; Sec. 4 +presents a few new simple models of ETGs growing with a SFR +depending on σ and accounts for the role of β. Finally, Sec. 6 +provides our discussion and conclusions. In all calculations we +used the parameters of the ΛCDM cosmology. +2. The samples of real and model galaxies +Observational data. The observational data used in this +work +are +the +same +of +D’Onofrio & Chiosi +(2021) +and +D’Onofrio & Chiosi (2022). The data for the real galaxies +are extracted from the WINGS and Omega-WINGS databases +(Fasano et al. +2006; +Varela et al. +2009; +Cava et al. +2009; +Valentinuzzi et al. 2009; Moretti et al. 2014; D’Onofrio et al. +2014; Gullieuszik et al. 2015; Moretti et al. 2017; Cariddi et al. +2018; Biviano et al. 2017). +The sample is not homogeneous because the spectroscopic +database is only a sub-sample of the whole optical sample. The +ETGs with available velocity dispersion σ, stellar mass Ms, and +star formation rate SFR, are less numerous than those extracted +from the photometric database (providing Re, Ie, n, LV, etc.). +In particular we used: 1) the velocity dispersion σ of ∼ 1700 +ETGs. The σ measurements come from the SDSS and NFPS +databases (Bernardi et al. 2003; Smith et al. 2004) and were +measured within a circular area of 3 arcsec around the center +of the galaxies; 2) the luminosity, effective radius and effective +surface brightness in the V-band of several thousand ETGs, de- +rived by D’Onofrio et al. (2014) with the software GASPHOT +(Pignatelli et al. 2006). The effective radius is determined from +the luminosity growth curve by considering the circle that con- +tains half the total luminosity. The effective surface intensity fol- +lows directly from the knowledge of L and Re; 3) the distance +of the galaxies derived from the redshift measured by Cava et al. +(2009) and Moretti et al. (2017); 4) the stellar mass obtained by +Fritz et al. (2007), only for the galaxies of the southern hemi- +sphere. +The cross-match between the spectroscopic and optical sam- +ples provides here only 480 ETGs with available stellar mass, lu- +minosity, velocity dispersion, Sérsic index, effective radius and +effective surface brightness. The error of these parameters is +≃ 20%. These are not shown in our plots, because they are much +lower than the observed range of variation of the structural pa- +rameters in the scaling relations and do not affect the whole dis- +tribution of ETGs. +Occasionally, we have also used the catalog by Burstein et al. +(1997) containing objects from Globular Clusters (GCs), Dwarf +Galaxies (DGs) of different types, to late and early type galax- +ies (LTGs and ETGs, respectively), and finally clusters of galax- +ies (GCGs). They are used to have a general idea of the Sc-Rs +for systems of different sizes, but dynamically close to the virial +condition. Limited to ETGs sometime we also used the sample +of Bernardi et al. (2010). +Simulated galaxies. The hydrodynamic simulations, are +probably the best galaxy models today available to compare +theory with observations despite the fact that several problems +still bias their results. There are several suites of galaxy simula- +tions in cosmological context among which we recall Illustris-1 +by Vogelsberger et al. (2014); Genel et al. (2014); Nelson et al. +(2015), recently superseded by Illustris-TNG by Springel et al. +(2018); Nelson et al. (2018a); Pillepich et al. (2018b), and EA- +GLE by Schaye et al. (2015). We decide to adopt here Illustris-1 +for two reasons: first chief the fact we want to be consistent with +the results shown in our previous papers on this same subject that +were based on the Illustris-1 models. Second, we have checked +that the main results of our analysis do not change passing from +Illustris-1 to Illustris-TNG. +The kind of analysis carried out here is indeed somehow in- +dependent of the level of precision reached by models from dif- +ferent sources, because we are mainly interested to present a new +method for deciphering the information encrypted in the obser- +vational data about the past history of ETGs. To this aim, we +have extracted from the Illustris-TNG database at redshift z = 0 +a sample of about thousand model galaxies of all possible masses +that are used to support the above statement. +Our data-set extracted from Illustris-1 consists of several +sub-sets of about ∼ 2400 galaxies each, sampled different at red- +shifts from to z = 0 to z = 4. A full description of these data +is given in Cariddi et al. (2018) and D’Onofrio et al. (2020). In +particular, we collected the effective radii, the total luminosity, +the stellar mass, and the velocity dispersion, the age, and the star +formation rate, together with radii, masses, and velocity disper- +sion of the dark matter component. +A detailed analysis of the differences between Illustris-1 and +Illustris-TNG data has been made by Pillepich et al. (2018b,a), +Rodriguez-Gomez et al. (2019), and Huertas-Company et al. +(2019). One of the issues of major tension between the two suites +of models concerns the radii of the low mass galaxies (roughly of +Ms ≤ 5 1010 M⊙ where the Illustris-TNG radii are about a factor +of two smaller that those of Illustris-1 while above it they are +nearly equal (Pillepich et al. 2018b,a; Rodriguez-Gomez et al. +2019). Huertas-Company et al. (2019) compared the log(Re) − +log(Ms) plane built with the two sources above and the SDSS +data of Meert et al. (2015) finding the same result (see their Fig. +11). +To better illustrate the difference in Fig. 1 we compare +the data of Illustris-1 with those of Illustris-TNG-100 and the +WINGS objects. The difference in the low mass range is con- +firmed, but the hockey-stick like shape of the distribution of +model galaxies in the two samples is the same (see also Fig. 9 +and 11 below). +In addition to this, there is the claim that Illustris-1 simu- +lations do not produce a realistic red sequence of galaxies due +to insufficient quenching of the star formation with too few red +galaxies (Snyder et al. 2015; Bottrell et al. 2017a,b; Nelson et al. +2018b; Rodriguez-Gomez et al. 2019), while the Illustris-TNG +simulations produce a much better red sequence (Nelson et al. +2018b; Rodriguez-Gomez et al. 2019). There is also the prob- +lem of the insufficient number of red galaxies with respect to the +Article number, page 2 of 25 + +M. D’Onofrio and C. Chiosi: A new framework for understanding the evolution of early type galaxies +Fig. 1. Left panel: The stellar mass versus radius relations for the Illustris-1 (open red squares) and the Illustris-TNG-100 (blue dots) samples at +z = 0 and comparison of the models with the WINGS data (black dots). There are 2400 objects for the Illustris-1 sample and about 600 objects for +the Illustris-TNG-100. The mean radii of Illustris-1 are smaller than about a factor of two for stellar masses smaller that about 6 1010 M⊙, while +they are nearly equal if not slightly larger above this limit. Right panel: The L − σ plane of the same data. The symbols and color codes are the +same as in the left panel. +observed population of ETGs. This is of little importance for our +analysis because we do not make use of colors but only of total +luminosities. +Concerning the internal structure of the Illustris-1 galaxies, +Bottrell et al. (2017b) measured the Sersic index, the axis ratio +and the radii of these galaxies and found that too few bulge- +dominated objects are produced in tension with observations. +In contrast the Illustris-TNG galaxies have much better inter- +nal structural parameters (Rodriguez-Gomez et al. 2019). Fortu- +nately, the point mass view of the Illustris-1 models we have +adopted secures that our analysis is not too much affected by +this problem. +Finally, the Illustris-1 data-set does not give information +about the morphology of the galaxies. This means that in our +comparison ETGs and late-type objects are mixed in our plots. +Again Fig. 11 of Huertas-Company et al. (2019) shows us that +ETGs and late-type objects follow very similar trends in the Sc- +Rs. The basic features of the Sc-Rs shown by ETGs are not de- +stroyed with the addition of late-type objects. +Anyway, since in our work we do not make any predic- +tion, but only qualitatively compare observations and simula- +tions. Looking at their behavior in the FP projections, the good +match between data and simulations, and the fact model galaxies +are able to reproduce some particular features visible in the FP +projections (like e.g. the position of the BCGs and the existence +of a ZoE) lend support to the scenario proposed here. All this +makes us confident that the simulations produce galaxies with +luminosity and primary structural parameters not too far from +those of real galaxies. +Given the large heterogeneity of data used here, we remark +that the completeness of the data sample is not fundamental for +the conclusions drawn in this work, because we neither make +any statistical analysis of the data nor we fit any distribution. +The data are only used qualitatively to show that our calculations +are in agreement with the observed distributions of ETGs in the +main Sc-Rs. +The purpose of this paper is only that of proposing a new +possible framework to analyze the evolution of ETGs. +3. The equations of galaxy evolution +The equations tracking the evolution of galaxies are based on +two hypotheses: 1) ETGs are always close to the virial equilib- +rium, a reasonable assumption since the dynamical time scale to +reach such condition is of the order of the free-fall time (< 300 +Myrs); 2) the L = L′ +0σβ law somehow mirrors the effects of +many internal and external events affecting luminosity and ve- +locity dispersion (i.e., mass). The two equations are: +σ2 += +G +kv +Ms +Re +(2) +σβ += +L +L′ +0 += 2πIeR2 +e +L′ +0 +(3) +where kv is the non homology parameter defined by Bertin et al. +(2002). The unknown variables of this system of equations to be +found are β and L′ +0. +Combining these two equations are together, one can write: +a1 log σ + b1 log Ie + c1 log Re + d1 = 0 +(4) +where the coefficients: +Article number, page 3 of 25 + +A&A proofs: manuscript no. FPaanda2 +a1 += +β − 2 +(5) +b1 += +−1 +c1 += +−3 +d1 += +log(Ms) − log(kv/G) − log(2π/L′ +0) +are written in terms of β and L′ +0. The similarity with the FP equa- +tion is clear. This is the equation of a plane in the log(σ) − +log(Ie) − log(Re) space. The novelty is that each galaxy fol- +lows independently an equation like this. In this case, since β +and L′ +0 are time dependent, the equation is telling us which +is the instantaneous direction of motion of an object in the +log(σ) − log(Ie) − log(Re) space and in its projections. +Before showing this, let us trace back the past history of the +reasoning presented in this section. Starting from the same ar- +guments and equations (3 and 4), after tedious algebraic manip- +ulations D’Onofrio & Chiosi (2022) arrived to a cubic equation +in the variable β (their eqn. 10), the coefficients of which where +function of σ, Ie, Re, Ms, and L. The cubic equation was ap- +plied to real galaxies of the WINGS list and model galaxies of +the Illustris-1 catalog. In most cases three real roots were found, +two of them positive and one negative. In some cases the so- +lutions were complex and this was attributed to insufficient ac- +curacy in the input parameters. The mutual agreement between +the two sets of data (WINGS and Illustris-1) was considered as +a strong hint for self consistency of the whole approach. This +agreement was indeed misleading because it masked first an al- +gebraic mistake made while carrying out the lengthy analytical +manipulations (i.e. a factor 0.5 missing in front of a group of +terms in logarithmic form), second that the agreement between +WINGS and Illustris-1 made via the cubic equation was in re- +ality a circular argument as in each case the results would have +been the same regardless of whether the equation was correct or +not. Furthermore, attempts to incorporate the cubic equation in +model galaxies did not lead to a clear understanding of the phys- +ical role and meaning played by the three L = L′ +0σβ relations +associated to each time step (the factor L′ +0 being derived from +the real luminosity by comparison). It was clear that some of the +βs changed sign in the course of evolution and also that complex +solutions could occur during the lifetime of a galaxy, the low +mass ones in particular. However, from these results the tantaliz- +ing suggestion came out that a solution of the puzzle could be +reached by changing strategy. All this led us to revise the whole +problem thus discovering the analytical mistake and putting the +mathematical formulation on the right track. The new version of +the problem is presented here below. The cubic is replaced by +a system of equations in the unknowns β and log L′ +0, thus fully +determining the L = L′ +0σβ and its evolutionary history. +Starting from eqs. 3 and 4, after some algebra it is possible +to write all the relations among the parameters of the FP projec- +tions. For the Ie − Re plane we have: +Ie = ΠRγ +e +(6) +where +γ = (2/β) − (1/2) +(1/2) − (1/β) +and Π is a factor that depends on kv, M/L, β, and L′ +0 and id de- +scribed by: +Fig. 2. Comparison between observed and calculated parameters. The +data of the WINGS database are used. The black solid line marks the +1:1 relationship. The red solid line is the bi-linear least square fit of the +distribution. +Π = + +�2π +L′ +0 +�1/β � L +Ms +�(1/2) � kv +2πG +�(1/2) +1 +1/2−1/β +. +For the Re − σ plane we have: +Re = +��kv +G +� � L′ +0 +2π +� � 1 +Ms +� � 1 +Ie +�� +σ(2+β), +(7) +for the Ie − σ plane: +Ie = +��G +kv +� +(Ms) +� L′ +0 +2π +� � 1 +R3e +�� +σ(β−2) +(8) +and for the Re–Ms plane: +Re = + +�G +kv +� � L′ +0 +2π +�2/β � 1 +Ie +�2/β +β/(β+4) +Mβ/(β+4) +s +. +(9) +It should be remarked here that these equations do not rep- +resent the true physical link between two variables because their +proportionality factor contains other variables as well. In other +words, they do not tell us how Re and Ie vary when σ changes. +They are intermediate mathematical expressions yielding the +structural parameters Re or Ie as functions of the others. Fig- +ure 2 gives an idea of the degree of precision in reproducing the +structural parameters when eqs. (6), (7), (8) and (9) are used. +The x-axis contains the measured parameters, while the y-axis +the values calculated on the basis of our equations. The scatter +in log units ranges from 0.3 − 0.4, so a factor of 2 − 2.5 uncer- +tainty is possible and likely attributable to the ∼ 20% errors of +the scaling parameters. +The importance of these equations is that, starting from them, +one can also write the following equations (in log form): +Article number, page 4 of 25 + +4.5 +4 +log(R) [calculated] +R-M" +[calculated] +4 +3 +3.5 +3 +log(1e) +2 +2.5 +I-R +2.53 +3.5 +4 +4.5 +1 +2 +3 +4 +log(R.) [observed] +log(I.) [observed] +4 +4.5 +[calculated] +R-0 +4 +3 +3.5 +3 +2.5 +1 +1 +2 +3 +4 +2.5 3 3.5 4 4.5 +log(I.)[observed] +log(R.)「observedlM. D’Onofrio and C. Chiosi: A new framework for understanding the evolution of early type galaxies +β[log(Ie) + log(G/kv) + log(Ms/L) + log(2π) + log(Re)] + +(10) ++2 log(L′ +0) − 2 log(2π) − 4 log(Re) = 0 +β log(σ) + log(L′ +0) + 2 log(σ) + log(kv/G) − log(Ms) + +(11) +− log(2π) − log(Ie) − log(Re) = 0 +and assuming: +A += +log(Ie) + log(G/kv) + log(Ms/L) + log(2π) + +(12) +log(Re) +B += +−2 log(2π) − 4 log(Re) +A′ += +log(σ) +B′ += +2 log(σ) − log(G/kv) − log(Ms) − log(2π) − +log(Ie) − log(Re) +write the following system: +Aβ + 2 log(L′ +0) + B = 0 +(13) +A′β + log(L′ +0) + B′ = 0 +with solutions: +β += +−2 log(L′ +0) − B +A +(14) +log(L′ +0) += +A′B/A − B′ +1 − 2A′/A . +In other words, it is possible to derive the values of β and L′ +0 +for each galaxy. This means that the knowledge of the structural +parameters reveals the basic step of galaxy evolution encoded in +the parameters β and L′ +0. +Fig. 3 shows the histograms of the distributions of the β pa- +rameter derived for the galaxies of the WINGS and Illustris-1 +samples respectively. In the upper panel the β values for the +Illustris-1 data are obtained from the ∆L/∆σ ratio measured on +the L − σ plane. This is possible by considering the values of L +and σ at two close redshift epochs (z = 0.2 and z = 0). In the +lower panel we have considered only the objects that are close to +the virial equilibrium, i.e. those for which: +2 log(σ) = log(G/kv) + log(Ms/L) + log(2π) + log(Ie) + log(Re) +(15) +within a 20% uncertainty, and calculated β using our new an- +alytical equations. When this condition is satisfied we get that +2A′/A = 1 and β and L′ +0 diverge. +Notably the values of β are both positive and negative and +there is a clear deficiency of objects with β close to 0. This is +true both for WINGS and Illustris-1. The average value of β is +−2.44 with a rms scatter of ∼ 178. The positive values range +from 1.05 to 1531, while the negative ones from −5.4 to −3860. +The importance of eqs. (15) is that we have now an empiri- +cal thermometer of the virial condition, realized when β and L′ +0 +diverge. +The meaning of Fig. 3 is that galaxies during their evolu- +tion can acquire either positive and negative values of β, depend- +ing on the particular events experienced (merging, stripping, star +Fig. 3. Upper panel: Histogram of the β solutions derived from eq. 15 +for the real ETGs (black line) compared with the distribution derived for +the galaxies of the Illustris-1 simulation when β is calculated looking at +the variation of luminosity and velocity dispersion in two close redshift +epochs (z = 0.2 and z = 0); Lower panel: Histogram of the β solutions +derived from eq. 15 for the real ETGs (black line) and the Illustris- +1 galaxies (red line) that are close to the virial equilibrium. The solid +black line marks the average value of β. +Fig. 4. The distribution of β as a function of the degree of virialization. +formation, etc.), and this has immediate effects on the structural +parameters in the Sc-Rs, that change accordingly. Consequently, +the Sc-Rs seen in their temporal framework become sources of +information for the global evolution of the stellar systems. +Figure 4 shows the distribution of β as a function of the de- +gree of virialization, expressed by the quantity 1 − 2A′/A de- +rived. The large values of β are attained by objects very close to +the virial condition. On the other hand, the small β’s belong to +objects still away from this condition. +Article number, page 5 of 25 + +200 +100 +* +0 +-100 +1: +1: +-200 +0.4 +-0.2 +0 +0.2 +0.4 +1-2A'/A0.25 +0.2 +8=4L/40 +WINGS +Illustris +0.15 +N +0.1 +0.05 +0 +-100 +0 +100 +0.25 +0.2 +Arnalytic B +WINGS +Illustris +0.15 +0.1 +0.05 +0 +-100 +0 +100A&A proofs: manuscript no. FPaanda2 +Fig. 5. Histogram of the β solutions derived from eq. 15 for the real +ETGs (black line) of WINGS, the Illustris-1 galaxies (red line) that are +close to the virial equilibrium, and the Illustris-TNG galaxies (blue line) +in the same conditions. The three histograms are qualitatively similar, +thus confirming that our analysis depends little on the choice between +the two theoretical data-bases. +In closing this section, we show the distribution of the βs we +would obtain with the galaxy models of the Illustris-TNG-100 +sample and compare it with those of Illustris-1 and the WINGS +data. The three histograms are shown in Fig. 5. The difference is +very small and largely due to the smaller number of galaxies in +the Illustris-TNG sample. +3.1. Trends in the FP projections +In this section we try to better explain the reasons why β can +change sign during the life of a galaxy or when passing from +one galaxy to another. The advantage of knowing β is much clear +when we look at the projections of the FP. Eqs. (6), (7), and (8) +can be further elaborated to eliminate the dependence on Ie and +Re present in their zero-points. We get: +Ie = +�G +kv +L′ +0 +2π MsΠ3/γ +� β−2 +1+3/γ +σ +β−2 +1+3/γ +(16) +Re = +�G +kv +L′ +0 +2π +Ms +Π +� +σ +β−2 +3+γ +(17) +Re = +� +(G +kv +)β/2 L′ +0 +2π +1 +Π +� +2(β−2) +β2−6β+12 +M +β2−2β +β2−6β+12 +s +(18) +These relations now better represent the mutual dependence +of the structural parameters (e.g. of Ie as a function of Re and σ +and of Re as function of σ) and clarify what is the role of β. When +a galaxy moves in the L−σ plane, according to the values of β, it +does the same also in the other FP projections, according to the +slopes reported in the last three columns of Tab. 1. These slopes +Fig. 6. The Ie − Re plane. The black and red arrows mark the direction +of motion of galaxies in this plane for large negative and positive values +of β. The black solid line gives the lsq fit of the data, while the broken +line represents the zone of exclusion. +depend on β and indicate the direction of motions (marked by +the arrows) that are visible in Figs. 6-11. +Figure 6 shows the case of the Ie − Re plane. The black and +red arrows mark the direction of motion of galaxies predicted on +the basis of their negative and positive values of β respectively. +Since the WINGS galaxies are well virialized, the values of β +are always very large, either positive and negative. Both such +slopes give consistently a direction of motion close to ∼ −1 in +the Ie − Re plane. The −1 slope is that predicted on the basis +of the VT (represented by the broken line which also marks the +ZoE) (see, D’Onofrio & Chiosi 2021)). We note that no galaxies +can cross the ZoE, because their motion is nearly parallel to the +ZoE. +These arguments demonstrate the reason why there is a ZoE +in the Ie−Re plane: the only possible direction of motion for well +virialized objects is that with slope ∼ −1 generated by the large +positive and negative values of β. +When β ∼ 0, i.e. when the galaxies are less virialized, they +can move in other directions in this plane. Unfortunately, our +sample does not include the dwarf ETGs, that are usually dis- +tributed in a cloud, below the ZoE with radii lower than 3-4 kpc +(see e.g. Capaccioli et al. 1992; D’Onofrio et al. 2020). For these +objects we predict values of the slopes in all possible directions. +The only way to check this is to make use of the model +galaxies either of Illustris-1 or Illustris-TNG. Figure 7 confirms +our prediction. Although the data of Illustris-1 are affected by +the well known problem of the systematically larger Re with re- +spect to the observed ones (D’Onofrio et al. 2020; Bottrell et al. +2017c), we can note that several objects have arrows nearly or- +thogonal to those of the well virialized galaxies. The expected +motions of the dwarf galaxies are in all possible directions, +thus giving rise to the cloud of the "ordinary" ETGs defined +by Capaccioli et al. (1992). Furthermore, when the distribution +curves its shape, we note a progressive variation of the arrow di- +rections. This means that the overall distribution in this plane is +Article number, page 6 of 25 + +B>0 +7 +([z-3d])60] +3 +3.5 +4 +4.5 +5 +log(R[pc])0.2 +WINGS +Illustris-1 +0.15 +0.1 +Tot +0.05 +Z +0 +N +0.2 +WINGS +Ilustris-TNG +0.15 +0.1 +0.05 +0 +100 +0 +100 +βM. D’Onofrio and C. Chiosi: A new framework for understanding the evolution of early type galaxies +Fig. 7. The Ie − Re plane for the WINGS and Illustris-1 data. The black +and red arrows mark the direction of motion of the WINGS galaxies +in this plane for large negative and positive values of β. The green and +blue arrows are those of Illustris-1 for negative and positive values of +β. In the plot we have used only 1/10 of the Illustris-1 galaxies in order +to permit to distinguish the objects with different β values moving in +different directions. +governed by the different movements of the galaxies in the L−σ +plane described empirically by the different values of β and L′ +0. +The direction of the arrows displayed in each figure visual- +izes the expected displacement of a galaxy based on the actual +value of β. However, the arrows only give the direction of mo- +tion, not the orientation of the future temporal evolution of a +galaxy. Furthermore, they do not indicate the path followed by +each galaxy to reach the current observed position in the dia- +grams. +The same can be said for the other two FP projections. Fig- +ures 8 and 10 represent the Re −σ and Ie −σ planes respectively. +Here, the role of β is much more clear. It is well evident in fact +that the galaxies with negative β’s move in different directions +with respect to those with positive β, originating the curvatures +observed in these diagrams. +Once more the addition of the Illustris-1 data (Fig. 7 and +9) confirms that the slopes derived from the βs are consistent +with the observed distribution of ETGs and demonstrates that +the observed curvatures originate from the different motion of +galaxies with positive and negative values of β. +Figures 12 and 13 are similar plots for the log(Re) − log(Ms) +plane. Even in this important diagram we observe a ZoE (marked +by the dashed line with slope equal to 1). Our calculations pre- +dict why this ZoE is here: the reason is that all the virialized +objects (with large β values) can only move in the direction with +slope equal to 1 (see Table 1). The Re of Illustris-1 are notori- +ously somewhat larger than those measured, but the general be- +havior is in good agreement with the observed distribution. The +galaxies with large positive and negative values of β move with +a slope close to 1, while in the cloud of points with small masses +we can note objects with different directions. +We conclude that all the positions of ETGs in the FP projec- +tions and in the log(Re) − log(Ms) plane depend on the motions +Fig. 8. The Re − σ plane. Symbols and colors as in Fig. 6. +Fig. 9. The Re −σ plane for WINGS and Illustris-1. Symbols and colors +as in Fig. 7. +occurred during the peculiar evolutionary path followed by each +galaxy. When the galaxies are well virialized these motions can +occur only in well fixed directions depending on the value of β. +This is a coherent and self-consistent explanation of all the +main scaling relations built with the structural parameters of +ETGs. It follows that even the FP, the father of the scaling re- +lations, must find a similar explanation. +Looking at Table 1 in detail we also note that: +– In all FP projections, when β becomes progressively nega- +tive, i.e. when the objects are rapidly declining in their lu- +minosity at nearly constant σ, the slopes either converge to +the values predicted by the VT (in the Ie − Re relation and +in the Re–Ms relation), or diverge toward large values (in the +Article number, page 7 of 25 + +5 +β>0 8>0 +4.5 +log(R[pc) +4 +3.5 +3 +2.5 +1.5 +2 +2.5 +log(α[km s-1)5 +R>0 +0>d +4.5 +log(R[pc]) +4 +3.5 +3 +2.5 +1.5 +2 +2.5 +log(α[km s-1l)B>0 β>0 +3 +0 +3 +3.5 +4 +4.5 +5 +log(R[pc])A&A proofs: manuscript no. FPaanda2 +Table 1. The slopes of the Ie − Re, Re − σ, Ie − σ and Re − Ms planes for different values of β. +β +Ie − Re +Re − σ a +Ie − σ b +Re-Ms c +Re − σ d +Ie − σ e +Re-Ms f +100.0 +-0.98 +102.0 +98.0 +0.96 +48.50 +-47.51 +1.04 +50.0 +-0.96 +52.0 +48.0 +0.92 +23.51 +-22.53 +1.08 +10.0 +-0.75 +12.0 +8.0 +0.71 +3.55 +-2.66 +1.54 +5.0 +-0.33 +7.0 +3.0 +0.55 +1.12 +-0.37 +2.14 +3.0 +1.00 +5.0 +1.0 +0.43 +0.25 +0.25 +1.00 +2.0 +0.00 +4.0 +0.0 +0.33 +0.0 +0.0 +0.00 +1.0 +-3.00 +3.0 +-1.0 +0.20 +0.0 +0.0 +-0.14 +0.5 +-2.33 +2.5 +-1.5 +0.11 +-2.25 +5.25 +-0.08 +0.0 +-2.00 +2.0 +-2.0 +0.00 +-2.00 +4.00 +0.28 +-0.5 +-1.80 +1.5 +-2.5 +-0.14 +-2.08 +3.74 +0.08 +-1.0 +-1.67 +1.0 +-3.0 +-0.33 +-2.25 +3.75 +0.16 +-2.0 +-1.50 +0.0 +-4.0 +-1.00 +-2.66 +4.00 +0.28 +-3.0 +-1.40 +-1.0 +-5.0 +-3.00 +-3.12 +4.37 +0.38 +-5.0 +-1.28 +-3.0 +-7.0 +5.00 +-4.08 +5.25 +0.52 +-10.0 +-1.16 +-8.0 +-12.0 +1.67 +-6.54 +7.63 +0.69 +-50.0 +-1.03 +-48.0 +-52.0 +1.08 +-26.51 +27.53 +0.92 +-100.0 +-1.02 +-98.0 +-102.0 +1.04 +-51.50 +52.51 +0.96 +Notes: a) Slope when kv, Ms and Ie are constant; b) Slope when kv, Ms and Re are constant; c) Slope when kv, and Ie are constant; +d) Slope when kv and Ms are constant: e) Slope when kv and Ms are constant; f) Slope when kv is constant. +Fig. 10. The Ie − σ plane. Symbols and colors as in Fig. 6. +Ie − σ and Re − σ relations), because the galaxy keeps its ve- +locity dispersion when the luminosity decreases (only Ie and +Revary). This offers a natural explanation of the ZoE. +– Positive and negative values of β are equally permitted, both +with real and simulated data. In general, the objects that are +still active in their star formation or have recently experi- +enced a merger, have positive values of β, while those pro- +gressively quenching their SF have increasing negative β. +– The "curvature" in the observed distributions (i.e. the transi- +tion from the large cloud of small galaxies to the much nar- +row tail of the brightest objects) is naturally explained by the +existence of positive and negative values of β. +A way to better understand the effects played by β is to think +at the possible variations of Re and Ie when L and σ vary in +Fig. 11. The Ie−σ plane for WINGS and Illustris-1. Symbols and colors +as in Fig. 7. +Table 2. Trends of the effective parameters as a consequence of changes +in σ and L. +β > 0 +L&σ ր +Re ր +Ie(const. or ց) +Ms ր +L&σ ց +Re ց +Ie(const. or ր) +Ms ց +β < 0 +L ց & σ ր +Re ց +Ie(const. or ր) +Ms(const. or ր) +L ր & σ ց +Re ր +Ie(const. or ց) +Ms(const. or ց) +Article number, page 8 of 25 + +B>0B≥0 +3 +log(Ie[Lopc-2]) +2 +0 +1.5 +2 +2.5 +log(αkm s-1])8>0 +3 +2 +1.5 +2 +2.5 +log(α[km s-1])M. D’Onofrio and C. Chiosi: A new framework for understanding the evolution of early type galaxies +Fig. 12. The log(Re) − log(Ms) plane for the WINGS galaxies. Symbols +and colors as in Fig. 6. +Fig. 13. The log(Re)−log(Ms) plane for WINGS and Illustris-1 galaxies. +Symbols and colors as in Fig. 7. +the L − σ plane. There are four possible changes of L and σ +in this plane. They are schematically shown in Table 2, which +displays, according to the values of β, the expected variations of +Reand Ie, when L, Ms and σ vary. Note that when β is negative, +not necessarily there is a decrease in luminosity, and when β is +positive, a decrease in luminosity might also occurs. +When the luminosity of a galaxy changes, both the effective +radius and the mean effective surface intensity Ie vary. This hap- +pens because Re is not a physical radius, like e.g. the virial radius +(which depends only on the total mass), but it is the radius of the +circle that encloses half of the galaxy total luminosity. Since the +ETGs have different stellar populations with different ages and +metallicity, it is highly improbable that the decrease in luminos- +ity does not change the whole appearance of the luminosity pro- +file2. Consequently the growth curve changes and determines a +variation of Re and Ie. If the luminosity decreases passively, in +general one could expect a decrease of Re and an increase of Ie. +On the other hand, if a shock induced by harassment or stripping +induces an increase of L (and a small decrease in σ), we might +expect an increase of Re and a decrease of Ie. +The observed variations of these parameters depend strongly +on the type of event that a galaxy is experiencing (stripping, +shocks, feedback, merging, etc.). In general, one should keep in +mind that these three variables L, Re and Ie are strongly coupled +each other and that even a small variation in L might result in am- +ple changes of Re and Ie. In this context, we begin to understand +that the Sc-Rs are useful tools for guessing both the dynamics +and the evolutionary state of the stellar content of a galaxy. +In summary, what we claim here is that all the above dia- +grams should be analyzed taking into account the effects of time +and should not be investigated separately. They are snapshots +of an evolving situation, and such temporal evolution cannot be +discarded. The L = L′ +0σβ law catches such evolution in the cor- +rect way by predicting the direction of the future motion of each +galaxy in the diagnostic planes (D’Onofrio and Chiosi, A&A +submitted). In principle, this way of reasoning should allow us +to understand why galaxies are in the positions observed today +in each diagram. As β gives only the present direction of motion +and not that of the motion in the past, the simultaneous use of +simulations and high redshift observations might help to infer +the possible precursors of the present day galaxies on the basis +of the physical properties and the distribution in the FP projec- +tions, indicated by the values of β. In other words these scaling +relations become a possible tool for inferring the evolutionary +path of each galaxy. +3.2. Origin of the FP and its tilt +The final step is that related to the question of the origin of the +FP. Equation (4) tells us that each galaxy follows its own FP-like +equation, whose coefficients are functions of β. Starting from eq. +(4) it is possible to derive the coefficients a, b and c of the plane +hosting each single ETG. To do this we adopt the notation that +is commonly used for the FP, in which ⟨µ⟩e is expressed in mag +arcsec−2 and Re in kpc: +log(Re) = a log(σ) + b < µ >e +c. +(19) +We get: +a += +a1/(−c1) +(20) +b += +b1/(−c1) +c′ += +d1/(−c1) +where a1, b1, c1, and d1 are from eq. (4), and c = (10.56 ∗ b1)/ − +c1) − 3 + c′. This transformation is necessary because in our no- +tation Ie is expressed in L⊙pc−2 and Re is in pc. +The distribution of these coefficients for all the WINGS sam- +ple of 479 galaxies is shown in Fig. 14. It is clear from the figure +that the values for the FP coefficients, derived from the fit of +the ETGs distribution and indicated by the dashed areas in each +panel, are very close to the average of the single coefficients cal- +culated by eq. 20 (the vertical black lines). The gray bands show +2 This could happen only in a coeval stellar system with the same type +of stars in any galaxy volume. +Article number, page 9 of 25 + +8>0 +8>0 +5 +0>9 +4.5 +log(Re[pc) +3.5 +3 +2.5 +6 +10 +1 1 +12 +13 +log(M.[M.)5 +8>0 +0>9 +4.5 +log(R[pc]) +4 +3.5 +3 +2.5 +9 +10 +11 +12 +13 +log(M.[M.l)A&A proofs: manuscript no. FPaanda2 +Fig. 14. Histograms of the values of the calculated FP coefficients a +(top panel), b (middle panel) and c (bottom panel). In each panel we +show the histogram of the coefficient (blue line), the average value (the +vertical black line) and the median (the vertical red line). In the case +of the b coefficient, that does not depend on β, average and median are +the same (the black and red vertical lines coincide). The dashed gray +regions mark the intervals of the FP coefficients found by fitting the +distribution of ETGs in the log(σ) − log(Ie) − log(Re) space. +indeed the interval of a, b and c obtained by D’Onofrio et al. +(2008) fitting the FP of ETGS separately for each cluster of the +WINGS data-set. +In other words, we have framed the FP and its tilt in a new +context in which each ETG follows its own eq. (4), namely FP, +and contributes to shape the global FP (both tilt and thickness) of +the ETG population. Since the FP coefficients are obtained from +a fit, it is clear that the final coefficients of the plane will be close +to the average of the single values valid for each object. Some +differences are expected because the final values will depend on +the sample adopted (each having its own average) and from the +technique used to perform the fit. +With this statement we do not mean to say that the various +mechanisms invoked in the literature to explain the tilt and thick- +ness are incorrect. Rather, we claim that all of them can actually +contribute to the average properties of the galaxy sample, giving +rise to a different β for each object. +Since its discovery, the FP has been the subject of several +studies aimed at understanding why the plane is tilted with re- +spect to the prediction of the VT (Ms ∝ Reσ2), and why its in- +trinsic scatter is so small (see e.g., Faber et al. 1987; Ciotti 1991; +Jorgensen et al. 1996; Cappellari et al. 2006; D’Onofrio et al. +2006; Bolton et al. 2007, among many others). While the VT +predicts a = 2 and b = −1, the values coming from the fit of +several samples of ETGs are systematically lower (a ∼ 1.2) and +higher (b ∼ −0.8), and vary according to the sample used and +the fitting strategy. +Among the physical mechanisms invoked to explain +the FP tilt we can find: 1) a progressive change of the +stellar +mass-to-light +ratio +(Ms/L) +(see +e.g., +Faber et al. +1987; +van Dokkum & Franx +1996; +Cappellari et al. +2006; +van Dokkum & van der Marel +2007; +Holden et al. +2010; +de Graaff et al. +2021); +2) +structural +and +dynamical +non- +homology (see e.g., Prugniel & Simien 1997; Busarello et al. +1998; Trujillo et al. 2004; D’Onofrio et al. 2008); 3) dark +matter (DM) content and distribution (see e.g., Ciotti et al. +1996; Borriello et al. 2003; Tortora et al. 2009; Taranu et al. +2015; de Graaff et al. 2021); 4) star formation history (SFH) +and initial mass function (IMF) (see e.g., Renzini & Ciotti +1993; Chiosi et al. 1998; Chiosi & Carraro 2002; Allanson et al. +2009); 5) the effects of environment (see e.g., Lucey et al. +1991; +de Carvalho & Djorgovski +1992; +Bernardi et al. +2003; +D’Onofrio et al. +2008; +La Barbera et al. +2010; +Ibarra-Medel & López-Cruz 2011; Samir et al. 2016). +Recent observational work has shown that variations in +the Ms/L ratio can account only for half of the tilt (see +D’Eugenio et al. 2021), with remainder being due to structural +variation and possibly variations in the galaxy-averaged initial +mass function of the stellar populations. Uncertainties in Ms +can affect the tilt if the error is mass-dependent, although this +systematic uncertainty is not large enough (see Leja et al. 2019; +Lower et al. 2020). Schechter et al. (2014) using strong lensing +measurement provides an independent estimate of Ms but still +finds a tilt of the FP. Finally, the tilt is found in cosmological +simulations (Rosito et al. 2019a,b; de Graaff et al. 2022). +All these effects are in practice "involved" in our view of +the problem. Indeed, since each sample of galaxies has its own +average value of β, because of the different history of mass ac- +cretion and luminosity evolution, it is easy to verify that sys- +tematic changes in the tilt could arise for the above mentioned +reasons. When the sample changes its average properties, a small +variation of the tilt of the FP follows. This explains for instance +why Robertson et al. (2006) found that star-forming and quies- +cent galaxies follow different Sc-Rs, i.e. a different FP tilt, due +to differences in the merger histories. +With the emerging of the hierarchical scenario of galaxy for- +mation and evolution, some additional mechanisms for the FP +tilt have been proposed: 1) the effects of dissipation-less merg- +ing (Nipoti et al. 2003); 2) the gas dissipation (Robertson et al. +2006); 3) the non regular sequence of mergers with progressively +decreasing mass ratios (Novak 2008); 4) the multiple dry merg- +ers of spiral galaxies (Taranu et al. 2015). +Since the galaxy properties change with time, the slope of +the FP is expected to change with redshift. This is confirmed by +the numerical models of single galaxies, large scale cosmologi- +cal simulations, and observational surveys at different redshifts, +among others see Beifiori et al. (2017); Rosito et al. (2019a,b); +Lu et al. (2019); Ferrero et al. (2021); de Graaff et al. (2022) and +references. +The most remarkable physical feature of the FP is the ob- +served very small scatter, which amounts to ≈ 0.05 dex in +the V-band. It seems to require a sort of fine tuning among +different physical processes. The scatter has been attributed +to: 1) the variation in the formation epoch; 2) the DM con- +tent; 3) the existence of metallicity or age trends; 4) the vari- +ations of the mass-to-light ratio M/L (see e.g., Faber et al. +1987; Gregg 1992; Guzman, R. et al. 1993; Forbes et al. 1998; +Bernardi et al. 2003; Reda et al. 2005; Cappellari et al. 2006; +Bolton et al. 2008; Graves et al. 2009; Graves & Faber 2010; +Auger et al. 2010; Magoulas et al. 2012). +Our approach cannot predict the scatter around the FP, be- +cause this does not depend on the structural parameters, but on +the properties of the stellar populations and the peculiar history +of mass accretion/stripping. +Although investigating the causes of the tilt and the small +dispersion around the FP is beyond the aims of this paper, let as +Article number, page 10 of 25 + +a +100 +50 +0 +20 +-10 +0 +10 +20 +b +100 +50 +0 +0.2 +0 +0.2 +0.4 +80 +60 +C +40 +20 +0 +-40 +-20 +0 +20 +40M. D’Onofrio and C. Chiosi: A new framework for understanding the evolution of early type galaxies +conclude this section with one consideration: it may be an amaz- +ing coincidence, but we note that going back to high redshift, +the numerical simulations of Illustris-1 and Illustris-TNG show +that the FP and the tail of the MRR persist until redshift z ≃ 1.6 +and then disappear or is no longer so well defined (D’Onofrio & +Chiosi, 2022; Chiosi et al., 2022, in preparation). The concomi- +tant appearance of the MRR and FP for ETGs more massive that +about 1010 M⊙ maybe is a mere coincidence, but surely it is a +question that must be investigated. +The galaxies on the tails of the FP and the MRR are mas- +sive ETGs whose stellar content is predominantly made of old +stars or in the case of mergers with objects of smaller mass the +percentage of younger stars does not alter significantly the lumi- +nosity and the colors of the basic stellar populations. To quantify +this statement let us make the following example. At proceed- +ing galaxy building via the hierarchical scenario, the probability +that a massive objects merge with another of similar mass be- +comes rarer and rarer as galaxies become more massive. There- +fore, massive ETGs tends to evolve in isolation or merging ob- +jects of much smaller mass. In general, the merger of two galax- +ies with very different masses (e.g., M1/M2 ≃ 1/10) and some +companion stellar activity leaves the mass and velocity disper- +sion nearly unchanged while the luminosity first undergoes a +burst of short duration and relative intensity proportional to the +luminosity ratio L1/L2 (only slightly higher than the previous +value). This should correspond to a nearly vertical shift on the +FP of small amplitude. The M/L ratio either remain unchanged +or slightly decrease thus causing a little scatter of the FP. +The opposite should occur in a merger between two galaxies +of nearly equal mass, typical situation in the range of low mass +galaxies. In this case, the mass and luminosity both change. If +additional star formation occurs, there should be an additional +increase of the total luminosity that depends on the amount +of mass converted into new stars. Therefore, the total lumi- +nosity should hardly recover the pre-burst value, the mass- +weighted mean of the two component galaxies (see Fig.11 in +Tantalo & Chiosi 2004b). So most likely the luminosity remains +higher than before, and the M/L ratio is expected to decrease. +This should generate a tilt of the FP in the right direction. It is +not easy to foresee the effect on the scatter. Better estimates re- +quire numerical simulations of burst of star formation. In any +case after a short time interval the maximum shift in luminosity +cannot overcome a factor of ∼ 2 (0.3 dex). +4. Application to model galaxies +To lend support to the picture outlined above concerning the +physical role and meaning of the L = L′ +0σβ relation and the +role of the parameters β and L′ +0 without resorting to the numeri- +cal simulations of Illustri-1 and Illustris-TNG of which we have +no control at all, we make use of very simple, almost analytical +models of galaxy formation and evolution. The ideal models of +this type suited to describe ETGs are those in which the total +mass increases by infall and the stars are formed according to +a simple law of star formation rate (SFR) that have been devel- +oped long ago by Chiosi (1980) and extended by Tantalo et al. +(1998a). The novelty here is that we have incorporated the equa- +tions for β and L′ +0 (eqs. 4) into the models once the luminos- +ity, the radius and the velocity dispersion are calculated. With +the aid of these models and eqs. (16) and (17) we have calcu- +lated the basic relationships Ie − σ and Re − σ, and finally made +a cross-test of mutual consistency between the results from the +galaxy models and the β and L′ +0 theory. These simple model of +galaxy formation and evolution were first proposed by Chiosi +(1980), much later extended by Tantalo et al. (1998a), and re- +cently used by Chiosi et al. (2017) to study the cosmic SFR +and by Sciarratta et al. (2019a) to investigate the galaxy color- +magnitude diagram. Although they may look too simplistic com- +pared to the numerical models of Illustris-1 and Illustris-TNG, +yet they catch the main features of these latter and are suitable to +our purposes. +In brief, a galaxy of total mass MG is made of baryonic (B) +and dark matter (D), with mass MB and MD respectively, and at +any time satisfies the equation: +MG(t) = MB(t) + MD(t). +(21) +At all times MB(t) and MD(t) are in cosmological proportion, +i.e. they satisfy the condition MD(t) = fcMB(t) where fc depends +on the adopted ΛCDM cosmological model of the Universe (fc ≃ +6.1 in our case). +The baryonic mass is supposed to be originally in form of +gas, to flow in at a suitable rate and, when physical conditions +allow it, to transform into stars. With the same rate also dark +matter is let flow in together with the baryonic matter to build up +the total gravitational potential. Suitable prescriptions of their +spatial distribution are needed to calculate the gravitational po- +tential (see Tantalo et al. 1998a, for more details). +This kind of galaxy model is named "infall model", the +essence of which resides in the gas accretion into the cen- +tral region of the proto-galaxy at a suitable rate (driven by the +timescale τ) and in the gas consumption by a Schmidt-like law +of star formation. The gas accretion and consumption coupled +together provide a time dependent SFR closely resembling the +one resulting from N-body simulations (e.g. Chiosi & Carraro +2002; Merlin & Chiosi 2006, 2007; Merlin et al. 2012). +At any time t the baryonic mass MB is given by the sum: +MB(t) = Mg(t) + Ms(t), +(22) +where Mg(t) is the gaseous mass and Ms(t) the mass in stars. At +the beginning, both the gas and the star mass in the proto-galaxy +are zero Mg(t = 0) = Ms(t = 0) = 0. The rate of baryonic mass +(and gas in turn) accretion is driven by the timescale τ according +to: +dMB(t) +dt += MB,τ exp(−t/τ), +(23) +where MB,τ is a constant with the dimensions of [Mass/Time] to +be determined by imposing that at the galaxy age TG the total +baryonic mass of the galaxy MB(TG) is reached: +MB,τ = +MB(TG) +τ[1 − exp(−TG/τ)]. +(24) +Therefore, by integrating the accretion law, the time dependence +of MB(t) is: +MB(t) = +MB(TG) +[1 − exp(−TG/τ)][1 − exp(−t/τ)]. +(25) +Since dark matter flows in at the same rate of the baryonic +matter, it obeys similar equations in which MB is replaced by +MD. However, since at any time MB and MD, are in cosmic pro- +portions, MD = fcMB, the equations for MD are superfluous and +Article number, page 11 of 25 + +A&A proofs: manuscript no. FPaanda2 +the normalization on MB is enough. The underlying hypothesis +is that the presence of dark matter does not affect the evolution +of the baryonic component, but for its effect on the gravitational +potential energy. To this aim, some assumptions about the spatial +distribution of MB and MD are needed. In other words, assuming +spherical symmetry, the radii RB and RD must be specified. +The timescale τ is related to the collapse time and the average +cooling rate of the gas. Therefore, it is expected to depend on the +mass of the system. At the same time, the gas mass increases by +infall and decreases by star formation. +The rate of star formation is modeled throughout the whole +life of the galaxy with the Schmidt (1959) law: +Ψ(t) ≡ dMs +dt += νMg(t)k, +(26) +where k regulates the dependency of the SFR on the gas content: +we assume k = 1. The quantity ν is the efficiency parameter of +the star formation process that must be specified (see below). +In the infall model, because of the interplay between gas ac- +cretion and consumption, the SFR starts low, reaches a peak after +a time approximately equal to τ and then declines. The func- +tional form that could mimic this behavior is the time delayed +exponentially declining law: +Ψ(t) ∝ t +τ exp +� +− t +τ +� +. +(27) +The Schmidt law in eq. 26 is therefore the link between gas ac- +cretion by infall and gas consumption by star formation. +As a whole, this kind of approach stands on a number of ob- +servational and theoretical arguments among which we recall: (i) +the parameters ν and τ can be related to morphology (Buzzoni +2002) and to the presence of ongoing star formation activity in- +side observed galaxies (Cassarà et al. 2016); (ii) the aforemen- +tioned quantities can be easily tuned in order to fit observational +data, and also complex phenomena that would affect the rate of +gas cooling, such as active galactic nuclei (AGN), can be em- +pirically taken into account without going into detail (see e.g. +Chiosi et al. 2017). +The infall models we have described may include many im- +portant physical phenomena, for instance gas heating by su- +pernova explosions (both type Ia and type II), stellar winds, +gas cooling by radiative emission, and the presence of galactic +winds. See the study by Tantalo et al. (1998a) for all details on +these topics. +4.1. Outline of the galaxy models +The complexity of real globular clusters, galaxies and galaxy +clusters and the history of their evolution are reduced here to +ideal systems of which we know the current masses M(t), MB(t), +MD(t), Ms(t), Mg(t) together with the mass abundances of some +important elements Xi(t) (where i stands for H, He, C, N, O, +Mg, ... Fe) and total abundance of heavy elements Z(t) 3, and +finally half-stellar mass (half-light) radius Re(t), and dark mass +radius RD(t). At each time, the system contains a manifold of +stellar populations of different metallicity and age which can +be approximated by single stellar populations (SSP) of mean +metallicity < Z(t) > and mean age T(t) defined by the relation +3 For more details on chemical enrichment, companion equations and +chemical yields per stellar generation see Tantalo et al. (1998a) +T(t) = Ms(t)/ < Ψ(t) > where < Ψ(t) > is the mean star for- +mation rate in the interval 0 ÷ t (with t the current age). This +value of the age T(t) will be used to infer the current luminosity +associated to the stellar content Ms(t) (see below). +The infall model of a galaxy must be completed with the radii +Re(t) and RD(t) that are necessary to calculate the velocity dis- +persion of the stellar component, and the gravitational potential +for the onset of galactic winds. To this aim we shortly discuss a +few items of interest here: +i) +The +MD-ML +and +RD-RL +relationships. Following +Bertin et al. (1992) and Saglia et al. (1992), the spatial distribu- +tion of the dark component with respect to the luminous one in +dynamical models is such that the mass and radius of the dark +component (MD, RD) are related to the luminous ones (ML, RL) +by +ML(t)(t) +MD(t) +≥ 1 +2π +RL(t) +RD(t) +� +1 + 1.37 RL(t) +RD(t) +� +(28) +where we can pose ML(t) ≃ Ms(t), MD(t) = fcMB(t) ≥ fcMs(t) +and RL(t) ≃ 2Re(t). Therefore knowing MD(t), Ms(t), and Re(t), +we can get an estimate of RD(t) to be used in the calculation of +the total gravitational potential. According Bertin et al. (1992) +and Saglia et al. (1992) typical values are ML/MD ≃ 0.2 and +RL/RD ≃ 0.2. Consequently within Re the mass of dark matter is +small with respect to the stellar mass and can be neglected. Fur- +thermore, the binding gravitational energy of the gas and stars is +given by +Ω j(t) = −αLG M j(t)ML(t) +RL(t) +− G M j(t)MD +RL(t) +Ω′ +LD +(29) +where j stands for g (gas) or s (stars), and αL is a numerical +factor = 0.5, and finally the term +Ω′ +LD = 1 +2π(RL(t) +RD +)[1 + 1.37(RL(t) +RD +)] +(30) +is the contribution to the gravitational energy given by the pres- +ence of dark matter. With the assumed ratios ML/MD and the +above replacements of ML, RL and RD, the term Ω′ +LD is about +0.04. Therefore in the evaluation of the velocity dispersion of the +stellar component via the VT the effect of DM can be neglected. +ii) Velocity dispersion. The velocity dispersion of an object +with MD(t), Ms(t), and radius Re(t) is derived from the scalar VT: +at each time an object is supposed to be very close to the condi- +tion of mechanical equilibrium and hence to satisfy the relation +σs(t) = +� +G +kv +Ms(t) +Re(t) +(31) +(iii) The Re(Ms) relation. The mass-radius relation (MRR) +suited to our models is the empirical law proposed by Fan et al. +(2010) in the context of the Λ-CDM cosmology. The expression +is: +Re = 0.9 +�S S (nS) +0.34 +� �25 +m +� �1.5 +fσ +�2 � +MD +1012M⊙ +�1/3 +4 +(1 + zf). +(32) +where MD, Ms, and Re have their usual meaning; Re is in kpc; +zf the redshift at which the collapse took place; S S (nS ) indicates +the shape of the baryonic component that in turn is related to +Article number, page 12 of 25 + +M. D’Onofrio and C. Chiosi: A new framework for understanding the evolution of early type galaxies +the Sérsic brightness profile from which Re is derived; nS is the +Sérsic index; fσ is the three dimensional stellar velocity disper- +sion as a function of the DM velocity dispersion, σs = fσσD; +and finally m is the ratio MD/Ms. We adopt here S S (nS) = 0.34 +and fσ = 1. For more details see Fan et al. (2010); Chiosi et al. +(2020) and references therein. The most important parameter of +eq.(32) is the ratio m = MD/Ms that is shortly discussed below. +The MRR of eq. 32 is the locus of galaxy models on the +MR-plane, the formation of which occurred at redshift zf . It +represents the position of model galaxies for different sources +(Chiosi & Carraro 2002; Merlin et al. 2012; Vogelsberger et al. +2014), however it does not correspond to the real MRR observed +for objects from GCs to ETGs and GCGs because cosmologi- +cal effects are also present (the subject has been thoroughly dis- +cussed by Chiosi et al. 2020, to whom the reader should refer +for all details). +(iv) The MD/Ms ratio. Basing on the Illustris-1 data +Chiosi et al. (2020) have investigated how this ratio varies in the +mass interval 108.5 < MD < 1013.5 (masses are in M⊙) and from +z = 0 to z = 4 and proposed the following relation: +m ≡ MD +Ms += (−0.223zf +0.375) log MD +(3.138zf −3.430) . (33) +In the present study, however, we follow a different strategy that +at each time step tightly correlates the mass in stars Ms(t) to +the total baryonic mass MB(t) and the total mass of dark matter +MD(t). At each time we have MD(t) = fc MB(t) where fc is the +cosmic proportion (fc ≃ 6). The mass in stars Ms(t) is deter- +mined by the efficiency of star formation and in any case it is a +fraction of the current baryonic mass. Therefore the parameter m +is given by the relation: +log m(t) = log MD(t) +Ms(t) . +(34) +At the beginning of a galaxy history the ratio m is very large +and then declines tending to the limit value fc ≃ 6, if the total +baryonic mass is eventually turned into stars. Examples of the +time behaviour of the ratio m will be shown when presenting our +model galaxies in some detail. +(v) The star formation rate. Thanks to the short time scale of +the energy input from massive stars (a few million years), com- +pared to the mass accretion time scale by infall (from hundred +to thousand million years) the galaxy is supposed not to differ +from an equilibrium state so that the Talbot & Arnett (1975) for- +malism can be applied. Chiosi (1980) and Chiosi & Matteucci +(1980) adapted the SFR of Talbot & Arnett (1975) to model disk +galaxies in which the surface mass density of stars, gas and total +baryonic mass are used and a suitable radial distance ˜r is intro- +duced. +We have adapted their formalism to our case (in which spher- +ical symmetry is implicitly assumed), +dMs(r, t) +dt += −dMg(r, t) +dt += ˜ν +� M(r, t)Mg(r, t) +M(˜r, t) +�κ−1 +Mg(r, t) +(35) +where Mg(r, t) and Ms(r, t) are the mean mass densities of gas +and stars within the generic sphere of radius r at the time t, re- +spectively. M(˜r, t) is the total mass density (gas and stars) within +a particular radial distance from the galaxy center, and finally ˜ν +is a parameter measuring the efficiency of star formation. The ra- +dius ˜r is a suitable radial scale controlling star formation. In the +Larson’s view they might be associated to the radial distance at +which the central spheroidal component exerts its tidal effect on +the residual external gas. As a consequence of it, at any time the +SFR is significantly inhibited at distances r > ˜r. Since our mod- +els do not include any geometrical description, but deal a galaxy +as a point-mass entity whose mass varies with time, we drop the +radial dependence of the SFR and the rate of star formation is +simply reduced to: +dMs(t) +dt += −dMg(t) +dt += ˜ν +� M(t)Mg(t) +M(t) +�κ−1 +Mg(t) = ˜νMg(t)κ +(36) +Since in all infall models Mg(t) increases by infall and de- +creases by star formation, the SFR starts low, reaches a peak af- +ter a time approximately equal to τ and then declines. By varying +τ (time scale of the galaxy formation process) one can recover +all types of star formation indicated by observational data going +from GCs to LTGs and ETGs. The infall scheme and companion +SFR have been widely used in many studies on the subject of +galactic chemical evolution (e.g. Matteucci 2016, for a review +and references). The infall galaxy model is very flexible and +can be adapted to a wide range of astrophysical problems. Suf- +fice it to recall that it has been adopted by Bressan et al. (1994) +to model the spectro-photometric evolution of ETGs reduced to +point mass objects, extended by Tantalo et al. (1998b) to the case +of spherical systems made of BM and DM mimicking ETGs, +adapted by Portinari & Chiosi (2000) to include radial flows of +gas in disk galaxies, and recently used by Chiosi et al. (2017) +to study the cosmic star formation rate and by Sciarratta et al. +(2019b) to investigate the color-magnitude diagram of galaxies +in general. +(vi) The SF efficiency ˜ν. In most galaxy models of this kind +the specific efficiency of star formation ˜ν is an external free pa- +rameter to be adjusted according to the case under investigation. +In this paper we follow a different strategy and derive ˜ν from +other properties of the models. Starting from the idea put for- +ward by Brosche (1970, 1973) that the efficiency of star forma- +tion is driven by the velocity dispersion, we suppose that ˜ν can +be written as: +˜ν = ν0 +� σt +σs +× σT +σs +�0.5 +(37) +where σs, σt, and σT are the velocity dispersion calculated using +only the stellar component and the total mass, both measured at +current time t and present day age T. The factor ν0 depends on +the choice made for κ and secures the correct dimensions to ˜ν. +For kappa = 1, ˜ν ≡ 1/t. Finally, the harmonic mean between +two different normalizations is meant to somehow cope with the +uncertainty affecting the whole procedure. Since the stellar mass +Ms and radius Re grow with time, the efficiency is large at young +ages and decreases with time toward the limit value of ν ≃ 1. +(vii) Luminosity and specific intensity from mean SSPs. In +order to calculate the B and V luminosities and the associated +specific intensities IeB and IeV of the stellar content of galaxy +models in the course of their evolution, we make use of the SSPs +with the Salpeter (1955) IMF (slope in number x=-2.35, lower +mass Ml = 0.1M⊙, upper mass Mu = 100M⊙, total SSP mass +Mssp = 5.82M⊙, metallicity from Z = 0.0004 to Z = 0.04, 6 +values in total, and age from 10 Myr to 14 Gyrs) of the library +by Bertelli et al. (2008, 2009); Tantalo (2005). The absolute MB +and MV magnitudes can be plotted against the logarithm of the +age in years, and for each pass-band the mean age-magnitude +Article number, page 13 of 25 + +A&A proofs: manuscript no. FPaanda2 +Fig. 15. The MB (top) and MV (bottom) magnitudes versus age relation- +ships for SSPs of different metallicity according to the color code. From +the top to the bottom the metallicity is Z=0.0001, 0.001, 0.010, 0.019, +0.040, and 0.070. The black dotted lines are the mean values of MB and +MV over the metallicity. +relation is derived. Owing to the nearly linear behavior of each +relationship, a linear fit is suited to get the relation between the +mean absolute magnitude and the age t. These are given by: +MB += +2.361 logt − 17.841 +(38) +MV += +1.975 logt − 14.886. +(39) +The age is expressed in years. The B and V magnitudes of the +original SSPs with different metallicity are shown in Fig. 15 to- +gether with the metallicity averaged SSP (full dots). The mean +values of the magnitudes are meant to mimic the mixture of +chemical compositions in a galaxy. At each time we know the +total mass made by stars of different age and chemical compo- +sition. In practice we assume that this complex situation can be +reduced to a single SSP of the same mass, mean chemical com- +position (metallicity) and mean age T. The mean age is evaluated +from the relation T(t) = Ms(t)/ < Ψ(t) > where < Ψ(t) > is the +mean SFR in the interval 0 ÷ t (with t the current age). Using +the mean age T(t), from eqs.(38) and (39) we derive the B/V +magnitudes (the luminosities) per unit mass of the SSP and then +re-scale them to the mass Ms of the galaxy. +(viii) Solution of the basic equation eq.(15). At each time +step of the evolutionary history of a model galaxy, known the +star mass Ms, the radius Re, the velocity dispersion σs, the lu- +minosities LB and LV (in solar units) and the specific intensities +IeB and IeV, the equation system eq.(15) is solved deriving β and +L′ +0 at each time step. These are the two physical quantities that +in our view drive the distribution of galaxies in the space of the +physical parameters L, Re, σ, and Ie, and determine the observed +FP. +Final remarks. The model age refers to the galaxy rest-frame +and goes from Tg = 0 at redshift zf, when the galaxy is supposed +to form, to Tg = TG at z = 0 (present time). The corresponding +ages of the Universe TU(z) are TU(zf) and TU(0). For the ΛCDM +cosmology with H0 =71 km/s/Mpc, ΩΛ = 0.71, Ωm = 0.23, +ΩΛ = 0.73, ΩmD/ΩmB ≃ 6, we obtain TU(zf ) = 0.484 Gyr for +zf = 10 and TU(0) = 13.67 Gyr and TG = 13.187 Gyr. When- +ever needed we will pass from one to the other. In order to min- +imize the number of free parameters in each model, we assume +that all galaxies are born at the same redshift zf = 10; the col- +lapse time scale of τ = 1 Gyr for all galaxies; the Salpeter initial +mass function (in number) with a slope x=-2.35 and a fraction +of stars more massive than 1 M⊙ equal to ζ = 0.30, and absence +of galactic winds. However, a few cases will be shown for differ- +ent values of τ, different values of zf , and in presence of galactic +winds. +4.2. Model results +In this section we discuss the galaxy models obtained with the +above prescription for the infall scheme and star formation in +particular. First, we present the reference case with τ = 1 for +all galaxy masses and the prescription for ˜ν given by eq. (37) +together with the corresponding case ˜ν = 1 (we refer to these +latter models as the reference case). Then we discuss some cases +in which the effect of galactic winds energized by supernova ex- +plosions (both Type Ia and Type II) are taken into account. Table +3 lists the models we have considered and presents some charac- +teristic features at the last stage with active star formation: this is +either the present age for the models without galactic wind or at +the onset of galactic wind. In the following, we mainly present +and discuss the models without galactic winds, limiting the dis- +cussion of those with galactic winds to some general remarks. +SFR and SF efficiency. In Fig. 16 we show the history of +star formation in M⊙/yr of galaxies with MB(TG) equal to 106, +108, 1010, and 1012 M⊙/yr (black, blue, green, and red in the +order) and variable ˜ν (solid lines) and ˜ν = 1 (dashed lines, the +reference case). As expected, the SFR starts small, reaches a +peak value and then declines to low values even though it never +extinguishes. The peak value is at an age nearly equal to the +infall time scale. Models with variable efficiency do not differ +from their corresponding reference case with constant ˜ν = 1. +The reason for it resides in the value of τ. The point will be clear +discussing the case in which τ is let change with the galaxy mass. +The SFR efficiency ˜ν is a dimensionless quantity and therefore is +the same for all galaxies; it varies with time from the initial top +value 4.04 to 1 as shown in Fig.17. +The advantage of our choice for the SF efficiency ˜ν is that +this important physical quantity is no longer a free parameter. +It is indeed deeply driven by the galaxy mass building process +and the time scale associated to it. With our choice for τ, the +SF efficiency very quickly reaches its asymptotic value (within +about 2 × τ). If τ is increased the time scale over which ˜ν goes to +the asymptotic value gets accordingly longer. +The ratio MD/Ms and the radius Re. The stellar radius Re +depends on the Dark Mass to stellar mass ratio MD/Ms. As al- +ready explained this ratio is determined at each time from the +current value of the stellar mass built up by star formation, the +current mass of baryonic mass MB(t) and the current mass of +Dark Matter associated to it given by MD(t) = fcMB(t). The ra- +tio MD/Ms is shown in Fig.18 as a function of the mass MD (top +panel) and age in Gyr (bottom panel). In each galaxy the ratio +starts very high and, as time increases, tends to the limit value +fc ≃ 6 as the whole baryonic gas mass is turned into stars by +star formation. The general behaviour of MD/Ms as a function +of MD and age is the same to the point that in the bottom panel +all the curves overlap. Also in this case the ratio MD/Ms is not an +external parameter, but it is determined in a self consistent way +by the internal properties of the models. +Article number, page 14 of 25 + +M. D’Onofrio and C. Chiosi: A new framework for understanding the evolution of early type galaxies +Table 3. Tables of galaxy models. MB(tG) in solar units is the present-day baryonic mass. Age is either the galaxy age at the present time or the +age at the onset of galactic winds (ages in Gyrs). Mg and Ms are the gas and stellar masses in solar units at the indicated age. Zg and < Zg > are the +metallicity at the indicated age and the mean metallicity reached by the gas. SFR is the star formation rate in solar masses per year at the indicated +age. Finally, Ωg and Eg are the gravitational energy and thermal energy of the gas at the onset of galactic winds. All energies are in units of 1030 +ergs. In the case of models without galactic winds Eg is not given. +MB(tG) +Age +Mg +Ms +Zg +< Zg > +SFR +Ωg +Eg +No Galactic Winds +1e6 +13.19 +0.35E-02 +0.94E+00 +0.109 +0.038 +3.46E-06 +1.98E-04 +1e8 +13.19 +0.35E-02 +0.94E+00 +0.109 +0.038 +3.46E-04 +1.98E+00 +1e10 +13.19 +0.35E-02 +0.94E+00 +0.109 +0.038 +3.46E-02 +1.91E+04 +1e12 +13.19 +0.35E-02 +0.94E+00 +0.109 +0.038 +3.46E+00 +4.02E+07 +5e12 +13.19 +0.35E-02 +0.94E+00 +0.109 +0.038 +1.73E+01 +2.43E+08 +Galactic Winds +1e6 +13.19 +0.63E-03 +0.99E+00 +0.038 +0.012 +6.28E-07 +3.60E-05 +2.2E-05 +1e8 +7.46 +0.96E-02 +0.98E+00 +0.056 +0.017 +9.55E-04 +5.46E+00 +5.7E+00 +1e10 +5.75 +0.44E-01 +0.92E+00 +0.089 +0.033 +4.35E-01 +2.39E+05 +2.5E+05 +1e12 +5.25 +0.74E-01 +0.88E+00 +0.109 +0.045 +7.35E+01 +8.50E+08 +8.8E+08 +Fig. 16. The SFR histories of models with MB(TG) equal to 106, 108, +1010, and 1012 M⊙ (black, blue, green, and red in the order) and variable +˜ν (solid lines) and ˜ν = 1 (dashed lines, the reference case). +With the aid of the m-ratio and the MRR of eq. (32) we derive +the radius Re of the stellar component Ms and build the mass- +radius relationship (MRR) of our model galaxies shown in Fig. +19 both along their evolutionary history (the black line drawn by +filled squares, one for each time step, where the present time is +at the top and the initial stage at the bottom). Each curve corre- +sponds to a model with a different final total baryonic mass MB, +namely 106 M⊙, 108 M⊙, 1010 M⊙, 1012 M⊙ from left to right. +Although the MRRs of the models are in fair agreement with +the bulk of observational data and other theoretical MRRs, the +closer inspection of the issue reveals that our theoretical radii +are likely overestimated by a factor that is difficult to assess. +Our best estimate is about a ∆logRe ≃ −0.6 to −0.8. The mean +radii should be a factor 4 to 6 smaller. There are many possi- +ble causes for this disagreement: first of all, in addition to the +m-ratio in the term (MD)1/3 eq. (32) contains other terms each +of which is affected by some uncertainty. The terms in ques- +tion are the ratio S S (nS )/0.34, the ratio (1.5fσ)2, and finally +Fig. 17. The temporal variation of the SF efficiency of the galaxy models +with with MB(TG) equal to 106, 108, 1010, and 1012 M⊙ (black, blue, +green, and red in the order). The efficiency is the same for all models +and goes from 4.04 to 1. Plotting the data log(˜) of each case has been +shifted by 0.01 with respect to the other ones. +the ratio (25/m). The first two are simply assumed to be equal +to one, while the last one contains the ratio m and deserves +some remarks. It is clear that it has been introduced as an ad- +justment factor based on some estimates of the m-ratio derived +from current theoretical N-Body Smoothed Particle Hydrody- +namic (NBTSPH) simulations of galaxy formation in which only +a small fraction of the available gas was used to form stars (e.g. +see for instance Chiosi & Carraro 2002), which explains the fac- +tor 25. The present infall models have a different behaviour be- +cause nearly all the gas is used up to form stars and the limit +value of the m-ratio is about fc ≃ 6. This implies that the above +adjustment factor should become (fc/m), and consequently a re- +duction of the estimated radius by a factor of about 4 to 6. How- +ever, instead of forcing the radius to strictly agree with the data, +thus introducing some ad hoc adjustments, we keep the radii as +they are but also keep in mind that in reality they could be 4 to 6 +Article number, page 15 of 25 + +A&A proofs: manuscript no. FPaanda2 +Fig. 18. The ratio m = MD/Ms as a function of the mass of Dark Matter +MD (top panel) and age (bottom panel) for our model galaxies. Masses +are in M⊙ and ages in Gyrs. The model galaxies are in different colors +and ranked according to their total baryonic mass MB reached at the +present time (black: 106; blue: 108; green: 1010, and red: 1012). In the +bottom panel all the lines overlap each other. +times smaller than estimated. This would immediately affect our +evaluation of the specific intensity Ie = L/(2πR2 +e) that could be a +factor 16 to 36 higher than our straight evaluation (see below). +Other important relationships: the L vs Re and the Re vs +σ. The uncertainty on the radius affects also other important re- +lationships such as the luminosity-radius relation (LRR) shown +in Fig 20 and the radius velocity dispersion relation (RSR) dis- +played in Fig. 21. The theoretical data are compared with the +observational ones by Burstein et al. (1997) and Bernardi et al. +(2010), for these latter the mean colour (B-V)=0.85 has been +applied to the MV magnitudes to get the B-luminosity. In both +cases the best results are for radii reduced by a factor of 4. +Remarks on the luminosity. Before proceeding further it +is worth commenting on the luminosity of the model galaxies. +As already explained, for the sake of a quick assessment of the +model galaxies luminosity in the B and V pass-bands, we have +used suitable linear relationships between the absolute B and/or +V magnitudes of the Johnson system and the mean age T based +on SSPs of mean metallicity. One may argue that the luminosi- +ties derived in this way are much different from those evaluated +by means of full population synthesis technique, i.e. by integrat- +ing the spectral energy distribution of SSPs over the star forma- +tion rate, initial mass, and the metallicity range spanned by the +stellar populations of galaxies at each time (see Bressan et al. +1994, for all details). This is done "a posteriori", once the whole +SFR(t), Ms(t), and metallicity Z(t) are known. The results are +shown in Fig.22 for the MB(TG) = 106, 1012 M⊙, where the solid +lines are the luminosity from the analytical relationships, and the +dotted lines the luminosity from full population synthesis. The +two luminosities differ from each other by a maximum factor of +3 back in the past when the SFR(t) was maximum (ages of about +1.5 Gyr), while they coincide in the less remote past (roughly +past 5-6 Gyr). Therefore our approximation that nicely speeds +up the model calculation is reasonable and leads to acceptable +Fig. 19. The mass-radius relations (MRRs) of our model galaxies la- +belled by their present day total baryonic mass MB(TG) equal to 106, +108, 1010, and 1012 M⊙ from left to right. Each line made by filled black +squares represents the whole evolutionary history of both Ms and Re +both increasing with time (the top is the present). These models are +compared both with observational and theoretical data from different +sources: (i) the observational data of Burstein et al. (1997) from GCs +to GCGs (powder-blue small dots) and the ETGs by Bernardi et al. +(2010) (red small dots); (ii) the Illustris-1 galaxies (light green small +dots); (iii) the low initial density models (blue squares and their best +fit) and the high initial density ones (red squares and their best- +fit) by Chiosi & Carraro (2002); (iv) the early hierarchical models by +Merlin et al. (2012) (black squares and their best-fit); (v) the Fan et al. +(2010) MRRs for different values of the formation redshift zf = 0 (top), +1, 5 10, and 20 (bottom); (v) finally and the MRR by Chiosi et al. (2020) +(dark golden line). See the text for more details. +results. Our luminosities can be safely used for the present pur- +poses. +The Ie vs Re plane. Together with the FP and the luminosity- +velocity dispersion relation, otherwise known as Faber-Jackson +(FJ) relation, the Ie − Re plane is one of the most studied pro- +jection of the FP. The uncertainty on the radius Re (a factor of +4) reflects on Ie as an increase of a factor of 16 at fixed stellar +mass of the galaxy. The results for our models (with no galactic +winds) are shown in Fig. 23 and are compared with the obser- +vational data of Burstein et al. (1997) from GCs to GCGs using +the same color code as in Fig.20. The evolutionary sequences on +display are for model galaxies with MB(TG) equal to 106, 108, +1010, and 1012 M⊙, from left to right. For each mass we display +two lines: the one with the original radii (dashed black line) and +the case with the radii decreased by a factor of 4 and the spe- +cific intensity Ie increased by a factor of 16 as explained in the +text (line made by filled black squares). The time evolution goes +from the top to the bottom of each line. The present day stage is +the last bottom point of each line. Finally the thick dashed line is +the border of the ZOE. Please note that no model at the present +time falls in the ZOE, but all are below it. +The present models cannot account for the data of GCs (as +expected). Even if the model with MB(TG) = 106 M⊙ crosses +the region of GCs it cannot reproduce these objects because the +present radius and specific intensity are too large and too low re- +spectively. The ongoing star formation yields too luminous and +Article number, page 16 of 25 + +2 +(Kpc) +0 +R +Log +-2 +DATA & MODELS +4 +6 +8 +10 +12 +14 +Log M, (Mo)M. D’Onofrio and C. Chiosi: A new framework for understanding the evolution of early type galaxies +Fig. 20. The B-luminosity-radius relation (LBRRs) of our model galax- +ies with MB(TG) equal to 106, 108, 1010, and 1012 M⊙, from bottom to +top. For each mass we display two lines: the one with the original radii +(dashed black line) and the case with the radii decreased by a factor of 4 +as explained in the text (line made by filled black squares). The models +are compared both with observational data from Burstein et al. (1997) +from GCs (magenta small dots) to Dwarf Galaxies (blue small dots), to +ETGs (red small dots), GCGs (powder-blue small dots), and the ETGs +by Bernardi et al. (2010) (red small dots, overlap the previous ones). +The agreement for the smaller radii case is soon evident. See the text +for more details. +Fig. 21. The Re-σ relation (RSR) of our model galaxies. In this figure +the same models, observational data, color code, and symbols used in +Fig.20 are adopted. +too large objects that do not match with general properties of +GCs. What would be needed are models in which star forma- +tion ceased and radius stopped growing long ago (short initial +episode followed by quiescence perhaps because of strong galac- +tic wind), or to take into account the important transformations +induced by the interaction with the Galaxy. +Fig. 22. The luminosity LV versus age (in Gyr) relation (LAR) of our +model galaxies. The cases MB(TG) = 106 (bottom) and 1012 M⊙ (top) +are shown. The solid lines are luminosities derived from the analyti- +cal relationships while the dotted lines are those from full population +synthesis. +Similar considerations can be applied to clusters of galaxies +for which different type of models should be set up. To develop +a suitable model for the formation and evolution of GCGs along +the same lines we have followed for single galaxies is beyond +the aims of this study and we leave the subject to a future inves- +tigation. +The β − L′ +0 space. With the aid of the equations from eq.(3) +to eq. (15) we derive the exponent β and proportionality factor +L′ +0 along the whole evolutionary sequence of our model galax- +ies evolved without galactic winds. In the left panel of Fig. 24 +we display all cases under consideration: models with large radii +and models with smaller radii (the factor of 4) for the two photo- +metric pass-bands in usage (B and V Johnson). Along each line +time increases from the bottom to the top where the last stage +at the present age is indicated by the mass label (total asymt- +potic baryon mass). For each galaxy mass (MB(tG)) the results +are nearly the same, all sequences overlap each other. See also +the entries of Table 4 containing the slope α and zero-point γ of +their linear best-fits. It turns out that the relationships in question +depend only on the galaxy mass. Remarkably β the exponent of +the L = L′ +0σβ relation is positive during the early stages and neg- +ative afterwards. The luminosity first increases with σ and then +decreases with it afterwards. Finally note that all curves cross +each other at β ≃ 3 and log L′ +0 ≃ 2.5, values very close to the +observed FJ relation. To confirm this picture, in the right panel +of Fig. 24 we plot the same relations for the Illustris-1 models +grouped at different redshifts from z = 4 to z = 0. Now the sit- +uation is not the same as before, because in each group with the +same redshift, mass and age vary from galaxy to galaxy. Fur- +thermore not all masses are present at each redshift: samples at +high redshifts, say ≥ 1, 6, are dominated by low mass objects +(masses lower than 108 M⊙ are missing anyway because of the +mass resolution), massive objects up to 1012 M⊙ are present at +lower redshifts. However, the resulting distributions in the β−L′ +0 +plane are much similar to that of the left panel, and remarkably +there is also some evidence of the β ≃ 3 cross-point. This fact +Article number, page 17 of 25 + +2 +(Kpc) +(R.) +Log +2 +0 +1 +2 +3 +4 +Log (a) (km/s)12 +10 +le10 +8 +le8 +6 +e6 +4 +-2 +0 +2 +4 +Log (Re) +(Kpc)A&A proofs: manuscript no. FPaanda2 +Fig. 23. The Ie-Re plane of our model galaxies compared with the obser- +vational data of Burstein et al. (1997). The color code of the data is the +same as in previous figures. There are two groups of models: the black +thin dashed lines are models with original radii, while the thick lines +made by filled black squares are those with the radius decreased by a +factor of 4 and the specific intensity Ie increased by a factor of 16. The +galaxy mass is MB(TG) equal to 106, 108, 1010, and 1012 M⊙, from left +to right. Along each line the time runs from zero to present age from the +top to the bottom. The formation redshift of all the models is zfor=10. +Table 4. The relationships between L′ +0 and β for the model galaxies +evolved without galactic wind. These relationships are the linear best +fits of the curves shown in Fig.24. All these relationships are nearly +identical passing from models with large radii to those with smaller +radii (by a factor of 4), and the corresponding solutions of the equations +for β and L′ +0, and finally changing only the photometric pass-band in +use. The relationships seem to depend indeed only on the galaxy mass. +logL = α ∗ β + γ +B-Band +V-band +MB/M⊙ +α +γ +α +γ +1e6 +-0.478 +4.369 +-0.508 +4.368 +1e8 +-1.137 +6.224 +-1.167 +6.287 +1e10 +-1.796 +8.046 +-1.760 +7.340 +1e12 +-2.453 +9.834 +-2.474 +9.888 +strongly supports the notion that infall models nicely mimic the +numerical hierarchical simulations. +The most important relation to look at and to examine in de- +tail is the luminosity versus velocity dispersion. This is shown +in Fig.25 which displays the log(LB/L⊙) vs log σ for the model +galaxies and compare it with observational data of Burstein et al. +(1997). In the main panel we display three possible relationships: +(i) the plain LB/L⊙ vs σ of the models with their original lumi- +nosity and radii (the thick black curves). Along each curve the +evolution starts at the bottom point of each line and proceeds +to the final stage indicated by the label MB(tG) in solar units. +Please note that during the galaxy lifetime the LB/L⊙ vs σ rela- +tion bends over past a certain age toward lower luminosities and +lower velocity dispersion. This roughly happens past the peak of +star formation. While the luminosity decrease can be easily un- +derstood, the decrease in velocity dispersion of the stellar com- +Fig. 24. Left panel: The L′ +0−β relation of our models (left panel) evolved +without galactic winds. All the relationships are nearly identical passing +from models with large radii to those with smaller radii (by a factor of +4), and the corresponding solutions of the equations for β and L′ +0, and +finally changing only the photometric pass-band in use. The relation- +ships seem to depend indeed only on the galaxy mass. Right panel: The +L′ +0 − β relation for the artificial galaxies of Illustris-1. Each color corre- +sponds to a different redshift epoch: green (z = 4), blue (z = 3), yellow +(z = 2.2), brown (z = 1.6), magenta (z = 1.0), dark gray (z = 0.6), red +(z = 0.2) and light gray (z = 0). +ponent needs some explanation. Stars during their lifetime can +explode as Type II and Type I supernovae: in the first case a +small remnant is left (neutron star or black hole), in the second +one no remnant at all. They can also lose lots of mass by stellar +winds. In any case the total mass in stars is expected to decrease +and so does the velocity dispersion. (ii) The second case is the +associated LB/L⊙ = L′ +0σβ relation in which the original β and L′ +0 +are used (the red curves together with the linear fit limited to the +descending branch of each curve, (the red solid lines). (iii) Fi- +nally, the LB/L⊙ = L′ +0σβ relation, in which the correction on the +radius has been applied and new values of β and L′ +0 are derived. +It is worth recalling that the L′ +0 vs β relation remains unchanged. +The results are shown by the green curves. The small insert in +Fig. 25 shows the case of the MB(tG) = 1010 M⊙ in more detail +for the sake of better understanding. Similar results are found for +the V pass-band that are not shown here. Two important features +are soon evident; first of all the relations log(LB/L⊙) vs log σ, +based on the model history past the star formation activity pe- +riod have a similar slope, but different zero point (that depends +on the galaxy mass). The manifold of these relations provides a +sort of natural width to the luminosity-sigma relationship. The +mean slope of the manifold agrees with the current value of the +observed FJ. Second, the theoretical relations marginally agree +with the body of ETGs, a steeper slope at luminosities above +log LB/L⊙ ≃ 9 would be more appropriate. +The simplicity of the current models cannot lead to better re- +sults. A possible improvement could be given by allowing small +secondary episodes of star formation. The argument is as fol- +lows. The luminosity is the product of the star mass times the +flux per unit mass: let us call Lo = foMo the original luminosity +and Ln = fnMn the expected luminosity including some recent +star forming activity; LnMn is in turn made by fyMy + foMo, +Article number, page 18 of 25 + +20 +10 +0 +-10 +-20 +-10 +0 +10 +2020 +10 +Log(L'o) +0 +-10 +-20 +-10 +0 +10 +20M. D’Onofrio and C. Chiosi: A new framework for understanding the evolution of early type galaxies +6 +6 +6 +6 +6 +6 +6 +0 +1 +2 +3 +6 +8 +10 +12 +6 + 1e10 +Fig. 25. The luminosity LB/L⊙ versus velocity dispersion σ in (km/s) re- +lation (LSR) of our model galaxies. For each mass (labelled by MB(tG) +as indicated ) three relations are shown: (1) the original models with +no revision of the radii (lines made by filled black squares); (2) mod- +els whose luminosity is derived from the LB/L⊙ = L′ +0σβ relation with +the original β and L′ +0 (the curves made by red squares) together with +the linear fit limited to the descending branch of each curve (the black +solid lines); (3) models in which the radii have been revised and new +values of β and L′ +0 are calculated (the green curves). Note how in each +case, the luminosity vs sigma relation bends past the stage that roughly +corresponds to the maximum stellar activity. From this stage the lumi- +nosity and velocity dispersion decrease (see the text for details). The +insert shows the case of the MB(tG) = 1010 M⊙ for the sake of better +illustration. The models are compared with the data by Burstein et al. +(1997) from GCs to GCGs (the same color code as in previous figures +is used). +where fyMy is the contribution by the episodic stellar activity; +it follows that (fyMy + foMo)/(foMo) = λ = Ln/Lo. Basing on +the current observations one would expect λ ≃ 2 or so. Now we +may also assume My << Mo so that the total stellar mass and ve- +locity dispersion in turn remain nearly constant. Indicating with +θ = My/Mo, one gets fy/fo = λ − 1/θ ≃ 5 − 10 which is not +impossible according to current population theories for Single +Stellar Populations. (iii) In the theoretical models the exponent +β of the LB/L⊙ = L′ +0σβ relationship (a generalization of the FJ) +can be either positive or negative depending on the particular +evolutionary stage of the galaxy. Therefore among the observa- +tional data both values of β are to be expected without violating +the trend indicated by the FJ, i.e. that the luminosity of galax- +ies increases with the velocity dispersion, hence the mass of the +galaxy. +Galactic Winds. Long ago Larson (1974) postulated that the +present-day Color-Magnitude Relation (CMR) of ETGs could be +the result of galactic winds powered by supernova explosions, +thus initiating a long series of chemo-spectro-photometric mod- +els of elliptical galaxies standing on this idea (see for instance +Tantalo et al. 1998a, and references). In brief, gas is let escape +from the galaxy and star formation is supposed to halt when the +total thermal energy of the gas equates its gravitational binding +energy. This idea has been extended including the effect of stel- +lar winds in the thermal energy budget of the gas. It was also +included in NBTSPH models of galaxies (see Merlin et al. 2012, +and references). +The same scheme proposed by (Tantalo et al. 1998a) is +adopted here, however with minor modifications because of the +much simpler present formalism. As already said, the thermal +energy of the gas is the sum of three contributions, namely type +I and II supernovae and stellar winds from massive stars: +Eth(t) = Eth(t)S NI + Eth(t)S NII + Eth(t)W +(40) +where each term has the generic expression +Eth(t) j = +� t +0 +ǫj(t − t′)R j(t′)MB(tG)dt′ +(41) +with j= SNI, SNII, W with obvious meaning of the symbols. The +normalization factor MB(tG) in the above equations is required +to calculate the energy in physical units. The time t′ is either the +SN explosion time or the time of ejection of the stellar winds +as appropriate. The functions ǫS N(t) and ǫW(t) are cooling laws +governing the energy content of supernova remnants and stellar +winds, respectively. Finally, star formation and chemical enrich- +ment are halted, and the remaining gas content is supposed to be +expelled out of the galaxy (winds) when the condition +Eth(t) ≥ Ωg(t) +(42) +is verified. For all other details concerning the above rates, the +evolution of SN remnants and stellar winds and how much of +the initial energy budget is shared with the gas to energize it, and +finally the expression for the gravitational energy of the gas in +presence of baryonic and dark mass and their space distribution +in a galaxy see Tantalo et al. (1998a). +A small sample of models with galactic winds are calcu- +lated and their main features are summarized in Table 3. It is +worth noting that the onset of galactic winds occurs at younger +and younger ages as the galaxy mass increases. Thanks to it, +these models obey the constraint imposed by observational data +on chemical elements like Carbon (C), Oxygen (O), Magne- +sium (Mg), also known as α-elements, and Iron (Fe) and their +ratios [α/Fe]: the high mass galaxies are more α-enhanced +([α/Fe] > 0) than the low-mass ones ([α/Fe] ≤ 0). This fact +cannot be easily reconciled with other properties of the same ob- +jects. See Chiosi et al. (1998) and Tantalo & Chiosi (2002) for +detailed discussions of this issue and possible ways out. In the +present models we have taken the suggestions by Chiosi et al. +(1998) and Tantalo & Chiosi (2002) into account. +The role of Galactic Winds. The main lines of the discussion +for models without galactic winds holds good also for the new +ones. Therefore we focus on key relations such as the Ie − Re +plane which is displayed in Fig. 26. The comparison with the +same plot of Fig. 23 shows that there is no visible difference +passing from models without to those with galactic winds. The +reason for that is the kind of star formation at work. Because of +the short infall time scale and the dependence of ˜ν on the inverse +of the velocity dispersion, most of the stars are in place before +the occurrence of galactic winds. To somewhat alter this trend +one should change the parameter τ and make it to depend on the +galaxy mass, for instance long in low mass galaxies and short in +the high mass ones. To further investigate this point is beyond +the aims of this study. +Role of the Initial Mass Function. To avoid misunderstand- +ing, we need to recall here that the present models are calculated +Article number, page 19 of 25 + +A&A proofs: manuscript no. FPaanda2 +Table 5. A few key quantities of the model galaxies at the present time. From left to right: age in Gyr, the logarithm of the stellar mass Ms in solar +units, the logarithm of the effective radius Re in kpc, the logarithm of the velocity dispersion σ in km/s, the logarithm of the B luminosity LB in +solar units, the logarithm of specific intensity IeB in LB/pc2, the logarithm of the mass to light ratio Ms/LB in solar units, LV, IeV, Ms/LV the same +for the V band, the redshift of galaxy formation zf , the asymptotic baryonic mass MB(tG) in solar units, the infall time scale τ in Gyr, and finally +the notes N where the asterisks mean that the models take all corrections to the Fan et al. (2010) radius into account. +age +Ms +Re +σ +LB +IeB +Ms/LB +LV +IeV +Ms/LV +zf +MB +τ +N +13.18 +5.975 +-0.245 +0.410 +4.981 +-1.326 +0.99 +5.096 +-1.211 +0.87 +10 +6 +1 +13.18 +7.975 +0.422 +1.077 +6.981 +-0.659 +0.99 +7.102 +-0.538 +0.87 +10 +8 +1 +13.18 +9.975 +1.088 +1.744 +8.981 +0.008 +0.99 +9.096 +0.123 +0.87 +10 +10 +1 +13.18 +11.975 +1.755 +2.410 +10.981 +0.674 +0.99 +11.102 +0.795 +0.87 +10 +12 +1 +13.18 +5.987 +-0.845 +0.716 +4.988 +-0.118 +0.99 +5.109 +0.003 +0.87 +10 +6 +1 +* +13.18 +7.987 +-0.179 +1.383 +6.988 +0.549 +0.99 +7.109 +0.670 +0.87 +10 +8 +1 +* +13.18 +9.987 +0.488 +2.050 +8.988 +1.215 +0.99 +9.109 +1.337 +0.87 +10 +10 +1 +* +13.18 +11.987 +1.155 +2.716 +10.988 +1.882 +0.99 +11.109 +2.003 +0.87 +10 +12 +1 +* +12.63 +5.989 +-0.845 +0.717 +5.018 +-0.088 +0.97 +5.135 +0.029 +0.85 +5 +6 +1 +* +12.63 +7.988 +-0.178 +1.383 +7.011 +0.570 +0.97 +7.129 +0.688 +0.85 +5 +8 +1 +* +12.63 +9.988 +0.489 +2.050 +9.011 +1.237 +0.97 +9.129 +1.354 +0.85 +5 +10 +1 +* +12.63 +11.988 +1.155 +2.716 +11.011 +1.903 +0.97 +11.129 +2.021 +0.85 +5 +12 +1 +* +11.58 +5.990 +-0.843 +0.716 +5.049 +-0.063 +0.94 +5.161 +0.049 +0.82 +3 +6 +1 +* +11.58 +7.990 +-0.176 +1.383 +7.049 +0.604 +0.94 +7.174 +0.750 +0.81 +3 +8 +1 +* +11.58 +9.990 +0.491 +2.050 +9.049 +1.271 +0.94 +9.161 +1.382 +0.82 +3 +10 +1 +* +11.58 +11.990 +1.157 +2.716 +11.049 +1.937 +0.94 +11.161 +2.049 +0.82 +3 +12 +1 +* +7.77 +5.989 +-0.844 +0.716 +5.211 +0.102 +0.77 +5.296 +0.187 +0.69 +1 +6 +1 +* +7.77 +7.989 +-0.341 +1.465 +7.211 +1.096 +0.77 +7.296 +1.181 +0.69 +1 +8 +1 +* +7.77 +9.989 +0.489 +2.050 +9.211 +1.436 +0.77 +9.296 +1.521 +0.69 +1 +10 +1 +* +7.77 +11.989 +1.156 +2.716 +11.211 +2.102 +0.77 +11.296 +2.187 +0.69 +1 +12 +1 +* +5.10 +5.975 +-0.858 +0.716 +5.370 +0.289 +0.60 +5.427 +0.346 +0.54 +0.5 +6 +1 +* +5.10 +7.975 +-0.191 +1.383 +7.370 +0.956 +0.60 +7.427 +1.013 +0.54 +0.5 +8 +1 +* +5.10 +9.975 +0.475 +2.050 +9.370 +1.622 +0.60 +9.427 +1.679 +0.54 +0.5 +10 +1 +* +5.10 +11.975 +1.142 +2.716 +11.370 +2.289 +0.60 +11.427 +2.346 +0.54 +0.5 +12 +1 +* +5.12 +9.929 +0.427 +2.051 +9.323 +1.671 +0.60 +9.380 +1.728 +0.55 +0.5 +10 +5 +* +1e6 +1e6 +1e8 +1e10 +1e12 +Fig. 26. The Ie-Re plane of our model galaxies with galactic winds pow- +ered by the energy input from supernova explosions and stellar winds. +There is no visible difference with respect to the same plane of models +without galactic wind. The same notation, symbols and color code of +Fig.23 is adopted here. +with the classical IMF of Salpeter (1955). Therefore the Ms/L +ratio based on these models has this fundamental limitation and +cannot by applied to investigate the problem of the FP tilt in a +very general way. Our infall models can easily be adapted to in- +clude popular IMFs in literature different from the Salpeter case, +see for instance Chiosi et al. (1998), where the IMF is let vary +with the physical condition (mean density, temperature and ve- +locity dispersion) of the gas inside a galaxy and therefore with +time for a galaxy of given mass and with time and mass in ob- +jects of different mass. However, for the aims of this study, in +order to simplify this we thought it wise to rely on the classi- +cal IMF of Salpeter. If the present models were applied to the +issue of the FP tilt, most likely they could account for only half +of the observational tilt. This subject was specifically addressed +in Chiosi et al. (1998) with good results for the tilt of the FP of +ETGs in the Virgo and Coma clusters. +Changing the galaxy mass and zf. An important feature +of the models is related to the formalism in use. According to +the formalism and equations widely described in Tantalo et al. +(1998a) all relevant physical quantities describing the model and +its temporal evolution are suitably normalized to the so-called +asymptotic baryonic mass MB(tG), for instance the gas mass at +time t is expressed as Gg(t) = Mg(t)/MB(tG), equally for the star +mass Gs(t) = Ms(t)/MB(tG), and the current total baryonic mass +GB(t) = MB(t)/MB(tG). The amount of dark matter at any time is +simply related to the current baryonic mass via the cosmic ratio +(the components are intimately mixed together so that they fall +together at the same rate). Furthermore, the accretion rate, the +star formation rate, etc. are all expressed using the same kind +of normalization. The advantage is that the time scale of mass +accretion τ, the cosmic ratio fc, and all the rest is parameter- +Article number, page 20 of 25 + +M. D’Onofrio and C. Chiosi: A new framework for understanding the evolution of early type galaxies +6 +12 +Fig. 27. The Ie-Re plane of our model galaxies with different formation +redshift zf, namely 10, 5, 3, 1 and 0.5. The four green points of differ- +ent colors are the present day stage of model galaxies whose existence +began at redshifts from 0.5 to 5. The effect is quite small. +free, so that the only free quantity is the asymptotic baryonic +mass MB(tG). This allows us to generate models for any value of +MB(tG). +All galaxy models discussed so far are calculated assuming +the redshift of galaxy formation zf = 10. Other values are of +course possible. Higher values are unlikely whereas lower values +are much plausible. Since the age of the Universe TU depends +only on the cosmological model in use and therefore is a fixed +quantity, the age of a galaxy TG expressed by TG = TU − TU(zf) +at decreasing zf becomes shorter. Consequently some features +of the models will change, such as total ages, radii, luminosities +and specific intensities. The following values of zf are consid- +ered: 5, 3, 1, and 0.5 in addition to the previous set with zf = 10. +The results are shown in Fig.27 limited to the cases zf = 5 +(black lines) and zf = 0.5 (red lines). The case zf = 10 runs +nearly over the case zf = 5. All the others are in between the +case zf = 5 and zf = 0.5. From left to right, the galaxy mass is +MB(TG) = 106108, 1010, 1012 M⊙. The age increases along each +line from the top to the bottom. The final age (in Gyr) decreases +from 13.19 for zf =10 to 12.47 for zf =5, 11.48 for zf =3, 7.73 +for zf =1, and 5.02 for zf =0.5. See Table 5 for more informa- +tion on the final stage of each model. In Fig 27 the final stages +are represented by the green circles (some of them overlap). +From these data we derive that variations in zf from 10 to 0.5 +yields variations in log(Ie) of about ∆ log(Ie) ≃ 0.5 while the ra- +dius does not change significantly. More efficient star formation +in recent times generates more luminosity and hence higher spe- +cific intensity Ie. This is achieved by changing τ from 1 to 5 Gyr +(in the case of the 1010 M⊙ galaxy) yielding ∆ log(Ie) ≃ 0.4. Re- +cent bursts of star formation either by internal causes or mergers +would also increase Ie. Analysing all implications of it is beyond +the aims of this study. What we can say with confidence is that +a significant scatter in the Ie − Re plane is likely to occur. In any +case, the gross distribution of galaxies in this plane (but for GCs +and GCGs) is accounted for by these models. +Finally, the homologous behaviour of the models and the +limited effect of the formation redshift on their evolutionary be- +Fig. 28. The comparison of Ie and Re derived for the model galax- +ies (indicated by the suffix [i] and those calculated with relations +(16) and (17) for galaxies with asymptotic baryonic mass MB(TG) = +106, 107, 108, 1010, and 1012 M⊙ from left to right. The redshift of galaxy +formation is zf =10. +haviour make it possible to generate simulations of the distribu- +tion of large number of galaxies in the parameter space we are +investigating in practice at no cost. +A test of consistency. The galaxy models we have presented +are based on physical assumptions such as the infall picture, the +star formation rate, the mass-radius relationship and the pop- +ulation synthesis governing their luminosity in different pass- +bands, that are not explicitly related to our interpretation of the +parameter space of galaxies (luminosity, stellar mass and ra- +dius, velocity dispersion, and specific intensity), the FP in the +multi-dimensional space and its possible projections onto differ- +ent planes that led us to the L = L′ +0σβ relationship with L′ +0 and β +changing from galaxy to galaxy and for each of them also with +time. On this ground we have made some detailed predictions +about L′ +0 and β and derived a number of equations whose solu- +tions on one hand yield L′ +0 and β as function of L, Ms, Re, σ, etc. +and on the other hand allows to construct the expected relation- +ships among pair of fundamental variables such as Ie vs Re, Ie vs +σ, Re vs σ etc.. Among these we choose here as an example the +variables Ie and Re and compare the values given by the models +with those derived from eqs. (16) and (17). The comparison is +shown in Fig. 28: the top panels are for Ie while the bottom pan- +els are for Re; the galaxy mass MB(TG) is 106, 107, 108, 1010, and +1012 M⊙ from left to right (the case MB(TG) = 107 ⊙ is added). +On the abscissa are the input values from the models (labelled +Ie[i] and Re[i]) and on the ordinate the values calculated from +eqs. (16) and (17) (labelled Ie[c] and Re[c]). In general there +is a surprisingly good agreement between [i] and [c] quantities, +but for some particular stages in which the [c]-values rapidly di- +verge and change sign. The cause of it resides in the analytical +relationships themselves that contain various exponents (e.g. γ, +[(β − 2)/(1 + 3/γ)], [(β − 2)/(3 + γ)] that in turn are functions +of β which varies in the course of evolution. In this narrow inter- +val the disagreement is of mathematical nature with no physical +implications. It simply means that these analytical relationships +cannot be used with safe to derive the corresponding variables. +Article number, page 21 of 25 + +6 +log(I_e[c]) +4 +2 +0 +2 +log(I_e[i]) log(I_e[i]) log(I_e[i]) log(I_e[i]) log(I_e[i]) +log(R_e[c]) +4 +2 +0 +一 +20 2 4 6-20 2 4 6-20 24 6-20 24 6-20 2 46 +log(R_e[i]) log(R_e[i]) log(R_e[i]) log(R_e[i]) log(R_e[i])A&A proofs: manuscript no. FPaanda2 +General remarks and preliminary conclusions. Since the +[i]- and [c]-values are nearly coincident, using the analytical re- +lationships would predict results in the various projection planes +we have examined identical to those obtained from using the nu- +merical galaxy model. The overall agreement between the model +and analytical approach lends strong support to the idea at the +base of the analytical view, i.e. that the relation between the +luminosity and velocity dispersion of a galaxy is governed by +L = L′ +0σβ in which both β and L′ +0 vary with the galaxy mass, +evolutionary stage (and hence time and redshift) and these quan- +tities in turn are intimately related to key physical parameters +such as the stellar mass and radius, the velocity dispersion (a +measure of the gravitational potential well), the star formation +rate, the infall time scale, and finally the ratio MD/Ms. The dis- +tribution of galaxies on the usual diagnostic planes such as FP, +FJ, Ie-Re, Ms-Re, L-Re, Re- σ, and finally the border of the exclu- +sion zone, mirror the mean behaviour of galaxies each of which +has its particular history and is observed in some evolutionary +stage. +5. The important role of β +In this section we cast light on the role of β. To this aim we +adopt the reference case (zf = 10 and τ = 1) and leave the is- +sue of galactic winds aside. For this case we present a few basic +relationships among β and other important parameters namely +the SFR (in M⊙/yr), the age (in Gyr), and the specific intensity +IeB or IeV (in L⊙/pc2). These relationships are shown in Fig. 29. +In the left panel, the homologous nature of the galaxy models is +evident: all curves have the same shape, but each one is sepa- +rated from all the others by the homology parameter, namely the +total baryonic mass at the present age MB(TG) annotated along +each curve. The temporal evolution occurs from the top to the +bottom of each curve. Identical behaviors are found between β +and the luminosity LB or LV (in L⊙), and the velocity dispersion +σ (in km/s). However, these relationships are not shown here for +the sake of brevity. The central panel of Fig. 29, showing the +variation of β with the age, still displays the dependence of the +results on the homology parameter and thus there are four dif- +ferent curves one for each value of the MB(TG). Finally, in the +right panel we show the dependence of β on the surface bright- +ness IeB; all curves collapse to a single relation, the homology is +destroyed by the underlying relationship between the mass and +the effective radius of the models. A similar relation is found be- +tween between β and IeV. The analytical relations between β and +Ie are given by +β += +3.159Log(IeB) − 2.003 +(43) +β += +3.159Log(IeV) − 1.900. +(44) +The evolution along each line is from top-right to bottom-left +and the present stage is the last point where MB(TG) is annotated. +The above relations indicate both the path followed by a single +galaxy in the course of its history and also the locus on the β-Ie +plane of galaxies of different mass observed at the present age. +There is no appreciable effect of different formation redshifts, +at least in the interval 10 ≥ zf ≥ 1 nor of different accretion +timescale τ in the interval 1 ≤ τ ≤ 5 Gyr. We estimate a total +effect on β by redshift zf and accretion time scale τ of the order +∆β ≃ 2 over the interval of IeV of interest here. +The linear relation between β and Ie shown by our models is +a very intriguing result that demands a thorough analysis because +observational data and numerical hierarchical models seem to in- +dicate a different picture. The situation is best illustrated by Fig. +30 comparing data and models from different sources. On the ob- +servational side we have three data-sets: Burstein et al. (1997), +WINGS, and Bernardi et al. (2010). The last two (mainly de- +voted to ETGs) are based on equivalent methods to estimate Re, +and therefore yield similar results as far as the β-Ie plane is con- +cerned. In contrast, the first one that contains objects going from +GCs to DGs, LTGs, ETGs and GCGs, differs in the method used +to derive the effective radius, and consequently yields different +relationships in the β-Ie plane. Owing to this, some preliminary +remarks are needed. First of all, the data of Burstein et al. (1997) +are in the B-band so that must be transformed into the V-band. +This is made by means of the relation +log LV = 0.4[(B − V)0 − 0.65] + log LB, +where the luminosities are in solar units, (B − V)0 is the colour, +and -0.65 is the difference between the B and V photometric +constants (5.48 and 4.83 respectively). Second, recalling that the +luminosity Le given by Burstein et al. (1997) is the amount of +light falling within the effective radius Re, where half the total +luminosity is found, we scale it by a factor of two to make it +consistent with the definition of Ie we have adopted. +The observational data for Re, LV, Ms, (Ms/LV), σ, and +IeV are fed to the system of equations (14) and the solutions β +and L′ +0 are derived. The powder-blue points n Fig. 30 are the +Burstein et al. (1997) data; three sequences are seen: the GCG- +GC sequence, the one of ETGs (no evidence of star formation), +and the one of LTGs and DGs (evidence of ongoing star for- +mation). By construction, the data of Burstein et al. (1997) are +well behaved with no evidence of dispersion. The red squares +are the WINGS data showing large dispersion in both coordi- +nates, log Ie is always positive, and β can be very large both +positive and negative. Much similar results are found with the +Bernardi et al. (2010) data, the open green circles. The Illustris-1 +model galaxies are indicated by the blue dots; their distribution +closely mimics that of the observational data. Finally, the long +dashed red line shows the present-day position on the β-Ie plane +of our models for the reference case (with τ = 1 Gyr, zf = 10 +and no galactic winds). This line coincides with the lower border +of the Illustris-1 distribution in the β > 0 hemi-plane. Choosing +different values of zf in the interval 0.5 ≤ zf ≤ 10 does not shift +significantly the line predicted at the present time. Equally for +the effect of galactic winds. Lumping all these effects together +we expect a typical width of this border line of about ∆β ≃ 10 +over the IeV interval of interest here. +From this preliminary comparison we may conclude that mu- +tual consistency among different sources of data and of these lat- +ter with models exists. +The major issue now is to understand the physical causes of +the large dispersion in β for all values of Ie ≥ 1. Looking at the +eqs. (15), providing the solutions β and log(L′ +0) of our equations, +one notes that under suitable conditions the term 1 − 2A′/A at +the denominator of log(L′ +0) gets very close to zero, consequently +β can be either very large and positive or large and negative. As +already discussed in Sect. 3 when this happens the system is in +conditions of strict virialization. The sign of β depends on the +particular history of the constituent variables (Ms, Re, L, and Ie), +in other words whether the term 2A′/A is tending to 1 from below +(β > 0) or above 1 (β < 0). From an operative point of view we +may define "state close to strict virialization" when |β| > 20. This +would account for the gap on the negative hemi-plane of Fig. 30. +Article number, page 22 of 25 + +M. D’Onofrio and C. Chiosi: A new framework for understanding the evolution of early type galaxies +Fig. 29. Left Panel: The relationship between β and the star formation rate (SFR). Each curve labelled by MB(TG), is identified by a different +colour according to the color-code adopted in previous figures. The total baryonic mass is the homology parameter separating each curve from +the others. Much similar trends are found for the luminosity LB and LV, and the velocity dispersion σ that are not displayed here for the sake of +brevity. In all three relations β mirrors the behavior of the SFR, the luminosity in turn, and finally the velocity dispersion. The SFR is in M⊙/yr. +Middle Panel: The relationship between β and age (in Gyr). Symbols and color-code have the same meaning as in the left panel. Right Panel: the +relationship between β and IeB in L⊙/pc2. The lines corresponding to different masses of galaxies have been displayed by shifting each of them +by 0.1; in reality they collapse to a unique curve given by β = 3.159Log(IeB) − 2.003. The long dashed line is the best fit of the theoretical data. +Identical relation can be found for IeV: same slope but slightly different zero point. +Do data and models ever reach the condition of full virializa- +tion indicated β ⇒ ±∞ or do they remain somewhat far it? The +answer is that both possibilities occur. +On the observational side, given any galaxy for which the +set of parameters (Ms, Re, L, Ie and σ) has been measured, it +is not granted that they would satisfy the virialization condi- +tion. The major uncertainties are with Ms and Re and Ie in turn. +Therefore, many of them crowd in the interval −1 ≤ β ≤ 20 +which implies deviations from virial equilibrium. However with +the present data, it is not possible to say whether this is due to in- +sufficient accuracy in the parameters determinations or real devi- +ations from virial equilibrium due to recent mergers, harassment, +loss of mass, interactions etc. However, many other galaxies in +both hemi-planes with |β| > 20 which is a strong indication that +they are close to virial equilibrium. +On the theoretical side, our model galaxies with infall (no +dynamics in them) seem to be in a state far from strict virializa- +tion. This is suggested by the small values of β reached at the +present time. The reason for that resides in the way the models +are built up. In brief, mass point description with no dynamics +is adopted, the total mass is assigned (via the accretion law), +the stellar mass is derived from star formation, the effective ra- +dius is estimated from a suitable relationship, the luminosity is +evaluated from the stellar mass and a mean luminosity-age re- +lationship for a fictitious SSP with mean metal content (the dif- +ference with respect to the luminosity correctly derived from the +theory of population synthesis via the history of star formation +and metal enrichment is not large but still significant), finally +the velocity dispersion is derived from the VT with the current +values of Ms and Re. The major uncertainties are in Re and L. +Therefore our set of basic parameters not necessarily can fulfill +all the requirements imposed by the VT. In consequence, our βs +are always quite small (say smaller than 25-30) implying that +full virialization is not reached. However, this failure is not as +severe as it appears because small adjustments of Re and L are +possible while the models are successful in many other aspects. +The situation is much better with the Illustris-1 models where +if a good number of galaxies have β < 25−30 as in the case of our +models, still a large number of objects is clearly seen in regime of +strict virialization because of their high positive and/or negative +βs. The inclusion of real dynamics and the hierarchical scenario +at work provide much better conditions to bring the action of +virialization into evidence. In the hierarchical scenario, mergers, +ablation of stars and gas, harassment, secondary star formation, +inflation of dimension by energy injections of various kinds, etc. +induce strong variations on the structural parameters and hence +strong temporary deviations from the virial conditions. However, +once this happened, the viral conditions can be soon recovered +over a suitable timescale. This can be short or long depending +on the amount of mass engaged in the secondary star forming +activity and the amount of time elapsed since the star forming +event took place (see the burst experiments in Chiosi & Carraro +2002; Tantalo & Chiosi 2004a). As a consequence of all this, de- +tecting systems on their way back to virial equilibrium is likely +a frequent event thus explaining the high dispersion seen on the +β-Ie plane. +In principle, the value of β evaluated for each galaxy could +provide a useful hint about the equilibrium state reached by the +system. Most likely, the condition of strict virial equilibrium is a +transient phenomenon that could occur several times during the +life of a galaxy. This is perhaps suggested by the high numbers +of galaxies with both low and positive values of β and high pos- +itive/negative values of β. +6. Discussion and conclusions +The aim of this paper is to prove that the difficulties encoun- +tered in understanding the distribution of galaxies on the FP in +the parameter space σ, Re Ie and its projections on the three co- +ordinate planes, can be removed by introducing the L = L′ +0σβ +relation as a proxy of evolution, in which β and L′ +0 vary from +galaxy to galaxy and for each of them in course of time (see +D’Onofrio et al. 2017a, 2019, 2020; D’Onofrio & Chiosi 2021, +for previous efforts along this line of thought).The continuous +variation of β and L′ +0 traces the path followed by each ETG in +the L − σ plane. The L = L′ +0σβ law together with the VT yield +a set of relations Re-σ, Ie-σ and Re-Ms that nicely reproduce the +data and suggest the existence of a system of two equations in +Article number, page 23 of 25 + +A&A proofs: manuscript no. FPaanda2 +Fig. 30. +The β - IeV plane: data and theoretical models. Data from +the different sources are plotted: (i) The powder-blue points from +Burstein et al. (1997); three sequences are seen: the sequence of GCGs +& GCs, the one of ETGs (no evidence of star formation), and the one of +LTGs and DGs (evidence of ongoing star formation). By construction, +the data of Burstein et al. (1997) are well behaved with no evidence +of dispersion. (ii) The red squares are the WINGS data showing large +dispersion in both coordinates, log Ie is always positive, and β can be +very large both positive and negative. (iii) the open green circles are +the ETGs of Bernardi et al. (2010) however limited to z ≤ 0.02. The +Illustris-1 model galaxies are indicated by the blue dots; their distri- +bution closely mimics that of the observational data. Finally, the long +dashed red line shows the present-day position on the β-Ie plane of our +models for the reference case (with τ = 1 Gyr, zf = 10 and no galac- +tic winds). This line coincides with the lower border of the Illustris-1 +distribution in the β > 0 hemi-plane. +the unknowns β and L′ +0 with coefficients functions of Ms, Re, L, +and Ie that for each galaxy determine the value of β and L′ +0. With +the aid of these relations we can determine the instantaneous po- +sition and direction of a galaxy on the FP and the projection +planes. +The analysis is made in two steps. In the first one, the prob- +lem is addressed from an observational point of view, inferring +from the data the expected position and evolutionary direction +of a galaxy in the various planes and owing to the large number +of galaxies in the samples the range of values spanned by β and +L′ +0 is determined. In the second step, simple models of galaxy +formation, structure and evolution are set up, the basic equations +are solved at each time step of a galaxy’s lifetime so that the his- +tory of β and L′ +0 is known. Basing on these results, the various +projection planes are examined finding consistency between ob- +servational data and theoretical models. The same procedure is +applied to literature galaxy models calculated in the framework +of the hierarchical scenario. The theoretical results are compared +with the observational data and good mutual agreement is found. +Basing on this, we conclude that the starting hypothesis about the +real existence of the L = L′ +0σβ relation is correct. In more detail, +the present analysis has clarified the following issues: +1. The FP can be understood as the average of the single FP- +like relations valid for each galaxy (eq. 4). The coefficients +of the FP-like relation are function of β. This means that the +FP must evolve with redshift and that its coefficients depend +on the adopted waveband (in which observations are taken) +and on the nature of the data sample (how many ETGs are +included) as confirmed by the current observational data; +2. All the features of the FP projections can be explained at the +same time. This includes: 1) the curvature of the relations, +that turns out to depend on the existence of positive and neg- +ative values of β, and 2), the existence of the ZoE, i.e. the line +marking the separation between the permitted and forbidden +regions in these planes. No galaxies can reside in the ZoE. +3. The FP and all its projections, such as for instance the clas- +sical Faber-Jackson relation, are instantaneous pictures of +the present-day situation. They should change with redshift +hence lifetime of galaxies. +4. The ZoE is obtained in a natural way as the only possible +evolutionary path for objects with large positive and neg- +ative β’s that are well virialized. These objects in general +have stopped their star formation long ago and their lumi- +nosity progressively decreases. When ETGs become passive +quenched objects, with a luminosity decreasing at nearly +constant σ, the galaxies can only move in one direction, that +given by the large positive and negative values of β. +5. The infall model galaxies built here, although lacking the dy- +namical component, are in reasonable agreement with obser- +vations and support the idea that β and L′ +0 vary across time, +and therefore that the L = L′ +0σβ law is a plausible empirical +relation accounting of the variation occurred in a galaxy. +6. All the diagrams built using the structural parameters are +sensitive to the temporal evolution of galaxies, simply be- +cause each individual object moves in a different way ac- +cording to the value of β. +7. Both observations and theory suggest that the L = L′ +0σβ re- +lation provides an empirical way of capturing the temporal +evolution of ETGs (and probably of late-type objects) be- +cause the values of β are related to the history of mass assem- +bly and luminosity evolution. Because of it we are tempted to +suggest that eqs. (14) are two important equations governing +the evolution of ETGs. +8. Finally, the large negative and positive values of β of some +galaxies can be considered as the signature that these sys- +tem are very close to the virial equilibrium, i.e. their basic +parameters Re, L∆λ, Ms, (Ms/L∆λ), σ, and Ie,∆λ are such that +the strict virial condition is verified. The situation is likely +transient because both internal and/or external events may +alter one or more parameter so that the strict virial condition +is no longer verified. 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P. 2007, ApJ, 655, 30 +Varela, J., D’Onofrio, M., Marmo, C., et al. 2009, A&A, 497, 667 +Vogelsberger, M., Genel, S., Springel, V., et al. 2014, Nature, 509, 177 +Article number, page 25 of 25 + diff --git a/u9E8T4oBgHgl3EQfaxWZ/content/tmp_files/load_file.txt b/u9E8T4oBgHgl3EQfaxWZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8073783ce34e00b059d151c81a3b6a0d0c2973af --- /dev/null +++ b/u9E8T4oBgHgl3EQfaxWZ/content/tmp_files/load_file.txt @@ -0,0 +1,2590 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf,len=2589 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='06953v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='GA] 17 Jan 2023 Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' FPaanda2 ©ESO 2023 January 18, 2023 A new framework for understanding the evolution of early type galaxies M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio,⋆1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi1 Department of Physics and Astronomy, University of Padua, Vicolo Osservatorio 3, I35122 Padova (Italy) e-mail: mauro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='donofrio@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='it e-mail: cesare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='chiosi@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='it Received December, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' accepted January, 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' We have recently suggested that the combination of the scalar virial theorem (Ms ∝ Reσ2) and the L = L′ 0σβ law, with L′ 0 and β changing from galaxy to galaxy (and with time), can provide a new set of equations valid for investigating the evolution of early-type galaxies (D’Onofrio & Chiosi 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' These equations are able to account for the tilt of the Fundamental Plane and to explain the observed distributions of early-type galaxies in all its projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In this paper we analyze the advantages offered by those equations, derive the β and L′ 0 parameters for real and simulated galaxies, and demonstrate that, according to the value of β, galaxies can move only along some permitted directions in the fundamental plane projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Then, we show that simple galaxy models that grow in mass by infall of gas and form stars with a star formation rate depending on the stellar velocity dispersion nicely reproduce the observed distributions of early-type galaxies in the Fundamental Plane projections and yield βs that agree with the measured ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' We derive the mutual relationships among the stellar mass, effective radius, velocity dispersion, and luminosity of early- type galaxies as a function of β and calculate the coefficients of the Fundamental Plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Then, using the simple infall models, we show that the star formation history of early-type galaxies is compatible with the σ-dependent star formation rate, and that both positive and negative values of β are possible in a standard theory of galaxy evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The parameter β(t) offers a new view of the evolution of early-type galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In brief, i) it gives a coherent interpretation of the Fundamental Plane and of the motions of galaxies in its projections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' ii) it is the fingerprint of their evolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' iii) it measures the degree of virialization of early-type galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' iv) and finally it allows us to infer their evolution in the near past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' galaxies: structure – galaxies: evolution – galaxies: ellipticals and lenticulars – galaxies: scaling relations 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Introduction This study is the latest of a series aimed at demonstrating that the scaling relations (Sc-Rs) for early-type galaxies (ETGs), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the mutual correlations between the main structural parameters of galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the stellar mass Ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the effective radius Re,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the effective surface intensity Ie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the luminosity L and the central velocity dispersion σ)1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' can be fully understood if we adopt a new perspective in which the Virial Theorem (VT) of the stellar systems is coupled to the galaxy luminosity taking into account that this latter can randomly vary with time as a result of accre- tion/depletion events associated to mergers/close encounters ex- pected in the hierarchical galaxy formation scenario in addition to the natural evolution of its stellar content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The new equation governing the luminosity is expressed by L(t) = L′ 0(t)σ(t)β(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1) in which L, L′ 0, σ and β are all functions of time and can vary from galaxy to galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This relation is formally equivalent to the Faber & Jackson relation for ETGs (Faber & Jackson 1976), but it has a profoundly different physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In this relation ⋆ Corresponding author: Mauro D’Onofrio 1 Therein after by structural parameters of a galaxy we mean those of the above list, and leave aside the parameters that define the internal structure such as the Sérsic index, the axial ratio, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Galaxies are con- sidered point mass objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' β and L′ 0 are free time-dependent parameters that can vary con- siderably from galaxy to galaxy, according to the mass assembly history and stellar evolution of each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This relation empir- ically encrypts the effects of all the above physical processes in terms of luminosity and velocity dispersion variations, param- eters that can both vary across time because galaxies evolve, merge, and interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In our previous works we tried to highlight some of the advantages offered by coupling the VT with the L = L′ 0σβ law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The first efforts were dedicated to understand the origin of the Fundamental Plane (FP) of ETGs and the distributions observed in its 2D projections (D’Onofrio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2017a, 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio & Chiosi 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' While discussing this prob- lems, D’Onofrio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2017b) and D’Onofrio & Chiosi (2022) advanced the idea that the explanation invoked for the origin of the FP tilt (and its small scatter) should also account for the ob- served distributions of galaxies in all the 2D projections of the FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The solution was found in the coupling of the VT with the time-dependent L = L′ 0σβ relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The key idea behind this approach is that the luminosity of galaxies is not simply related to the total stellar mass, but also to random variations caused by mergers and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This im- plies that, accepting the L = L′ 0σβ law as an empirical descriptor of the possible changes occurring in σ and L, one can describe a galaxy with two different independent equations: the classical scalar VT on the notion that galaxies are always very close to Article number, page 1 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' FPaanda2 the mechanical equilibrium, and the L = L′ 0σβ law that fully ac- counts for all possible processes taking place during the lifetime of a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Our previous studies have successfully shown that a β param- eter changing with time and assuming either positive and nega- tive values, can easily explain the movements and distribution in the planes of the Sc-Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This approach is in fact able to explain in a natural way the tilt of the FP, the existence of the Zone of Exclusions (ZoE) observed in many Sc-Rs, and the direction of motion derived from the changes in σ and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In this work we aim to provide evidences that such an ap- proach gives a global interpretation of the Sc-Rs observed for ETGs and that even the classical monolithic view of mass as- sembly is in agreement with the idea of a variable β parameter thus confirming the L = L′ 0(t)σβ(t) law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The paper is organized as follows: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2 gives a short de- scription of the samples of galaxies (both real and simulated) used in this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 3 is dedicated to the derivation of the new equations of galaxy evolution and to the different relations among the structural parameters in all FP projections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 4 presents a few new simple models of ETGs growing with a SFR depending on σ and accounts for the role of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Finally, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 6 provides our discussion and conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In all calculations we used the parameters of the ΛCDM cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The samples of real and model galaxies Observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The observational data used in this work are the same of D’Onofrio & Chiosi (2021) and D’Onofrio & Chiosi (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The data for the real galaxies are extracted from the WINGS and Omega-WINGS databases (Fasano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Varela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Cava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Valentinuzzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Gullieuszik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Cariddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Biviano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The sample is not homogeneous because the spectroscopic database is only a sub-sample of the whole optical sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The ETGs with available velocity dispersion σ, stellar mass Ms, and star formation rate SFR, are less numerous than those extracted from the photometric database (providing Re, Ie, n, LV, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In particular we used: 1) the velocity dispersion σ of ∼ 1700 ETGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The σ measurements come from the SDSS and NFPS databases (Bernardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2004) and were measured within a circular area of 3 arcsec around the center of the galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2) the luminosity, effective radius and effective surface brightness in the V-band of several thousand ETGs, de- rived by D’Onofrio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2014) with the software GASPHOT (Pignatelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The effective radius is determined from the luminosity growth curve by considering the circle that con- tains half the total luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The effective surface intensity fol- lows directly from the knowledge of L and Re;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 3) the distance of the galaxies derived from the redshift measured by Cava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2009) and Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 4) the stellar mass obtained by Fritz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2007), only for the galaxies of the southern hemi- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The cross-match between the spectroscopic and optical sam- ples provides here only 480 ETGs with available stellar mass, lu- minosity, velocity dispersion, Sérsic index, effective radius and effective surface brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The error of these parameters is ≃ 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' These are not shown in our plots, because they are much lower than the observed range of variation of the structural pa- rameters in the scaling relations and do not affect the whole dis- tribution of ETGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Occasionally, we have also used the catalog by Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997) containing objects from Globular Clusters (GCs), Dwarf Galaxies (DGs) of different types, to late and early type galax- ies (LTGs and ETGs, respectively), and finally clusters of galax- ies (GCGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' They are used to have a general idea of the Sc-Rs for systems of different sizes, but dynamically close to the virial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Limited to ETGs sometime we also used the sample of Bernardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Simulated galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The hydrodynamic simulations, are probably the best galaxy models today available to compare theory with observations despite the fact that several problems still bias their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' There are several suites of galaxy simula- tions in cosmological context among which we recall Illustris-1 by Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Genel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2015), recently superseded by Illustris-TNG by Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2018a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2018b), and EA- GLE by Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' We decide to adopt here Illustris-1 for two reasons: first chief the fact we want to be consistent with the results shown in our previous papers on this same subject that were based on the Illustris-1 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Second, we have checked that the main results of our analysis do not change passing from Illustris-1 to Illustris-TNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The kind of analysis carried out here is indeed somehow in- dependent of the level of precision reached by models from dif- ferent sources, because we are mainly interested to present a new method for deciphering the information encrypted in the obser- vational data about the past history of ETGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' To this aim, we have extracted from the Illustris-TNG database at redshift z = 0 a sample of about thousand model galaxies of all possible masses that are used to support the above statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Our data-set extracted from Illustris-1 consists of several sub-sets of about ∼ 2400 galaxies each, sampled different at red- shifts from to z = 0 to z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' A full description of these data is given in Cariddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2018) and D’Onofrio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In particular, we collected the effective radii, the total luminosity, the stellar mass, and the velocity dispersion, the age, and the star formation rate, together with radii, masses, and velocity disper- sion of the dark matter component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' A detailed analysis of the differences between Illustris-1 and Illustris-TNG data has been made by Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2018b,a), Rodriguez-Gomez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2019), and Huertas-Company et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' One of the issues of major tension between the two suites of models concerns the radii of the low mass galaxies (roughly of Ms ≤ 5 1010 M⊙ where the Illustris-TNG radii are about a factor of two smaller that those of Illustris-1 while above it they are nearly equal (Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2018b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Rodriguez-Gomez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Huertas-Company et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2019) compared the log(Re) − log(Ms) plane built with the two sources above and the SDSS data of Meert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2015) finding the same result (see their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' To better illustrate the difference in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1 we compare the data of Illustris-1 with those of Illustris-TNG-100 and the WINGS objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The difference in the low mass range is con- firmed, but the hockey-stick like shape of the distribution of model galaxies in the two samples is the same (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 9 and 11 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In addition to this, there is the claim that Illustris-1 simu- lations do not produce a realistic red sequence of galaxies due to insufficient quenching of the star formation with too few red galaxies (Snyder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Bottrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2017a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Rodriguez-Gomez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2019), while the Illustris-TNG simulations produce a much better red sequence (Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Rodriguez-Gomez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' There is also the prob- lem of the insufficient number of red galaxies with respect to the Article number, page 2 of 25 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi: A new framework for understanding the evolution of early type galaxies Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Left panel: The stellar mass versus radius relations for the Illustris-1 (open red squares) and the Illustris-TNG-100 (blue dots) samples at z = 0 and comparison of the models with the WINGS data (black dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' There are 2400 objects for the Illustris-1 sample and about 600 objects for the Illustris-TNG-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The mean radii of Illustris-1 are smaller than about a factor of two for stellar masses smaller that about 6 1010 M⊙, while they are nearly equal if not slightly larger above this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Right panel: The L − σ plane of the same data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The symbols and color codes are the same as in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' observed population of ETGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This is of little importance for our analysis because we do not make use of colors but only of total luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Concerning the internal structure of the Illustris-1 galaxies, Bottrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2017b) measured the Sersic index, the axis ratio and the radii of these galaxies and found that too few bulge- dominated objects are produced in tension with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In contrast the Illustris-TNG galaxies have much better inter- nal structural parameters (Rodriguez-Gomez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Fortu- nately, the point mass view of the Illustris-1 models we have adopted secures that our analysis is not too much affected by this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Finally, the Illustris-1 data-set does not give information about the morphology of the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This means that in our comparison ETGs and late-type objects are mixed in our plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Again Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 11 of Huertas-Company et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2019) shows us that ETGs and late-type objects follow very similar trends in the Sc- Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The basic features of the Sc-Rs shown by ETGs are not de- stroyed with the addition of late-type objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Anyway, since in our work we do not make any predic- tion, but only qualitatively compare observations and simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Looking at their behavior in the FP projections, the good match between data and simulations, and the fact model galaxies are able to reproduce some particular features visible in the FP projections (like e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the position of the BCGs and the existence of a ZoE) lend support to the scenario proposed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' All this makes us confident that the simulations produce galaxies with luminosity and primary structural parameters not too far from those of real galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Given the large heterogeneity of data used here, we remark that the completeness of the data sample is not fundamental for the conclusions drawn in this work, because we neither make any statistical analysis of the data nor we fit any distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The data are only used qualitatively to show that our calculations are in agreement with the observed distributions of ETGs in the main Sc-Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The purpose of this paper is only that of proposing a new possible framework to analyze the evolution of ETGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The equations of galaxy evolution The equations tracking the evolution of galaxies are based on two hypotheses: 1) ETGs are always close to the virial equilib- rium, a reasonable assumption since the dynamical time scale to reach such condition is of the order of the free-fall time (< 300 Myrs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2) the L = L′ 0σβ law somehow mirrors the effects of many internal and external events affecting luminosity and ve- locity dispersion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The two equations are: σ2 = G kv Ms Re (2) σβ = L L′ 0 = 2πIeR2 e L′ 0 (3) where kv is the non homology parameter defined by Bertin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The unknown variables of this system of equations to be found are β and L′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Combining these two equations are together, one can write: a1 log σ + b1 log Ie + c1 log Re + d1 = 0 (4) where the coefficients: Article number, page 3 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' FPaanda2 a1 = β − 2 (5) b1 = −1 c1 = −3 d1 = log(Ms) − log(kv/G) − log(2π/L′ 0) are written in terms of β and L′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The similarity with the FP equa- tion is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This is the equation of a plane in the log(σ) − log(Ie) − log(Re) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The novelty is that each galaxy fol- lows independently an equation like this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In this case, since β and L′ 0 are time dependent, the equation is telling us which is the instantaneous direction of motion of an object in the log(σ) − log(Ie) − log(Re) space and in its projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Before showing this, let us trace back the past history of the reasoning presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Starting from the same ar- guments and equations (3 and 4), after tedious algebraic manip- ulations D’Onofrio & Chiosi (2022) arrived to a cubic equation in the variable β (their eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 10), the coefficients of which where function of σ, Ie, Re, Ms, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The cubic equation was ap- plied to real galaxies of the WINGS list and model galaxies of the Illustris-1 catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In most cases three real roots were found, two of them positive and one negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In some cases the so- lutions were complex and this was attributed to insufficient ac- curacy in the input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The mutual agreement between the two sets of data (WINGS and Illustris-1) was considered as a strong hint for self consistency of the whole approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This agreement was indeed misleading because it masked first an al- gebraic mistake made while carrying out the lengthy analytical manipulations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' a factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 missing in front of a group of terms in logarithmic form), second that the agreement between WINGS and Illustris-1 made via the cubic equation was in re- ality a circular argument as in each case the results would have been the same regardless of whether the equation was correct or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Furthermore, attempts to incorporate the cubic equation in model galaxies did not lead to a clear understanding of the phys- ical role and meaning played by the three L = L′ 0σβ relations associated to each time step (the factor L′ 0 being derived from the real luminosity by comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' It was clear that some of the βs changed sign in the course of evolution and also that complex solutions could occur during the lifetime of a galaxy, the low mass ones in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' However, from these results the tantaliz- ing suggestion came out that a solution of the puzzle could be reached by changing strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' All this led us to revise the whole problem thus discovering the analytical mistake and putting the mathematical formulation on the right track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The new version of the problem is presented here below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The cubic is replaced by a system of equations in the unknowns β and log L′ 0, thus fully determining the L = L′ 0σβ and its evolutionary history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Starting from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 3 and 4, after some algebra it is possible to write all the relations among the parameters of the FP projec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' For the Ie − Re plane we have: Ie = ΠRγ e (6) where γ = (2/β) − (1/2) (1/2) − (1/β) and Π is a factor that depends on kv, M/L, β, and L′ 0 and id de- scribed by: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Comparison between observed and calculated parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The data of the WINGS database are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The black solid line marks the 1:1 relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The red solid line is the bi-linear least square fit of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Π = \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 �2π L′ 0 �1/β � L Ms �(1/2) � kv 2πG �(1/2)\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb 1 1/2−1/β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' For the Re − σ plane we have: Re = ��kv G � � L′ 0 2π � � 1 Ms � � 1 Ie �� σ(2+β), (7) for the Ie − σ plane: Ie = ��G kv � (Ms) � L′ 0 2π � � 1 R3e �� σ(β−2) (8) and for the Re–Ms plane: Re = \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 �G kv � � L′ 0 2π �2/β � 1 Ie �2/β\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb β/(β+4) Mβ/(β+4) s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (9) It should be remarked here that these equations do not rep- resent the true physical link between two variables because their proportionality factor contains other variables as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In other words, they do not tell us how Re and Ie vary when σ changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' They are intermediate mathematical expressions yielding the structural parameters Re or Ie as functions of the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Fig- ure 2 gives an idea of the degree of precision in reproducing the structural parameters when eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (6), (7), (8) and (9) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The x-axis contains the measured parameters, while the y-axis the values calculated on the basis of our equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The scatter in log units ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='4, so a factor of 2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 uncer- tainty is possible and likely attributable to the ∼ 20% errors of the scaling parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The importance of these equations is that, starting from them, one can also write the following equations (in log form): Article number, page 4 of 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 4 log(R) [calculated] R-M" [calculated] 4 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 3 log(1e) 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 I-R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 1 2 3 4 log(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=') [observed] log(I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=') [observed] 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 [calculated] R-0 4 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 1 1 2 3 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 log(I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' )[observed] log(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=')「observedlM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi: A new framework for understanding the evolution of early type galaxies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='β[log(Ie) + log(G/kv) + log(Ms/L) + log(2π) + log(Re)] + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='(10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='+2 log(L′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='0) − 2 log(2π) − 4 log(Re) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='β log(σ) + log(L′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='0) + 2 log(σ) + log(kv/G) − log(Ms) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='(11) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='− log(2π) − log(Ie) − log(Re) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='and assuming: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='log(Ie) + log(G/kv) + log(Ms/L) + log(2π) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='(12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='log(Re) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='−2 log(2π) − 4 log(Re) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='A′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='log(σ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='B′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2 log(σ) − log(G/kv) − log(Ms) − log(2π) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='log(Ie) − log(Re) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='write the following system: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='Aβ + 2 log(L′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='0) + B = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='(13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='A′β + log(L′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='0) + B′ = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='with solutions: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='−2 log(L′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='0) − B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='(14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='log(L′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='A′B/A − B′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='1 − 2A′/A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In other words, it is possible to derive the values of β and L′ 0 for each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This means that the knowledge of the structural parameters reveals the basic step of galaxy evolution encoded in the parameters β and L′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 3 shows the histograms of the distributions of the β pa- rameter derived for the galaxies of the WINGS and Illustris-1 samples respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the upper panel the β values for the Illustris-1 data are obtained from the ∆L/∆σ ratio measured on the L − σ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This is possible by considering the values of L and σ at two close redshift epochs (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2 and z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the lower panel we have considered only the objects that are close to the virial equilibrium, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' those for which: 2 log(σ) = log(G/kv) + log(Ms/L) + log(2π) + log(Ie) + log(Re) (15) within a 20% uncertainty, and calculated β using our new an- alytical equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' When this condition is satisfied we get that 2A′/A = 1 and β and L′ 0 diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Notably the values of β are both positive and negative and there is a clear deficiency of objects with β close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This is true both for WINGS and Illustris-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The average value of β is −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='44 with a rms scatter of ∼ 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The positive values range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='05 to 1531, while the negative ones from −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='4 to −3860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The importance of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (15) is that we have now an empiri- cal thermometer of the virial condition, realized when β and L′ 0 diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The meaning of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 3 is that galaxies during their evolu- tion can acquire either positive and negative values of β, depend- ing on the particular events experienced (merging, stripping, star Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Upper panel: Histogram of the β solutions derived from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 15 for the real ETGs (black line) compared with the distribution derived for the galaxies of the Illustris-1 simulation when β is calculated looking at the variation of luminosity and velocity dispersion in two close redshift epochs (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2 and z = 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Lower panel: Histogram of the β solutions derived from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 15 for the real ETGs (black line) and the Illustris- 1 galaxies (red line) that are close to the virial equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The solid black line marks the average value of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The distribution of β as a function of the degree of virialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' formation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' ), and this has immediate effects on the structural parameters in the Sc-Rs, that change accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Consequently, the Sc-Rs seen in their temporal framework become sources of information for the global evolution of the stellar systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Figure 4 shows the distribution of β as a function of the de- gree of virialization, expressed by the quantity 1 − 2A′/A de- rived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The large values of β are attained by objects very close to the virial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' On the other hand, the small β’s belong to objects still away from this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Article number, page 5 of 25 200 100 0 100 1: 1: 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content="4 1-2A'/A0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2 8=4L/40 WINGS Illustris 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='15 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='05 0 100 0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2 Arnalytic B WINGS Illustris 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='05 0 100 0 100A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' FPaanda2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Histogram of the β solutions derived from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 15 for the real ETGs (black line) of WINGS, the Illustris-1 galaxies (red line) that are close to the virial equilibrium, and the Illustris-TNG galaxies (blue line) in the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The three histograms are qualitatively similar, thus confirming that our analysis depends little on the choice between the two theoretical data-bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In closing this section, we show the distribution of the βs we would obtain with the galaxy models of the Illustris-TNG-100 sample and compare it with those of Illustris-1 and the WINGS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The three histograms are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The difference is very small and largely due to the smaller number of galaxies in the Illustris-TNG sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Trends in the FP projections In this section we try to better explain the reasons why β can change sign during the life of a galaxy or when passing from one galaxy to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The advantage of knowing β is much clear when we look at the projections of the FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (6), (7), and (8) can be further elaborated to eliminate the dependence on Ie and Re present in their zero-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' We get: Ie = �G kv L′ 0 2π MsΠ3/γ � β−2 1+3/γ σ β−2 1+3/γ (16) Re = �G kv L′ 0 2π Ms Π � σ β−2 3+γ (17) Re = � (G kv )β/2 L′ 0 2π 1 Π � 2(β−2) β2−6β+12 M β2−2β β2−6β+12 s (18) These relations now better represent the mutual dependence of the structural parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' of Ie as a function of Re and σ and of Re as function of σ) and clarify what is the role of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' When a galaxy moves in the L−σ plane, according to the values of β, it does the same also in the other FP projections, according to the slopes reported in the last three columns of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' These slopes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The Ie − Re plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The black and red arrows mark the direction of motion of galaxies in this plane for large negative and positive values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The black solid line gives the lsq fit of the data, while the broken line represents the zone of exclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' depend on β and indicate the direction of motions (marked by the arrows) that are visible in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 6-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Figure 6 shows the case of the Ie − Re plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The black and red arrows mark the direction of motion of galaxies predicted on the basis of their negative and positive values of β respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Since the WINGS galaxies are well virialized, the values of β are always very large, either positive and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Both such slopes give consistently a direction of motion close to ∼ −1 in the Ie − Re plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The −1 slope is that predicted on the basis of the VT (represented by the broken line which also marks the ZoE) (see, D’Onofrio & Chiosi 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' We note that no galaxies can cross the ZoE, because their motion is nearly parallel to the ZoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' These arguments demonstrate the reason why there is a ZoE in the Ie−Re plane: the only possible direction of motion for well virialized objects is that with slope ∼ −1 generated by the large positive and negative values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' When β ∼ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' when the galaxies are less virialized, they can move in other directions in this plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Unfortunately, our sample does not include the dwarf ETGs, that are usually dis- tributed in a cloud, below the ZoE with radii lower than 3-4 kpc (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Capaccioli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' For these objects we predict values of the slopes in all possible directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The only way to check this is to make use of the model galaxies either of Illustris-1 or Illustris-TNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Figure 7 confirms our prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Although the data of Illustris-1 are affected by the well known problem of the systematically larger Re with re- spect to the observed ones (D’Onofrio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Bottrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2017c), we can note that several objects have arrows nearly or- thogonal to those of the well virialized galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The expected motions of the dwarf galaxies are in all possible directions, thus giving rise to the cloud of the "ordinary" ETGs defined by Capaccioli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Furthermore, when the distribution curves its shape, we note a progressive variation of the arrow di- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This means that the overall distribution in this plane is Article number, page 6 of 25 B>0 7 ([z-3d])60] 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 5 log(R[pc])0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2 WINGS Illustris-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='1 Tot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='05 Z 0 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2 WINGS Ilustris-TNG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='05 0 100 0 100 βM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi: A new framework for understanding the evolution of early type galaxies Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The Ie − Re plane for the WINGS and Illustris-1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The black and red arrows mark the direction of motion of the WINGS galaxies in this plane for large negative and positive values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The green and blue arrows are those of Illustris-1 for negative and positive values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the plot we have used only 1/10 of the Illustris-1 galaxies in order to permit to distinguish the objects with different β values moving in different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' governed by the different movements of the galaxies in the L−σ plane described empirically by the different values of β and L′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The direction of the arrows displayed in each figure visual- izes the expected displacement of a galaxy based on the actual value of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' However, the arrows only give the direction of mo- tion, not the orientation of the future temporal evolution of a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Furthermore, they do not indicate the path followed by each galaxy to reach the current observed position in the dia- grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The same can be said for the other two FP projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Fig- ures 8 and 10 represent the Re −σ and Ie −σ planes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Here, the role of β is much more clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' It is well evident in fact that the galaxies with negative β’s move in different directions with respect to those with positive β, originating the curvatures observed in these diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Once more the addition of the Illustris-1 data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 7 and 9) confirms that the slopes derived from the βs are consistent with the observed distribution of ETGs and demonstrates that the observed curvatures originate from the different motion of galaxies with positive and negative values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Figures 12 and 13 are similar plots for the log(Re) − log(Ms) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Even in this important diagram we observe a ZoE (marked by the dashed line with slope equal to 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Our calculations pre- dict why this ZoE is here: the reason is that all the virialized objects (with large β values) can only move in the direction with slope equal to 1 (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The Re of Illustris-1 are notori- ously somewhat larger than those measured, but the general be- havior is in good agreement with the observed distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The galaxies with large positive and negative values of β move with a slope close to 1, while in the cloud of points with small masses we can note objects with different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' We conclude that all the positions of ETGs in the FP projec- tions and in the log(Re) − log(Ms) plane depend on the motions Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The Re − σ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Symbols and colors as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The Re −σ plane for WINGS and Illustris-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Symbols and colors as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' occurred during the peculiar evolutionary path followed by each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' When the galaxies are well virialized these motions can occur only in well fixed directions depending on the value of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This is a coherent and self-consistent explanation of all the main scaling relations built with the structural parameters of ETGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' It follows that even the FP, the father of the scaling re- lations, must find a similar explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Looking at Table 1 in detail we also note that: – In all FP projections, when β becomes progressively nega- tive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' when the objects are rapidly declining in their lu- minosity at nearly constant σ, the slopes either converge to the values predicted by the VT (in the Ie − Re relation and in the Re–Ms relation), or diverge toward large values (in the Article number, page 7 of 25 5 β>0 8>0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 log(R[pc) 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 log(α[km s-1)5 R>0 0>d 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 log(R[pc]) 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 log(α[km s-1l)B>0 β>0 3 0 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 5 log(R[pc])A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' FPaanda2 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The slopes of the Ie − Re, Re − σ, Ie − σ and Re − Ms planes for different values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' β Ie − Re Re − σ a Ie − σ b Re-Ms c Re − σ d Ie − σ e Re-Ms f 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='98 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='0 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='04 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='50 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='96 Notes: a) Slope when kv, Ms and Ie are constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' b) Slope when kv, Ms and Re are constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' c) Slope when kv, and Ie are constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' d) Slope when kv and Ms are constant: e) Slope when kv and Ms are constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' f) Slope when kv is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The Ie − σ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Symbols and colors as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Ie − σ and Re − σ relations), because the galaxy keeps its ve- locity dispersion when the luminosity decreases (only Ie and Revary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This offers a natural explanation of the ZoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' – Positive and negative values of β are equally permitted, both with real and simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In general, the objects that are still active in their star formation or have recently experi- enced a merger, have positive values of β, while those pro- gressively quenching their SF have increasing negative β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' – The "curvature" in the observed distributions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the transi- tion from the large cloud of small galaxies to the much nar- row tail of the brightest objects) is naturally explained by the existence of positive and negative values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' A way to better understand the effects played by β is to think at the possible variations of Re and Ie when L and σ vary in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The Ie−σ plane for WINGS and Illustris-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Symbols and colors as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Trends of the effective parameters as a consequence of changes in σ and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' β > 0 L&σ ր Re ր Ie(const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' or ց) Ms ր L&σ ց Re ց Ie(const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' or ր) Ms ց β < 0 L ց & σ ր Re ց Ie(const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' or ր) Ms(const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' or ր) L ր & σ ց Re ր Ie(const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' or ց) Ms(const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' or ց) Article number, page 8 of 25 B>0B≥0 3 log(Ie[Lopc-2]) 2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 log(αkm s-1])8>0 3 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 log(α[km s-1])M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi: A new framework for understanding the evolution of early type galaxies Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The log(Re) − log(Ms) plane for the WINGS galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Symbols and colors as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The log(Re)−log(Ms) plane for WINGS and Illustris-1 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Symbols and colors as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the L − σ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' There are four possible changes of L and σ in this plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' They are schematically shown in Table 2, which displays, according to the values of β, the expected variations of Reand Ie, when L, Ms and σ vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Note that when β is negative, not necessarily there is a decrease in luminosity, and when β is positive, a decrease in luminosity might also occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' When the luminosity of a galaxy changes, both the effective radius and the mean effective surface intensity Ie vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This hap- pens because Re is not a physical radius, like e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the virial radius (which depends only on the total mass), but it is the radius of the circle that encloses half of the galaxy total luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Since the ETGs have different stellar populations with different ages and metallicity, it is highly improbable that the decrease in luminos- ity does not change the whole appearance of the luminosity pro- file2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Consequently the growth curve changes and determines a variation of Re and Ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' If the luminosity decreases passively, in general one could expect a decrease of Re and an increase of Ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' On the other hand, if a shock induced by harassment or stripping induces an increase of L (and a small decrease in σ), we might expect an increase of Re and a decrease of Ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The observed variations of these parameters depend strongly on the type of event that a galaxy is experiencing (stripping, shocks, feedback, merging, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In general, one should keep in mind that these three variables L, Re and Ie are strongly coupled each other and that even a small variation in L might result in am- ple changes of Re and Ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In this context, we begin to understand that the Sc-Rs are useful tools for guessing both the dynamics and the evolutionary state of the stellar content of a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In summary, what we claim here is that all the above dia- grams should be analyzed taking into account the effects of time and should not be investigated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' They are snapshots of an evolving situation, and such temporal evolution cannot be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The L = L′ 0σβ law catches such evolution in the cor- rect way by predicting the direction of the future motion of each galaxy in the diagnostic planes (D’Onofrio and Chiosi, A&A submitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In principle, this way of reasoning should allow us to understand why galaxies are in the positions observed today in each diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' As β gives only the present direction of motion and not that of the motion in the past, the simultaneous use of simulations and high redshift observations might help to infer the possible precursors of the present day galaxies on the basis of the physical properties and the distribution in the FP projec- tions, indicated by the values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In other words these scaling relations become a possible tool for inferring the evolutionary path of each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Origin of the FP and its tilt The final step is that related to the question of the origin of the FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Equation (4) tells us that each galaxy follows its own FP-like equation, whose coefficients are functions of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Starting from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (4) it is possible to derive the coefficients a, b and c of the plane hosting each single ETG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' To do this we adopt the notation that is commonly used for the FP, in which ⟨µ⟩e is expressed in mag arcsec−2 and Re in kpc: log(Re) = a log(σ) + b < µ >e +c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (19) We get: a = a1/(−c1) (20) b = b1/(−c1) c′ = d1/(−c1) where a1, b1, c1, and d1 are from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (4), and c = (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='56 ∗ b1)/ − c1) − 3 + c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This transformation is necessary because in our no- tation Ie is expressed in L⊙pc−2 and Re is in pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The distribution of these coefficients for all the WINGS sam- ple of 479 galaxies is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' It is clear from the figure that the values for the FP coefficients, derived from the fit of the ETGs distribution and indicated by the dashed areas in each panel, are very close to the average of the single coefficients cal- culated by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 20 (the vertical black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The gray bands show 2 This could happen only in a coeval stellar system with the same type of stars in any galaxy volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Article number, page 9 of 25 8>0 8>0 5 0>9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 log(Re[pc) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 6 10 1 1 12 13 log(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='[M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=')5 8>0 0>9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 log(R[pc]) 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 9 10 11 12 13 log(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='[M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='l)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' FPaanda2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Histograms of the values of the calculated FP coefficients a (top panel), b (middle panel) and c (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In each panel we show the histogram of the coefficient (blue line), the average value (the vertical black line) and the median (the vertical red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the case of the b coefficient, that does not depend on β, average and median are the same (the black and red vertical lines coincide).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The dashed gray regions mark the intervals of the FP coefficients found by fitting the distribution of ETGs in the log(σ) − log(Ie) − log(Re) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' indeed the interval of a, b and c obtained by D’Onofrio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2008) fitting the FP of ETGS separately for each cluster of the WINGS data-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In other words, we have framed the FP and its tilt in a new context in which each ETG follows its own eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (4), namely FP, and contributes to shape the global FP (both tilt and thickness) of the ETG population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Since the FP coefficients are obtained from a fit, it is clear that the final coefficients of the plane will be close to the average of the single values valid for each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Some differences are expected because the final values will depend on the sample adopted (each having its own average) and from the technique used to perform the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' With this statement we do not mean to say that the various mechanisms invoked in the literature to explain the tilt and thick- ness are incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Rather, we claim that all of them can actually contribute to the average properties of the galaxy sample, giving rise to a different β for each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Since its discovery, the FP has been the subject of several studies aimed at understanding why the plane is tilted with re- spect to the prediction of the VT (Ms ∝ Reσ2), and why its in- trinsic scatter is so small (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Faber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Ciotti 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Jorgensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Cappellari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Bolton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2007, among many others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' While the VT predicts a = 2 and b = −1, the values coming from the fit of several samples of ETGs are systematically lower (a ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2) and higher (b ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='8), and vary according to the sample used and the fitting strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Among the physical mechanisms invoked to explain the FP tilt we can find: 1) a progressive change of the stellar mass-to-light ratio (Ms/L) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Faber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' van Dokkum & Franx 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Cappellari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' van Dokkum & van der Marel 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Holden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' de Graaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2) structural and dynamical non- homology (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Prugniel & Simien 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Busarello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Trujillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 3) dark matter (DM) content and distribution (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Ciotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Borriello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Tortora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Taranu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' de Graaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 4) star formation history (SFH) and initial mass function (IMF) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Renzini & Ciotti 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi & Carraro 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Allanson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 5) the effects of environment (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Lucey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' de Carvalho & Djorgovski 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Bernardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' La Barbera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Ibarra-Medel & López-Cruz 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Samir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Recent observational work has shown that variations in the Ms/L ratio can account only for half of the tilt (see D’Eugenio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2021), with remainder being due to structural variation and possibly variations in the galaxy-averaged initial mass function of the stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Uncertainties in Ms can affect the tilt if the error is mass-dependent, although this systematic uncertainty is not large enough (see Leja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Lower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Schechter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2014) using strong lensing measurement provides an independent estimate of Ms but still finds a tilt of the FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Finally, the tilt is found in cosmological simulations (Rosito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2019a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' de Graaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' All these effects are in practice "involved" in our view of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Indeed, since each sample of galaxies has its own average value of β, because of the different history of mass ac- cretion and luminosity evolution, it is easy to verify that sys- tematic changes in the tilt could arise for the above mentioned reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' When the sample changes its average properties, a small variation of the tilt of the FP follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This explains for instance why Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2006) found that star-forming and quies- cent galaxies follow different Sc-Rs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' a different FP tilt, due to differences in the merger histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' With the emerging of the hierarchical scenario of galaxy for- mation and evolution, some additional mechanisms for the FP tilt have been proposed: 1) the effects of dissipation-less merg- ing (Nipoti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2) the gas dissipation (Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 3) the non regular sequence of mergers with progressively decreasing mass ratios (Novak 2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 4) the multiple dry merg- ers of spiral galaxies (Taranu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Since the galaxy properties change with time, the slope of the FP is expected to change with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This is confirmed by the numerical models of single galaxies, large scale cosmologi- cal simulations, and observational surveys at different redshifts, among others see Beifiori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Rosito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2019a,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Ferrero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' de Graaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2022) and references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The most remarkable physical feature of the FP is the ob- served very small scatter, which amounts to ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='05 dex in the V-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' It seems to require a sort of fine tuning among different physical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The scatter has been attributed to: 1) the variation in the formation epoch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2) the DM con- tent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 3) the existence of metallicity or age trends;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 4) the vari- ations of the mass-to-light ratio M/L (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Faber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Gregg 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Guzman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Forbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Bernardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Reda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Cappellari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Bolton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Graves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Graves & Faber 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Auger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Magoulas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Our approach cannot predict the scatter around the FP, be- cause this does not depend on the structural parameters, but on the properties of the stellar populations and the peculiar history of mass accretion/stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Although investigating the causes of the tilt and the small dispersion around the FP is beyond the aims of this paper, let as Article number, page 10 of 25 a 100 50 0 20 10 0 10 20 b 100 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='4 80 60 C 40 20 0 40 20 0 20 40M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi: A new framework for understanding the evolution of early type galaxies conclude this section with one consideration: it may be an amaz- ing coincidence, but we note that going back to high redshift, the numerical simulations of Illustris-1 and Illustris-TNG show that the FP and the tail of the MRR persist until redshift z ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='6 and then disappear or is no longer so well defined (D’Onofrio & Chiosi, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', 2022, in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The concomi- tant appearance of the MRR and FP for ETGs more massive that about 1010 M⊙ maybe is a mere coincidence, but surely it is a question that must be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The galaxies on the tails of the FP and the MRR are mas- sive ETGs whose stellar content is predominantly made of old stars or in the case of mergers with objects of smaller mass the percentage of younger stars does not alter significantly the lumi- nosity and the colors of the basic stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' To quantify this statement let us make the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' At proceed- ing galaxy building via the hierarchical scenario, the probability that a massive objects merge with another of similar mass be- comes rarer and rarer as galaxies become more massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' There- fore, massive ETGs tends to evolve in isolation or merging ob- jects of much smaller mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In general, the merger of two galax- ies with very different masses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', M1/M2 ≃ 1/10) and some companion stellar activity leaves the mass and velocity disper- sion nearly unchanged while the luminosity first undergoes a burst of short duration and relative intensity proportional to the luminosity ratio L1/L2 (only slightly higher than the previous value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This should correspond to a nearly vertical shift on the FP of small amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The M/L ratio either remain unchanged or slightly decrease thus causing a little scatter of the FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The opposite should occur in a merger between two galaxies of nearly equal mass, typical situation in the range of low mass galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In this case, the mass and luminosity both change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' If additional star formation occurs, there should be an additional increase of the total luminosity that depends on the amount of mass converted into new stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Therefore, the total lumi- nosity should hardly recover the pre-burst value, the mass- weighted mean of the two component galaxies (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='11 in Tantalo & Chiosi 2004b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' So most likely the luminosity remains higher than before, and the M/L ratio is expected to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This should generate a tilt of the FP in the right direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' It is not easy to foresee the effect on the scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Better estimates re- quire numerical simulations of burst of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In any case after a short time interval the maximum shift in luminosity cannot overcome a factor of ∼ 2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='3 dex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Application to model galaxies To lend support to the picture outlined above concerning the physical role and meaning of the L = L′ 0σβ relation and the role of the parameters β and L′ 0 without resorting to the numeri- cal simulations of Illustri-1 and Illustris-TNG of which we have no control at all, we make use of very simple, almost analytical models of galaxy formation and evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The ideal models of this type suited to describe ETGs are those in which the total mass increases by infall and the stars are formed according to a simple law of star formation rate (SFR) that have been devel- oped long ago by Chiosi (1980) and extended by Tantalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1998a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The novelty here is that we have incorporated the equa- tions for β and L′ 0 (eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 4) into the models once the luminos- ity, the radius and the velocity dispersion are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' With the aid of these models and eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (16) and (17) we have calcu- lated the basic relationships Ie − σ and Re − σ, and finally made a cross-test of mutual consistency between the results from the galaxy models and the β and L′ 0 theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' These simple model of galaxy formation and evolution were first proposed by Chiosi (1980), much later extended by Tantalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1998a), and re- cently used by Chiosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2017) to study the cosmic SFR and by Sciarratta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2019a) to investigate the galaxy color- magnitude diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Although they may look too simplistic com- pared to the numerical models of Illustris-1 and Illustris-TNG, yet they catch the main features of these latter and are suitable to our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In brief, a galaxy of total mass MG is made of baryonic (B) and dark matter (D), with mass MB and MD respectively, and at any time satisfies the equation: MG(t) = MB(t) + MD(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (21) At all times MB(t) and MD(t) are in cosmological proportion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' they satisfy the condition MD(t) = fcMB(t) where fc depends on the adopted ΛCDM cosmological model of the Universe (fc ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='1 in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The baryonic mass is supposed to be originally in form of gas, to flow in at a suitable rate and, when physical conditions allow it, to transform into stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' With the same rate also dark matter is let flow in together with the baryonic matter to build up the total gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Suitable prescriptions of their spatial distribution are needed to calculate the gravitational po- tential (see Tantalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1998a, for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This kind of galaxy model is named "infall model", the essence of which resides in the gas accretion into the cen- tral region of the proto-galaxy at a suitable rate (driven by the timescale τ) and in the gas consumption by a Schmidt-like law of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The gas accretion and consumption coupled together provide a time dependent SFR closely resembling the one resulting from N-body simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi & Carraro 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Merlin & Chiosi 2006, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Merlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' At any time t the baryonic mass MB is given by the sum: MB(t) = Mg(t) + Ms(t), (22) where Mg(t) is the gaseous mass and Ms(t) the mass in stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' At the beginning, both the gas and the star mass in the proto-galaxy are zero Mg(t = 0) = Ms(t = 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The rate of baryonic mass (and gas in turn) accretion is driven by the timescale τ according to: dMB(t) dt = MB,τ exp(−t/τ), (23) where MB,τ is a constant with the dimensions of [Mass/Time] to be determined by imposing that at the galaxy age TG the total baryonic mass of the galaxy MB(TG) is reached: MB,τ = MB(TG) τ[1 − exp(−TG/τ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (24) Therefore, by integrating the accretion law, the time dependence of MB(t) is: MB(t) = MB(TG) [1 − exp(−TG/τ)][1 − exp(−t/τ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (25) Since dark matter flows in at the same rate of the baryonic matter, it obeys similar equations in which MB is replaced by MD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' However, since at any time MB and MD, are in cosmic pro- portions, MD = fcMB, the equations for MD are superfluous and Article number, page 11 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' FPaanda2 the normalization on MB is enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The underlying hypothesis is that the presence of dark matter does not affect the evolution of the baryonic component, but for its effect on the gravitational potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' To this aim, some assumptions about the spatial distribution of MB and MD are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In other words, assuming spherical symmetry, the radii RB and RD must be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The timescale τ is related to the collapse time and the average cooling rate of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Therefore, it is expected to depend on the mass of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' At the same time, the gas mass increases by infall and decreases by star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The rate of star formation is modeled throughout the whole life of the galaxy with the Schmidt (1959) law: Ψ(t) ≡ dMs dt = νMg(t)k, (26) where k regulates the dependency of the SFR on the gas content: we assume k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The quantity ν is the efficiency parameter of the star formation process that must be specified (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the infall model, because of the interplay between gas ac- cretion and consumption, the SFR starts low, reaches a peak after a time approximately equal to τ and then declines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The func- tional form that could mimic this behavior is the time delayed exponentially declining law: Ψ(t) ∝ t τ exp � − t τ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (27) The Schmidt law in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 26 is therefore the link between gas ac- cretion by infall and gas consumption by star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' As a whole, this kind of approach stands on a number of ob- servational and theoretical arguments among which we recall: (i) the parameters ν and τ can be related to morphology (Buzzoni 2002) and to the presence of ongoing star formation activity in- side observed galaxies (Cassarà et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (ii) the aforemen- tioned quantities can be easily tuned in order to fit observational data, and also complex phenomena that would affect the rate of gas cooling, such as active galactic nuclei (AGN), can be em- pirically taken into account without going into detail (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The infall models we have described may include many im- portant physical phenomena, for instance gas heating by su- pernova explosions (both type Ia and type II), stellar winds, gas cooling by radiative emission, and the presence of galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' See the study by Tantalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1998a) for all details on these topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Outline of the galaxy models The complexity of real globular clusters, galaxies and galaxy clusters and the history of their evolution are reduced here to ideal systems of which we know the current masses M(t), MB(t), MD(t), Ms(t), Mg(t) together with the mass abundances of some important elements Xi(t) (where i stands for H, He, C, N, O, Mg, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Fe) and total abundance of heavy elements Z(t) 3, and finally half-stellar mass (half-light) radius Re(t), and dark mass radius RD(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' At each time, the system contains a manifold of stellar populations of different metallicity and age which can be approximated by single stellar populations (SSP) of mean metallicity < Z(t) > and mean age T(t) defined by the relation 3 For more details on chemical enrichment, companion equations and chemical yields per stellar generation see Tantalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1998a) T(t) = Ms(t)/ < Ψ(t) > where < Ψ(t) > is the mean star for- mation rate in the interval 0 ÷ t (with t the current age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This value of the age T(t) will be used to infer the current luminosity associated to the stellar content Ms(t) (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The infall model of a galaxy must be completed with the radii Re(t) and RD(t) that are necessary to calculate the velocity dis- persion of the stellar component, and the gravitational potential for the onset of galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' To this aim we shortly discuss a few items of interest here: i) The MD-ML and RD-RL relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Following Bertin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1992) and Saglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1992), the spatial distribu- tion of the dark component with respect to the luminous one in dynamical models is such that the mass and radius of the dark component (MD, RD) are related to the luminous ones (ML, RL) by ML(t)(t) MD(t) ≥ 1 2π RL(t) RD(t) � 1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='37 RL(t) RD(t) � (28) where we can pose ML(t) ≃ Ms(t), MD(t) = fcMB(t) ≥ fcMs(t) and RL(t) ≃ 2Re(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Therefore knowing MD(t), Ms(t), and Re(t), we can get an estimate of RD(t) to be used in the calculation of the total gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' According Bertin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1992) and Saglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1992) typical values are ML/MD ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2 and RL/RD ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Consequently within Re the mass of dark matter is small with respect to the stellar mass and can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Fur- thermore, the binding gravitational energy of the gas and stars is given by Ω j(t) = −αLG M j(t)ML(t) RL(t) − G M j(t)MD RL(t) Ω′ LD (29) where j stands for g (gas) or s (stars), and αL is a numerical factor = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5, and finally the term Ω′ LD = 1 2π(RL(t) RD )[1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='37(RL(t) RD )] (30) is the contribution to the gravitational energy given by the pres- ence of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' With the assumed ratios ML/MD and the above replacements of ML, RL and RD, the term Ω′ LD is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Therefore in the evaluation of the velocity dispersion of the stellar component via the VT the effect of DM can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' ii) Velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The velocity dispersion of an object with MD(t), Ms(t), and radius Re(t) is derived from the scalar VT: at each time an object is supposed to be very close to the condi- tion of mechanical equilibrium and hence to satisfy the relation σs(t) = � G kv Ms(t) Re(t) (31) (iii) The Re(Ms) relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The mass-radius relation (MRR) suited to our models is the empirical law proposed by Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2010) in the context of the Λ-CDM cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The expression is: Re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='9 �S S (nS) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='34 � �25 m � �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 fσ �2 � MD 1012M⊙ �1/3 4 (1 + zf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (32) where MD, Ms, and Re have their usual meaning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Re is in kpc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' zf the redshift at which the collapse took place;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' S S (nS ) indicates the shape of the baryonic component that in turn is related to Article number, page 12 of 25 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi: A new framework for understanding the evolution of early type galaxies the Sérsic brightness profile from which Re is derived;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' nS is the Sérsic index;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' fσ is the three dimensional stellar velocity disper- sion as a function of the DM velocity dispersion, σs = fσσD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' and finally m is the ratio MD/Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' We adopt here S S (nS) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='34 and fσ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' For more details see Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2020) and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The most important parameter of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (32) is the ratio m = MD/Ms that is shortly discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The MRR of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 32 is the locus of galaxy models on the MR-plane, the formation of which occurred at redshift zf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' It represents the position of model galaxies for different sources (Chiosi & Carraro 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Merlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2014), however it does not correspond to the real MRR observed for objects from GCs to ETGs and GCGs because cosmologi- cal effects are also present (the subject has been thoroughly dis- cussed by Chiosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2020, to whom the reader should refer for all details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (iv) The MD/Ms ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Basing on the Illustris-1 data Chiosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2020) have investigated how this ratio varies in the mass interval 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 < MD < 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 (masses are in M⊙) and from z = 0 to z = 4 and proposed the following relation: m ≡ MD Ms = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='223zf +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='375) log MD +(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='138zf −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='430) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (33) In the present study, however, we follow a different strategy that at each time step tightly correlates the mass in stars Ms(t) to the total baryonic mass MB(t) and the total mass of dark matter MD(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' At each time we have MD(t) = fc MB(t) where fc is the cosmic proportion (fc ≃ 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The mass in stars Ms(t) is deter- mined by the efficiency of star formation and in any case it is a fraction of the current baryonic mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Therefore the parameter m is given by the relation: log m(t) = log MD(t) Ms(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (34) At the beginning of a galaxy history the ratio m is very large and then declines tending to the limit value fc ≃ 6, if the total baryonic mass is eventually turned into stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Examples of the time behaviour of the ratio m will be shown when presenting our model galaxies in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (v) The star formation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Thanks to the short time scale of the energy input from massive stars (a few million years), com- pared to the mass accretion time scale by infall (from hundred to thousand million years) the galaxy is supposed not to differ from an equilibrium state so that the Talbot & Arnett (1975) for- malism can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi (1980) and Chiosi & Matteucci (1980) adapted the SFR of Talbot & Arnett (1975) to model disk galaxies in which the surface mass density of stars, gas and total baryonic mass are used and a suitable radial distance ˜r is intro- duced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' We have adapted their formalism to our case (in which spher- ical symmetry is implicitly assumed), dMs(r, t) dt = −dMg(r, t) dt = ˜ν � M(r, t)Mg(r, t) M(˜r, t) �κ−1 Mg(r, t) (35) where Mg(r, t) and Ms(r, t) are the mean mass densities of gas and stars within the generic sphere of radius r at the time t, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' M(˜r, t) is the total mass density (gas and stars) within a particular radial distance from the galaxy center, and finally ˜ν is a parameter measuring the efficiency of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The ra- dius ˜r is a suitable radial scale controlling star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the Larson’s view they might be associated to the radial distance at which the central spheroidal component exerts its tidal effect on the residual external gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' As a consequence of it, at any time the SFR is significantly inhibited at distances r > ˜r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Since our mod- els do not include any geometrical description, but deal a galaxy as a point-mass entity whose mass varies with time, we drop the radial dependence of the SFR and the rate of star formation is simply reduced to: dMs(t) dt = −dMg(t) dt = ˜ν � M(t)Mg(t) M(t) �κ−1 Mg(t) = ˜νMg(t)κ (36) Since in all infall models Mg(t) increases by infall and de- creases by star formation, the SFR starts low, reaches a peak af- ter a time approximately equal to τ and then declines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' By varying τ (time scale of the galaxy formation process) one can recover all types of star formation indicated by observational data going from GCs to LTGs and ETGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The infall scheme and companion SFR have been widely used in many studies on the subject of galactic chemical evolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Matteucci 2016, for a review and references).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The infall galaxy model is very flexible and can be adapted to a wide range of astrophysical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Suf- fice it to recall that it has been adopted by Bressan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1994) to model the spectro-photometric evolution of ETGs reduced to point mass objects, extended by Tantalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1998b) to the case of spherical systems made of BM and DM mimicking ETGs, adapted by Portinari & Chiosi (2000) to include radial flows of gas in disk galaxies, and recently used by Chiosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2017) to study the cosmic star formation rate and by Sciarratta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2019b) to investigate the color-magnitude diagram of galaxies in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (vi) The SF efficiency ˜ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In most galaxy models of this kind the specific efficiency of star formation ˜ν is an external free pa- rameter to be adjusted according to the case under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In this paper we follow a different strategy and derive ˜ν from other properties of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Starting from the idea put for- ward by Brosche (1970, 1973) that the efficiency of star forma- tion is driven by the velocity dispersion, we suppose that ˜ν can be written as: ˜ν = ν0 � σt σs × σT σs �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 (37) where σs, σt, and σT are the velocity dispersion calculated using only the stellar component and the total mass, both measured at current time t and present day age T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The factor ν0 depends on the choice made for κ and secures the correct dimensions to ˜ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' For kappa = 1, ˜ν ≡ 1/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Finally, the harmonic mean between two different normalizations is meant to somehow cope with the uncertainty affecting the whole procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Since the stellar mass Ms and radius Re grow with time, the efficiency is large at young ages and decreases with time toward the limit value of ν ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (vii) Luminosity and specific intensity from mean SSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In order to calculate the B and V luminosities and the associated specific intensities IeB and IeV of the stellar content of galaxy models in the course of their evolution, we make use of the SSPs with the Salpeter (1955) IMF (slope in number x=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='35, lower mass Ml = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='1M⊙, upper mass Mu = 100M⊙, total SSP mass Mssp = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='82M⊙, metallicity from Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='0004 to Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='04, 6 values in total, and age from 10 Myr to 14 Gyrs) of the library by Bertelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2008, 2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Tantalo (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The absolute MB and MV magnitudes can be plotted against the logarithm of the age in years, and for each pass-band the mean age-magnitude Article number, page 13 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' FPaanda2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The MB (top) and MV (bottom) magnitudes versus age relation- ships for SSPs of different metallicity according to the color code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' From the top to the bottom the metallicity is Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='0001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='010, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='019, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='040, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The black dotted lines are the mean values of MB and MV over the metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' relation is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Owing to the nearly linear behavior of each relationship, a linear fit is suited to get the relation between the mean absolute magnitude and the age t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' These are given by: MB = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='361 logt − 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='841 (38) MV = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='975 logt − 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (39) The age is expressed in years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The B and V magnitudes of the original SSPs with different metallicity are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 15 to- gether with the metallicity averaged SSP (full dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The mean values of the magnitudes are meant to mimic the mixture of chemical compositions in a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' At each time we know the total mass made by stars of different age and chemical compo- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In practice we assume that this complex situation can be reduced to a single SSP of the same mass, mean chemical com- position (metallicity) and mean age T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The mean age is evaluated from the relation T(t) = Ms(t)/ < Ψ(t) > where < Ψ(t) > is the mean SFR in the interval 0 ÷ t (with t the current age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Using the mean age T(t), from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (38) and (39) we derive the B/V magnitudes (the luminosities) per unit mass of the SSP and then re-scale them to the mass Ms of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (viii) Solution of the basic equation eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='(15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' At each time step of the evolutionary history of a model galaxy, known the star mass Ms, the radius Re, the velocity dispersion σs, the lu- minosities LB and LV (in solar units) and the specific intensities IeB and IeV, the equation system eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (15) is solved deriving β and L′ 0 at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' These are the two physical quantities that in our view drive the distribution of galaxies in the space of the physical parameters L, Re, σ, and Ie, and determine the observed FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Final remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The model age refers to the galaxy rest-frame and goes from Tg = 0 at redshift zf, when the galaxy is supposed to form, to Tg = TG at z = 0 (present time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The corresponding ages of the Universe TU(z) are TU(zf) and TU(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' For the ΛCDM cosmology with H0 =71 km/s/Mpc, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='71, Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='23, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='73, ΩmD/ΩmB ≃ 6, we obtain TU(zf ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='484 Gyr for zf = 10 and TU(0) = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='67 Gyr and TG = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='187 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' When- ever needed we will pass from one to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In order to min- imize the number of free parameters in each model, we assume that all galaxies are born at the same redshift zf = 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the col- lapse time scale of τ = 1 Gyr for all galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the Salpeter initial mass function (in number) with a slope x=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='35 and a fraction of stars more massive than 1 M⊙ equal to ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='30, and absence of galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' However, a few cases will be shown for differ- ent values of τ, different values of zf , and in presence of galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Model results In this section we discuss the galaxy models obtained with the above prescription for the infall scheme and star formation in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' First, we present the reference case with τ = 1 for all galaxy masses and the prescription for ˜ν given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (37) together with the corresponding case ˜ν = 1 (we refer to these latter models as the reference case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Then we discuss some cases in which the effect of galactic winds energized by supernova ex- plosions (both Type Ia and Type II) are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Table 3 lists the models we have considered and presents some charac- teristic features at the last stage with active star formation: this is either the present age for the models without galactic wind or at the onset of galactic wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the following, we mainly present and discuss the models without galactic winds, limiting the dis- cussion of those with galactic winds to some general remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' SFR and SF efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 16 we show the history of star formation in M⊙/yr of galaxies with MB(TG) equal to 106, 108, 1010, and 1012 M⊙/yr (black, blue, green, and red in the order) and variable ˜ν (solid lines) and ˜ν = 1 (dashed lines, the reference case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' As expected, the SFR starts small, reaches a peak value and then declines to low values even though it never extinguishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The peak value is at an age nearly equal to the infall time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Models with variable efficiency do not differ from their corresponding reference case with constant ˜ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The reason for it resides in the value of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The point will be clear discussing the case in which τ is let change with the galaxy mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The SFR efficiency ˜ν is a dimensionless quantity and therefore is the same for all galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' it varies with time from the initial top value 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='04 to 1 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The advantage of our choice for the SF efficiency ˜ν is that this important physical quantity is no longer a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' It is indeed deeply driven by the galaxy mass building process and the time scale associated to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' With our choice for τ, the SF efficiency very quickly reaches its asymptotic value (within about 2 × τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' If τ is increased the time scale over which ˜ν goes to the asymptotic value gets accordingly longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The ratio MD/Ms and the radius Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The stellar radius Re depends on the Dark Mass to stellar mass ratio MD/Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' As al- ready explained this ratio is determined at each time from the current value of the stellar mass built up by star formation, the current mass of baryonic mass MB(t) and the current mass of Dark Matter associated to it given by MD(t) = fcMB(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The ra- tio MD/Ms is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='18 as a function of the mass MD (top panel) and age in Gyr (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In each galaxy the ratio starts very high and, as time increases, tends to the limit value fc ≃ 6 as the whole baryonic gas mass is turned into stars by star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The general behaviour of MD/Ms as a function of MD and age is the same to the point that in the bottom panel all the curves overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Also in this case the ratio MD/Ms is not an external parameter, but it is determined in a self consistent way by the internal properties of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Article number, page 14 of 25 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi: A new framework for understanding the evolution of early type galaxies Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Tables of galaxy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' MB(tG) in solar units is the present-day baryonic mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Age is either the galaxy age at the present time or the age at the onset of galactic winds (ages in Gyrs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Mg and Ms are the gas and stellar masses in solar units at the indicated age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Zg and < Zg > are the metallicity at the indicated age and the mean metallicity reached by the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' SFR is the star formation rate in solar masses per year at the indicated age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Finally, Ωg and Eg are the gravitational energy and thermal energy of the gas at the onset of galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' All energies are in units of 1030 ergs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the case of models without galactic winds Eg is not given.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The SFR histories of models with MB(TG) equal to 106, 108, 1010, and 1012 M⊙ (black, blue, green, and red in the order) and variable ˜ν (solid lines) and ˜ν = 1 (dashed lines, the reference case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' With the aid of the m-ratio and the MRR of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (32) we derive the radius Re of the stellar component Ms and build the mass- radius relationship (MRR) of our model galaxies shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 19 both along their evolutionary history (the black line drawn by filled squares, one for each time step, where the present time is at the top and the initial stage at the bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Each curve corre- sponds to a model with a different final total baryonic mass MB, namely 106 M⊙, 108 M⊙, 1010 M⊙, 1012 M⊙ from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Although the MRRs of the models are in fair agreement with the bulk of observational data and other theoretical MRRs, the closer inspection of the issue reveals that our theoretical radii are likely overestimated by a factor that is difficult to assess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Our best estimate is about a ∆logRe ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='6 to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The mean radii should be a factor 4 to 6 smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' There are many possi- ble causes for this disagreement: first of all, in addition to the m-ratio in the term (MD)1/3 eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (32) contains other terms each of which is affected by some uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The terms in ques- tion are the ratio S S (nS )/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='34, the ratio (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5fσ)2, and finally Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The temporal variation of the SF efficiency of the galaxy models with with MB(TG) equal to 106, 108, 1010, and 1012 M⊙ (black, blue, green, and red in the order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The efficiency is the same for all models and goes from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='04 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Plotting the data log(˜) of each case has been shifted by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='01 with respect to the other ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the ratio (25/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The first two are simply assumed to be equal to one, while the last one contains the ratio m and deserves some remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' It is clear that it has been introduced as an ad- justment factor based on some estimates of the m-ratio derived from current theoretical N-Body Smoothed Particle Hydrody- namic (NBTSPH) simulations of galaxy formation in which only a small fraction of the available gas was used to form stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' see for instance Chiosi & Carraro 2002), which explains the fac- tor 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The present infall models have a different behaviour be- cause nearly all the gas is used up to form stars and the limit value of the m-ratio is about fc ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This implies that the above adjustment factor should become (fc/m), and consequently a re- duction of the estimated radius by a factor of about 4 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' How- ever, instead of forcing the radius to strictly agree with the data, thus introducing some ad hoc adjustments, we keep the radii as they are but also keep in mind that in reality they could be 4 to 6 Article number, page 15 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' FPaanda2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The ratio m = MD/Ms as a function of the mass of Dark Matter MD (top panel) and age (bottom panel) for our model galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Masses are in M⊙ and ages in Gyrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The model galaxies are in different colors and ranked according to their total baryonic mass MB reached at the present time (black: 106;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' blue: 108;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' green: 1010, and red: 1012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the bottom panel all the lines overlap each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' times smaller than estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This would immediately affect our evaluation of the specific intensity Ie = L/(2πR2 e) that could be a factor 16 to 36 higher than our straight evaluation (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Other important relationships: the L vs Re and the Re vs σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The uncertainty on the radius affects also other important re- lationships such as the luminosity-radius relation (LRR) shown in Fig 20 and the radius velocity dispersion relation (RSR) dis- played in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The theoretical data are compared with the observational ones by Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997) and Bernardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2010), for these latter the mean colour (B-V)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='85 has been applied to the MV magnitudes to get the B-luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In both cases the best results are for radii reduced by a factor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Remarks on the luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Before proceeding further it is worth commenting on the luminosity of the model galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' As already explained, for the sake of a quick assessment of the model galaxies luminosity in the B and V pass-bands, we have used suitable linear relationships between the absolute B and/or V magnitudes of the Johnson system and the mean age T based on SSPs of mean metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' One may argue that the luminosi- ties derived in this way are much different from those evaluated by means of full population synthesis technique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' by integrat- ing the spectral energy distribution of SSPs over the star forma- tion rate, initial mass, and the metallicity range spanned by the stellar populations of galaxies at each time (see Bressan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1994, for all details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This is done "a posteriori", once the whole SFR(t), Ms(t), and metallicity Z(t) are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='22 for the MB(TG) = 106, 1012 M⊙, where the solid lines are the luminosity from the analytical relationships, and the dotted lines the luminosity from full population synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The two luminosities differ from each other by a maximum factor of 3 back in the past when the SFR(t) was maximum (ages of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 Gyr), while they coincide in the less remote past (roughly past 5-6 Gyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Therefore our approximation that nicely speeds up the model calculation is reasonable and leads to acceptable Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The mass-radius relations (MRRs) of our model galaxies la- belled by their present day total baryonic mass MB(TG) equal to 106, 108, 1010, and 1012 M⊙ from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Each line made by filled black squares represents the whole evolutionary history of both Ms and Re both increasing with time (the top is the present).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' These models are compared both with observational and theoretical data from different sources: (i) the observational data of Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997) from GCs to GCGs (powder-blue small dots) and the ETGs by Bernardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2010) (red small dots);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (ii) the Illustris-1 galaxies (light green small dots);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (iii) the low initial density models (blue squares and their best fit) and the high initial density ones (red squares and their best- fit) by Chiosi & Carraro (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (iv) the early hierarchical models by Merlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2012) (black squares and their best-fit);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (v) the Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2010) MRRs for different values of the formation redshift zf = 0 (top), 1, 5 10, and 20 (bottom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (v) finally and the MRR by Chiosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2020) (dark golden line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' See the text for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Our luminosities can be safely used for the present pur- poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The Ie vs Re plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Together with the FP and the luminosity- velocity dispersion relation, otherwise known as Faber-Jackson (FJ) relation, the Ie − Re plane is one of the most studied pro- jection of the FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The uncertainty on the radius Re (a factor of 4) reflects on Ie as an increase of a factor of 16 at fixed stellar mass of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The results for our models (with no galactic winds) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 23 and are compared with the obser- vational data of Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997) from GCs to GCGs using the same color code as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The evolutionary sequences on display are for model galaxies with MB(TG) equal to 106, 108, 1010, and 1012 M⊙, from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' For each mass we display two lines: the one with the original radii (dashed black line) and the case with the radii decreased by a factor of 4 and the spe- cific intensity Ie increased by a factor of 16 as explained in the text (line made by filled black squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The time evolution goes from the top to the bottom of each line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The present day stage is the last bottom point of each line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Finally the thick dashed line is the border of the ZOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Please note that no model at the present time falls in the ZOE, but all are below it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The present models cannot account for the data of GCs (as expected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Even if the model with MB(TG) = 106 M⊙ crosses the region of GCs it cannot reproduce these objects because the present radius and specific intensity are too large and too low re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The ongoing star formation yields too luminous and Article number, page 16 of 25 2 (Kpc) 0 R Log 2 DATA & MODELS 4 6 8 10 12 14 Log M, (Mo)M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi: A new framework for understanding the evolution of early type galaxies Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The B-luminosity-radius relation (LBRRs) of our model galax- ies with MB(TG) equal to 106, 108, 1010, and 1012 M⊙, from bottom to top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' For each mass we display two lines: the one with the original radii (dashed black line) and the case with the radii decreased by a factor of 4 as explained in the text (line made by filled black squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The models are compared both with observational data from Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997) from GCs (magenta small dots) to Dwarf Galaxies (blue small dots), to ETGs (red small dots), GCGs (powder-blue small dots), and the ETGs by Bernardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2010) (red small dots, overlap the previous ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The agreement for the smaller radii case is soon evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' See the text for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The Re-σ relation (RSR) of our model galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In this figure the same models, observational data, color code, and symbols used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='20 are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' too large objects that do not match with general properties of GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' What would be needed are models in which star forma- tion ceased and radius stopped growing long ago (short initial episode followed by quiescence perhaps because of strong galac- tic wind), or to take into account the important transformations induced by the interaction with the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The luminosity LV versus age (in Gyr) relation (LAR) of our model galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The cases MB(TG) = 106 (bottom) and 1012 M⊙ (top) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The solid lines are luminosities derived from the analyti- cal relationships while the dotted lines are those from full population synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Similar considerations can be applied to clusters of galaxies for which different type of models should be set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' To develop a suitable model for the formation and evolution of GCGs along the same lines we have followed for single galaxies is beyond the aims of this study and we leave the subject to a future inves- tigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The β − L′ 0 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' With the aid of the equations from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (3) to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (15) we derive the exponent β and proportionality factor L′ 0 along the whole evolutionary sequence of our model galax- ies evolved without galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 24 we display all cases under consideration: models with large radii and models with smaller radii (the factor of 4) for the two photo- metric pass-bands in usage (B and V Johnson).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Along each line time increases from the bottom to the top where the last stage at the present age is indicated by the mass label (total asymt- potic baryon mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' For each galaxy mass (MB(tG)) the results are nearly the same, all sequences overlap each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' See also the entries of Table 4 containing the slope α and zero-point γ of their linear best-fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' It turns out that the relationships in question depend only on the galaxy mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Remarkably β the exponent of the L = L′ 0σβ relation is positive during the early stages and neg- ative afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The luminosity first increases with σ and then decreases with it afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Finally note that all curves cross each other at β ≃ 3 and log L′ 0 ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5, values very close to the observed FJ relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' To confirm this picture, in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 24 we plot the same relations for the Illustris-1 models grouped at different redshifts from z = 4 to z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Now the sit- uation is not the same as before, because in each group with the same redshift, mass and age vary from galaxy to galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Fur- thermore not all masses are present at each redshift: samples at high redshifts, say ≥ 1, 6, are dominated by low mass objects (masses lower than 108 M⊙ are missing anyway because of the mass resolution), massive objects up to 1012 M⊙ are present at lower redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' However, the resulting distributions in the β−L′ 0 plane are much similar to that of the left panel, and remarkably there is also some evidence of the β ≃ 3 cross-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This fact Article number, page 17 of 25 2 (Kpc) (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=') Log 2 0 1 2 3 4 Log (a) (km/s)12 10 le10 8 le8 6 e6 4 2 0 2 4 Log (Re) (Kpc)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' FPaanda2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The Ie-Re plane of our model galaxies compared with the obser- vational data of Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The color code of the data is the same as in previous figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' There are two groups of models: the black thin dashed lines are models with original radii, while the thick lines made by filled black squares are those with the radius decreased by a factor of 4 and the specific intensity Ie increased by a factor of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The galaxy mass is MB(TG) equal to 106, 108, 1010, and 1012 M⊙, from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Along each line the time runs from zero to present age from the top to the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The formation redshift of all the models is zfor=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The relationships between L′ 0 and β for the model galaxies evolved without galactic wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' These relationships are the linear best fits of the curves shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' All these relationships are nearly identical passing from models with large radii to those with smaller radii (by a factor of 4), and the corresponding solutions of the equations for β and L′ 0, and finally changing only the photometric pass-band in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The relationships seem to depend indeed only on the galaxy mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' logL = α ∗ β + γ B-Band V-band MB/M⊙ α γ α γ 1e6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='478 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='369 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='508 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='368 1e8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='137 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='224 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='167 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='287 1e10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='796 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='046 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='760 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='340 1e12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='453 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='834 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='474 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='888 strongly supports the notion that infall models nicely mimic the numerical hierarchical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The most important relation to look at and to examine in de- tail is the luminosity versus velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='25 which displays the log(LB/L⊙) vs log σ for the model galaxies and compare it with observational data of Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the main panel we display three possible relationships: (i) the plain LB/L⊙ vs σ of the models with their original lumi- nosity and radii (the thick black curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Along each curve the evolution starts at the bottom point of each line and proceeds to the final stage indicated by the label MB(tG) in solar units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Please note that during the galaxy lifetime the LB/L⊙ vs σ rela- tion bends over past a certain age toward lower luminosities and lower velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This roughly happens past the peak of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' While the luminosity decrease can be easily un- derstood, the decrease in velocity dispersion of the stellar com- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Left panel: The L′ 0−β relation of our models (left panel) evolved without galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' All the relationships are nearly identical passing from models with large radii to those with smaller radii (by a factor of 4), and the corresponding solutions of the equations for β and L′ 0, and finally changing only the photometric pass-band in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The relation- ships seem to depend indeed only on the galaxy mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Right panel: The L′ 0 − β relation for the artificial galaxies of Illustris-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Each color corre- sponds to a different redshift epoch: green (z = 4), blue (z = 3), yellow (z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2), brown (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='6), magenta (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='0), dark gray (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='6), red (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='2) and light gray (z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' ponent needs some explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Stars during their lifetime can explode as Type II and Type I supernovae: in the first case a small remnant is left (neutron star or black hole), in the second one no remnant at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' They can also lose lots of mass by stellar winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In any case the total mass in stars is expected to decrease and so does the velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (ii) The second case is the associated LB/L⊙ = L′ 0σβ relation in which the original β and L′ 0 are used (the red curves together with the linear fit limited to the descending branch of each curve, (the red solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (iii) Fi- nally, the LB/L⊙ = L′ 0σβ relation, in which the correction on the radius has been applied and new values of β and L′ 0 are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' It is worth recalling that the L′ 0 vs β relation remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The results are shown by the green curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The small insert in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 25 shows the case of the MB(tG) = 1010 M⊙ in more detail for the sake of better understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Similar results are found for the V pass-band that are not shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Two important features are soon evident;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' first of all the relations log(LB/L⊙) vs log σ, based on the model history past the star formation activity pe- riod have a similar slope, but different zero point (that depends on the galaxy mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The manifold of these relations provides a sort of natural width to the luminosity-sigma relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The mean slope of the manifold agrees with the current value of the observed FJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Second, the theoretical relations marginally agree with the body of ETGs, a steeper slope at luminosities above log LB/L⊙ ≃ 9 would be more appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The simplicity of the current models cannot lead to better re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' A possible improvement could be given by allowing small secondary episodes of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The argument is as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The luminosity is the product of the star mass times the flux per unit mass: let us call Lo = foMo the original luminosity and Ln = fnMn the expected luminosity including some recent star forming activity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=" LnMn is in turn made by fyMy + foMo, Article number, page 18 of 25 20 10 0 10 20 10 0 10 2020 10 Log(L'o) 0 10 20 10 0 10 20M." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi: A new framework for understanding the evolution of early type galaxies 6 6 6 6 6 6 6 0 1 2 3 6 8 10 12 6 1e10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The luminosity LB/L⊙ versus velocity dispersion σ in (km/s) re- lation (LSR) of our model galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' For each mass (labelled by MB(tG) as indicated ) three relations are shown: (1) the original models with no revision of the radii (lines made by filled black squares);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2) mod- els whose luminosity is derived from the LB/L⊙ = L′ 0σβ relation with the original β and L′ 0 (the curves made by red squares) together with the linear fit limited to the descending branch of each curve (the black solid lines);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (3) models in which the radii have been revised and new values of β and L′ 0 are calculated (the green curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Note how in each case, the luminosity vs sigma relation bends past the stage that roughly corresponds to the maximum stellar activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' From this stage the lumi- nosity and velocity dispersion decrease (see the text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The insert shows the case of the MB(tG) = 1010 M⊙ for the sake of better illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The models are compared with the data by Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997) from GCs to GCGs (the same color code as in previous figures is used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' where fyMy is the contribution by the episodic stellar activity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' it follows that (fyMy + foMo)/(foMo) = λ = Ln/Lo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Basing on the current observations one would expect λ ≃ 2 or so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Now we may also assume My << Mo so that the total stellar mass and ve- locity dispersion in turn remain nearly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Indicating with θ = My/Mo, one gets fy/fo = λ − 1/θ ≃ 5 − 10 which is not impossible according to current population theories for Single Stellar Populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (iii) In the theoretical models the exponent β of the LB/L⊙ = L′ 0σβ relationship (a generalization of the FJ) can be either positive or negative depending on the particular evolutionary stage of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Therefore among the observa- tional data both values of β are to be expected without violating the trend indicated by the FJ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' that the luminosity of galax- ies increases with the velocity dispersion, hence the mass of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Galactic Winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Long ago Larson (1974) postulated that the present-day Color-Magnitude Relation (CMR) of ETGs could be the result of galactic winds powered by supernova explosions, thus initiating a long series of chemo-spectro-photometric mod- els of elliptical galaxies standing on this idea (see for instance Tantalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1998a, and references).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In brief, gas is let escape from the galaxy and star formation is supposed to halt when the total thermal energy of the gas equates its gravitational binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This idea has been extended including the effect of stel- lar winds in the thermal energy budget of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' It was also included in NBTSPH models of galaxies (see Merlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2012, and references).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The same scheme proposed by (Tantalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1998a) is adopted here, however with minor modifications because of the much simpler present formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' As already said, the thermal energy of the gas is the sum of three contributions, namely type I and II supernovae and stellar winds from massive stars: Eth(t) = Eth(t)S NI + Eth(t)S NII + Eth(t)W (40) where each term has the generic expression Eth(t) j = � t 0 ǫj(t − t′)R j(t′)MB(tG)dt′ (41) with j= SNI, SNII, W with obvious meaning of the symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The normalization factor MB(tG) in the above equations is required to calculate the energy in physical units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The time t′ is either the SN explosion time or the time of ejection of the stellar winds as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The functions ǫS N(t) and ǫW(t) are cooling laws governing the energy content of supernova remnants and stellar winds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Finally, star formation and chemical enrich- ment are halted, and the remaining gas content is supposed to be expelled out of the galaxy (winds) when the condition Eth(t) ≥ Ωg(t) (42) is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' For all other details concerning the above rates, the evolution of SN remnants and stellar winds and how much of the initial energy budget is shared with the gas to energize it, and finally the expression for the gravitational energy of the gas in presence of baryonic and dark mass and their space distribution in a galaxy see Tantalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1998a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' A small sample of models with galactic winds are calcu- lated and their main features are summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' It is worth noting that the onset of galactic winds occurs at younger and younger ages as the galaxy mass increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Thanks to it, these models obey the constraint imposed by observational data on chemical elements like Carbon (C), Oxygen (O), Magne- sium (Mg), also known as α-elements, and Iron (Fe) and their ratios [α/Fe]: the high mass galaxies are more α-enhanced ([α/Fe] > 0) than the low-mass ones ([α/Fe] ≤ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This fact cannot be easily reconciled with other properties of the same ob- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' See Chiosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1998) and Tantalo & Chiosi (2002) for detailed discussions of this issue and possible ways out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the present models we have taken the suggestions by Chiosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1998) and Tantalo & Chiosi (2002) into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The role of Galactic Winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The main lines of the discussion for models without galactic winds holds good also for the new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Therefore we focus on key relations such as the Ie − Re plane which is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The comparison with the same plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 23 shows that there is no visible difference passing from models without to those with galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The reason for that is the kind of star formation at work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Because of the short infall time scale and the dependence of ˜ν on the inverse of the velocity dispersion, most of the stars are in place before the occurrence of galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' To somewhat alter this trend one should change the parameter τ and make it to depend on the galaxy mass, for instance long in low mass galaxies and short in the high mass ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' To further investigate this point is beyond the aims of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Role of the Initial Mass Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' To avoid misunderstand- ing, we need to recall here that the present models are calculated Article number, page 19 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' FPaanda2 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' A few key quantities of the model galaxies at the present time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' From left to right: age in Gyr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the logarithm of the stellar mass Ms in solar units,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the logarithm of the effective radius Re in kpc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the logarithm of the velocity dispersion σ in km/s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the logarithm of the B luminosity LB in solar units,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the logarithm of specific intensity IeB in LB/pc2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the logarithm of the mass to light ratio Ms/LB in solar units,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' LV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' IeV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Ms/LV the same for the V band,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the redshift of galaxy formation zf ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the asymptotic baryonic mass MB(tG) in solar units,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the infall time scale τ in Gyr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' and finally the notes N where the asterisks mean that the models take all corrections to the Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2010) radius into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' age Ms Re σ LB IeB Ms/LB LV IeV Ms/LV zf MB τ N 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 10 5 1e6 1e6 1e8 1e10 1e12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The Ie-Re plane of our model galaxies with galactic winds pow- ered by the energy input from supernova explosions and stellar winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' There is no visible difference with respect to the same plane of models without galactic wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The same notation, symbols and color code of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='23 is adopted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' with the classical IMF of Salpeter (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Therefore the Ms/L ratio based on these models has this fundamental limitation and cannot by applied to investigate the problem of the FP tilt in a very general way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Our infall models can easily be adapted to in- clude popular IMFs in literature different from the Salpeter case, see for instance Chiosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1998), where the IMF is let vary with the physical condition (mean density, temperature and ve- locity dispersion) of the gas inside a galaxy and therefore with time for a galaxy of given mass and with time and mass in ob- jects of different mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' However, for the aims of this study, in order to simplify this we thought it wise to rely on the classi- cal IMF of Salpeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' If the present models were applied to the issue of the FP tilt, most likely they could account for only half of the observational tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This subject was specifically addressed in Chiosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1998) with good results for the tilt of the FP of ETGs in the Virgo and Coma clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Changing the galaxy mass and zf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' An important feature of the models is related to the formalism in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' According to the formalism and equations widely described in Tantalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1998a) all relevant physical quantities describing the model and its temporal evolution are suitably normalized to the so-called asymptotic baryonic mass MB(tG), for instance the gas mass at time t is expressed as Gg(t) = Mg(t)/MB(tG), equally for the star mass Gs(t) = Ms(t)/MB(tG), and the current total baryonic mass GB(t) = MB(t)/MB(tG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The amount of dark matter at any time is simply related to the current baryonic mass via the cosmic ratio (the components are intimately mixed together so that they fall together at the same rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Furthermore, the accretion rate, the star formation rate, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' are all expressed using the same kind of normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The advantage is that the time scale of mass accretion τ, the cosmic ratio fc, and all the rest is parameter- Article number, page 20 of 25 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi: A new framework for understanding the evolution of early type galaxies 6 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The Ie-Re plane of our model galaxies with different formation redshift zf, namely 10, 5, 3, 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The four green points of differ- ent colors are the present day stage of model galaxies whose existence began at redshifts from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The effect is quite small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' free, so that the only free quantity is the asymptotic baryonic mass MB(tG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This allows us to generate models for any value of MB(tG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' All galaxy models discussed so far are calculated assuming the redshift of galaxy formation zf = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Other values are of course possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Higher values are unlikely whereas lower values are much plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Since the age of the Universe TU depends only on the cosmological model in use and therefore is a fixed quantity, the age of a galaxy TG expressed by TG = TU − TU(zf) at decreasing zf becomes shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Consequently some features of the models will change, such as total ages, radii, luminosities and specific intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The following values of zf are consid- ered: 5, 3, 1, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 in addition to the previous set with zf = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='27 limited to the cases zf = 5 (black lines) and zf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 (red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The case zf = 10 runs nearly over the case zf = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' All the others are in between the case zf = 5 and zf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' From left to right, the galaxy mass is MB(TG) = 106108, 1010, 1012 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The age increases along each line from the top to the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The final age (in Gyr) decreases from 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='19 for zf =10 to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='47 for zf =5, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='48 for zf =3, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='73 for zf =1, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='02 for zf =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' See Table 5 for more informa- tion on the final stage of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In Fig 27 the final stages are represented by the green circles (some of them overlap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' From these data we derive that variations in zf from 10 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 yields variations in log(Ie) of about ∆ log(Ie) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 while the ra- dius does not change significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' More efficient star formation in recent times generates more luminosity and hence higher spe- cific intensity Ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This is achieved by changing τ from 1 to 5 Gyr (in the case of the 1010 M⊙ galaxy) yielding ∆ log(Ie) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Re- cent bursts of star formation either by internal causes or mergers would also increase Ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Analysing all implications of it is beyond the aims of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' What we can say with confidence is that a significant scatter in the Ie − Re plane is likely to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In any case, the gross distribution of galaxies in this plane (but for GCs and GCGs) is accounted for by these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Finally, the homologous behaviour of the models and the limited effect of the formation redshift on their evolutionary be- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The comparison of Ie and Re derived for the model galax- ies (indicated by the suffix [i] and those calculated with relations (16) and (17) for galaxies with asymptotic baryonic mass MB(TG) = 106, 107, 108, 1010, and 1012 M⊙ from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The redshift of galaxy formation is zf =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' haviour make it possible to generate simulations of the distribu- tion of large number of galaxies in the parameter space we are investigating in practice at no cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' A test of consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The galaxy models we have presented are based on physical assumptions such as the infall picture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the star formation rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the mass-radius relationship and the pop- ulation synthesis governing their luminosity in different pass- bands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' that are not explicitly related to our interpretation of the parameter space of galaxies (luminosity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' stellar mass and ra- dius,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' velocity dispersion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' and specific intensity),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the FP in the multi-dimensional space and its possible projections onto differ- ent planes that led us to the L = L′ 0σβ relationship with L′ 0 and β changing from galaxy to galaxy and for each of them also with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' On this ground we have made some detailed predictions about L′ 0 and β and derived a number of equations whose solu- tions on one hand yield L′ 0 and β as function of L, Ms, Re, σ, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' and on the other hand allows to construct the expected relation- ships among pair of fundamental variables such as Ie vs Re, Ie vs σ, Re vs σ etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='. Among these we choose here as an example the variables Ie and Re and compare the values given by the models with those derived from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (16) and (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The comparison is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 28: the top panels are for Ie while the bottom pan- els are for Re;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the galaxy mass MB(TG) is 106, 107, 108, 1010, and 1012 M⊙ from left to right (the case MB(TG) = 107 ⊙ is added).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' On the abscissa are the input values from the models (labelled Ie[i] and Re[i]) and on the ordinate the values calculated from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (16) and (17) (labelled Ie[c] and Re[c]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In general there is a surprisingly good agreement between [i] and [c] quantities, but for some particular stages in which the [c]-values rapidly di- verge and change sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The cause of it resides in the analytical relationships themselves that contain various exponents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' γ, [(β − 2)/(1 + 3/γ)], [(β − 2)/(3 + γ)] that in turn are functions of β which varies in the course of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In this narrow inter- val the disagreement is of mathematical nature with no physical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' It simply means that these analytical relationships cannot be used with safe to derive the corresponding variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Article number, page 21 of 25 6 log(I_e[c]) 4 2 0 2 log(I_e[i]) log(I_e[i]) log(I_e[i]) log(I_e[i]) log(I_e[i]) log(R_e[c]) 4 2 0 一 20 2 4 6-20 2 4 6-20 24 6-20 24 6-20 2 46 log(R_e[i]) log(R_e[i]) log(R_e[i]) log(R_e[i]) log(R_e[i])A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' FPaanda2 General remarks and preliminary conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Since the [i]- and [c]-values are nearly coincident, using the analytical re- lationships would predict results in the various projection planes we have examined identical to those obtained from using the nu- merical galaxy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The overall agreement between the model and analytical approach lends strong support to the idea at the base of the analytical view, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' that the relation between the luminosity and velocity dispersion of a galaxy is governed by L = L′ 0σβ in which both β and L′ 0 vary with the galaxy mass, evolutionary stage (and hence time and redshift) and these quan- tities in turn are intimately related to key physical parameters such as the stellar mass and radius, the velocity dispersion (a measure of the gravitational potential well), the star formation rate, the infall time scale, and finally the ratio MD/Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The dis- tribution of galaxies on the usual diagnostic planes such as FP, FJ, Ie-Re, Ms-Re, L-Re, Re- σ, and finally the border of the exclu- sion zone, mirror the mean behaviour of galaxies each of which has its particular history and is observed in some evolutionary stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The important role of β In this section we cast light on the role of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' To this aim we adopt the reference case (zf = 10 and τ = 1) and leave the is- sue of galactic winds aside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' For this case we present a few basic relationships among β and other important parameters namely the SFR (in M⊙/yr), the age (in Gyr), and the specific intensity IeB or IeV (in L⊙/pc2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' These relationships are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the left panel, the homologous nature of the galaxy models is evident: all curves have the same shape, but each one is sepa- rated from all the others by the homology parameter, namely the total baryonic mass at the present age MB(TG) annotated along each curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The temporal evolution occurs from the top to the bottom of each curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Identical behaviors are found between β and the luminosity LB or LV (in L⊙), and the velocity dispersion σ (in km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' However, these relationships are not shown here for the sake of brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The central panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 29, showing the variation of β with the age, still displays the dependence of the results on the homology parameter and thus there are four dif- ferent curves one for each value of the MB(TG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Finally, in the right panel we show the dependence of β on the surface bright- ness IeB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' all curves collapse to a single relation, the homology is destroyed by the underlying relationship between the mass and the effective radius of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' A similar relation is found be- tween between β and IeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The analytical relations between β and Ie are given by β = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='159Log(IeB) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='003 (43) β = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='159Log(IeV) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (44) The evolution along each line is from top-right to bottom-left and the present stage is the last point where MB(TG) is annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The above relations indicate both the path followed by a single galaxy in the course of its history and also the locus on the β-Ie plane of galaxies of different mass observed at the present age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' There is no appreciable effect of different formation redshifts, at least in the interval 10 ≥ zf ≥ 1 nor of different accretion timescale τ in the interval 1 ≤ τ ≤ 5 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' We estimate a total effect on β by redshift zf and accretion time scale τ of the order ∆β ≃ 2 over the interval of IeV of interest here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The linear relation between β and Ie shown by our models is a very intriguing result that demands a thorough analysis because observational data and numerical hierarchical models seem to in- dicate a different picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The situation is best illustrated by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 30 comparing data and models from different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' On the ob- servational side we have three data-sets: Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997), WINGS, and Bernardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The last two (mainly de- voted to ETGs) are based on equivalent methods to estimate Re, and therefore yield similar results as far as the β-Ie plane is con- cerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In contrast, the first one that contains objects going from GCs to DGs, LTGs, ETGs and GCGs, differs in the method used to derive the effective radius, and consequently yields different relationships in the β-Ie plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Owing to this, some preliminary remarks are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' First of all, the data of Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997) are in the B-band so that must be transformed into the V-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This is made by means of the relation log LV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='4[(B − V)0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='65] + log LB, where the luminosities are in solar units, (B − V)0 is the colour, and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='65 is the difference between the B and V photometric constants (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='48 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='83 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Second, recalling that the luminosity Le given by Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997) is the amount of light falling within the effective radius Re, where half the total luminosity is found, we scale it by a factor of two to make it consistent with the definition of Ie we have adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The observational data for Re, LV, Ms, (Ms/LV), σ, and IeV are fed to the system of equations (14) and the solutions β and L′ 0 are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The powder-blue points n Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 30 are the Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997) data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' three sequences are seen: the GCG- GC sequence, the one of ETGs (no evidence of star formation), and the one of LTGs and DGs (evidence of ongoing star for- mation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' By construction, the data of Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997) are well behaved with no evidence of dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The red squares are the WINGS data showing large dispersion in both coordi- nates, log Ie is always positive, and β can be very large both positive and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Much similar results are found with the Bernardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2010) data, the open green circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The Illustris-1 model galaxies are indicated by the blue dots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' their distribution closely mimics that of the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Finally, the long dashed red line shows the present-day position on the β-Ie plane of our models for the reference case (with τ = 1 Gyr, zf = 10 and no galactic winds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This line coincides with the lower border of the Illustris-1 distribution in the β > 0 hemi-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Choosing different values of zf in the interval 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='5 ≤ zf ≤ 10 does not shift significantly the line predicted at the present time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Equally for the effect of galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Lumping all these effects together we expect a typical width of this border line of about ∆β ≃ 10 over the IeV interval of interest here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' From this preliminary comparison we may conclude that mu- tual consistency among different sources of data and of these lat- ter with models exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The major issue now is to understand the physical causes of the large dispersion in β for all values of Ie ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Looking at the eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (15), providing the solutions β and log(L′ 0) of our equations, one notes that under suitable conditions the term 1 − 2A′/A at the denominator of log(L′ 0) gets very close to zero, consequently β can be either very large and positive or large and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' As already discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 3 when this happens the system is in conditions of strict virialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The sign of β depends on the particular history of the constituent variables (Ms, Re, L, and Ie), in other words whether the term 2A′/A is tending to 1 from below (β > 0) or above 1 (β < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' From an operative point of view we may define "state close to strict virialization" when |β| > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This would account for the gap on the negative hemi-plane of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Article number, page 22 of 25 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Chiosi: A new framework for understanding the evolution of early type galaxies Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Left Panel: The relationship between β and the star formation rate (SFR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Each curve labelled by MB(TG), is identified by a different colour according to the color-code adopted in previous figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The total baryonic mass is the homology parameter separating each curve from the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Much similar trends are found for the luminosity LB and LV, and the velocity dispersion σ that are not displayed here for the sake of brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In all three relations β mirrors the behavior of the SFR, the luminosity in turn, and finally the velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The SFR is in M⊙/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Middle Panel: The relationship between β and age (in Gyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Symbols and color-code have the same meaning as in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Right Panel: the relationship between β and IeB in L⊙/pc2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The lines corresponding to different masses of galaxies have been displayed by shifting each of them by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' in reality they collapse to a unique curve given by β = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='159Log(IeB) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The long dashed line is the best fit of the theoretical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Identical relation can be found for IeV: same slope but slightly different zero point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Do data and models ever reach the condition of full virializa- tion indicated β ⇒ ±∞ or do they remain somewhat far it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The answer is that both possibilities occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' On the observational side, given any galaxy for which the set of parameters (Ms, Re, L, Ie and σ) has been measured, it is not granted that they would satisfy the virialization condi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The major uncertainties are with Ms and Re and Ie in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Therefore, many of them crowd in the interval −1 ≤ β ≤ 20 which implies deviations from virial equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' However with the present data, it is not possible to say whether this is due to in- sufficient accuracy in the parameters determinations or real devi- ations from virial equilibrium due to recent mergers, harassment, loss of mass, interactions etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' However, many other galaxies in both hemi-planes with |β| > 20 which is a strong indication that they are close to virial equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' On the theoretical side, our model galaxies with infall (no dynamics in them) seem to be in a state far from strict virializa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This is suggested by the small values of β reached at the present time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The reason for that resides in the way the models are built up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In brief,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' mass point description with no dynamics is adopted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the total mass is assigned (via the accretion law),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the stellar mass is derived from star formation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the effective ra- dius is estimated from a suitable relationship,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the luminosity is evaluated from the stellar mass and a mean luminosity-age re- lationship for a fictitious SSP with mean metal content (the dif- ference with respect to the luminosity correctly derived from the theory of population synthesis via the history of star formation and metal enrichment is not large but still significant),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' finally the velocity dispersion is derived from the VT with the current values of Ms and Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The major uncertainties are in Re and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Therefore our set of basic parameters not necessarily can fulfill all the requirements imposed by the VT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In consequence, our βs are always quite small (say smaller than 25-30) implying that full virialization is not reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' However, this failure is not as severe as it appears because small adjustments of Re and L are possible while the models are successful in many other aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The situation is much better with the Illustris-1 models where if a good number of galaxies have β < 25−30 as in the case of our models, still a large number of objects is clearly seen in regime of strict virialization because of their high positive and/or negative βs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The inclusion of real dynamics and the hierarchical scenario at work provide much better conditions to bring the action of virialization into evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the hierarchical scenario, mergers, ablation of stars and gas, harassment, secondary star formation, inflation of dimension by energy injections of various kinds, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' induce strong variations on the structural parameters and hence strong temporary deviations from the virial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' However, once this happened, the viral conditions can be soon recovered over a suitable timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This can be short or long depending on the amount of mass engaged in the secondary star forming activity and the amount of time elapsed since the star forming event took place (see the burst experiments in Chiosi & Carraro 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Tantalo & Chiosi 2004a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' As a consequence of all this, de- tecting systems on their way back to virial equilibrium is likely a frequent event thus explaining the high dispersion seen on the β-Ie plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In principle, the value of β evaluated for each galaxy could provide a useful hint about the equilibrium state reached by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Most likely, the condition of strict virial equilibrium is a transient phenomenon that could occur several times during the life of a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This is perhaps suggested by the high numbers of galaxies with both low and positive values of β and high pos- itive/negative values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Discussion and conclusions The aim of this paper is to prove that the difficulties encoun- tered in understanding the distribution of galaxies on the FP in the parameter space σ, Re Ie and its projections on the three co- ordinate planes, can be removed by introducing the L = L′ 0σβ relation as a proxy of evolution, in which β and L′ 0 vary from galaxy to galaxy and for each of them in course of time (see D’Onofrio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2017a, 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' D’Onofrio & Chiosi 2021, for previous efforts along this line of thought).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='The continuous variation of β and L′ 0 traces the path followed by each ETG in the L − σ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The L = L′ 0σβ law together with the VT yield a set of relations Re-σ, Ie-σ and Re-Ms that nicely reproduce the data and suggest the existence of a system of two equations in Article number, page 23 of 25 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' FPaanda2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The β - IeV plane: data and theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Data from the different sources are plotted: (i) The powder-blue points from Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' three sequences are seen: the sequence of GCGs & GCs, the one of ETGs (no evidence of star formation), and the one of LTGs and DGs (evidence of ongoing star formation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' By construction, the data of Burstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (1997) are well behaved with no evidence of dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (ii) The red squares are the WINGS data showing large dispersion in both coordinates, log Ie is always positive, and β can be very large both positive and negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (iii) the open green circles are the ETGs of Bernardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (2010) however limited to z ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The Illustris-1 model galaxies are indicated by the blue dots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' their distri- bution closely mimics that of the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Finally, the long dashed red line shows the present-day position on the β-Ie plane of our models for the reference case (with τ = 1 Gyr, zf = 10 and no galac- tic winds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This line coincides with the lower border of the Illustris-1 distribution in the β > 0 hemi-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the unknowns β and L′ 0 with coefficients functions of Ms, Re, L, and Ie that for each galaxy determine the value of β and L′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' With the aid of these relations we can determine the instantaneous po- sition and direction of a galaxy on the FP and the projection planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The analysis is made in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the first one, the prob- lem is addressed from an observational point of view, inferring from the data the expected position and evolutionary direction of a galaxy in the various planes and owing to the large number of galaxies in the samples the range of values spanned by β and L′ 0 is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In the second step, simple models of galaxy formation, structure and evolution are set up, the basic equations are solved at each time step of a galaxy’s lifetime so that the his- tory of β and L′ 0 is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Basing on these results, the various projection planes are examined finding consistency between ob- servational data and theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The same procedure is applied to literature galaxy models calculated in the framework of the hierarchical scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The theoretical results are compared with the observational data and good mutual agreement is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Basing on this, we conclude that the starting hypothesis about the real existence of the L = L′ 0σβ relation is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' In more detail, the present analysis has clarified the following issues: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The FP can be understood as the average of the single FP- like relations valid for each galaxy (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The coefficients of the FP-like relation are function of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This means that the FP must evolve with redshift and that its coefficients depend on the adopted waveband (in which observations are taken) and on the nature of the data sample (how many ETGs are included) as confirmed by the current observational data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' All the features of the FP projections can be explained at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' This includes: 1) the curvature of the relations, that turns out to depend on the existence of positive and neg- ative values of β, and 2), the existence of the ZoE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' the line marking the separation between the permitted and forbidden regions in these planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' No galaxies can reside in the ZoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The FP and all its projections, such as for instance the clas- sical Faber-Jackson relation, are instantaneous pictures of the present-day situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' They should change with redshift hence lifetime of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The ZoE is obtained in a natural way as the only possible evolutionary path for objects with large positive and neg- ative β’s that are well virialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' These objects in general have stopped their star formation long ago and their lumi- nosity progressively decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' When ETGs become passive quenched objects, with a luminosity decreasing at nearly constant σ, the galaxies can only move in one direction, that given by the large positive and negative values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The infall model galaxies built here, although lacking the dy- namical component, are in reasonable agreement with obser- vations and support the idea that β and L′ 0 vary across time, and therefore that the L = L′ 0σβ law is a plausible empirical relation accounting of the variation occurred in a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' All the diagrams built using the structural parameters are sensitive to the temporal evolution of galaxies, simply be- cause each individual object moves in a different way ac- cording to the value of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Both observations and theory suggest that the L = L′ 0σβ re- lation provides an empirical way of capturing the temporal evolution of ETGs (and probably of late-type objects) be- cause the values of β are related to the history of mass assem- bly and luminosity evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Because of it we are tempted to suggest that eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' (14) are two important equations governing the evolution of ETGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Finally, the large negative and positive values of β of some galaxies can be considered as the signature that these sys- tem are very close to the virial equilibrium, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' their basic parameters Re, L∆λ, Ms, (Ms/L∆λ), σ, and Ie,∆λ are such that the strict virial condition is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The situation is likely transient because both internal and/or external events may alter one or more parameter so that the strict virial condition is no longer verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Since the recovery time can vary a lot from galaxy to galaxy, this ideal situation has an ample range of occurrence probabilities, from frequent in some galaxies to never in others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The parameter β can be taken as the sig- nature of how far is the system from full virialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' The authors thank the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Lanzoni, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Capaccioli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1998, Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Ital- iana, 69, 217 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2006, MNRAS, 366, 1126 Cariddi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', D’Onofrio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Fasano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2018, A&A, 609, A133 Cassarà, L.' metadata={'source': 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Poggianti, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2009, A&A, 495, 707 Chiosi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1980, A&A, 83, 206 Chiosi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Bressan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Portinari, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', & Tantalo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1998, A&A, 339, 355 Chiosi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' & Carraro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2002, MNRAS, 335, 335 Chiosi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', D’Onofrio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Merlin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Piovan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', & Marziani, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2020, A&A, 643, A136 Chiosi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' & Matteucci, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1980, Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' Italiana, 51, 107 Chiosi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Sciarratta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', D’Onofrio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2017, ApJ, 851, 44 Ciotti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1991, A&A, 249, 99 Ciotti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Lanzoni, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', & Renzini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1996, MNRAS, 282, 1 de Carvalho, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' & Djorgovski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1992, ApJ, 389, L49 de Graaff, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Bezanson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Franx, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2021, ApJ, 913, 103 de Graaff, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Franx, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Bell, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2021, MNRAS, 504, 5098 D’Onofrio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Bindoni, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Fasano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2014, A&A, 572, A87 D’Onofrio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Cariddi, S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Chiosi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Sciarratta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', & Marziani, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2020, A&A, 641, A94 D’Onofrio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Fasano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Varela, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2008, ApJ, 685, 875 D’Onofrio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Sciarratta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Cariddi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Marziani, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', & Chiosi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} 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J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', & Mast, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2021, A&A, 648, A124 Forbes, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Ponman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Hernquist, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2018a, MNRAS, 475, 648 Pillepich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Springel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Nelson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', et al.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Blackburne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', & Wambsganss, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2014, ApJ, 793, 96 Schmidt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 1959, The Astrophysical Journal, 129, 243 Sciarratta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=', Chiosi, C.' metadata={'source': 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+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} +page_content=' 2014, Nature, 509, 177 Article number, page 25 of 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E8T4oBgHgl3EQfaxWZ/content/2301.06953v1.pdf'} diff --git a/uNE1T4oBgHgl3EQf3wXV/content/tmp_files/2301.03494v1.pdf.txt b/uNE1T4oBgHgl3EQf3wXV/content/tmp_files/2301.03494v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..85239bc42d42fa505d65cf0e5150568d7d853dad --- /dev/null +++ b/uNE1T4oBgHgl3EQf3wXV/content/tmp_files/2301.03494v1.pdf.txt @@ -0,0 +1,1728 @@ +arXiv:2301.03494v1 [math.GR] 9 Jan 2023 +ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT +LIE GROUP, REVISITED +PIERRE DE LA HARPE +Abstract. Let X be a closed smooth manifold, G be a simple connected +compact real Lie group, M(G) be the group of all smooth maps from X to G, +and M0(G) be its connected component for the C∞-compact open topology. +It is shown that maximal normal subgroups of M0(G) are precisely the +inverse images of the centre Z(G) of G by the evaluation homomorphisms +M0(G) → G, γ �→ γ(a), for a ∈ X. This in turn is a consequence of a result +on the group C∞ +n,G of germs at the origin O of Rn of smooth maps Rn → G: +this group has a unique maximal normal subgroup, which is the inverse image +of Z(G) by the evaluation homomorphism C∞ +n,G → G, γ �→ γ(O). +This article provides corrections for part of an earlier article [Harp–88]. +Introduction +An earlier article [Harp–88] was found to contain several mistakes. A list of +local corrections would have been confusing, and we rather repeat with complete +proofs Propositions I and II below (II and III in [Harp–88]), about maximal +subgroups. Theorem I of [Harp–88], about automorphisms, is replaced here by a +conjectural statement only. +Let X be a closed smooth manifold and let G be a simple connected compact +real Lie group, with Lie algebra g. Let M(G) be the group of all smooth maps +from X to G, and let M0(G) be its connected component for the C∞-compact +open topology (the topology of uniform convergence of the maps and all their +partial derivatives of all orders). +Then M0(G) is an important example of a +well behaved infinite dimensional Lie group; see [Miln–84, PrSe–86, Neeb–06, +KhWe–09]. However, M0(G) is viewed here as an abstract group. +Proposition I provides the classification of all maximal normal subgroups of +M0(G). For a ∈ X, let εa : M0(G) → G denote the evaluation map γ �→ γ(a), +and let +NaM0(G) = ε−1 +a (Z(G)) +be the inverse image by εa of the centre Z(G) of G. Since G/Z(G) is simple, +NaM0(G) is a maximal normal subgroup of M0(G). Conversely: +Date: 9 January 2023. +2000 Mathematics Subject Classification. 22E65. +Key words and phrases. Group of smooth maps into compact Lie group, maximal subgroups. +1 + +2 +PIERRE DE LA HARPE +Proposition I. Any proper normal subgroup of M0(G) is contained in NaM0(G) +for some a ∈ X. +One may think of Proposition I as a global result, of which the proof uses +Proposition II which is a related local result. Denote by n the dimension of X, +by C∞ +n,G the set of germs at the origin O of Rn of smooth maps from Rn to G, +and by ε : C∞ +n,G → G the evaluation map γ �→ γ(O). Let +NOC∞ +n,G = ε−1(Z(G)) +be the inverse image of Z(G) by ε. Then C∞ +n,G has a unique maximal normal +subgroup; in other words: +Proposition II. Any proper normal subgroup of C∞ +n,G is contained in NOC∞ +n,G. +Everything here works equally well for maps and germs which are of class Ck for +some k ≥ 0. The proof of Proposition II works also in the real analytic setting. +Though a result analogous to Proposition I for real analytic maps looks plausible, +it is not covered by our proof, which uses partitions of unity. Also, I guess that +Proposition II holds for a simple connected Lie group which is not necessarily +compact. +Let M(Aut(G)) be the group of smooth maps from X to the group Aut(G) of +automorphisms of G (recall that any automorphism in Aut(G) is automatically +continuous [Cart–30], indeed analytic). Let D(X) be the group of smooth diffeo- +morphisms of X. Consider the natural action of D(X) on M(Aut(G)), defined +by +ϕ(β) = β ◦ ϕ−1 +for ϕ ∈ D(X) and β ∈ M(Aut(G)), +and the associated semi-direct product +M(Aut(G)) ⋊ D(X), +with multiplication (α, ϕ)(β, ψ) = (αϕ(β), ϕψ). +This acts on M0(G) by automorphisms: +(α, ϕ)(γ)(x) = α(x) +� +γ +� +ϕ−1(x) +� � +for α ∈ M(Aut(G)), ϕ ∈ D(X), γ ∈ M0(G), and x ∈ X. We believe that any au- +tomorphism of the abstract group M0(G) is of this form, hence in particular that +any automorphism of M0(G) is continuous for the C∞-compact open topology. +In other terms, we can formulate the following statement, which is the alleged +Theorem I in [Harp–88]: +Conjectural Theorem. With the notation above, the group of all abstract group +automorphisms of M0(G) coincides with the group M(Aut(G)) ⋊ D(X). +In Proposition 3.4.2 of [PrSe–86], Pressley and Segal claim that the group +M(Aut(G)) ⋊ D(X) is the group of all bicontinuous automorphisms of the topo- +logical group M0(G). + +ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP +3 +The article [Harp–88] has clearly not been much read. However, in November +2009, Michael Murray and Daniel Stevenson pointed out a serious flaw in the so- +called proof of Lemma 14 — and consequently also in the proof of Proposition I. +(A problematic step is the sentence “By Lemma 12 again one has NW ⊂ ker(π)”; +this Lemma 14 of [Harp–88] has been replaced by Lemma 22 below.) At the time, +I was not smart enough (or stubborn enough) to write a correction. I discussed +the matter with Georges Skandalis, who convinced me that repairing Lemma 14 +(Lemma 22 below) was feasible, and I made a good resolution to work on this as +soon as possible, but this was delayed by several years. +As I finally came back to the problem in 2022, I found other gaps and defects +in proofs of other lemmas. (In particular, it is not appropriate to define as there +a topology on a space of smooth germs.) Short of an erratum restoring all claims +and in particular Theorem I of [Harp–88], I decided to write up a complete proof +of Propositions I and II. Moreover, the so-called Theorem I of [Harp–88] is now +degraded to a “Conjectural Theorem” about the group of automorphims of the +abstract group of M0(G), discussed in the introduction. +The following concordance table indicates how lemmas below correspond to +lemmas in [Harp–88]: +[Harp–88] +Lem 1 +Lem 2 +− +Lem 3 +Lem 4 +Lem 5 +Below +Lem 1 +Lem 2 +Remark 3 +Lem 4 +Lem 5 +Lem 6 +[Harp–88] +Prop 6 +− +− +Lem 7 +Lem 8 +Lem 9 +Below +Prop 7 +Remind 8 +Reduct 9 +Lem 10 +Lem 11 +Lem 12 +[Harp–88] +− +− +Lem 10 +Lem 11 +Lem 13 +Lem 12 +Below +Remind 13 +Ex 14 +Lem 15 +Lem 16 +Lem 17 +Lem 18 +[Harp–88] +− − − +Lem 14 +Lem 15 & 16 +Below +Lem 19 & 20 & 21 +Lem 22 +−− +(Lemmas 15 and 16 in [Harp–88] are not correct and should now be ignored). +1. Proof of Proposition II when G = SU(2) +Note that the result of Section 1 for this particular case G = SU(2) will be +used in the proof of the result for the general case, see the proof of Lemma 15 in +Section 2. +We adopt the following notation. The multiplicative group of complex numbers +of modulus 1 is denoted by U. The algebra of linear endomorphisms of C2 is the +algebra M2(C) of 2-by-2 matrices with complex coefficients; x∗ is the conjugate +transpose of a matrix x ∈ M2(C), and id2 is the unit matrix. +The space of +orthogonal projections from C2 onto lines is the projective line +P1 +C = {p ∈ M2(C) | p∗ = p = p2 and trace(p) = 1}. + +4 +PIERRE DE LA HARPE +In this section, until Proposition 7 included, we use G for the simple connected +compact Lie group +SU(2) = {g ∈ M2(C) | g∗g = id2 and det g = 1} += +�� +ρ +σ +−σ +ρ +� +∈ M2(C) +���� ρ, σ ∈ C, |ρ|2 + |σ|2 = 1 +� +. +An element g ∈ G is regular if g /∈ {id2, −id2}; we denote by Greg the open +subset of G of regular elements. +Any g ∈ Greg has two eigenvalues zg, zg ∈ U; the notation is such that +ℑ(zg) > 0. Set +A := ]0, π[. +Let sg ∈ A be the number such that zg = exp(isg). Let pg ∈ P1 +C be the projection +of C2 onto the eigenspace ker(zgid2 − g) and p′ +G ∈ P1 +C the projection of C2 onto +ker(zgid2 − g). We have +g = exp(isg)pg + exp(−isg)p′ +g +for all g ∈ Greg. +The group G acts by conjugation on both Greg and P1 +C, trivially on A, and by +the product action on P1 +C × A. +Lemma 1. The map +ψA : +� +Greg → P1 +C × A +g �→ (pg, sg) +is a smooth diffeomorphism, and +it is equivariant for the action of G. +Proof. Let h ∈ Greg. Let Dr be a closed disc of centre exp(ish) and some ra- +dius r > 0, contained in the half-plane {z ∈ C | ℑz > 0}, and let Γr be the +circle ∂Dr, with the positive orientation. Since eigenvalues depend continuously +on matrices, there exists a neighbourhood V of h in Greg such that exp(isg) is in +the interior of Dr for all g ∈ V. For g ∈ V, we have the explicit formula +pg = +1 +2iπ +� +Γr +(z − g)−1dz +by holomorphic functional calculus. It follows that pg depends smoothly on g. +We have also +(∗) +sg = arccos +�1 +2trace(g) +� +, +hence sg depends smoothly on g. +For the continuity of eigenvalues and for holomorphic functional calculus, see +for example [DuSc–58, Chapter VII, § 3, no. 19, Corollary 21], or [BoTS, Chap. +1, § 4, no 11, Proposition 16], or [Serr–10, 5.2.3 & 5.5], or [Texi–18]. +The equivariance of ψA is straightforward. +□ + +ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP +5 +The complex projective line can also be described as +P1 +C = +��a +b +b +1 − a +� ��� a ∈ R, b ∈ C, 0 ≤ a ≤ 1, a2 + |b|2 = a +� += +��1 +2 +� +1 + +√ +1 − 4r2� +reiϕ +re−iϕ +1 +2 +� +1 − +√ +1 − 4r2� +� ����� 0 ≤ r ≤ 1 +2, ϕ ∈ [0, 2π[ +� +∪ +��1 +2 +� +1 − +√ +1 − 4r2� +reiϕ +re−iϕ +1 +2 +� +1 + +√ +1 − 4r2� +� ����� +1 +2 ≥ r ≥ 0, ϕ ∈ [0, 2π[ +� +. +The second parametrization makes it clear that P1 +C is diffeomorphic to the 2- +sphere, shown as two hemispheres glued along the equator. We define +U = P1 +C ∖ +�� +0 +0 +0 +1 +�� += +��a +b +b +1 − a +� ��� a ∈ R, b ∈ C, 0 < a ≤ 1, a2 + |b|2 = a +� +, +which is an open neighbourhood of +� +1 +0 +0 +0 +� +in P1 +C. +Lemma 2. There exists a smooth map +�U → G +u �→ gu +such that gu +� +1 +0 +0 +0 +� +g−1 +u += u +for all u ∈ U. +Proof. Given a rank one projection u = +�a +b +b +1 − a +� +∈ U, set ρ = √a, σ = −b/ρ, +and define gu = +� +ρ +σ +−σ +ρ +� +∈ SU(2). +Then gu +� +1 +0 +0 +0 +� +g−1 +u += u, by a simple +computation. +□ +Remark 3. The diffeomorphism of Lemma 1 and the map of Lemma 2 are not +only smooth, they are indeed real analytic. +Similarly, the map of Lemma 10 +below is real analytic. +Let O be a neighbourhood of the origin in Rn and let γ : O → G be a smooth +map with values in Greg. Denote by γ ∈ C∞ +n,G the germ defined by γ. Then sγ(x) +depends smoothly on x, by Lemma 1, and defines at the origin a germ sγ of A- +valued smooth map. Similarly pγ(x) defines at the origin a germ pγ of P1 +C-valued +smooth map. We write simply s and p when the reference to γ is clear. +Lemma 4. Let γ and δ be two germs in C∞ +n,G such that γ(O), δ(O) ∈ Greg. Let +(p, s) and (q, t) be the associated germs at O of (P1 +C × A)-valued smooth maps. +Then γ and δ are conjugate in C∞ +n,G if and only if s = t. + +6 +PIERRE DE LA HARPE +Proof. Consider ζ ∈ C∞ +n,G. +We may choose representatives γ, p, s, δ, q, t, ζ of +γ, p, s, δ, q, t, ζ defined in a common neighbourhood O of the origin in Rn, and +such that γ(x), δ(x) ∈ Greg for all x ∈ O. +Suppose first that ζ γ ζ−1 = δ. Upon replacing O by a smaller neighbourhood, +we may assume that ζ(x)γ(x)ζ(x)−1 = δ(x) for all x ∈ O, hence that s(x) = t(x) +for all x ∈ O. It follows that s = t. +Suppose now that s = t. We may again assume that s(x) = t(x) for all x ∈ O. +Suppose first that, moreover, the range of p(O) and the range of q(O) are not +orthogonal in C2. In appropriate orthogonal coordinates: +p(O) = +� +1 +0 +0 +0 +� +and q(O) = +�c +d +d +1 − c +� +with c > 0, d ∈ C, c2 + |d|2 = c. +Upon shrinking O once more if necessary, we may assume that p and q are of the +form +p(x) = +�a(x) +b(x) +b(x) +1 − a(x) +� +where a(x) > 0 for all x ∈ O, with a(O) = 1, +q(x) = +�c(x) +d(x) +d(x) +1 − c(x) +� +where c(x) > 0 for all x ∈ O, with c(O) = c. +By Lemma 2, there exists a smooth map ζ : O → G such that ζ(x)p(x)ζ(x)−1 = +q(x) for all x ∈ O. Since s = t, it follows that ζγζ−1 = δ, hence ζ γ ζ−1 = δ. +Suppose now that the range of p(O) and the range of q(O) are orthogonal in +P1 +C. Define a third germ η ∈ C∞ +n,G with eigenvalue germ equal to s (hence also +equal to t), and with η(O) orthogonal neither to γ(O) nor to δ(O). The previous +argument shows that γ and δ are both conjugate to η, hence are conjugate one +to the other. +□ +Lemma 5. Let O be a neighbourhood of the origin in Rn and s, t two smooth +maps O → A such that sin(t(x)/2) < sin(s(x)) for all x ∈ O. Define smooth +maps ζ, η, δ : O → G by +ζ(x) = + + + +� +1 − sin2(t(x)/2) +sin2(s(x)) +�1/2 +sin(t(x)/2) +sin(s(x)) +− sin(t(x)/2) +sin(s(x)) +� +1 − sin2(t(x)/2) +sin2(s(x)) +�1/2 + + + ∈ SO(2) ⊂ G, +η(x) = +� +exp(is(x)) +0 +0 +exp(−is(x)) +� +ζ(x) +� +exp(−is(x)) +0 +0 +exp(is(x)) +� +ζ(x)−1, +δ(x) = +� +exp(it(x)) +0 +0 +exp(−it(x)) +� +, +for all x ∈ O. +Then η(x) ∈ Greg for all x ∈ O, and the germs of η and δ are conjugate in +C∞ +n,G. + +ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP +7 +Proof. Note that s(x), t(x) ∈ A implies ζ(x), δ(x) ∈ Greg for all x ∈ O. +A +straightforward computation shows that, for all x ∈ O, +trace(η(x)) = 2 sin2(s(x)) − (1 − cos(2s(x))) sin2(t(x)/2) +sin2 s(x) += 2(1 − 2 sin2(t(x)/2)) = 2 cos(t(x)) = trace(δ(x)). +A first consequence is that η(x) ∈ Greg, because 2 cos(t(x)) ̸= ±2. (Remember +Formula (*) in the proof of Lemma 1.) A second consequence is that the A-valued +germs at O associated to η and δ are equal. It follows from Lemma 4 that the +germs at O of η and δ are conjugate in C∞ +n,G. +□ +Lemma 6. Let N be a normal subgroup of C∞ +n,G containing a germ γ such that +γ(O) ∈ Greg. +There exists a symmetric neighbourhood V of the identity in G (depending on +γ) such that every δ ∈ C∞ +n,G with δ(O) ∈ V ∩ Greg is in N. +(A neighbourhood V of id2 in G is symmetric if g−1 ∈ V for all g ∈ V.) +Proof. Let (p, s) be the (P1 +C×A)-valued germ at O associated to γ, as in Lemma 4. +Let γ, p, s be representatives of γ, p, s defined in a neighbourhood O of the origin +in Rn. By this Lemma 4, there is no loss of generality if we assume that +γ(x) = +� +exp(is(x)) +0 +0 +exp(−is(x)) +� +and that s(x) is bounded away from {0, π} for all x ∈ O, more precisely that +there exists c > 0 such that c < s(x) < π − c for all x ∈ O. +Let V• be the open subset of Greg of elements g such that sin(sg/2) < sin(c), +where sg is defined by ψA(g) = (pg, sg), where ψA is as in Lemma 1. Set V = +V• ∪ {id2}; it is a symmetric open neighbourhood of id2 in G. Let δ ∈ C∞ +n,G be +such that δ(O) ∈ V•; we have to show that δ ∈ N. +Let (q, t) be the (P1 +C × A)-valued germ associated to δ. Upon replacing O by +a smaller neighbourhood of the origin in Rn, we can find representatives δ, q, t of +δ, q, t such that δ(x) ∈ V•, i.e., such that 0 < sin(t(x)/2) < sin(x), for all x ∈ O. +By Lemma 4 again, it suffices to consider the element δ defined by +δ(x) = +� +exp(it(x)) +0 +0 +exp(−it(x)) +� +for all x ∈ O. +By Lemma 5, there exists a smooth map ζ : O → G such that δ and γ ζ γ−1 ζ−1 +are conjugate in C∞ +n,G, and therefore auch that δ is in the normal subgroup gen- +erated by γ. +□ +Proposition 7. Proposition II of the Introduction holds for G = SU(2): +any proper normal subgroup of C∞ +n,G is contained in NOC∞ +n,G. + +8 +PIERRE DE LA HARPE +Proof. Let N be a normal subgroup of C∞ +n,G which is not contained in NOC∞ +n,G = +ε−1(Z(G)). Then ε(N) = G, because a normal subgroup of G not contained in +Z(G) is G itself (see Reminder 8). In particular N contains a germ γ such that +γ(O) is regular. Let ζ ∈ C∞ +n,G; we have to show that ζ ∈ N. +Let O be a neighbourhood of the origin in Rn and let γ, ζ : O → G be rep- +resentatives of γ, ζ. By Lemma 6, there exists a symmetric neighbourhood V +of id2 in G such that every δ ∈ C∞ +n,G with δ(O) ∈ V ∩ Greg is in N. Since G is +connected, there exists an integer k ≥ 1 and elements g1, g2, . . . gk ∈ V ∩Greg such +that ζ(O) = g1g2 · · · gk. For any g ∈ G, denote by ηg : O → G the constant map +of value g. Set δ = ζ(ηg2 · · · ηgk)−1, so that ζ = δηg2 · · · ηgk. Then the k matrices +δ(O) = g1, ηg2(O) = g2, . . . , ηgk(O) = gk are all in V ∩Greg. Lemma 6 implies that +the germs δ, ηg2, . . . ηgk are all in N, so that their product ζ is also in N. +□ +Reminder 8. Let G be a connected compact Lie group G which is simple as a +Lie group, i.e., such that its Lie algebra is simple. The following results are due +to Cartan and van der Waerden [Cart–30, vdWa–33]. +(i) Any non-trivial normal subgroup of G is contained in the centre Z(G) of +G, and the quotient G/Z(G) is simple as an abstract group. +(ii) Any abstract group homomorphism with bounded image from G to a Lie +group is continuous, and therefore smooth. +In particular any abstract group +endomorphism φ of G is necessarily smooth. Since the Lie algebra of G is simple, +the derivative of φ is either an isomorphism, in which case φ is an automorphism, +or zero, in which case φ(g) = 1G for all g ∈ G. +For the historical context of these articles by Cartan and van der Waerden, see +[Bore–01, Chap. VI, § 6]. For hints of proofs, see [BoLG2-3, Chapitre III, § 4 +exercices 8 & 9, and § 9 exercice 25]. Recall that any continuous homomorphism +between Lie groups is analytic, and in particular is smooth; see for example +[BoLG2-3, Chapitre III, § 8, no. 1]. +We end this section by a digression. Germs which are regular can be diagonal- +ized by Lemma 4, but the regularity condition cannot be removed. +The following example illustrates this; it is a minor adaptation of one in +[Rell–69, Chap. 1, § 3]. Set n = 1. Define first a map θ from R to the space of +2-by-2 matrices by +θ(x) = + + + + + + + +exp(−x−2) +� +cos(2/x) +sin(2/x) +sin(2/x) +− cos(2/x) +� +if x ̸= 0, +� +0 +0 +0 +0 +� +if x = 0. +Define then a map γ : R → SU(2) by γ(x) = exp(iθ(x)) for all x ∈ R. Outside +the origin, γ(x) has eigenvectors +� +cos(1/x) +sin(1/x) +� +and +� +sin(1/x) +− cos(1/x) +� +with eigenvalues + +ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP +9 +exp(i exp(−x−2)) and exp(−i exp(−x−2)). +But there is no germ ζ such that +ζ γ ζ−1 is diagonal. +Even though it is not relevant here, we mention that one may diagonalize γ +with a Borel map [Azof–74]. +2. Proof of Proposition II in the general case +Let G be simple connected compact Lie group, g its Lie algebra, exp : g → G +the exponential map, and Ad : G → GL(g⊗R C) the adjoint representation of G. +Reduction 9. It is sufficient to prove Proposition II when G is simply connected. +Proof. Indeed, suppose that Proposition II holds for the universal cover �G of G +(which is still compact by Weyl’s theorem [BoLG9, § 1, no 4]). The short exact +sequence +{1} → π1(G) → �G → G → {1} +induces a sequence +{1} → π1(G) → C∞ +n, �G +p→ C∞ +n,G → {1} +which is again exact (in these sequences, elements of π1(G) are viewed as elements +of the centre of �G, and also as germs of constant maps from Rn to the centre of +�G). We denote here by εG : C∞ +n,G → G and ε � +G : C∞ +n, �G → �G the evaluation maps at +the origin of Rn. +Let N be a normal subgroup of C∞ +n,G which is not contained in ε−1 +G (Z(G)). As +�G/Z( �G) = G/Z(G), the normal subgroup � +N := p−1(N) of C∞ +n, �G is not contained +in ε−1 +�G (Z( �G)). If Proposition II holds for �G, then � +N = C∞ +n, �G, hence N = C∞ +n,G; +thus Proposition II holds for G. +□ +For g ∈ G, denote by ZG(g)0 the connected component of the centralizer +ZG(g) = {h ∈ G | hg = gh} of g. The element g is regular if ZG(g)0 is a +maximal torus in G, equivalently if there exists a unique maximal torus in G +containing g. When g is regular and h ∈ G is not, dim ZG(g)0 < dim ZG(h)0 and +there are infinitely many maximal tori in G containing h. We denote by Greg the +set of regular elements in G. Conjugates of regular elements are regular, so that +G acts on Greg by conjugation. The subset Greg is symmetric, open, and dense +in G. More on regular elements in [BoLG7-8, Chap. VII, § 4] and [BoLG9, § 5, +no 1]. We +choose a maximal torus T in G +and we denote by t its Lie algebra. We choose an alcove A in t, namely a connected +component of the subset of t consisting of those ξ ∈ t with exp(ξ) regular. The +group G acts as usual on G/T, trivially on A, and by conjugation on the subset +Greg of regular elements. Because of Reduction 9: + +10 +PIERRE DE LA HARPE +from now on in this section, we assume that G is simply connected. +Lemmas 10 and 12 can be seen as generalizations of Lemmas 1 and 4 respec- +tively. +Lemma 10. The map ϕA : +� +G/T × A → +Greg +(gT, s) �→ g(exp(s))g−1 +is a smooth diffeomor- +phism, and it is equivariant for the action of G. +Proof. The map ϕA is a diffeomorphism, see [BoLG9, Proposition 4b of Page 51]; +the group denoted there by HA is here reduced to {1}, because G is simply +connected, see [BoLG9, Remark 1 of Page 45]. The equivariance is clear. +□ +We denote by π : G → G/T the canonical projection. +Lemma 11. There exist an integer k ≥ 1 and +open subsets U1, U2, . . . , Uk in G/T, +smooth maps µj : Uj → G for j = 1, 2, . . . , k, +such that +π +� +µj(u) +� += u for each u ∈ Uj, for j = 1, 2, . . . , k. +Proof. This is a way to say that π : G → G/T is the projection of a locally trivial +bundle with compact base G/T and fibre T. +□ +Lemma 12. Let γ and δ be two germs in C∞ +n,G such that γ(O), δ(O) ∈ Greg. +Let (p, s) and (q, t) be the associated germs at O of (G/T × A)-valued smooth +maps obtained by composition of γ and δ with the map ϕ−1 +A , where ϕA is as in +Lemma 10. +Then γ and δ are conjugate in C∞ +n,G if and only if s = t. +Proof. We prove the non-trivial implication only. We assume that s = t and we +have to show that γ and δ are conjugate. We use the notation of Lemma 11. We +distinguish two cases. +In the first case, there exists j ∈ {1, . . . , k} such that p(O), q(O) are in Uj. +We may choose representatives γ, p, s, δ, q, t of γ, p, s, δ, q, t defined in a common +neighbourhood O of the origin in Rn, such that p(x), q(x) ∈ Uj and s(x) = t(x) +for all x ∈ O. Define a smooth map ζ : O → G by ζ(x) = µj(q(x)) +� +µj(p(x)) +�−1; +then ζ(x)p(x) = q(x) for all x ∈ O. (Note that ζ(x) is an element of G which +acts on G/T, and that ζ(x)p(x) = q(x) is an equality between elements of G/T.) +By Lemma 10 we have ζ(x)γ(x)ζ(x)−1 = δ(x) for all x ∈ O, so that γ and δ are +conjugate. +In the second case, there exist j, j′ ∈ {1, . . . , k} with j ̸= j′ such that p(O) ∈ +Uj and q(O) ∈ Uj′. We may choose representatives γ, p, s, δ, q, t of γ, p, s, δ, q, t +defined in a common neighbourhood O of the origin in Rn, such that p(x) ∈ Uj, +q(x) ∈ Uj′, and s(x) = t(x) for all x ∈ O. There exist an integer m ≥ 1 and + +ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP +11 +a sequence q0 = p(O), q1, . . . , qm−1, qm = q(O) of elements of G/T such that, for +each i ∈ {1, . . . , m}, there exists j(i) ∈ {1, . . . , k} with both qi−1 and qi in Uj(i). +For i ∈ {1, . . . , m − 1}, define a smooth map γi : O → G by γi(x) = ϕA(qi, s(x)) +for all x ∈ O; write γ0 for γ and γm for δ. By the argument for the first case, +there exists for i ∈ {1, . . . , m} a smooth map ζi : O → G such that ζi(x)qi−1(x) = +qi(x), and therefore ζi(x)γi−1(x)ζi(x)−1 = γi(x), for all x ∈ O. It follows that +ζmζm−1 · · · ζ1 conjugates γ0 = γ to γm = δ, and therefore that γ is conjugate +to δ. +□ +As we have not been able to generalize Lemma 5, we proceed in the next +lemmas by reduction to the case of SU(2). Before this, we recall the following +facts on the structure of simple connected compact Lie groups. +Reminder 13. Let X(T) denote the group of continuous homomorphisms from +the maximal torus T to the group U of complex numbers of modulus 1. For +α ∈ X(T), set +gα +C = {ξ ∈ g ⊗R C | Ad(t)ξ = α(t)ξ for all t ∈ T}. +The group X(T) is written additively, so that α = 0 is the constant homomor- +phism T → U of value 1. A root of G with respect to T is an element α ̸= 0 in +X(T) such that dim gα +C > 0. If α is a root, it is known that dim gα +C = 1 and that +−α is also a root. Let R(G, T) denote the set of roots. An element t ∈ T is regular +if and only if t is not contained in ker(α) for all α ∈ R(G, T), so that the alcove A +chosen above is a connected component of t∖� +α∈R(G,T){ξ ∈ t | exp(ξ) ∈ ker(α)}. +Choose a basis of the set of roots R(G, T) and let R+(G, T) be the corresponding +set of positive roots. +Let α be a root. +There exists a connected closed subgroup Sα of G and a +surjective morphism of Lie groups +να : SU(2) → Sα +such that +(i) Sα and the kernel of α : T → U commute, +(ii) να +� +a +0 +0 +a +� +∈ T and α +� +να +� +a +0 +0 +a +�� += a2 for all a ∈ U, +(iii) T = +� +Sα ∩ T +� +· +� +ker(α) +� +, +(iv) να is injective, i.e., να : SU(2) → Sα is an isomorphism. +The group Sα is the Lie subgroup of G of Lie algebra +su(2)α = g ∩ +� +gα +C ⊕ g−α +C ⊕ [gα +C, g−α +C ] +� +, +which is a subalgebra of g isomorphic to su(2). +For the existence of να, and (i) and (ii), see [BoLG9, § 4, no 5, Page 31]. + +12 +PIERRE DE LA HARPE +For (iii), consider t ∈ T. Choose a square root a of α(t). Set t1 = να +� +a +0 +0 +a +� +and t2 = t−1 +1 t. Then t = t1t2 with t1 ∈ Sα ∩ T and t2 ∈ ker(α). +For (iv), see the Remarque in [BoLG9, § 4, no 5, Page 32], and recall that G +is simply connected. (For G = SO(2n + 1), which is not simply connected, and +for α a short root, the image of να is isomorphic to SO(3) = SU(2)/{±id2}.) +Let ψα denote the automorphism of G of conjugation by να +� +2−1/2 +2−1/2 +−2−1/2 +2−1/2 +� +. +Then +(v) ψα +� +να +� +a +0 +0 +a +�� += να +� a+a +2 +−a+a +2 +−a+a +2 +a+a +2 +� +for all να +� +a +0 +0 +a +� +∈ Im(να) ∩ T, +(vi) ψα(t) = t for all t ∈ ker(α). +(vii) tψα(t)−1 ∈ να (SU(2)reg) for all t ∈ T ∩ Greg. +Claim (v) follows from the computation +� +2−1/2 +2−1/2 +−2−1/2 +2−1/2 +� � +a +0 +0 +a +� � +2−1/2 +−2−1/2 +2−1/2 +2−1/2 +� += +� a+a +2 +−a+a +2 +−a+a +2 +a+a +2 +� +and Claim (vi) follows from (i). +(Compare (v) and (vi) with [BoLG9, § 4, +no 5, Page 35], where the relevant inner automorphism of G is that defined +by να +� +0 +1 +−1 +0 +� +.) +Let us show (vii). Let t ∈ T. There exist by (iii) commuting elements +t1 = να +� +a +0 +0 +a +� +∈ Im(να) ∩ T +and +t2 ∈ ker(α) +such that +t = t1t2. +Then +tψα(t)−1 = t1t2να +� +2−1/2 +2−1/2 +−2−1/2 +2−1/2 +� +t−1 +2 t−1 +1 να +� +2−1/2 +−2−1/2 +2−1/2 +2−1/2 +� +by (i) += +t1να +� +2−1/2 +2−1/2 +−2−1/2 +2−1/2 +� +t−1 +1 να +� +2−1/2 +−2−1/2 +2−1/2 +2−1/2 +� += να +� +1 +2 +� +a +0 +0 +a +� � +1 +1 +−1 +1 +� � +a +0 +0 +a +� � +1 +−1 +1 +1 +� � += να +�1+a2 +2 +−1+a2 +2 +1−a2 +2 +1+a2 +2 +� +. +. +If t ∈ T ∩ Greg, then t /∈ ker(α), hence t1 /∈ ker(α), hence a2 ̸= 1 by (ii), and +therefore +�1+a2 +2 +−1+a2 +2 +1−a2 +2 +1+a2 +2 +� +∈ SU(2)reg. +Finally, we have the following, used below in the proof of Lemma 17. + +ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP +13 +(viii) There exist a neighbourhood V of 1G in G and for each α ∈ R+(G, T) a +smooth map λα : V → Sα such that � +α∈R+(G,T) να(λα(g)) = g. +Proof of (viii). We write R+ for R+(G, T). For α ∈ R+, recall that su(2)α is +the Lie algebra of Sα. Consider the surjective linear map � +α∈R+ su(2)α → g +which applies (ξα)α∈R+ onto � +α∈R+ ξα. Choose for each α ∈ R+ a linear map +κα : g → su(2)α such that � +α∈R+ κα(ξ) = ξ for all ξ ∈ g. +Let V′ be a neighbourhood of 1G in G and W be a neighbourhood of the +origin O in g such that the exponential map is a diffeomorphism exp : W → V′; +denote by log the inverse diffeomorphism. For each α ∈ R+, define a smooth +map λ′ +α : V′ → Sα by λ′ +α(g) = exp +� +κα +� +log(g) +�� +. Define next a smooth map +λ′ : V′ → G by λ′(g) = � +α∈R+ να(λ′ +α(g)); the terms of this product are arranged +according to some chosen order, and the same order is used for the next product +� +α∈R+(· · · ) below. Observe that λ′(1G) = 1G and that the derivative of λ′ at +1G is the identity map g → g. Upon replacing V′ by a smaller neighbourhood +of 1G, one may therefore assume that λ′ is a diffeomorphism from V′ to some +neighbourhood V of 1G in G. We denote by ν the inverse diffeomorphism. For +each α ∈ R+, define a smooth map λα : V → Sα by λα(g) = λ′ +α(ν(g)). Then +� +λ∈R+ να(λα(g)) = � +λ∈R+ να(λ′ +α(ν(g))) = λ′(ν(g)) = g for all g ∈ G, and this +concludes the proof. +□ +Example 14. Let G = SU(n), for n ≥ 3, and let T be the torus of diago- +nal matrices in G. The open dense subset T ∩ Greg of T consists of matrices +diag(z1, z2, . . . , zn) such that z1, z2, . . . , zn are all distinct and z1z2 · · · zn = 1. Let +α be the homomorphism G → U mapping diag(z1, z2, . . . , zn) to z1z−1 +2 . Then α is +a root of G with respect to T, the image of να is +� +SU(2) +0 +0 +idn−2 +� +and the kernel +of α is +� +±id2 +0 +0 +SU(n − 2) +� +. +Lemma 15. Let N be a normal subgroup of C∞ +n,G which contains a germ γ such +that γ(O) /∈ Z(G). +Then N contains anll constant germs. +Proof. Step one. We can assume that γ(O) ∈ Greg. +Indeed, set g = γ(O). Denote by k the dimension of G; define +M(g) = +� +h ∈ G | h = +k +� +j=1 +ajbjgb−1 +j g−1a−1 +j +for some a1, b1, . . . , ak, bk ∈ G +� +. +Since g is not central, M(g) is a neighbourhood of 1G in G (this is the essential +point in van der Waerden’s proof of the simplicity of the abstract group G/Z(G) + +14 +PIERRE DE LA HARPE +[vdWa–33]). Choose a1, b1, . . . , ak, bk ∈ G such that g′ ≑ �k +j=1 ajbjgb−1 +j g−1a−1 +j +∈ +Greg. Recall that ηa denote the constant map Rn → G of value a. Define +γ′ = +k +� +j=1 +ηaj ηbj γ η−1 +bj γ−1 η−1 +aj . +Then γ′ is in N and g′ = γ′(O) ∈ Greg. So that we can indeed assume from the +start that γ(O) ∈ Greg. +Step two. We can assume that γ(x) ∈ T ∩ Greg for all x ∈ O. +Denote by (p, s) the germ at O of (G/T × A)-valued smooth maps associated +to γ as in Lemma 12. Let γ : O → Greg and s : O → A be representatives of γ +and s. By Lemma 12, the germ at O of the map γ′ : O → Greg, x �→ exp(s(x)) +is conjugate to γ, hence is in N. Therefore, we may assume from now on that +γ is itself the germ of the map γ : O → Greg, x �→ exp(s(x)), in particular that +γ(x) ∈ T ∩ Greg for all x ∈ O. +Step three. The group N contains the germ of one constant map with value +not in Z(G). +Let α be a root of G with respect to T. Let να : SU(2) → G and ψα ∈ Aut(G) +be as in Reminder 13. Let Iα : C∞ +n,SU(2) → C∞ +n,G be the map which applies the +germ at O of a smooth map O → SU(2) to the germ at O of the composition +O → SU(2) +να +→ G; then Iα is an injective homomorphism of groups. Define the +map δ : O → G by δ(x) = γ(x)ψα(γ(x))−1 for all x ∈ O and let δ denote its germ +at O ∈ O. Since δ is the commutator of γ with something, δ is in N. It follows +from (vii) in Reminder 13 that δ(x) ∈ να (SU(2)reg) for all x ∈ O and that δ is in +the image of Iα. Therefore I−1 +α (N) is a normal subgroup of C∞ +n,SU(2) which contains +the germ I−1 +α (δ), and I−1 +α (δ)(O) ∈ SU(2)reg. It follows from Proposition 7 that N +contains all germs at O of smooth maps O → να(SU(2)). +In particular, N contains the germ of a constant map O → G with value in +να(SU(2)reg), in particular with value in G and not in Z(G). +Final step. Since any normal subgroup of G not inside Z(G) is G itself (see +Reminder 8), N contains the germs of all constant maps O → G. +□ +The following lemma is the last step in the proof of Proposition II. +Lemma 16. Let N be a normal subgroup of C∞ +n,G which contains a germ γ such +that γ(O) /∈ Z(G). +Then N = C∞ +n,G. +Proof. Step one. Let W be the Weyl group of G with respect to T (it is the +quotient of the normalizer NormG(T) by T), let c ∈ W be a Coxeter element, +and let m ∈ NormG(T) be an element of class c. Recall that W acts on T and t, +in particular c acts on T by t �→ mtm−1 and on t by ξ �→ Ad(m)ξ. Consider the + +ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP +15 +smooth map +f : T → T, +t �→ c(t)t−1 = mtm−1t−1. +The derivative of f at the identity is +L(f) : t → t, +ξ �→ c(ξ) − ξ = Ad(m)ξ − ξ. +Now L(f) is a linear automorphism, because Coxeter elements do not have 1 as +an eigenvalue (see [BoLG9, Page 33], referring itself to [BoLG4-6, Chap. V, § 6, +no 2]). By the implicit function theorem, there exist a symmetric neighbour- +hood V of the identity in T and a smooth map χ : V → T such that +(♯) +t = mχ(t)m−1χ(t)−1 +for all t ∈ V. +Observe that � +g∈G gVg−1 is a neighbourhood of 1G in G. +Step two. Let δ ∈ C∞ +n,G be such that δ(O) ∈ +� � +g∈G gVg−1� +∩ Greg; then δ ∈ N. +Let O be a neighbourhood of the origin in Rn and let δ : O → +� � +g∈G gVg−1� +∩ +Greg be a smooth map of germ δ. By Lemma 12, there exists a smooth map +ζ : O → V ∩ Greg of which the germ ζ is conjugate to δ in C∞ +n,G, so that it suffices +to show that ζ ∈ N. Let ηm be the constant map O → G of value m; by (♯), we +have +ζ(x) = ηm(x)χ(ζ(x))ηm(x)−1χ(ζ(x))−1 +for all x ∈ O. +Since the germ of ηm is in N by Lemma 15, if follows that ζ ∈ N. +Final step. Let now δ be arbitrary in C∞ +n,G, and let δ : O → G be a represen- +tative of δ. Since +� � +g∈G gVg−1� +∩ Greg is a nonempty symmetric open subset of +G and since G is connected, there exist g1, . . . , gk ∈ +� � +g∈G gVg−1� +∩ Greg such +that δ(O) = g1 · · ·gk. +For j ∈ {1, . . . , k}, let ηj : O → � +g∈G gVg−1 be the +constant map of value gj. Set ϕ = δ(η2 · · · ηk)−1, so that δ = ϕη2 · · · ηk. Then +ϕ(O) = g1, η2(O) = g2, . . . , ηk(O) = gk are all in (� +g∈G gVg−1� +∩ Greg, hence +ϕ, η2, . . . , ηk are all in N by Step two. It follows that δ = ϕη2 · · · ηk is in N. +□ +3. Proof of Proposition I +Let X be a closed smooth manifold, n its dimension, G a simple connected com- +pact real Lie group, 1G the identity in G, and M0(G) the connected component +of the group M(G) of all smooth maps from X to G, as in Proposition I. +In the following lemma, G is identified with the group of constant maps in +M0(G). Given subsets A, B in M0(G), we write (A, B) for the subgroup of M0(G) +generated by the commutators γδγ−1δ−1 with γ ∈ A and δ ∈ B. +Lemma 17. We have M0(G) = (G, M0(G). In particular, the group M0(G) is +perfect. +Note. The group M(G) need not be perfect. For example, when X is the circle +S1 and G is not simply connected, there is a natural epimorphism from M(G) to + +16 +PIERRE DE LA HARPE +the fundamental group of G, which is abelian and not trivial, so that M(G) is +not perfect. +There are cases where M0(G) = M(G). When G is simply connected, this +holds for X the circle (obvious) and for X the 2-sphere (because the second +homotopy group of G is trivial, by a theorem of Weyl). When G = SU(2), this +holds whenever the third cohomotopy set of X has one element, in particular +when X is the circle or a closed surface. +Proof. We follow essentially the proof of Proposition 3.4.1 in [PrSe–86]. +We +consider first the case of G = SU(2). +Step one. Let T1 be the torus of diagonal matrices in G. Let γ ∈ M0(T1). +Then γ ∈ (G, M0(G)). +Indeed, let U1 = +�� +eis +0 +0 +e−is +� +∈ T1 +��� − π < s < π +� +. Let +V1 = {γ ∈ M(T1) | γ(x) ∈ U1 for all x ∈ X}, +which is a neighbourhood of the identity in M0(T1). Since there is a smooth +retraction by deformation of U1 to {1T1} in T1, any smooth map γ ∈ V1 is in +M0(T1). Suppose first that γ ∈ V1. Then γ is of the form x �→ +� +eis(x) +0 +0 +e−is(x) +� +for some smooth map s : X → ]−π, π[. Since +� +eis(x) +0 +0 +e−is(x) +� += +� +0 +1 +−1 +0 +� � +e−is(x)/2 +0 +0 +eis(x)/2 +� � +0 +−1 +1 +0 +� � +eis(x)/2 +0 +0 +e−is(x)/2 +� +for all x ∈ X, we have γ ∈ (G, M0(G)). Let now γ be arbitrary in M0(T1). +Since M0(T1) is connected, there exist γ1, . . . , γk ∈ V1 such that γ is the product +γ1 · · ·γk. As each γj is contained in (G, M0(G)) by the previous argument, γ ∈ +(G, M0(G)). +Step two. Let +T2 = +�� +cos(t) +i sin(t) +i sin(t) +cos(t) +� ��� t ∈ [0, 2π[ +� +, +T3 = +�� +cos(u) +sin(u) +− sin(u) +cos(u) +� ��� u ∈ [0, 2π[ +� +. +Similarly, if γ is in M0(T2) or in M0(T3), then γ ∈ (G, M0(G)). +This follows from the Step one because maximal tori in SU(2) are conjugate. +Step three. There is a neighbourhood of the identity W in M0(G) such that +every γ ∈ W is in (G, M0(G)). +For j ∈ {1, 2, 3}, let tj denote the Lie algebra of Tj. Consider the multiplication +map +µ : T1 × T2 × T3 → G, (g1, g2, g3) �→ g1g2g3. + +ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP +17 +Its derivative at the identity +t1 × t2 × t3 → g, (ξ1, ξ2, ξ3) �→ ξ1 + ξ2 + ξ3 +is a linear isomorphism. By the implicit function theorem, there exist a neigh- +bourhood VT of the identity in T1 × T2 × T3 and a neighbourhood VG of the +identity in G such that the restriction of µ is a homeomorphism VT → VG. +Upon restricting these neighbourhoods, we can assume moreover that there exist +smooth retractions by deformation of VT to {1T} in T1 × T2 × T3 and of VG to +{1G} in G. Define W = {γ ∈ M(G) | γ(x) ∈ VG for all x ∈ X}; any smooth +map γ ∈ W is in M0(G), and W is a neighbourhood of the identity in M0(G). +Then every γ ∈ W is a product γ1γ2γ3 with γj ∈ M0(Tj) for j ∈ {1, 2, 3}, hence +γ ∈ (G, M0(G)) by the first two steps. +Step four. Every γ ∈ M0(G) is in (G, M0(G)). +Since M0(G) is connected, this follows from Step three by the usual argument. +(Compare with the end of the proof of Step one.) +The part in which G stands for SU(2) is now finished. From now G denotes +again a general simple connected compact real Lie group, as in the statement of +Lemma 17. +Step five. Choose a maximal torus T of G and a basis of the set R = R(G, T) +of roots of G with respect to T. Let R+ be the corresponding set of positive roots. +Let V be a neighbourhood of 1G in G and, for each α ∈ R+, let λα : V → Sα +be a smooth map such that � +α∈R+ να +� +λα(g) +� += g for all g ∈ V. +(See (viii) +in Reminder 13; recall that να is the inclusion homomorphism of Sα ≈ SU(2) +into G.) +Let γ ∈ M0(G) be such that γ(X) ⊂ V. For each α ∈ R+, the map λα◦γ : X → +Sα is in the commutator subgroup (Sα, M0(Sα)), by Step four. It follows that +γ = � +α∈R+ να +� +µα ◦ γ +� +is in the commutator subgroup (G, M0(G)). +Let now γ be arbitrary in M0(G). Since M0(G) is connected, γ is a product of +elements with images in V, hence γ ∈ (G, M0(G)). +□ +The support of an element γ of M(G) is the closure supp(γ) of the subset of X +of points x such that γ(x) ̸= 1G. +Lemma 18. Let (Ui)1≤i≤k be an open covering of X and let γ ∈ M0(G). +There exist a finite sequence of smooth maps (γj)1≤j≤ℓ in M0(G) and a sequence +of indices (i(j))1≤j≤ℓ in {1, . . . , k} with supp(γj) ⊂ Ui(j) for j ∈ {1, . . . , ℓ} such +that γ = γ1 · · · γℓ. +Proof. Let C∞(X, g) denote the space of all smooth maps from X to the Lie +algebra g of G. Then M0(G) is a Fr´echet Lie group with Lie algebra C∞(X, g); +there is an exponential map +EXP : C∞(X, g) → M0(G) + +18 +PIERRE DE LA HARPE +given by +(EXP(ζ))(x) = exp(ζ(x)) +for all ζ ∈ C∞(X, g) and x ∈ X, +where exp : g → G is the exponential map of G, and this EXP is a local dif- +feomorphism. Since M0(G) is connected, there exist ζ1, . . . ζm ∈ C∞(X, g) such +that +γ = EXP(ζ1) · · ·EXP(ζm). +Let (λi)1≤i≤k be a smooth partition of unity subordinated to (Ui)1≤i≤k. Since the +λiζ1 ’s commute with each other, we have +EXP(ζ1) = +k +� +i=1 +EXP(λiζ1) +and supp(λiζ1) ⊂ Ui for i ∈ {1, . . . , k}. The same holds for each of EXP(ζ2), . . ., +EXP(ζm), and the lemma follows, with N = km. +□ +For a ∈ X, recall that εa denotes the evaluation map M0(G) → G, γ �→ γ(a). +The kernel ker(εa) is a closed normal subgroup of M0(G). For an open subset U +of X, define +NU = {γ ∈ M0(G) | supp(γ) ⊂ U} , +which is a normal subgroup of M0(G). For a ∈ X, we write NX∖a for NX∖{a}. +Lemma 19. Let V be an open neighbourhood of 1G in G and W an open neigh- +bourhood of O in g such that exp : W → V is a diffeomorphism. Let a ∈ X. +(i) The normal closed subgroup ker(εa) of M0(G) is connected. +Let γ ∈ NX∖a +(ii) There exist an open neighbourhood U of a in X and a homotopy (γt)0,≤t≤1 +in M0(G) from γ = γ0 to the constant map γ1 : X → {1G} such that +γt(x) = 1G for all t ∈ [0, 1] and all x ∈ U. +(iii) Moreover, there exists a sequence ξ1, . . . , ξk ∈ C∞(X, g) with the following +properties +ξi(x) ∈ W +for all i ∈ {1, . . . , k} and x ∈ X, +ξi(x) = 0 +for all i ∈ {1, . . . , k} and x ∈ U, +γ(x) = exp +� +ξ1(x) +� +exp +� +ξ2(x) +� +· · · exp +� +ξk(x) +� +for all x ∈ X. +Proof. (i) Let γ ∈ M0(G). +Since M0(G) is by definition a connected group, there exists a continuous map +Γγ : X × [0, 1] → G +such that +(11) +Γγ(·, t) : X → G +is smooth for all t ∈ [0, 1], +(12) +Γγ(x, 0) = γ(x) +for all x ∈ X, +(13) +Γγ(x, 1) = γ(a) +for all +x ∈ X. + +ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP +19 +When γ ∈ ker(εa), note that Γγ(·, t) need not be in ker(εa) for t ∈ ]0, 1[, so that +Claim (i) is not yet proved. +Define +∆′ +γ : X × [0, 1] → G +by +∆′ +γ(x, t) = Γγ(x, t)Γγ(a, t)−1γ(a) +It follows from (11), (12), and (13), that +(21) +∆′ +γ(·, t) : X → G +is smooth for all t ∈ [0, 1], +(22) +∆′ +γ(x, 0) = γ(x)γ(a)−1γ(a) = γ(x) +for all x ∈ X, +(23) +∆′ +γ(x, 1) = γ(a)γ(a)−1γ(a) = γ(a) +for all x ∈ X, +(24) +∆′ +γ(a, t) = Γγ(a, t)Γγ(a, t)−1γ(a) = γ(a) +for all t ∈ [0, 1]. +Suppose now that γ ∈ ker(εa). +Then (x �→ ∆′ +γ(x, t))t∈[0,1] is a continuous +family of maps connecting γ to the constant map X → {γ(a)} = {1G} in the +group M0(G), indeed in the group ker(εa) by (24). This proves Claim (i). +(ii) Let U′ be an open neighbourhood of a in X such that γ(x) = γ(a) for +all x ∈ U′. +Upon replacing U′ by a smaller neighbourhood, we can assume +that U′ is diffeomorphic to an open ball and that U′ has spherical coordinates +(r, σ) ∈ [0, 1[ × Sn−1; set U = {(r, σ) ∈ U′ | r < 1/3}. +Let λ : [0, 1[ → [0, 1[ be a smooth function such that λ(t) = 0 for t ≤ 1/3 and +λ(t) = 1 for t ≥ 2/3. Define a smooth map h : X → X by h(x) = h(λ(r), σ) for +all x = (r, σ) ∈ U′ and h(x) = x for all x ∈ X ∖ U′. Then +(31) +h(x) = x +for all x ∈ X ∖ U′, +(32) +h(U′) ⊂ U′, +(33) +h(x) = a +for all x ∈ U. +(34) +γ ◦ h = γ. +[Check of (34): if x ∈ X ∖ U′ we have γ(h(x)) = γ(x) by (31) and if x ∈ U′ we +have γ(h(x)) = γ(a) = γ(x) by (33).] +Let ∆′ +γ be as in (i). Define +∆γ : X × [0, 1] → G +by +∆γ(x, t) = ∆′ +γ(h(x), t). +We have +(41) +∆γ(·, t) : X → G +is smooth for all t ∈ [0, 1], +(42) +∆γ(x, 0) = γ(h(x)) = γ(x) +for all x ∈ X, by (22) and (34), +(43) +∆γ(x, 1) = γ(a) = 1G +for all x ∈ X, by (23), +(44) +∆γ(a, t) = γ(a) = 1G +for all t ∈ [0, 1], by (33) and (24). +With γt = ∆γ(·, t), this shows Claim (ii). + +20 +PIERRE DE LA HARPE +(iii) By uniform continuity of ∆γ, for all t′, t′′ ∈ [0, 1] such that t′ < t′′ and +t′′ − t′ is small enough, we have +∆γ(x, t′)−1∆γ(x, t′′) ∈ V +for all x ∈ X. +We can therefore find a sequence t0, t1, . . . , tk ∈ [0, 1] such that +0 = t0 < t1 < · · · < tk−1 < tk = 1 +∆γ(x, ti−1)−1∆γ(x, ti) ∈ V +for all x ∈ X and i ∈ {1, . . . , k}. +Let log : V → W denote the inverse of the diffeomorphism exp : W → V. For +i ∈ {1, . . . , k}, define +γi(x) = ∆γ(x, ti−1)−1∆γ(x, ti) +ξi(x) = log(γi(x)) +for all x ∈ X. Then +γ(x) = exp +� +ξ1(x) +� +exp +� +ξ2(x) +� +· · · exp +� +ξk(x) +� +for all x ∈ X +and ξ1, . . . , ξk have the properties stated in (iii). +□ +Whenever useful below, we consider the Lie algebra g furnished with a scalar +product, for example that given by minus the Killing form, so that the notion of +ball makes sense in g. +The next lemma shows that elements in M(G) which are constant outside a +small open subset of X are in M0(G). More precisely: +Lemma 20. Let a ∈ X and g ∈ G. Let U be an open neighbourhood of a in X. +Let V be an open neighbourhood of g in G and let W be an open ball centred at the +origin O in g such that the map ϕ : W → V, ξ �→ g exp(ξ) is a diffeomorphism. +Let δ ∈ M(G) be an element such that +δ(a) = g, +δ(U) ∈ V, +δ(x) = g for all x in some neighbourhood of X ∖ U. +Then δ ∈ M0(G). +Proof. Let λ : [0, 1] → [0, 1] be a smooth function such that λ(t) = 1 for t ≤ 1/3 +and λ(t) = 0 for t ≥ 2/3. +Consider W with polar coordinates (r, σ), where +r ∈ [0, 1[ and σ in the unit sphere of g. Define a continuous map +h : W × [0, 1] → W, (r, σ, t) �→ (λ(t)r, σ) +for all (r, σ) ∈ W and t ∈ [0, 1]. Then: +W → W, ξ �→ h(ξ, t), is smooth for all t ∈ [0, 1], +h(ξ, 0) = ξ for all ξ ∈ W, +h(ξ, 1) = O for all ξ ∈ W, +h(O, t) = O for all t ∈ [0, 1]. + +ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP +21 +For t ∈ [0, 1], define δt : X → G by +δt(x) = g exp +� +h +� +ϕ−1(δ(x)), t +� � +for all x ∈ U, +δt(x) = g for all x ∈ X ∖ U. +Then (t �→ δt)t∈[0,1] is a continuous family in M(G) of maps connecting δ0 = δ to +the constant map δ1 of value g. It follows that δ and δ1 are in the same connected +component of M(G). Since G is connected, the constant map δ1 is in M0(G), +hence δ ∈ M0(G). +□ +For a point a ∈ X, denote by C∞ +a,G the group of germs at a of smooth maps +from X to G. Recall that C∞ +n,G denotes the group of germs at the origin of smooth +maps from Rn to G. +Lemma 21. Let a ∈ X. +(i) The groups C∞ +a,G and C∞ +n,G are isomorphic. +(ii) The quotient group M0(G)/NX∖a is isomorphic to the group of germs C∞ +a,G, +and therefore also to C∞ +n,G. +Proof. Claim (i) is straightforward; indeed, any chart of M around a provides an +isomorphism of C∞ +a,G onto C∞ +n,G. +Consider the homomorphism +ρa : M0(G) → C∞ +a,G +which associates to a globally defined smooth map X → G in M0(X) its local +germ at a. The kernel of ρa is NX∖a. For Claim (ii), it remains to show that the +homomorphism ρa is surjective. +Let γ ∈ C∞ +a,G; set g = γ(a). Let U′ be an open neighbourhood of a in X such +that there exists a representative γa : U′ → G of γ. As in the proof of Lemma 19, +we can assume that U′ is diffeomorphic to an open ball and that U′ has spherical +coordinates (r, σ) ∈ [0, 1[ × Sn−1; set U = {(r, σ) ∈ U′ | r < 1/3}. We can also +assume that there exist V ⊂ G and W ∈ g as in Lemma 20, such that γ(U′) ⊂ V. +Let λ : [0, 1[ → [0, 1[ be a smooth function such that λ(t) = t for t ≤ 1/3 and +λ(t) = 0 for t ≥ 2/3. Define a smooth map h : X → X by h(x) = h(λ(r), σ) for +all x = (r, σ) ∈ U′ and h(x) = a for all x ∈ X ∖ U′. Define δ : X → G in M(G) +by δ(x) = γ(h(x)) for all x ∈ X. Then +δ(x) = γ(x) +for all x ∈ U +and therefore +δ = γ, +δ(x) = a +for all x ∈ U′, +and δ ∈ M0(G) by Lemma 20. +This concludes the proof of the surjectivity +of ρa. +□ +Lemma 22. Let N be a maximal normal subgroup in M0(G). +There exists a ∈ X such that N contains NX∖a. + +22 +PIERRE DE LA HARPE +Proof. Step one. Set S = M0(G)/N. This quotient group S is simple, because N +is maximal, and perfect, by Lemma 17. For an open subset U of X, recall that +NU has been defined as {γ ∈ M0(G) | supp(γ) ⊂ U}. Define +Y = +� +x ∈ X +���� +there exists an open neighbourhood +U(x) of x in X such that NU(x) ⊂ N +� +. +It follows from the definition that Y is open in X. The purpose of Step one is to +show that +Y = X ∖ a +for some point +a ∈ X. +The inclusion Y ⊂ X is strict. Indeed, suppose by contradiction that Y = X. +For each x ∈ X, there exists an open neighbourhood U(x) such that NU(x) ⊂ N. +Let {x1, . . . , xk} be a subset of X such that �k +i=1 U(xi) = X. Let γ be an arbitrary +element in M0(G). By Lemma 18, there exist a finite sequence of smooth maps +(γj)1≤j≤ℓ in M0(G) and a sequence of indices (i(j))1≤j≤ℓ in {1, . . . , k} such that +γ = γ1 · · · γℓ, and γj ∈ NU(xi(j)) for all j ∈ {1, . . . , ℓ}. This implies that γj ∈ N +for all j ∈ {1, . . . , ℓ}, hence that γ ∈ N. Since γ is arbitrary, this shows that +N = M0(G), but this is impossible because N is maximal normal in M0(G). +There exists a single point a ∈ X such that Y = X ∖ {a}. Indeed, suppose by +contradiction that there are two distinct points a and b in X outside Y . There +exist disjoint neighbourhoods A of a and B of b in X; observe that NA and NB +commute. Consider the canonical projection π : M0(G) → S; the definition of Y +implies that NA ̸⊂ N, hence π(NA) = S, and similarly π(NB) = S. Since S is +perfect, we have +S = (S, S) = (π(NA), π(NB)) = π(NA, NB) = π({1G}) = {1S}, +and this is impossible because S is not the one element group. +Step two. It remains to show that N ⊃ NX∖a (= NY ). +Let γ ∈ NX∖a. Let U ⊂ X and ξ1, . . . , ξk ∈ C∞(X, g) be as in Lemma 19 (ii) +and (iii). +Let x ∈ supp(ξ1). +Since x ∈ X ∖ {a}, there exists by Step one an open +neighbourhood U(x) of x in X such that NU(x) ⊂ N. Since supp(ξ1) is com- +pact, there exists a finite subset {x1, . . . , xℓ} of supp(ξ1) such that supp(ξ1) ⊂ +� +1≤j≤ℓ U(xj). Let {λ0, λ1, . . . , λℓ} be a smooth partition of unity subordinated to +(X ∖ supp(ξ1), U(x1), . . . , U(xℓ)}. For j ∈ {1, . . . , ℓ}, the λjξ1 ’s commute with +each other, and λ0ξ1 = 0, therefore we have +exp(ξ1(x)) = +ℓ� +j=1 +exp(λj(x)ξ1(x)) +for all x ∈ X. +By definition of the U(xj) ’s, the maps x �→ exp(λj(x)ξ1(x)) are in NU(xj), and +therefore in N, hence the map x �→ exp(ξ1(x)) is in N. +Similarly, x �→ exp(ξi(x)) is in N for all i ∈ {1, . . . , k}. It follows that their +product, γ, is also in N, hence that NX∖a ⊂ N. +□ + +ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP +23 +End of proof of Proposition I. Let N be a maximal normal subgroup in +M0(G). By Lemma 22, there exists a ∈ X such that N contains NX∖a. Let +ρa : M0(G) → C∞ +n,G be the epimorphism of Lemma 21, of kernel NX∖a. Then +ρa(N) is a maximal normal subgroup of C∞ +n,G, hence ρa(N) = NOC∞ +n,G by Propo- +sition II, hence N = NaM0(G). +□ +References +[Azof–74] +E.A. Azoff, Borel measurability in linear algebra. Proc. Amer. Math. Soc. 42 (1974), +346–350. +[Bore–01] +A. Borel, Essays in the history of Lie groups and algebraic groups. Amer. Math. +Soc. and London Math. Soc., 2001. +[BoLG2-3] N. Bourbaki, Groupes et alg`ebres de Lie. Chapitre II: Alg`ebres de Lie libres. +Chapitre III: Groupes de Lie. Hermann, 1972. +[BoLG4-6] N. Bourbaki, Groupes et alg`ebres de Lie. Chapitre IV: groupes de Coxeter et +syst`emes de Tits. Chapitre V: groupes engendr´es par des r´eflexions. Chapitre VI: +syst`emes de racines. Hermann, 1968. +[BoLG7-8] N. Bourbaki, Groupes et alg`ebres de Lie. Chapitre VII: sous-alg`ebres de Cartan, +´el´ements r´eguliers. Chapitre VIII: Alg`ebres de Lie semi-simples d´eploy´ees. Hermann, +1975 +[BoLG9] +N. Bourbaki, Groupes et alg`ebres de Lie. Chapitre 9. Groupes de Lie r´eels compacts. +Masson, 1982. +[BoTS] +N. Bourbaki, Th´eories spectrales. Chapitre I: Alg`ebres norm´ees. Chapitre II: +Groupes localement compacts commutatifs. Hermann, 1967. +[Cart–30] +E. Cartan, Sur les repr´esentations lin´eaires des groupes clos. Comment. Math. 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Theory and applications. Second edition. Graduate Texts in +Mathematics 216, Springer, 2010 [First edition 2001]. +[Texi–18] +B. Texier, Basic matrix perturbation theory. Enseign. Math. 64 (2018), no. 3–4, +249–263. +[vdWa–33] B.L. van der Waerden, Stetigkeitss¨atze f¨ur halbeinfache Liesche Gruppen. Math. Z. +36 (1933), no. 1, 780–786. +Pierre de la Harpe, Section de math´ematiques, Universit´e de Gen`eve, +Uni Dufour, 24 rue du G´en´eral Dufour, Case postale 64, CH–1211 Gen`eve 4. +Email address: Pierre.delaHarpe@unige.ch + diff --git a/uNE1T4oBgHgl3EQf3wXV/content/tmp_files/load_file.txt b/uNE1T4oBgHgl3EQf3wXV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d3b19a4c04b506a43046f2f5278b1101ac75e01 --- /dev/null +++ b/uNE1T4oBgHgl3EQf3wXV/content/tmp_files/load_file.txt @@ -0,0 +1,867 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf,len=866 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='03494v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='GR] 9 Jan 2023 ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP, REVISITED PIERRE DE LA HARPE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let X be a closed smooth manifold, G be a simple connected compact real Lie group, M(G) be the group of all smooth maps from X to G, and M0(G) be its connected component for the C∞-compact open topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' It is shown that maximal normal subgroups of M0(G) are precisely the inverse images of the centre Z(G) of G by the evaluation homomorphisms M0(G) → G, γ �→ γ(a), for a ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' This in turn is a consequence of a result on the group C∞ n,G of germs at the origin O of Rn of smooth maps Rn → G: this group has a unique maximal normal subgroup, which is the inverse image of Z(G) by the evaluation homomorphism C∞ n,G → G, γ �→ γ(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' This article provides corrections for part of an earlier article [Harp–88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Introduction An earlier article [Harp–88] was found to contain several mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' A list of local corrections would have been confusing, and we rather repeat with complete proofs Propositions I and II below (II and III in [Harp–88]), about maximal subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Theorem I of [Harp–88], about automorphisms, is replaced here by a conjectural statement only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let X be a closed smooth manifold and let G be a simple connected compact real Lie group, with Lie algebra g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let M(G) be the group of all smooth maps from X to G, and let M0(G) be its connected component for the C∞-compact open topology (the topology of uniform convergence of the maps and all their partial derivatives of all orders).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then M0(G) is an important example of a well behaved infinite dimensional Lie group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' see [Miln–84, PrSe–86, Neeb–06, KhWe–09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' However, M0(G) is viewed here as an abstract group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proposition I provides the classification of all maximal normal subgroups of M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For a ∈ X, let εa : M0(G) → G denote the evaluation map γ �→ γ(a), and let NaM0(G) = ε−1 a (Z(G)) be the inverse image by εa of the centre Z(G) of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since G/Z(G) is simple, NaM0(G) is a maximal normal subgroup of M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Conversely: Date: 9 January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 22E65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Group of smooth maps into compact Lie group, maximal subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 1 2 PIERRE DE LA HARPE Proposition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Any proper normal subgroup of M0(G) is contained in NaM0(G) for some a ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' One may think of Proposition I as a global result, of which the proof uses Proposition II which is a related local result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Denote by n the dimension of X, by C∞ n,G the set of germs at the origin O of Rn of smooth maps from Rn to G, and by ε : C∞ n,G → G the evaluation map γ �→ γ(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let NOC∞ n,G = ε−1(Z(G)) be the inverse image of Z(G) by ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then C∞ n,G has a unique maximal normal subgroup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' in other words: Proposition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Any proper normal subgroup of C∞ n,G is contained in NOC∞ n,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Everything here works equally well for maps and germs which are of class Ck for some k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The proof of Proposition II works also in the real analytic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Though a result analogous to Proposition I for real analytic maps looks plausible, it is not covered by our proof, which uses partitions of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Also, I guess that Proposition II holds for a simple connected Lie group which is not necessarily compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let M(Aut(G)) be the group of smooth maps from X to the group Aut(G) of automorphisms of G (recall that any automorphism in Aut(G) is automatically continuous [Cart–30], indeed analytic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let D(X) be the group of smooth diffeo- morphisms of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Consider the natural action of D(X) on M(Aut(G)), defined by ϕ(β) = β ◦ ϕ−1 for ϕ ∈ D(X) and β ∈ M(Aut(G)), and the associated semi-direct product M(Aut(G)) ⋊ D(X), with multiplication (α, ϕ)(β, ψ) = (αϕ(β), ϕψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' This acts on M0(G) by automorphisms: (α, ϕ)(γ)(x) = α(x) � γ � ϕ−1(x) � � for α ∈ M(Aut(G)), ϕ ∈ D(X), γ ∈ M0(G), and x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We believe that any au- tomorphism of the abstract group M0(G) is of this form, hence in particular that any automorphism of M0(G) is continuous for the C∞-compact open topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' In other terms, we can formulate the following statement, which is the alleged Theorem I in [Harp–88]: Conjectural Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' With the notation above, the group of all abstract group automorphisms of M0(G) coincides with the group M(Aut(G)) ⋊ D(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='2 of [PrSe–86], Pressley and Segal claim that the group M(Aut(G)) ⋊ D(X) is the group of all bicontinuous automorphisms of the topo- logical group M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP 3 The article [Harp–88] has clearly not been much read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' However, in November 2009, Michael Murray and Daniel Stevenson pointed out a serious flaw in the so- called proof of Lemma 14 — and consequently also in the proof of Proposition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (A problematic step is the sentence “By Lemma 12 again one has NW ⊂ ker(π)”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' this Lemma 14 of [Harp–88] has been replaced by Lemma 22 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=') At the time, I was not smart enough (or stubborn enough) to write a correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' I discussed the matter with Georges Skandalis, who convinced me that repairing Lemma 14 (Lemma 22 below) was feasible, and I made a good resolution to work on this as soon as possible, but this was delayed by several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' As I finally came back to the problem in 2022, I found other gaps and defects in proofs of other lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (In particular, it is not appropriate to define as there a topology on a space of smooth germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=') Short of an erratum restoring all claims and in particular Theorem I of [Harp–88], I decided to write up a complete proof of Propositions I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Moreover, the so-called Theorem I of [Harp–88] is now degraded to a “Conjectural Theorem” about the group of automorphims of the abstract group of M0(G), discussed in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='The following concordance table indicates how lemmas below correspond to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='lemmas in [Harp–88]: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='[Harp–88] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Below ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Remark 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='[Harp–88] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Prop 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Below ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Prop 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Remind 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Reduct 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='[Harp–88] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Below ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Remind 13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Ex 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='[Harp–88] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='− − − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 15 & 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Below ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 19 & 20 & 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='Lem 22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='(Lemmas 15 and 16 in [Harp–88] are not correct and should now be ignored).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof of Proposition II when G = SU(2) Note that the result of Section 1 for this particular case G = SU(2) will be used in the proof of the result for the general case, see the proof of Lemma 15 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We adopt the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The multiplicative group of complex numbers of modulus 1 is denoted by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The algebra of linear endomorphisms of C2 is the algebra M2(C) of 2-by-2 matrices with complex coefficients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' x∗ is the conjugate transpose of a matrix x ∈ M2(C), and id2 is the unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The space of orthogonal projections from C2 onto lines is the projective line P1 C = {p ∈ M2(C) | p∗ = p = p2 and trace(p) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 4 PIERRE DE LA HARPE In this section, until Proposition 7 included, we use G for the simple connected compact Lie group SU(2) = {g ∈ M2(C) | g∗g = id2 and det g = 1} = �� ρ σ −σ ρ � ∈ M2(C) ���� ρ, σ ∈ C, |ρ|2 + |σ|2 = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' An element g ∈ G is regular if g /∈ {id2, −id2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' we denote by Greg the open subset of G of regular elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Any g ∈ Greg has two eigenvalues zg, zg ∈ U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' the notation is such that ℑ(zg) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Set A := ]0, π[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let sg ∈ A be the number such that zg = exp(isg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let pg ∈ P1 C be the projection of C2 onto the eigenspace ker(zgid2 − g) and p′ G ∈ P1 C the projection of C2 onto ker(zgid2 − g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We have g = exp(isg)pg + exp(−isg)p′ g for all g ∈ Greg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The group G acts by conjugation on both Greg and P1 C, trivially on A, and by the product action on P1 C × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The map ψA : � Greg → P1 C × A g �→ (pg, sg) is a smooth diffeomorphism, and it is equivariant for the action of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let h ∈ Greg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let Dr be a closed disc of centre exp(ish) and some ra- dius r > 0, contained in the half-plane {z ∈ C | ℑz > 0}, and let Γr be the circle ∂Dr, with the positive orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since eigenvalues depend continuously on matrices, there exists a neighbourhood V of h in Greg such that exp(isg) is in the interior of Dr for all g ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For g ∈ V, we have the explicit formula pg = 1 2iπ � Γr (z − g)−1dz by holomorphic functional calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' It follows that pg depends smoothly on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We have also (∗) sg = arccos �1 2trace(g) � , hence sg depends smoothly on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For the continuity of eigenvalues and for holomorphic functional calculus, see for example [DuSc–58, Chapter VII, § 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 19, Corollary 21], or [BoTS, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 1, § 4, no 11, Proposition 16], or [Serr–10, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='3 & 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='5], or [Texi–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The equivariance of ψA is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP 5 The complex projective line can also be described as P1 C = ��a b b 1 − a � ��� a ∈ R, b ∈ C, 0 ≤ a ≤ 1, a2 + |b|2 = a � = ��1 2 � 1 + √ 1 − 4r2� reiϕ re−iϕ 1 2 � 1 − √ 1 − 4r2� � ����� 0 ≤ r ≤ 1 2, ϕ ∈ [0, 2π[ � ∪ ��1 2 � 1 − √ 1 − 4r2� reiϕ re−iϕ 1 2 � 1 + √ 1 − 4r2� � ����� 1 2 ≥ r ≥ 0, ϕ ∈ [0, 2π[ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The second parametrization makes it clear that P1 C is diffeomorphic to the 2- sphere, shown as two hemispheres glued along the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We define U = P1 C ∖ �� 0 0 0 1 �� = ��a b b 1 − a � ��� a ∈ R, b ∈ C, 0 < a ≤ 1, a2 + |b|2 = a � , which is an open neighbourhood of � 1 0 0 0 � in P1 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' There exists a smooth map �U → G u �→ gu such that gu � 1 0 0 0 � g−1 u = u for all u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Given a rank one projection u = �a b b 1 − a � ∈ U, set ρ = √a, σ = −b/ρ, and define gu = � ρ σ −σ ρ � ∈ SU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then gu � 1 0 0 0 � g−1 u = u, by a simple computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The diffeomorphism of Lemma 1 and the map of Lemma 2 are not only smooth, they are indeed real analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Similarly, the map of Lemma 10 below is real analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let O be a neighbourhood of the origin in Rn and let γ : O → G be a smooth map with values in Greg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Denote by γ ∈ C∞ n,G the germ defined by γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then sγ(x) depends smoothly on x, by Lemma 1, and defines at the origin a germ sγ of A- valued smooth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Similarly pγ(x) defines at the origin a germ pγ of P1 C-valued smooth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We write simply s and p when the reference to γ is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let γ and δ be two germs in C∞ n,G such that γ(O), δ(O) ∈ Greg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let (p, s) and (q, t) be the associated germs at O of (P1 C × A)-valued smooth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then γ and δ are conjugate in C∞ n,G if and only if s = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 6 PIERRE DE LA HARPE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Consider ζ ∈ C∞ n,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We may choose representatives γ, p, s, δ, q, t, ζ of γ, p, s, δ, q, t, ζ defined in a common neighbourhood O of the origin in Rn, and such that γ(x), δ(x) ∈ Greg for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Suppose first that ζ γ ζ−1 = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Upon replacing O by a smaller neighbourhood, we may assume that ζ(x)γ(x)ζ(x)−1 = δ(x) for all x ∈ O, hence that s(x) = t(x) for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' It follows that s = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Suppose now that s = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We may again assume that s(x) = t(x) for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Suppose first that, moreover, the range of p(O) and the range of q(O) are not orthogonal in C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' In appropriate orthogonal coordinates: p(O) = � 1 0 0 0 � and q(O) = �c d d 1 − c � with c > 0, d ∈ C, c2 + |d|2 = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Upon shrinking O once more if necessary, we may assume that p and q are of the form p(x) = �a(x) b(x) b(x) 1 − a(x) � where a(x) > 0 for all x ∈ O, with a(O) = 1, q(x) = �c(x) d(x) d(x) 1 − c(x) � where c(x) > 0 for all x ∈ O, with c(O) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' By Lemma 2, there exists a smooth map ζ : O → G such that ζ(x)p(x)ζ(x)−1 = q(x) for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since s = t, it follows that ζγζ−1 = δ, hence ζ γ ζ−1 = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Suppose now that the range of p(O) and the range of q(O) are orthogonal in P1 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define a third germ η ∈ C∞ n,G with eigenvalue germ equal to s (hence also equal to t), and with η(O) orthogonal neither to γ(O) nor to δ(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The previous argument shows that γ and δ are both conjugate to η, hence are conjugate one to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let O be a neighbourhood of the origin in Rn and s, t two smooth maps O → A such that sin(t(x)/2) < sin(s(x)) for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define smooth maps ζ, η, δ : O → G by ζ(x) = \uf8eb \uf8ec \uf8ed � 1 − sin2(t(x)/2) sin2(s(x)) �1/2 sin(t(x)/2) sin(s(x)) − sin(t(x)/2) sin(s(x)) � 1 − sin2(t(x)/2) sin2(s(x)) �1/2 \uf8f6 \uf8f7 \uf8f8 ∈ SO(2) ⊂ G, η(x) = � exp(is(x)) 0 0 exp(−is(x)) � ζ(x) � exp(−is(x)) 0 0 exp(is(x)) � ζ(x)−1, δ(x) = � exp(it(x)) 0 0 exp(−it(x)) � , for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then η(x) ∈ Greg for all x ∈ O, and the germs of η and δ are conjugate in C∞ n,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Note that s(x), t(x) ∈ A implies ζ(x), δ(x) ∈ Greg for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' A straightforward computation shows that, for all x ∈ O, trace(η(x)) = 2 sin2(s(x)) − (1 − cos(2s(x))) sin2(t(x)/2) sin2 s(x) = 2(1 − 2 sin2(t(x)/2)) = 2 cos(t(x)) = trace(δ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' A first consequence is that η(x) ∈ Greg, because 2 cos(t(x)) ̸= ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (Remember Formula (*) in the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=') A second consequence is that the A-valued germs at O associated to η and δ are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' It follows from Lemma 4 that the germs at O of η and δ are conjugate in C∞ n,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let N be a normal subgroup of C∞ n,G containing a germ γ such that γ(O) ∈ Greg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' There exists a symmetric neighbourhood V of the identity in G (depending on γ) such that every δ ∈ C∞ n,G with δ(O) ∈ V ∩ Greg is in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (A neighbourhood V of id2 in G is symmetric if g−1 ∈ V for all g ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let (p, s) be the (P1 C×A)-valued germ at O associated to γ, as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let γ, p, s be representatives of γ, p, s defined in a neighbourhood O of the origin in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' By this Lemma 4, there is no loss of generality if we assume that γ(x) = � exp(is(x)) 0 0 exp(−is(x)) � and that s(x) is bounded away from {0, π} for all x ∈ O, more precisely that there exists c > 0 such that c < s(x) < π − c for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let V• be the open subset of Greg of elements g such that sin(sg/2) < sin(c), where sg is defined by ψA(g) = (pg, sg), where ψA is as in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Set V = V• ∪ {id2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' it is a symmetric open neighbourhood of id2 in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let δ ∈ C∞ n,G be such that δ(O) ∈ V•;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' we have to show that δ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let (q, t) be the (P1 C × A)-valued germ associated to δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Upon replacing O by a smaller neighbourhood of the origin in Rn, we can find representatives δ, q, t of δ, q, t such that δ(x) ∈ V•, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=', such that 0 < sin(t(x)/2) < sin(x), for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' By Lemma 4 again, it suffices to consider the element δ defined by δ(x) = � exp(it(x)) 0 0 exp(−it(x)) � for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' By Lemma 5, there exists a smooth map ζ : O → G such that δ and γ ζ γ−1 ζ−1 are conjugate in C∞ n,G, and therefore auch that δ is in the normal subgroup gen- erated by γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proposition II of the Introduction holds for G = SU(2): any proper normal subgroup of C∞ n,G is contained in NOC∞ n,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 8 PIERRE DE LA HARPE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let N be a normal subgroup of C∞ n,G which is not contained in NOC∞ n,G = ε−1(Z(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then ε(N) = G, because a normal subgroup of G not contained in Z(G) is G itself (see Reminder 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' In particular N contains a germ γ such that γ(O) is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let ζ ∈ C∞ n,G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' we have to show that ζ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let O be a neighbourhood of the origin in Rn and let γ, ζ : O → G be rep- resentatives of γ, ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' By Lemma 6, there exists a symmetric neighbourhood V of id2 in G such that every δ ∈ C∞ n,G with δ(O) ∈ V ∩ Greg is in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since G is connected, there exists an integer k ≥ 1 and elements g1, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' gk ∈ V ∩Greg such that ζ(O) = g1g2 · · · gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For any g ∈ G, denote by ηg : O → G the constant map of value g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Set δ = ζ(ηg2 · · · ηgk)−1, so that ζ = δηg2 · · · ηgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then the k matrices δ(O) = g1, ηg2(O) = g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , ηgk(O) = gk are all in V ∩Greg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Lemma 6 implies that the germs δ, ηg2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' ηgk are all in N, so that their product ζ is also in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ Reminder 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let G be a connected compact Lie group G which is simple as a Lie group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=', such that its Lie algebra is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The following results are due to Cartan and van der Waerden [Cart–30, vdWa–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (i) Any non-trivial normal subgroup of G is contained in the centre Z(G) of G, and the quotient G/Z(G) is simple as an abstract group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (ii) Any abstract group homomorphism with bounded image from G to a Lie group is continuous, and therefore smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' In particular any abstract group endomorphism φ of G is necessarily smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since the Lie algebra of G is simple, the derivative of φ is either an isomorphism, in which case φ is an automorphism, or zero, in which case φ(g) = 1G for all g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For the historical context of these articles by Cartan and van der Waerden, see [Bore–01, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' VI, § 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For hints of proofs, see [BoLG2-3, Chapitre III, § 4 exercices 8 & 9, and § 9 exercice 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Recall that any continuous homomorphism between Lie groups is analytic, and in particular is smooth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' see for example [BoLG2-3, Chapitre III, § 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We end this section by a digression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Germs which are regular can be diagonal- ized by Lemma 4, but the regularity condition cannot be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The following example illustrates this;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' it is a minor adaptation of one in [Rell–69, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 1, § 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Set n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define first a map θ from R to the space of 2-by-2 matrices by θ(x) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 exp(−x−2) � cos(2/x) sin(2/x) sin(2/x) − cos(2/x) � if x ̸= 0, � 0 0 0 0 � if x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define then a map γ : R → SU(2) by γ(x) = exp(iθ(x)) for all x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Outside the origin, γ(x) has eigenvectors � cos(1/x) sin(1/x) � and � sin(1/x) − cos(1/x) � with eigenvalues ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP 9 exp(i exp(−x−2)) and exp(−i exp(−x−2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' But there is no germ ζ such that ζ γ ζ−1 is diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Even though it is not relevant here, we mention that one may diagonalize γ with a Borel map [Azof–74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof of Proposition II in the general case Let G be simple connected compact Lie group, g its Lie algebra, exp : g → G the exponential map, and Ad : G → GL(g⊗R C) the adjoint representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Reduction 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' It is sufficient to prove Proposition II when G is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Indeed, suppose that Proposition II holds for the universal cover �G of G (which is still compact by Weyl’s theorem [BoLG9, § 1, no 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The short exact sequence {1} → π1(G) → �G → G → {1} induces a sequence {1} → π1(G) → C∞ n, �G p→ C∞ n,G → {1} which is again exact (in these sequences, elements of π1(G) are viewed as elements of the centre of �G, and also as germs of constant maps from Rn to the centre of �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We denote here by εG : C∞ n,G → G and ε � G : C∞ n, �G → �G the evaluation maps at the origin of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let N be a normal subgroup of C∞ n,G which is not contained in ε−1 G (Z(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' As �G/Z( �G) = G/Z(G), the normal subgroup � N := p−1(N) of C∞ n, �G is not contained in ε−1 �G (Z( �G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' If Proposition II holds for �G, then � N = C∞ n, �G, hence N = C∞ n,G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' thus Proposition II holds for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ For g ∈ G, denote by ZG(g)0 the connected component of the centralizer ZG(g) = {h ∈ G | hg = gh} of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The element g is regular if ZG(g)0 is a maximal torus in G, equivalently if there exists a unique maximal torus in G containing g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' When g is regular and h ∈ G is not, dim ZG(g)0 < dim ZG(h)0 and there are infinitely many maximal tori in G containing h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We denote by Greg the set of regular elements in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Conjugates of regular elements are regular, so that G acts on Greg by conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The subset Greg is symmetric, open, and dense in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' More on regular elements in [BoLG7-8, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' VII, § 4] and [BoLG9, § 5, no 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We choose a maximal torus T in G and we denote by t its Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We choose an alcove A in t, namely a connected component of the subset of t consisting of those ξ ∈ t with exp(ξ) regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The group G acts as usual on G/T, trivially on A, and by conjugation on the subset Greg of regular elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Because of Reduction 9: 10 PIERRE DE LA HARPE from now on in this section, we assume that G is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Lemmas 10 and 12 can be seen as generalizations of Lemmas 1 and 4 respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The map ϕA : � G/T × A → Greg (gT, s) �→ g(exp(s))g−1 is a smooth diffeomor- phism, and it is equivariant for the action of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The map ϕA is a diffeomorphism, see [BoLG9, Proposition 4b of Page 51];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' the group denoted there by HA is here reduced to {1}, because G is simply connected, see [BoLG9, Remark 1 of Page 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The equivariance is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ We denote by π : G → G/T the canonical projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' There exist an integer k ≥ 1 and open subsets U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , Uk in G/T, smooth maps µj : Uj → G for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , k, such that π � µj(u) � = u for each u ∈ Uj, for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' This is a way to say that π : G → G/T is the projection of a locally trivial bundle with compact base G/T and fibre T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let γ and δ be two germs in C∞ n,G such that γ(O), δ(O) ∈ Greg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let (p, s) and (q, t) be the associated germs at O of (G/T × A)-valued smooth maps obtained by composition of γ and δ with the map ϕ−1 A , where ϕA is as in Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then γ and δ are conjugate in C∞ n,G if and only if s = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We prove the non-trivial implication only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We assume that s = t and we have to show that γ and δ are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We use the notation of Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' In the first case, there exists j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , k} such that p(O), q(O) are in Uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We may choose representatives γ, p, s, δ, q, t of γ, p, s, δ, q, t defined in a common neighbourhood O of the origin in Rn, such that p(x), q(x) ∈ Uj and s(x) = t(x) for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define a smooth map ζ : O → G by ζ(x) = µj(q(x)) � µj(p(x)) �−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' then ζ(x)p(x) = q(x) for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (Note that ζ(x) is an element of G which acts on G/T, and that ζ(x)p(x) = q(x) is an equality between elements of G/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=') By Lemma 10 we have ζ(x)γ(x)ζ(x)−1 = δ(x) for all x ∈ O, so that γ and δ are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' In the second case, there exist j, j′ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , k} with j ̸= j′ such that p(O) ∈ Uj and q(O) ∈ Uj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We may choose representatives γ, p, s, δ, q, t of γ, p, s, δ, q, t defined in a common neighbourhood O of the origin in Rn, such that p(x) ∈ Uj, q(x) ∈ Uj′, and s(x) = t(x) for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' There exist an integer m ≥ 1 and ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP 11 a sequence q0 = p(O), q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , qm−1, qm = q(O) of elements of G/T such that, for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , m}, there exists j(i) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , k} with both qi−1 and qi in Uj(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , m − 1}, define a smooth map γi : O → G by γi(x) = ϕA(qi, s(x)) for all x ∈ O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' write γ0 for γ and γm for δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' By the argument for the first case, there exists for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , m} a smooth map ζi : O → G such that ζi(x)qi−1(x) = qi(x), and therefore ζi(x)γi−1(x)ζi(x)−1 = γi(x), for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' It follows that ζmζm−1 · · · ζ1 conjugates γ0 = γ to γm = δ, and therefore that γ is conjugate to δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ As we have not been able to generalize Lemma 5, we proceed in the next lemmas by reduction to the case of SU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Before this, we recall the following facts on the structure of simple connected compact Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Reminder 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let X(T) denote the group of continuous homomorphisms from the maximal torus T to the group U of complex numbers of modulus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For α ∈ X(T), set gα C = {ξ ∈ g ⊗R C | Ad(t)ξ = α(t)ξ for all t ∈ T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The group X(T) is written additively, so that α = 0 is the constant homomor- phism T → U of value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' A root of G with respect to T is an element α ̸= 0 in X(T) such that dim gα C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' If α is a root, it is known that dim gα C = 1 and that −α is also a root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let R(G, T) denote the set of roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' An element t ∈ T is regular if and only if t is not contained in ker(α) for all α ∈ R(G, T), so that the alcove A chosen above is a connected component of t∖� α∈R(G,T){ξ ∈ t | exp(ξ) ∈ ker(α)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Choose a basis of the set of roots R(G, T) and let R+(G, T) be the corresponding set of positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let α be a root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' There exists a connected closed subgroup Sα of G and a surjective morphism of Lie groups να : SU(2) → Sα such that (i) Sα and the kernel of α : T → U commute, (ii) να � a 0 0 a � ∈ T and α � να � a 0 0 a �� = a2 for all a ∈ U, (iii) T = � Sα ∩ T � � ker(α) � , (iv) να is injective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=', να : SU(2) → Sα is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The group Sα is the Lie subgroup of G of Lie algebra su(2)α = g ∩ � gα C ⊕ g−α C ⊕ [gα C, g−α C ] � , which is a subalgebra of g isomorphic to su(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For the existence of να, and (i) and (ii), see [BoLG9, § 4, no 5, Page 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 12 PIERRE DE LA HARPE For (iii), consider t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Choose a square root a of α(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Set t1 = να � a 0 0 a � and t2 = t−1 1 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then t = t1t2 with t1 ∈ Sα ∩ T and t2 ∈ ker(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For (iv), see the Remarque in [BoLG9, § 4, no 5, Page 32], and recall that G is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (For G = SO(2n + 1), which is not simply connected, and for α a short root, the image of να is isomorphic to SO(3) = SU(2)/{±id2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=') Let ψα denote the automorphism of G of conjugation by να � 2−1/2 2−1/2 −2−1/2 2−1/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then (v) ψα � να � a 0 0 a �� = να � a+a 2 −a+a 2 −a+a 2 a+a 2 � for all να � a 0 0 a � ∈ Im(να) ∩ T, (vi) ψα(t) = t for all t ∈ ker(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (vii) tψα(t)−1 ∈ να (SU(2)reg) for all t ∈ T ∩ Greg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Claim (v) follows from the computation � 2−1/2 2−1/2 −2−1/2 2−1/2 � � a 0 0 a � � 2−1/2 −2−1/2 2−1/2 2−1/2 � = � a+a 2 −a+a 2 −a+a 2 a+a 2 � and Claim (vi) follows from (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (Compare (v) and (vi) with [BoLG9, § 4, no 5, Page 35], where the relevant inner automorphism of G is that defined by να � 0 1 −1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=') Let us show (vii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' There exist by (iii) commuting elements t1 = να � a 0 0 a � ∈ Im(να) ∩ T and t2 ∈ ker(α) such that t = t1t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then tψα(t)−1 = t1t2να � 2−1/2 2−1/2 −2−1/2 2−1/2 � t−1 2 t−1 1 να � 2−1/2 −2−1/2 2−1/2 2−1/2 � by (i) = t1να � 2−1/2 2−1/2 −2−1/2 2−1/2 � t−1 1 να � 2−1/2 −2−1/2 2−1/2 2−1/2 � = να � 1 2 � a 0 0 a � � 1 1 −1 1 � � a 0 0 a � � 1 −1 1 1 � � = να �1+a2 2 −1+a2 2 1−a2 2 1+a2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' If t ∈ T ∩ Greg, then t /∈ ker(α), hence t1 /∈ ker(α), hence a2 ̸= 1 by (ii), and therefore �1+a2 2 −1+a2 2 1−a2 2 1+a2 2 � ∈ SU(2)reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Finally, we have the following, used below in the proof of Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP 13 (viii) There exist a neighbourhood V of 1G in G and for each α ∈ R+(G, T) a smooth map λα : V → Sα such that � α∈R+(G,T) να(λα(g)) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof of (viii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We write R+ for R+(G, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For α ∈ R+, recall that su(2)α is the Lie algebra of Sα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Consider the surjective linear map � α∈R+ su(2)α → g which applies (ξα)α∈R+ onto � α∈R+ ξα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Choose for each α ∈ R+ a linear map κα : g → su(2)α such that � α∈R+ κα(ξ) = ξ for all ξ ∈ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let V′ be a neighbourhood of 1G in G and W be a neighbourhood of the origin O in g such that the exponential map is a diffeomorphism exp : W → V′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' denote by log the inverse diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For each α ∈ R+, define a smooth map λ′ α : V′ → Sα by λ′ α(g) = exp � κα � log(g) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define next a smooth map λ′ : V′ → G by λ′(g) = � α∈R+ να(λ′ α(g));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' the terms of this product are arranged according to some chosen order, and the same order is used for the next product � α∈R+(· · · ) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Observe that λ′(1G) = 1G and that the derivative of λ′ at 1G is the identity map g → g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Upon replacing V′ by a smaller neighbourhood of 1G, one may therefore assume that λ′ is a diffeomorphism from V′ to some neighbourhood V of 1G in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We denote by ν the inverse diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For each α ∈ R+, define a smooth map λα : V → Sα by λα(g) = λ′ α(ν(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then � λ∈R+ να(λα(g)) = � λ∈R+ να(λ′ α(ν(g))) = λ′(ν(g)) = g for all g ∈ G, and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ Example 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let G = SU(n), for n ≥ 3, and let T be the torus of diago- nal matrices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The open dense subset T ∩ Greg of T consists of matrices diag(z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , zn) such that z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , zn are all distinct and z1z2 · · · zn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let α be the homomorphism G → U mapping diag(z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , zn) to z1z−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then α is a root of G with respect to T, the image of να is � SU(2) 0 0 idn−2 � and the kernel of α is � ±id2 0 0 SU(n − 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let N be a normal subgroup of C∞ n,G which contains a germ γ such that γ(O) /∈ Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then N contains anll constant germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Step one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We can assume that γ(O) ∈ Greg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Indeed, set g = γ(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Denote by k the dimension of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' define M(g) = � h ∈ G | h = k � j=1 ajbjgb−1 j g−1a−1 j for some a1, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , ak, bk ∈ G � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since g is not central, M(g) is a neighbourhood of 1G in G (this is the essential point in van der Waerden’s proof of the simplicity of the abstract group G/Z(G) 14 PIERRE DE LA HARPE [vdWa–33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Choose a1, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , ak, bk ∈ G such that g′ ≑ �k j=1 ajbjgb−1 j g−1a−1 j ∈ Greg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Recall that ηa denote the constant map Rn → G of value a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define γ′ = k � j=1 ηaj ηbj γ η−1 bj γ−1 η−1 aj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then γ′ is in N and g′ = γ′(O) ∈ Greg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' So that we can indeed assume from the start that γ(O) ∈ Greg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Step two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We can assume that γ(x) ∈ T ∩ Greg for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Denote by (p, s) the germ at O of (G/T × A)-valued smooth maps associated to γ as in Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let γ : O → Greg and s : O → A be representatives of γ and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' By Lemma 12, the germ at O of the map γ′ : O → Greg, x �→ exp(s(x)) is conjugate to γ, hence is in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Therefore, we may assume from now on that γ is itself the germ of the map γ : O → Greg, x �→ exp(s(x)), in particular that γ(x) ∈ T ∩ Greg for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Step three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The group N contains the germ of one constant map with value not in Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let α be a root of G with respect to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let να : SU(2) → G and ψα ∈ Aut(G) be as in Reminder 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let Iα : C∞ n,SU(2) → C∞ n,G be the map which applies the germ at O of a smooth map O → SU(2) to the germ at O of the composition O → SU(2) να → G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' then Iα is an injective homomorphism of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define the map δ : O → G by δ(x) = γ(x)ψα(γ(x))−1 for all x ∈ O and let δ denote its germ at O ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since δ is the commutator of γ with something, δ is in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' It follows from (vii) in Reminder 13 that δ(x) ∈ να (SU(2)reg) for all x ∈ O and that δ is in the image of Iα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Therefore I−1 α (N) is a normal subgroup of C∞ n,SU(2) which contains the germ I−1 α (δ), and I−1 α (δ)(O) ∈ SU(2)reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' It follows from Proposition 7 that N contains all germs at O of smooth maps O → να(SU(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' In particular, N contains the germ of a constant map O → G with value in να(SU(2)reg), in particular with value in G and not in Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Final step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since any normal subgroup of G not inside Z(G) is G itself (see Reminder 8), N contains the germs of all constant maps O → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ The following lemma is the last step in the proof of Proposition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let N be a normal subgroup of C∞ n,G which contains a germ γ such that γ(O) /∈ Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then N = C∞ n,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Step one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let W be the Weyl group of G with respect to T (it is the quotient of the normalizer NormG(T) by T), let c ∈ W be a Coxeter element, and let m ∈ NormG(T) be an element of class c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Recall that W acts on T and t, in particular c acts on T by t �→ mtm−1 and on t by ξ �→ Ad(m)ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Consider the ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP 15 smooth map f : T → T, t �→ c(t)t−1 = mtm−1t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The derivative of f at the identity is L(f) : t → t, ξ �→ c(ξ) − ξ = Ad(m)ξ − ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Now L(f) is a linear automorphism, because Coxeter elements do not have 1 as an eigenvalue (see [BoLG9, Page 33], referring itself to [BoLG4-6, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' V, § 6, no 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' By the implicit function theorem, there exist a symmetric neighbour- hood V of the identity in T and a smooth map χ : V → T such that (♯) t = mχ(t)m−1χ(t)−1 for all t ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Observe that � g∈G gVg−1 is a neighbourhood of 1G in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Step two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let δ ∈ C∞ n,G be such that δ(O) ∈ � � g∈G gVg−1� ∩ Greg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' then δ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let O be a neighbourhood of the origin in Rn and let δ : O → � � g∈G gVg−1� ∩ Greg be a smooth map of germ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' By Lemma 12, there exists a smooth map ζ : O → V ∩ Greg of which the germ ζ is conjugate to δ in C∞ n,G, so that it suffices to show that ζ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let ηm be the constant map O → G of value m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' by (♯), we have ζ(x) = ηm(x)χ(ζ(x))ηm(x)−1χ(ζ(x))−1 for all x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since the germ of ηm is in N by Lemma 15, if follows that ζ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Final step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let now δ be arbitrary in C∞ n,G, and let δ : O → G be a represen- tative of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since � � g∈G gVg−1� ∩ Greg is a nonempty symmetric open subset of G and since G is connected, there exist g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , gk ∈ � � g∈G gVg−1� ∩ Greg such that δ(O) = g1 · · ·gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , k}, let ηj : O → � g∈G gVg−1 be the constant map of value gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Set ϕ = δ(η2 · · · ηk)−1, so that δ = ϕη2 · · · ηk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then ϕ(O) = g1, η2(O) = g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , ηk(O) = gk are all in (� g∈G gVg−1� ∩ Greg, hence ϕ, η2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , ηk are all in N by Step two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' It follows that δ = ϕη2 · · · ηk is in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof of Proposition I Let X be a closed smooth manifold, n its dimension, G a simple connected com- pact real Lie group, 1G the identity in G, and M0(G) the connected component of the group M(G) of all smooth maps from X to G, as in Proposition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' In the following lemma, G is identified with the group of constant maps in M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Given subsets A, B in M0(G), we write (A, B) for the subgroup of M0(G) generated by the commutators γδγ−1δ−1 with γ ∈ A and δ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We have M0(G) = (G, M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' In particular, the group M0(G) is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The group M(G) need not be perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For example, when X is the circle S1 and G is not simply connected, there is a natural epimorphism from M(G) to 16 PIERRE DE LA HARPE the fundamental group of G, which is abelian and not trivial, so that M(G) is not perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' There are cases where M0(G) = M(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' When G is simply connected, this holds for X the circle (obvious) and for X the 2-sphere (because the second homotopy group of G is trivial, by a theorem of Weyl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' When G = SU(2), this holds whenever the third cohomotopy set of X has one element, in particular when X is the circle or a closed surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We follow essentially the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='1 in [PrSe–86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We consider first the case of G = SU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Step one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let T1 be the torus of diagonal matrices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let γ ∈ M0(T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then γ ∈ (G, M0(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Indeed, let U1 = �� eis 0 0 e−is � ∈ T1 ��� − π < s < π � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let V1 = {γ ∈ M(T1) | γ(x) ∈ U1 for all x ∈ X}, which is a neighbourhood of the identity in M0(T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since there is a smooth retraction by deformation of U1 to {1T1} in T1, any smooth map γ ∈ V1 is in M0(T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Suppose first that γ ∈ V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then γ is of the form x �→ � eis(x) 0 0 e−is(x) � for some smooth map s : X → ]−π, π[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since � eis(x) 0 0 e−is(x) � = � 0 1 −1 0 � � e−is(x)/2 0 0 eis(x)/2 � � 0 −1 1 0 � � eis(x)/2 0 0 e−is(x)/2 � for all x ∈ X, we have γ ∈ (G, M0(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let now γ be arbitrary in M0(T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since M0(T1) is connected, there exist γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , γk ∈ V1 such that γ is the product γ1 · · ·γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' As each γj is contained in (G, M0(G)) by the previous argument, γ ∈ (G, M0(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Step two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let T2 = �� cos(t) i sin(t) i sin(t) cos(t) � ��� t ∈ [0, 2π[ � , T3 = �� cos(u) sin(u) − sin(u) cos(u) � ��� u ∈ [0, 2π[ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Similarly, if γ is in M0(T2) or in M0(T3), then γ ∈ (G, M0(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' This follows from the Step one because maximal tori in SU(2) are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Step three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' There is a neighbourhood of the identity W in M0(G) such that every γ ∈ W is in (G, M0(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For j ∈ {1, 2, 3}, let tj denote the Lie algebra of Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Consider the multiplication map µ : T1 × T2 × T3 → G, (g1, g2, g3) �→ g1g2g3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP 17 Its derivative at the identity t1 × t2 × t3 → g, (ξ1, ξ2, ξ3) �→ ξ1 + ξ2 + ξ3 is a linear isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' By the implicit function theorem, there exist a neigh- bourhood VT of the identity in T1 × T2 × T3 and a neighbourhood VG of the identity in G such that the restriction of µ is a homeomorphism VT → VG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Upon restricting these neighbourhoods, we can assume moreover that there exist smooth retractions by deformation of VT to {1T} in T1 × T2 × T3 and of VG to {1G} in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define W = {γ ∈ M(G) | γ(x) ∈ VG for all x ∈ X};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' any smooth map γ ∈ W is in M0(G), and W is a neighbourhood of the identity in M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then every γ ∈ W is a product γ1γ2γ3 with γj ∈ M0(Tj) for j ∈ {1, 2, 3}, hence γ ∈ (G, M0(G)) by the first two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Step four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Every γ ∈ M0(G) is in (G, M0(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since M0(G) is connected, this follows from Step three by the usual argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (Compare with the end of the proof of Step one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=') The part in which G stands for SU(2) is now finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' From now G denotes again a general simple connected compact real Lie group, as in the statement of Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Step five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Choose a maximal torus T of G and a basis of the set R = R(G, T) of roots of G with respect to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let R+ be the corresponding set of positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let V be a neighbourhood of 1G in G and, for each α ∈ R+, let λα : V → Sα be a smooth map such that � α∈R+ να � λα(g) � = g for all g ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (See (viii) in Reminder 13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' recall that να is the inclusion homomorphism of Sα ≈ SU(2) into G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=') Let γ ∈ M0(G) be such that γ(X) ⊂ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For each α ∈ R+, the map λα◦γ : X → Sα is in the commutator subgroup (Sα, M0(Sα)), by Step four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' It follows that γ = � α∈R+ να � µα ◦ γ � is in the commutator subgroup (G, M0(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let now γ be arbitrary in M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since M0(G) is connected, γ is a product of elements with images in V, hence γ ∈ (G, M0(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ The support of an element γ of M(G) is the closure supp(γ) of the subset of X of points x such that γ(x) ̸= 1G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let (Ui)1≤i≤k be an open covering of X and let γ ∈ M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' There exist a finite sequence of smooth maps (γj)1≤j≤ℓ in M0(G) and a sequence of indices (i(j))1≤j≤ℓ in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , k} with supp(γj) ⊂ Ui(j) for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , ℓ} such that γ = γ1 · · · γℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let C∞(X, g) denote the space of all smooth maps from X to the Lie algebra g of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then M0(G) is a Fr´echet Lie group with Lie algebra C∞(X, g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' there is an exponential map EXP : C∞(X, g) → M0(G) 18 PIERRE DE LA HARPE given by (EXP(ζ))(x) = exp(ζ(x)) for all ζ ∈ C∞(X, g) and x ∈ X, where exp : g → G is the exponential map of G, and this EXP is a local dif- feomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since M0(G) is connected, there exist ζ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' ζm ∈ C∞(X, g) such that γ = EXP(ζ1) · · ·EXP(ζm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let (λi)1≤i≤k be a smooth partition of unity subordinated to (Ui)1≤i≤k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since the λiζ1 ’s commute with each other, we have EXP(ζ1) = k � i=1 EXP(λiζ1) and supp(λiζ1) ⊂ Ui for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The same holds for each of EXP(ζ2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=', EXP(ζm), and the lemma follows, with N = km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ For a ∈ X, recall that εa denotes the evaluation map M0(G) → G, γ �→ γ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The kernel ker(εa) is a closed normal subgroup of M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For an open subset U of X, define NU = {γ ∈ M0(G) | supp(γ) ⊂ U} , which is a normal subgroup of M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For a ∈ X, we write NX∖a for NX∖{a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Lemma 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let V be an open neighbourhood of 1G in G and W an open neigh- bourhood of O in g such that exp : W → V is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let a ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (i) The normal closed subgroup ker(εa) of M0(G) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let γ ∈ NX∖a (ii) There exist an open neighbourhood U of a in X and a homotopy (γt)0,≤t≤1 in M0(G) from γ = γ0 to the constant map γ1 : X → {1G} such that γt(x) = 1G for all t ∈ [0, 1] and all x ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (iii) Moreover, there exists a sequence ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , ξk ∈ C∞(X, g) with the following properties ξi(x) ∈ W for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , k} and x ∈ X, ξi(x) = 0 for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , k} and x ∈ U, γ(x) = exp � ξ1(x) � exp � ξ2(x) � · · exp � ξk(x) � for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (i) Let γ ∈ M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since M0(G) is by definition a connected group, there exists a continuous map Γγ : X × [0, 1] → G such that (11) Γγ(·, t) : X → G is smooth for all t ∈ [0, 1], (12) Γγ(x, 0) = γ(x) for all x ∈ X, (13) Γγ(x, 1) = γ(a) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP 19 When γ ∈ ker(εa), note that Γγ(·, t) need not be in ker(εa) for t ∈ ]0, 1[, so that Claim (i) is not yet proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define ∆′ γ : X × [0, 1] → G by ∆′ γ(x, t) = Γγ(x, t)Γγ(a, t)−1γ(a) It follows from (11), (12), and (13), that (21) ∆′ γ(·, t) : X → G is smooth for all t ∈ [0, 1], (22) ∆′ γ(x, 0) = γ(x)γ(a)−1γ(a) = γ(x) for all x ∈ X, (23) ∆′ γ(x, 1) = γ(a)γ(a)−1γ(a) = γ(a) for all x ∈ X, (24) ∆′ γ(a, t) = Γγ(a, t)Γγ(a, t)−1γ(a) = γ(a) for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Suppose now that γ ∈ ker(εa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then (x �→ ∆′ γ(x, t))t∈[0,1] is a continuous family of maps connecting γ to the constant map X → {γ(a)} = {1G} in the group M0(G), indeed in the group ker(εa) by (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' This proves Claim (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (ii) Let U′ be an open neighbourhood of a in X such that γ(x) = γ(a) for all x ∈ U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Upon replacing U′ by a smaller neighbourhood, we can assume that U′ is diffeomorphic to an open ball and that U′ has spherical coordinates (r, σ) ∈ [0, 1[ × Sn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' set U = {(r, σ) ∈ U′ | r < 1/3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let λ : [0, 1[ → [0, 1[ be a smooth function such that λ(t) = 0 for t ≤ 1/3 and λ(t) = 1 for t ≥ 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define a smooth map h : X → X by h(x) = h(λ(r), σ) for all x = (r, σ) ∈ U′ and h(x) = x for all x ∈ X ∖ U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then (31) h(x) = x for all x ∈ X ∖ U′, (32) h(U′) ⊂ U′, (33) h(x) = a for all x ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (34) γ ◦ h = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' [Check of (34): if x ∈ X ∖ U′ we have γ(h(x)) = γ(x) by (31) and if x ∈ U′ we have γ(h(x)) = γ(a) = γ(x) by (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='] Let ∆′ γ be as in (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define ∆γ : X × [0, 1] → G by ∆γ(x, t) = ∆′ γ(h(x), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We have (41) ∆γ(·, t) : X → G is smooth for all t ∈ [0, 1], (42) ∆γ(x, 0) = γ(h(x)) = γ(x) for all x ∈ X, by (22) and (34), (43) ∆γ(x, 1) = γ(a) = 1G for all x ∈ X, by (23), (44) ∆γ(a, t) = γ(a) = 1G for all t ∈ [0, 1], by (33) and (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' With γt = ∆γ(·, t), this shows Claim (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 20 PIERRE DE LA HARPE (iii) By uniform continuity of ∆γ, for all t′, t′′ ∈ [0, 1] such that t′ < t′′ and t′′ − t′ is small enough, we have ∆γ(x, t′)−1∆γ(x, t′′) ∈ V for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We can therefore find a sequence t0, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , tk ∈ [0, 1] such that 0 = t0 < t1 < · · · < tk−1 < tk = 1 ∆γ(x, ti−1)−1∆γ(x, ti) ∈ V for all x ∈ X and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let log : V → W denote the inverse of the diffeomorphism exp : W → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , k}, define γi(x) = ∆γ(x, ti−1)−1∆γ(x, ti) ξi(x) = log(γi(x)) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then γ(x) = exp � ξ1(x) � exp � ξ2(x) � · · exp � ξk(x) � for all x ∈ X and ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , ξk have the properties stated in (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ Whenever useful below, we consider the Lie algebra g furnished with a scalar product, for example that given by minus the Killing form, so that the notion of ball makes sense in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The next lemma shows that elements in M(G) which are constant outside a small open subset of X are in M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' More precisely: Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let a ∈ X and g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let U be an open neighbourhood of a in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let V be an open neighbourhood of g in G and let W be an open ball centred at the origin O in g such that the map ϕ : W → V, ξ �→ g exp(ξ) is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let δ ∈ M(G) be an element such that δ(a) = g, δ(U) ∈ V, δ(x) = g for all x in some neighbourhood of X ∖ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then δ ∈ M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let λ : [0, 1] → [0, 1] be a smooth function such that λ(t) = 1 for t ≤ 1/3 and λ(t) = 0 for t ≥ 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Consider W with polar coordinates (r, σ), where r ∈ [0, 1[ and σ in the unit sphere of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define a continuous map h : W × [0, 1] → W, (r, σ, t) �→ (λ(t)r, σ) for all (r, σ) ∈ W and t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then: W → W, ξ �→ h(ξ, t), is smooth for all t ∈ [0, 1], h(ξ, 0) = ξ for all ξ ∈ W, h(ξ, 1) = O for all ξ ∈ W, h(O, t) = O for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP 21 For t ∈ [0, 1], define δt : X → G by δt(x) = g exp � h � ϕ−1(δ(x)), t � � for all x ∈ U, δt(x) = g for all x ∈ X ∖ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then (t �→ δt)t∈[0,1] is a continuous family in M(G) of maps connecting δ0 = δ to the constant map δ1 of value g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' It follows that δ and δ1 are in the same connected component of M(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since G is connected, the constant map δ1 is in M0(G), hence δ ∈ M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ For a point a ∈ X, denote by C∞ a,G the group of germs at a of smooth maps from X to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Recall that C∞ n,G denotes the group of germs at the origin of smooth maps from Rn to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let a ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (i) The groups C∞ a,G and C∞ n,G are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' (ii) The quotient group M0(G)/NX∖a is isomorphic to the group of germs C∞ a,G, and therefore also to C∞ n,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Claim (i) is straightforward;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' indeed, any chart of M around a provides an isomorphism of C∞ a,G onto C∞ n,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Consider the homomorphism ρa : M0(G) → C∞ a,G which associates to a globally defined smooth map X → G in M0(X) its local germ at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The kernel of ρa is NX∖a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For Claim (ii), it remains to show that the homomorphism ρa is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let γ ∈ C∞ a,G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' set g = γ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let U′ be an open neighbourhood of a in X such that there exists a representative γa : U′ → G of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' As in the proof of Lemma 19, we can assume that U′ is diffeomorphic to an open ball and that U′ has spherical coordinates (r, σ) ∈ [0, 1[ × Sn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' set U = {(r, σ) ∈ U′ | r < 1/3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' We can also assume that there exist V ⊂ G and W ∈ g as in Lemma 20, such that γ(U′) ⊂ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let λ : [0, 1[ → [0, 1[ be a smooth function such that λ(t) = t for t ≤ 1/3 and λ(t) = 0 for t ≥ 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define a smooth map h : X → X by h(x) = h(λ(r), σ) for all x = (r, σ) ∈ U′ and h(x) = a for all x ∈ X ∖ U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define δ : X → G in M(G) by δ(x) = γ(h(x)) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then δ(x) = γ(x) for all x ∈ U and therefore δ = γ, δ(x) = a for all x ∈ U′, and δ ∈ M0(G) by Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' This concludes the proof of the surjectivity of ρa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let N be a maximal normal subgroup in M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' There exists a ∈ X such that N contains NX∖a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 22 PIERRE DE LA HARPE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Step one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Set S = M0(G)/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' This quotient group S is simple, because N is maximal, and perfect, by Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For an open subset U of X, recall that NU has been defined as {γ ∈ M0(G) | supp(γ) ⊂ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Define Y = � x ∈ X ���� there exists an open neighbourhood U(x) of x in X such that NU(x) ⊂ N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' It follows from the definition that Y is open in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The purpose of Step one is to show that Y = X ∖ a for some point a ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' The inclusion Y ⊂ X is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Indeed, suppose by contradiction that Y = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For each x ∈ X, there exists an open neighbourhood U(x) such that NU(x) ⊂ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , xk} be a subset of X such that �k i=1 U(xi) = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let γ be an arbitrary element in M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' By Lemma 18, there exist a finite sequence of smooth maps (γj)1≤j≤ℓ in M0(G) and a sequence of indices (i(j))1≤j≤ℓ in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , k} such that γ = γ1 · · · γℓ, and γj ∈ NU(xi(j)) for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' This implies that γj ∈ N for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , ℓ}, hence that γ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since γ is arbitrary, this shows that N = M0(G), but this is impossible because N is maximal normal in M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' There exists a single point a ∈ X such that Y = X ∖ {a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Indeed, suppose by contradiction that there are two distinct points a and b in X outside Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' There exist disjoint neighbourhoods A of a and B of b in X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' observe that NA and NB commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Consider the canonical projection π : M0(G) → S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' the definition of Y implies that NA ̸⊂ N, hence π(NA) = S, and similarly π(NB) = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since S is perfect, we have S = (S, S) = (π(NA), π(NB)) = π(NA, NB) = π({1G}) = {1S}, and this is impossible because S is not the one element group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Step two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' It remains to show that N ⊃ NX∖a (= NY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let γ ∈ NX∖a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let U ⊂ X and ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , ξk ∈ C∞(X, g) be as in Lemma 19 (ii) and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let x ∈ supp(ξ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since x ∈ X ∖ {a}, there exists by Step one an open neighbourhood U(x) of x in X such that NU(x) ⊂ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Since supp(ξ1) is com- pact, there exists a finite subset {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , xℓ} of supp(ξ1) such that supp(ξ1) ⊂ � 1≤j≤ℓ U(xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let {λ0, λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , λℓ} be a smooth partition of unity subordinated to (X ∖ supp(ξ1), U(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , U(xℓ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' For j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , ℓ}, the λjξ1 ’s commute with each other, and λ0ξ1 = 0, therefore we have exp(ξ1(x)) = ℓ� j=1 exp(λj(x)ξ1(x)) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' By definition of the U(xj) ’s, the maps x �→ exp(λj(x)ξ1(x)) are in NU(xj), and therefore in N, hence the map x �→ exp(ξ1(x)) is in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Similarly, x �→ exp(ξi(x)) is in N for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' It follows that their product, γ, is also in N, hence that NX∖a ⊂ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ ON GROUPS OF SMOOTH MAPS INTO A SIMPLE COMPACT LIE GROUP 23 End of proof of Proposition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let N be a maximal normal subgroup in M0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' By Lemma 22, there exists a ∈ X such that N contains NX∖a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Let ρa : M0(G) → C∞ n,G be the epimorphism of Lemma 21, of kernel NX∖a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Then ρa(N) is a maximal normal subgroup of C∞ n,G, hence ρa(N) = NOC∞ n,G by Propo- sition II, hence N = NaM0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' □ References [Azof–74] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Azoff, Borel measurability in linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 42 (1974), 346–350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' [Bore–01] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Borel, Essays in the history of Lie groups and algebraic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' and London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=', 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' [BoLG2-3] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Bourbaki, Groupes et alg`ebres de Lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Chapitre II: Alg`ebres de Lie libres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Chapitre III: Groupes de Lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Hermann, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' [BoLG4-6] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Bourbaki, Groupes et alg`ebres de Lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Chapitre IV: groupes de Coxeter et syst`emes de Tits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Chapitre V: groupes engendr´es par des r´eflexions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Chapitre VI: syst`emes de racines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Hermann, 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' [BoLG7-8] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Bourbaki, Groupes et alg`ebres de Lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Chapitre VII: sous-alg`ebres de Cartan, ´el´ements r´eguliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Chapitre VIII: Alg`ebres de Lie semi-simples d´eploy´ees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Hermann, 1975 [BoLG9] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Bourbaki, Groupes et alg`ebres de Lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Chapitre 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Groupes de Lie r´eels compacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Masson, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' [BoTS] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Bourbaki, Th´eories spectrales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Chapitre I: Alg`ebres norm´ees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Chapitre II: Groupes localement compacts commutatifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Hermann, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' [Cart–30] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Cartan, Sur les repr´esentations lin´eaires des groupes clos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Helv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 2 (1930), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 1, 269–283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' [DuSc–58] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Dunford and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Schwartz, Linear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Part I: General theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Inter- science, 1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' [Harp–88] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' de la Harpe, On groups of smooth maps into a simple compact Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Com- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Helv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 63 (1988), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 3, 450–463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' [KhWe–09] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Khesin and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Wendt, The 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Neeb, Towards a Lie theory of locally convex groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 1 (2006), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 2, 291–468.' metadata={'source': 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216, Springer, 2010 [First edition 2001].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' [Texi–18] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Texier, Basic matrix perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Enseign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 64 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 3–4, 249–263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' [vdWa–33] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' van der Waerden, Stetigkeitss¨atze f¨ur halbeinfache Liesche Gruppen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 36 (1933), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' 1, 780–786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Pierre de la Harpe, Section de math´ematiques, Universit´e de Gen`eve, Uni Dufour, 24 rue du G´en´eral Dufour, Case postale 64, CH–1211 Gen`eve 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content=' Email address: Pierre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='delaHarpe@unige.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} +page_content='ch' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQf3wXV/content/2301.03494v1.pdf'} diff --git 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